% Mizar ND problem: t1_birkhoff,birkhoff,60,29 
fof(dh_c1_2__birkhoff,definition,
    ( ( ( ~ v3_struct_0(c1_2__birkhoff)
        & ~ v2_msualg_1(c1_2__birkhoff)
        & l1_msualg_1(c1_2__birkhoff) )
     => ! [A] : 
          ( ( v5_msualg_1(A,c1_2__birkhoff)
            & l3_msualg_1(A,c1_2__birkhoff) )
         => ! [B] : 
              ( ( v2_relat_1(B)
                & m1_pboole(B,u1_struct_0(c1_2__birkhoff)) )
             => ! [C] : 
                  ( m3_pboole(C,u1_struct_0(c1_2__birkhoff),B,u4_msualg_1(c1_2__birkhoff,A))
                 => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),C),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,B,A,C))) ) ) ) )
   => ! [D] : 
        ( ( ~ v3_struct_0(D)
          & ~ v2_msualg_1(D)
          & l1_msualg_1(D) )
       => ! [E] : 
            ( ( v5_msualg_1(E,D)
              & l3_msualg_1(E,D) )
           => ! [F] : 
                ( ( v2_relat_1(F)
                  & m1_pboole(F,u1_struct_0(D)) )
               => ! [G] : 
                    ( m3_pboole(G,u1_struct_0(D),F,u4_msualg_1(D,E))
                   => r2_pboole(u1_struct_0(D),k2_mssubfam(u1_struct_0(D),G),k2_mssubfam(u1_struct_0(D),k1_birkhoff(D,F,E,G))) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_2__birkhoff),file(birkhoff,c1_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c1_2__birkhoff)]).

fof(dh_c2_2__birkhoff,definition,
    ( ( ( v5_msualg_1(c2_2__birkhoff,c1_2__birkhoff)
        & l3_msualg_1(c2_2__birkhoff,c1_2__birkhoff) )
     => ! [A] : 
          ( ( v2_relat_1(A)
            & m1_pboole(A,u1_struct_0(c1_2__birkhoff)) )
         => ! [B] : 
              ( m3_pboole(B,u1_struct_0(c1_2__birkhoff),A,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
             => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),B),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,A,c2_2__birkhoff,B))) ) ) )
   => ! [C] : 
        ( ( v5_msualg_1(C,c1_2__birkhoff)
          & l3_msualg_1(C,c1_2__birkhoff) )
       => ! [D] : 
            ( ( v2_relat_1(D)
              & m1_pboole(D,u1_struct_0(c1_2__birkhoff)) )
           => ! [E] : 
                ( m3_pboole(E,u1_struct_0(c1_2__birkhoff),D,u4_msualg_1(c1_2__birkhoff,C))
               => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),E),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,D,C,E))) ) ) ) ),
    introduced(definition,[new_symbol(c2_2__birkhoff),file(birkhoff,c2_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c2_2__birkhoff)]).

fof(dh_c3_2__birkhoff,definition,
    ( ( ( v2_relat_1(c3_2__birkhoff)
        & m1_pboole(c3_2__birkhoff,u1_struct_0(c1_2__birkhoff)) )
     => ! [A] : 
          ( m3_pboole(A,u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
         => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),A),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,A))) ) )
   => ! [B] : 
        ( ( v2_relat_1(B)
          & m1_pboole(B,u1_struct_0(c1_2__birkhoff)) )
       => ! [C] : 
            ( m3_pboole(C,u1_struct_0(c1_2__birkhoff),B,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
           => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),C),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,B,c2_2__birkhoff,C))) ) ) ),
    introduced(definition,[new_symbol(c3_2__birkhoff),file(birkhoff,c3_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c3_2__birkhoff)]).

fof(dh_c4_2__birkhoff,definition,
    ( ( m3_pboole(c4_2__birkhoff,u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
     => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff))) )
   => ! [A] : 
        ( m3_pboole(A,u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
       => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),A),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,A))) ) ),
    introduced(definition,[new_symbol(c4_2__birkhoff),file(birkhoff,c4_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c4_2__birkhoff)]).

fof(rc1_closure2,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A)
      & v1_fraenkel(A) ) ),
    file(closure2,rc1_closure2),
    [interesting(0.9),axiom,file(closure2,rc1_closure2)]).

fof(rc5_msafree2,theorem,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ? [D] : 
          ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
          & ~ v1_xboole_0(D)
          & v1_relat_1(D)
          & v1_funct_1(D)
          & v1_finset_1(D) ) ) ),
    file(msafree2,rc5_msafree2),
    [interesting(0.9),axiom,file(msafree2,rc5_msafree2)]).

fof(rc6_msafree2,theorem,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ? [D] : 
          ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
          & ~ v1_xboole_0(D)
          & v1_relat_1(D)
          & v1_funct_1(D)
          & v1_finset_1(D)
          & v3_trees_2(D) ) ) ),
    file(msafree2,rc6_msafree2),
    [interesting(0.9),axiom,file(msafree2,rc6_msafree2)]).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(cc10_membered,theorem,(
    ! [A] : 
      ( v1_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => v1_xcmplx_0(B) ) ) ),
    file(membered,cc10_membered),
    [interesting(0.9),axiom,file(membered,cc10_membered)]).

fof(cc11_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B) ) ) ) ),
    file(membered,cc11_membered),
    [interesting(0.9),axiom,file(membered,cc11_membered)]).

fof(cc12_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc12_membered),
    [interesting(0.9),axiom,file(membered,cc12_membered)]).

fof(cc13_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B)
            & v1_int_1(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc13_membered),
    [interesting(0.9),axiom,file(membered,cc13_membered)]).

fof(cc14_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v4_ordinal2(B)
            & v1_xreal_0(B)
            & v1_int_1(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc14_membered),
    [interesting(0.9),axiom,file(membered,cc14_membered)]).

fof(cc16_membered,theorem,(
    ! [A] : 
      ( v1_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_membered(B) ) ) ),
    file(membered,cc16_membered),
    [interesting(0.9),axiom,file(membered,cc16_membered)]).

fof(cc17_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B) ) ) ) ),
    file(membered,cc17_membered),
    [interesting(0.9),axiom,file(membered,cc17_membered)]).

fof(cc18_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B) ) ) ) ),
    file(membered,cc18_membered),
    [interesting(0.9),axiom,file(membered,cc18_membered)]).

fof(cc19_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B)
            & v4_membered(B) ) ) ) ),
    file(membered,cc19_membered),
    [interesting(0.9),axiom,file(membered,cc19_membered)]).

fof(cc1_relset_1,theorem,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => v1_relat_1(C) ) ),
    file(relset_1,cc1_relset_1),
    [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]).

fof(cc20_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B)
            & v4_membered(B)
            & v5_membered(B) ) ) ) ),
    file(membered,cc20_membered),
    [interesting(0.9),axiom,file(membered,cc20_membered)]).

fof(cc3_msafree2,theorem,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ! [D] : 
          ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
         => ( ~ v1_xboole_0(D)
            & v1_relat_1(D)
            & v1_funct_1(D)
            & v1_finset_1(D) ) ) ) ),
    file(msafree2,cc3_msafree2),
    [interesting(0.9),axiom,file(msafree2,cc3_msafree2)]).

fof(cc4_msafree2,theorem,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ! [D] : 
          ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
         => ( ( v1_relat_1(D)
              & v1_funct_1(D) )
           => ( ~ v1_xboole_0(D)
              & v1_relat_1(D)
              & v1_funct_1(D)
              & v1_finset_1(D)
              & v3_trees_2(D) ) ) ) ) ),
    file(msafree2,cc4_msafree2),
    [interesting(0.9),axiom,file(msafree2,cc4_msafree2)]).

fof(fc2_pboole,theorem,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & v2_relat_1(B)
        & m1_pboole(B,A)
        & m1_subset_1(C,A) )
     => ~ v1_xboole_0(k1_funct_1(B,C)) ) ),
    file(pboole,fc2_pboole),
    [interesting(0.9),axiom,file(pboole,fc2_pboole)]).

fof(rc5_struct_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B) ) ) ),
    file(struct_0,rc5_struct_0),
    [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]).

fof(redefinition_m2_relset_1,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    [interesting(0.9),axiom,file(relset_1,m2_relset_1)]).

fof(dt_k13_finseq_1,axiom,(
    $true ),
    file(finseq_1,k13_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,k13_finseq_1)]).

fof(dt_k5_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k5_relat_1(A,B)) ) ),
    file(relat_1,k5_relat_1),
    [interesting(0.9),axiom,file(relat_1,k5_relat_1)]).

fof(dt_m1_finseq_2,axiom,(
    $true ),
    file(finseq_2,m1_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]).

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    [interesting(0.9),axiom,file(relset_1,m1_relset_1)]).

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ),
    file(relset_1,m2_relset_1),
    [interesting(0.9),axiom,file(relset_1,m2_relset_1)]).

fof(fc1_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k5_relat_1(A,B))
        & v1_funct_1(k5_relat_1(A,B)) ) ) ),
    file(funct_1,fc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc1_funct_1)]).

fof(fc8_funcop_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k5_relat_1(B,A))
        & v1_funct_1(k5_relat_1(B,A))
        & v1_funcop_1(k5_relat_1(B,A)) ) ) ),
    file(funcop_1,fc8_funcop_1),
    [interesting(0.9),axiom,file(funcop_1,fc8_funcop_1)]).

fof(redefinition_k3_finseq_2,definition,(
    ! [A] : k3_finseq_2(A) = k13_finseq_1(A) ),
    file(finseq_2,k3_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,k3_finseq_2)]).

fof(redefinition_k7_pboole,definition,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B)
        & m1_pboole(D,B) )
     => k7_pboole(A,B,C,D) = k5_relat_1(C,D) ) ),
    file(pboole,k7_pboole),
    [interesting(0.9),axiom,file(pboole,k7_pboole)]).

fof(dt_k3_finseq_2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k3_finseq_2(A))
      & m1_finseq_2(k3_finseq_2(A),A) ) ),
    file(finseq_2,k3_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,k3_finseq_2)]).

fof(dt_k6_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => m1_pboole(k6_pboole(A,B),k3_finseq_2(A)) ) ),
    file(pboole,k6_pboole),
    [interesting(0.9),axiom,file(pboole,k6_pboole)]).

fof(dt_k7_pboole,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B)
        & m1_pboole(D,B) )
     => m1_pboole(k7_pboole(A,B,C,D),A) ) ),
    file(pboole,k7_pboole),
    [interesting(0.9),axiom,file(pboole,k7_pboole)]).

fof(dt_u1_msualg_1,axiom,(
    $true ),
    file(msualg_1,u1_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,u1_msualg_1)]).

fof(dt_u2_msualg_1,axiom,(
    ! [A] : 
      ( l1_msualg_1(A)
     => ( v1_funct_1(u2_msualg_1(A))
        & v1_funct_2(u2_msualg_1(A),u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)))
        & m2_relset_1(u2_msualg_1(A),u1_msualg_1(A),k3_finseq_2(u1_struct_0(A))) ) ) ),
    file(msualg_1,u2_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,u2_msualg_1)]).

fof(dt_u3_msualg_1,axiom,(
    ! [A] : 
      ( l1_msualg_1(A)
     => ( v1_funct_1(u3_msualg_1(A))
        & v1_funct_2(u3_msualg_1(A),u1_msualg_1(A),u1_struct_0(A))
        & m2_relset_1(u3_msualg_1(A),u1_msualg_1(A),u1_struct_0(A)) ) ) ),
    file(msualg_1,u3_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,u3_msualg_1)]).

fof(fc1_pboole,theorem,(
    ! [A,B] : 
      ( ( v2_relat_1(B)
        & m1_pboole(B,A) )
     => ( v1_relat_1(k6_pboole(A,B))
        & v2_relat_1(k6_pboole(A,B))
        & ~ v3_relat_1(k6_pboole(A,B))
        & v1_funct_1(k6_pboole(A,B)) ) ) ),
    file(pboole,fc1_pboole),
    [interesting(0.9),axiom,file(pboole,fc1_pboole)]).

fof(free_g3_msualg_1,definition,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & m1_pboole(B,u1_struct_0(A))
        & m3_pboole(C,u1_msualg_1(A),k7_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),B)),k7_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),B)) )
     => ! [D,E,F] : 
          ( g3_msualg_1(A,B,C) = g3_msualg_1(D,E,F)
         => ( A = D
            & B = E
            & C = F ) ) ) ),
    file(msualg_1,g3_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,g3_msualg_1)]).

fof(dt_g3_msualg_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & m1_pboole(B,u1_struct_0(A))
        & m3_pboole(C,u1_msualg_1(A),k7_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),B)),k7_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),B)) )
     => ( v4_msualg_1(g3_msualg_1(A,B,C),A)
        & l3_msualg_1(g3_msualg_1(A,B,C),A) ) ) ),
    file(msualg_1,g3_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,g3_msualg_1)]).

fof(dt_u5_msualg_1,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & l3_msualg_1(B,A) )
     => m3_pboole(u5_msualg_1(A,B),u1_msualg_1(A),k7_pboole(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),k6_pboole(u1_struct_0(A),u4_msualg_1(A,B))),k7_pboole(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A),u4_msualg_1(A,B))) ) ),
    file(msualg_1,u5_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,u5_msualg_1)]).

fof(cc1_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => v4_membered(A) ) ),
    file(membered,cc1_membered),
    [interesting(0.9),axiom,file(membered,cc1_membered)]).

fof(cc2_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => v3_membered(A) ) ),
    file(membered,cc2_membered),
    [interesting(0.9),axiom,file(membered,cc2_membered)]).

fof(cc2_pboole,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_pboole(B,A)
         => ( v3_relat_1(B)
           => ~ v2_relat_1(B) ) ) ) ),
    file(pboole,cc2_pboole),
    [interesting(0.9),axiom,file(pboole,cc2_pboole)]).

fof(cc3_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => v2_membered(A) ) ),
    file(membered,cc3_membered),
    [interesting(0.9),axiom,file(membered,cc3_membered)]).

fof(cc4_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => v1_membered(A) ) ),
    file(membered,cc4_membered),
    [interesting(0.9),axiom,file(membered,cc4_membered)]).

fof(fc6_msualg_2,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( v4_msualg_1(g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B)),A)
        & v5_msualg_1(g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B)),A) ) ) ),
    file(msualg_2,fc6_msualg_2),
    [interesting(0.9),axiom,file(msualg_2,fc6_msualg_2)]).

fof(rc1_membered,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_membered(A)
      & v2_membered(A)
      & v3_membered(A)
      & v4_membered(A)
      & v5_membered(A) ) ),
    file(membered,rc1_membered),
    [interesting(0.9),axiom,file(membered,rc1_membered)]).

fof(rc1_pboole,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(pboole,rc1_pboole),
    [interesting(0.9),axiom,file(pboole,rc1_pboole)]).

fof(rc2_pboole,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v3_relat_1(B)
      & v1_funct_1(B) ) ),
    file(pboole,rc2_pboole),
    [interesting(0.9),axiom,file(pboole,rc2_pboole)]).

fof(rc3_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]).

fof(rc4_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc4_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]).

fof(abstractness_v4_msualg_1,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & l3_msualg_1(B,A) )
     => ( v4_msualg_1(B,A)
       => B = g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B)) ) ) ),
    file(msualg_1,v4_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,v4_msualg_1)]).

fof(dt_l2_msualg_1,axiom,(
    $true ),
    file(msualg_1,l2_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,l2_msualg_1)]).

fof(cc15_membered,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_membered(A)
        & v2_membered(A)
        & v3_membered(A)
        & v4_membered(A)
        & v5_membered(A) ) ) ),
    file(membered,cc15_membered),
    [interesting(0.9),axiom,file(membered,cc15_membered)]).

fof(cc1_closure2,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_fraenkel(A) ) ),
    file(closure2,cc1_closure2),
    [interesting(0.9),axiom,file(closure2,cc1_closure2)]).

fof(cc1_funct_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]).

fof(cc1_pboole,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_pboole(B,A)
         => ( v2_relat_1(B)
           => ~ v3_relat_1(B) ) ) ) ),
    file(pboole,cc1_pboole),
    [interesting(0.9),axiom,file(pboole,cc1_pboole)]).

fof(cc2_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_xboole_0(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) ) ) ),
    file(funct_1,cc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]).

fof(fc1_msualg_1,theorem,(
    ! [A,B] : 
      ( ( l1_struct_0(A)
        & v5_msualg_1(B,A)
        & l2_msualg_1(B,A) )
     => ( v1_relat_1(u4_msualg_1(A,B))
        & v2_relat_1(u4_msualg_1(A,B))
        & v1_funct_1(u4_msualg_1(A,B)) ) ) ),
    file(msualg_1,fc1_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,fc1_msualg_1)]).

fof(rc2_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]).

fof(rc5_msualg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ? [B] : 
          ( l3_msualg_1(B,A)
          & v4_msualg_1(B,A) ) ) ),
    file(msualg_1,rc5_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,rc5_msualg_1)]).

fof(rc6_msualg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ? [B] : 
          ( l3_msualg_1(B,A)
          & v4_msualg_1(B,A)
          & v5_msualg_1(B,A) ) ) ),
    file(msualg_1,rc6_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,rc6_msualg_1)]).

fof(dt_k11_msafree,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => l3_msualg_1(k11_msafree(A,B),A) ) ),
    file(msafree,k11_msafree),
    [interesting(0.9),axiom,file(msafree,k11_msafree)]).

fof(dt_k3_funct_6,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k3_funct_6(A))
        & v1_funct_1(k3_funct_6(A)) ) ) ),
    file(funct_6,k3_funct_6),
    [interesting(0.9),axiom,file(funct_6,k3_funct_6)]).

fof(dt_l1_msualg_1,axiom,(
    ! [A] : 
      ( l1_msualg_1(A)
     => l1_struct_0(A) ) ),
    file(msualg_1,l1_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,l1_msualg_1)]).

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    [interesting(0.9),axiom,file(struct_0,l1_struct_0)]).

fof(dt_l3_msualg_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] : 
          ( l3_msualg_1(B,A)
         => l2_msualg_1(B,A) ) ) ),
    file(msualg_1,l3_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,l3_msualg_1)]).

fof(dt_m3_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ! [D] : 
          ( m3_pboole(D,A,B,C)
         => m1_pboole(D,A) ) ) ),
    file(pboole,m3_pboole),
    [interesting(0.9),axiom,file(pboole,m3_pboole)]).

fof(dt_u4_msualg_1,axiom,(
    ! [A,B] : 
      ( ( l1_struct_0(A)
        & l2_msualg_1(B,A) )
     => m1_pboole(u4_msualg_1(A,B),u1_struct_0(A)) ) ),
    file(msualg_1,u4_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,u4_msualg_1)]).

fof(cc1_msafree2,theorem,(
    ! [A] : 
      ( l1_msualg_1(A)
     => ( ( ~ v3_struct_0(A)
          & v2_msualg_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_msafree2(A) ) ) ) ),
    file(msafree2,cc1_msafree2),
    [interesting(0.9),axiom,file(msafree2,cc1_msafree2)]).

fof(cc3_pboole,theorem,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ! [D] : 
          ( m3_pboole(D,A,B,C)
         => v1_funcop_1(D) ) ) ),
    file(pboole,cc3_pboole),
    [interesting(0.9),axiom,file(pboole,cc3_pboole)]).

fof(fc1_struct_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ~ v1_xboole_0(u1_struct_0(A)) ) ),
    file(struct_0,fc1_struct_0),
    [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]).

fof(fc2_msualg_9,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => ( v4_msualg_1(k11_msafree(A,B),A)
        & v5_msualg_1(k11_msafree(A,B),A)
        & v2_msafree(k11_msafree(A,B),A) ) ) ),
    file(msualg_9,fc2_msualg_9),
    [interesting(0.9),axiom,file(msualg_9,fc2_msualg_9)]).

fof(fc7_funcop_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A) )
     => ( v1_relat_1(k1_funct_1(A,B))
        & v1_funct_1(k1_funct_1(A,B)) ) ) ),
    file(funcop_1,fc7_funcop_1),
    [interesting(0.9),axiom,file(funcop_1,fc7_funcop_1)]).

fof(rc1_funcop_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_funcop_1(A) ) ),
    file(funcop_1,rc1_funcop_1),
    [interesting(0.9),axiom,file(funcop_1,rc1_funcop_1)]).

fof(rc1_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]).

fof(rc3_pboole,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v2_relat_1(B)
      & v1_funct_1(B) ) ),
    file(pboole,rc3_pboole),
    [interesting(0.9),axiom,file(pboole,rc3_pboole)]).

fof(rc3_struct_0,theorem,(
    ? [A] : 
      ( l1_struct_0(A)
      & ~ v3_struct_0(A) ) ),
    file(struct_0,rc3_struct_0),
    [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]).

fof(rc4_pboole,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v1_funcop_1(B) ) ),
    file(pboole,rc4_pboole),
    [interesting(0.9),axiom,file(pboole,rc4_pboole)]).

fof(rc5_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v2_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc5_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc5_funct_1)]).

fof(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

fof(antisymmetry_r2_hidden,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    [interesting(0.9),axiom,file(hidden,r2_hidden)]).

fof(reflexivity_r2_pboole,theorem,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => r2_pboole(A,B,B) ) ),
    file(pboole,r2_pboole),
    [interesting(0.9),axiom,file(pboole,r2_pboole)]).

fof(redefinition_k2_mssubfam,definition,(
    ! [A,B] : 
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A) )
     => k2_mssubfam(A,B) = k3_funct_6(B) ) ),
    file(mssubfam,k2_mssubfam),
    [interesting(0.9),axiom,file(mssubfam,k2_mssubfam)]).

fof(dt_k1_birkhoff,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & v5_msualg_1(C,A)
        & l3_msualg_1(C,A)
        & m3_pboole(D,u1_struct_0(A),B,u4_msualg_1(A,C)) )
     => m3_pboole(k1_birkhoff(A,B,C,D),u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,C)) ) ),
    file(birkhoff,k1_birkhoff),
    [interesting(0.9),axiom,file(birkhoff,k1_birkhoff)]).

fof(dt_k1_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    [interesting(0.9),axiom,file(funct_1,k1_funct_1)]).

fof(dt_k2_mssubfam,axiom,(
    ! [A,B] : 
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A) )
     => m1_pboole(k2_mssubfam(A,B),A) ) ),
    file(mssubfam,k2_mssubfam),
    [interesting(0.9),axiom,file(mssubfam,k2_mssubfam)]).

fof(dt_m1_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v1_relat_1(B)
        & v1_funct_1(B) ) ) ),
    file(pboole,m1_pboole),
    [interesting(0.9),axiom,file(pboole,m1_pboole)]).

fof(dt_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    [interesting(0.9),axiom,file(struct_0,u1_struct_0)]).

fof(dt_c1_2__birkhoff,assumption,
    ( ~ v3_struct_0(c1_2__birkhoff)
    & ~ v2_msualg_1(c1_2__birkhoff)
    & l1_msualg_1(c1_2__birkhoff) ),
    introduced(assumption,[file(birkhoff,c1_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c1_2__birkhoff)]).

fof(dt_c2_2__birkhoff,assumption,
    ( v5_msualg_1(c2_2__birkhoff,c1_2__birkhoff)
    & l3_msualg_1(c2_2__birkhoff,c1_2__birkhoff) ),
    introduced(assumption,[file(birkhoff,c2_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c2_2__birkhoff)]).

fof(dt_c3_2__birkhoff,assumption,
    ( v2_relat_1(c3_2__birkhoff)
    & m1_pboole(c3_2__birkhoff,u1_struct_0(c1_2__birkhoff)) ),
    introduced(assumption,[file(birkhoff,c3_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c3_2__birkhoff)]).

fof(dt_c4_2__birkhoff,assumption,(
    m3_pboole(c4_2__birkhoff,u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff)) ),
    introduced(assumption,[file(birkhoff,c4_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c4_2__birkhoff)]).

fof(dt_c6_2__birkhoff,assumption,(
    $true ),
    introduced(assumption,[file(birkhoff,c6_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c6_2__birkhoff)]).

fof(d5_pboole,definition,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ( r2_pboole(A,B,C)
          <=> ! [D] : 
                ( r2_hidden(D,A)
               => r1_tarski(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ) ),
    file(pboole,d5_pboole),
    [interesting(0.9),axiom,file(pboole,d5_pboole)]).

fof(dh_c6_2__birkhoff,definition,
    ( ~ ( r2_hidden(c6_2__birkhoff,u1_struct_0(c1_2__birkhoff))
        & ~ r1_tarski(k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff),k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) )
   => ! [A] : 
        ~ ( r2_hidden(A,u1_struct_0(c1_2__birkhoff))
          & ~ r1_tarski(k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),A),k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),A)) ) ),
    introduced(definition,[new_symbol(c6_2__birkhoff),file(birkhoff,c6_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c6_2__birkhoff)]).

fof(e1_2__birkhoff,assumption,(
    r2_hidden(c6_2__birkhoff,u1_struct_0(c1_2__birkhoff)) ),
    introduced(assumption,[file(birkhoff,e1_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,e1_2__birkhoff)]).

fof(dt_c8_2__birkhoff,assumption,(
    $true ),
    introduced(assumption,[file(birkhoff,c8_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c8_2__birkhoff)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.9),axiom,file(tarski,d3_tarski)]).

fof(dh_c8_2__birkhoff,definition,
    ( ~ ( r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff))
        & ~ r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) )
   => ! [A] : 
        ~ ( r2_hidden(A,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff))
          & ~ r2_hidden(A,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) ) ),
    introduced(definition,[new_symbol(c8_2__birkhoff),file(birkhoff,c8_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c8_2__birkhoff)]).

fof(e3_2__birkhoff,assumption,(
    r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff)) ),
    introduced(assumption,[file(birkhoff,e3_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,e3_2__birkhoff)]).

fof(existence_m1_finseq_2,axiom,(
    ! [A] : 
    ? [B] : m1_finseq_2(B,A) ),
    file(finseq_2,m1_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]).

fof(cc2_msualg_1,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ! [C] : 
          ( m1_subset_1(C,k2_relat_1(k6_pboole(u1_struct_0(A),u4_msualg_1(A,B))))
         => ~ v1_xboole_0(C) ) ) ),
    file(msualg_1,cc2_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,cc2_msualg_1)]).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

fof(fc6_membered,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & v1_membered(k1_xboole_0)
    & v2_membered(k1_xboole_0)
    & v3_membered(k1_xboole_0)
    & v4_membered(k1_xboole_0)
    & v5_membered(k1_xboole_0) ),
    file(membered,fc6_membered),
    [interesting(0.9),axiom,file(membered,fc6_membered)]).

fof(existence_l1_msualg_1,axiom,(
    ? [A] : l1_msualg_1(A) ),
    file(msualg_1,l1_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,l1_msualg_1)]).

fof(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    [interesting(0.9),axiom,file(struct_0,l1_struct_0)]).

fof(existence_l2_msualg_1,axiom,(
    ! [A] : 
      ( l1_struct_0(A)
     => ? [B] : l2_msualg_1(B,A) ) ),
    file(msualg_1,l2_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,l2_msualg_1)]).

fof(existence_l3_msualg_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ? [B] : l3_msualg_1(B,A) ) ),
    file(msualg_1,l3_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,l3_msualg_1)]).

fof(existence_m1_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    [interesting(0.9),axiom,file(relset_1,m1_relset_1)]).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    [interesting(0.9),axiom,file(relset_1,m2_relset_1)]).

fof(existence_m3_pboole,axiom,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ? [D] : m3_pboole(D,A,B,C) ) ),
    file(pboole,m3_pboole),
    [interesting(0.9),axiom,file(pboole,m3_pboole)]).

fof(cc1_msualg_1,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ! [C] : 
          ( m1_subset_1(C,k2_relat_1(u4_msualg_1(A,B)))
         => ~ v1_xboole_0(C) ) ) ),
    file(msualg_1,cc1_msualg_1),
    [interesting(0.9),axiom,file(msualg_1,cc1_msualg_1)]).

fof(fc6_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A) )
     => v1_setfam_1(k2_relat_1(A)) ) ),
    file(funct_1,fc6_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc6_funct_1)]).

fof(t2_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    [interesting(0.9),axiom,file(subset,t2_subset)]).

fof(t3_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    [interesting(0.9),axiom,file(subset,t3_subset)]).

fof(t4_subset,theorem,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    [interesting(0.9),axiom,file(subset,t4_subset)]).

fof(t5_subset,theorem,(
    ! [A,B,C] : 
      ~ ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C))
        & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    [interesting(0.9),axiom,file(subset,t5_subset)]).

fof(t6_boole,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    [interesting(0.9),axiom,file(boole,t6_boole)]).

fof(t8_boole,theorem,(
    ! [A,B] : 
      ~ ( v1_xboole_0(A)
        & A != B
        & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    [interesting(0.9),axiom,file(boole,t8_boole)]).

fof(existence_m1_pboole,axiom,(
    ! [A] : 
    ? [B] : m1_pboole(B,A) ),
    file(pboole,m1_pboole),
    [interesting(0.9),axiom,file(pboole,m1_pboole)]).

fof(redefinition_k1_msualg_3,definition,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(A)
        & m1_pboole(B,A)
        & m1_pboole(C,A)
        & m3_pboole(D,A,B,C)
        & m1_subset_1(E,A) )
     => k1_msualg_3(A,B,C,D,E) = k1_funct_1(D,E) ) ),
    file(msualg_3,k1_msualg_3),
    [interesting(0.9),axiom,file(msualg_3,k1_msualg_3)]).

fof(redefinition_k5_relset_1,definition,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => k5_relset_1(A,B,C) = k2_relat_1(C) ) ),
    file(relset_1,k5_relset_1),
    [interesting(0.9),axiom,file(relset_1,k5_relset_1)]).

fof(dt_k1_msualg_3,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(A)
        & m1_pboole(B,A)
        & m1_pboole(C,A)
        & m3_pboole(D,A,B,C)
        & m1_subset_1(E,A) )
     => ( v1_funct_1(k1_msualg_3(A,B,C,D,E))
        & v1_funct_2(k1_msualg_3(A,B,C,D,E),k1_funct_1(B,E),k1_funct_1(C,E))
        & m2_relset_1(k1_msualg_3(A,B,C,D,E),k1_funct_1(B,E),k1_funct_1(C,E)) ) ) ),
    file(msualg_3,k1_msualg_3),
    [interesting(0.9),axiom,file(msualg_3,k1_msualg_3)]).

fof(dt_k2_relat_1,axiom,(
    $true ),
    file(relat_1,k2_relat_1),
    [interesting(0.9),axiom,file(relat_1,k2_relat_1)]).

fof(dt_k5_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) ),
    file(relset_1,k5_relset_1),
    [interesting(0.9),axiom,file(relset_1,k5_relset_1)]).

fof(t1_subset,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    [interesting(0.9),axiom,file(subset,t1_subset)]).

fof(t7_boole,theorem,(
    ! [A,B] : 
      ~ ( r2_hidden(A,B)
        & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    [interesting(0.9),axiom,file(boole,t7_boole)]).

fof(de_c7_2__birkhoff,definition,(
    c7_2__birkhoff = c6_2__birkhoff ),
    introduced(definition,[new_symbol(c7_2__birkhoff),file(birkhoff,c7_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c7_2__birkhoff)]).

fof(e2_2__birkhoff,plain,(
    m1_subset_1(c6_2__birkhoff,u1_struct_0(c1_2__birkhoff)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff])],[dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,rc1_membered,t8_boole,existence_l1_msualg_1,existence_l1_struct_0,dt_l1_msualg_1,dt_l1_struct_0,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,fc1_struct_0,rc3_struct_0,t2_subset,t6_boole,antisymmetry_r2_hidden,existence_m1_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_2__birkhoff,dt_c6_2__birkhoff,t1_subset,t7_boole,e1_2__birkhoff]),
    [interesting(0.8),file(birkhoff,e2_2__birkhoff),[file(birkhoff,e2_2__birkhoff)]]).

fof(dt_c7_2__birkhoff,plain,(
    m1_subset_1(c7_2__birkhoff,u1_struct_0(c1_2__birkhoff)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff])],[antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,rc1_membered,t1_subset,cc15_membered,cc1_closure2,cc1_funct_1,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_msualg_1,existence_l1_struct_0,dt_l1_msualg_1,dt_l1_struct_0,cc1_msafree2,fc1_struct_0,rc3_struct_0,existence_m1_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_2__birkhoff,dt_c6_2__birkhoff,de_c7_2__birkhoff,e2_2__birkhoff]),
    [interesting(0.8),file(birkhoff,c7_2__birkhoff),[file(birkhoff,c7_2__birkhoff)]]).

fof(existence_m4_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ? [C] : m4_pboole(C,A,B) ) ),
    file(pboole,m4_pboole),
    [interesting(0.9),axiom,file(pboole,m4_pboole)]).

fof(dt_m4_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m4_pboole(C,A,B)
         => m1_pboole(C,A) ) ) ),
    file(pboole,m4_pboole),
    [interesting(0.9),axiom,file(pboole,m4_pboole)]).

fof(rc1_extens_1,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ? [C] : 
          ( m1_msafree(C,A,B)
          & v1_relat_1(C)
          & v2_relat_1(C)
          & ~ v3_relat_1(C)
          & v1_funct_1(C) ) ) ),
    file(extens_1,rc1_extens_1),
    [interesting(0.9),axiom,file(extens_1,rc1_extens_1)]).

fof(rc5_pboole,theorem,(
    ! [A,B] : 
      ( ( v2_relat_1(B)
        & m1_pboole(B,A) )
     => ? [C] : 
          ( m4_pboole(C,A,B)
          & v1_relat_1(C)
          & v2_relat_1(C)
          & v1_funct_1(C) ) ) ),
    file(pboole,rc5_pboole),
    [interesting(0.9),axiom,file(pboole,rc5_pboole)]).

fof(existence_m1_msafree,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & l3_msualg_1(B,A) )
     => ? [C] : m1_msafree(C,A,B) ) ),
    file(msafree,m1_msafree),
    [interesting(0.9),axiom,file(msafree,m1_msafree)]).

fof(dt_m1_msafree,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & l3_msualg_1(B,A) )
     => ! [C] : 
          ( m1_msafree(C,A,B)
         => m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) ) ) ),
    file(msafree,m1_msafree),
    [interesting(0.9),axiom,file(msafree,m1_msafree)]).

fof(fc1_msualg_9,theorem,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => ( v1_relat_1(k13_msafree(A,B))
        & v2_relat_1(k13_msafree(A,B))
        & ~ v3_relat_1(k13_msafree(A,B))
        & v1_funct_1(k13_msafree(A,B))
        & v1_msafree(k13_msafree(A,B),A,k11_msafree(A,B)) ) ) ),
    file(msualg_9,fc1_msualg_9),
    [interesting(0.9),axiom,file(msualg_9,fc1_msualg_9)]).

fof(redefinition_k4_relset_1,definition,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => k4_relset_1(A,B,C) = k1_relat_1(C) ) ),
    file(relset_1,k4_relset_1),
    [interesting(0.9),axiom,file(relset_1,k4_relset_1)]).

fof(dt_k13_msafree,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => m1_msafree(k13_msafree(A,B),A,k11_msafree(A,B)) ) ),
    file(msafree,k13_msafree),
    [interesting(0.9),axiom,file(msafree,k13_msafree)]).

fof(dt_k15_msafree,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => m3_pboole(k15_msafree(A,B),u1_struct_0(A),k13_msafree(A,B),B) ) ),
    file(msafree,k15_msafree),
    [interesting(0.9),axiom,file(msafree,k15_msafree)]).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    [interesting(0.9),axiom,file(relat_1,k1_relat_1)]).

fof(dt_k4_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(relset_1,k4_relset_1),
    [interesting(0.9),axiom,file(relset_1,k4_relset_1)]).

fof(dh_c10_2__birkhoff,definition,
    ( ? [A] : 
        ( r2_hidden(A,k4_relset_1(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff)))
        & c9_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),A) )
   => ( r2_hidden(c10_2__birkhoff,k4_relset_1(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff)))
      & c9_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),c10_2__birkhoff) ) ),
    introduced(definition,[new_symbol(c10_2__birkhoff),file(birkhoff,c10_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c10_2__birkhoff)]).

fof(dh_c9_2__birkhoff,definition,
    ( ? [A] : 
        ( r2_hidden(A,k4_relset_1(k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff)))
        & c8_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff),A) )
   => ( r2_hidden(c9_2__birkhoff,k4_relset_1(k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff)))
      & c8_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff),c9_2__birkhoff) ) ),
    introduced(definition,[new_symbol(c9_2__birkhoff),file(birkhoff,c9_2__birkhoff)]),
    [interesting(0.8),axiom,file(birkhoff,c9_2__birkhoff)]).

fof(t13_mssubfam,theorem,(
    ! [A,B,C] : 
      ( ( v1_funcop_1(C)
        & m1_pboole(C,B) )
     => ! [D] : 
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ( ( r2_hidden(A,B)
              & D = k1_funct_1(C,A) )
           => k1_funct_1(k2_mssubfam(B,C),A) = k2_relat_1(D) ) ) ) ),
    file(mssubfam,t13_mssubfam),
    [interesting(0.9),axiom,file(mssubfam,t13_mssubfam)]).

fof(e4_2__birkhoff,plain,(
    r2_hidden(c8_2__birkhoff,k5_relset_1(k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m3_pboole,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k3_funct_6,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_msualg_1,cc1_pboole,cc2_funct_1,cc3_pboole,fc1_msualg_1,fc1_struct_0,fc2_pboole,fc6_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_pboole,redefinition_k1_msualg_3,redefinition_k2_mssubfam,redefinition_k5_relset_1,dt_k1_funct_1,dt_k1_msualg_3,dt_k2_mssubfam,dt_k2_relat_1,dt_k5_relset_1,dt_m1_pboole,dt_u1_struct_0,dt_u4_msualg_1,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,dt_c7_2__birkhoff,dt_c8_2__birkhoff,de_c7_2__birkhoff,fc7_funcop_1,rc1_funcop_1,rc1_funct_1,rc4_pboole,t1_subset,t7_boole,e3_2__birkhoff,t13_mssubfam]),
    [interesting(0.8),file(birkhoff,e4_2__birkhoff),[file(birkhoff,e4_2__birkhoff)]]).

fof(d5_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( B = k2_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : 
                  ( r2_hidden(D,k1_relat_1(A))
                  & C = k1_funct_1(A,D) ) ) ) ) ),
    file(funct_1,d5_funct_1),
    [interesting(0.9),axiom,file(funct_1,d5_funct_1)]).

fof(e5_2__birkhoff,plain,(
    ? [A] : 
      ( r2_hidden(A,k4_relset_1(k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff)))
      & c8_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff),A) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,fc7_funcop_1,rc1_funcop_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m3_pboole,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_msualg_1,cc1_pboole,cc2_funct_1,cc3_pboole,fc1_msualg_1,fc1_struct_0,fc2_pboole,fc6_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k1_msualg_3,redefinition_k4_relset_1,redefinition_k5_relset_1,dt_k1_funct_1,dt_k1_msualg_3,dt_k1_relat_1,dt_k2_relat_1,dt_k4_relset_1,dt_k5_relset_1,dt_u1_struct_0,dt_u4_msualg_1,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,dt_c8_2__birkhoff,de_c7_2__birkhoff,rc1_funct_1,t1_subset,t7_boole,e4_2__birkhoff,d5_funct_1]),
    [interesting(0.8),file(birkhoff,e5_2__birkhoff),[file(birkhoff,e5_2__birkhoff)]]).

fof(dt_c9_2__birkhoff,plain,(
    $true ),
    inference(consider,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[dh_c9_2__birkhoff,e5_2__birkhoff]),
    [interesting(0.8),file(birkhoff,c9_2__birkhoff),[file(birkhoff,c9_2__birkhoff)]]).

fof(symmetry_r6_pboole,theorem,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ( r6_pboole(A,B,C)
       => r6_pboole(A,C,B) ) ) ),
    file(pboole,r6_pboole),
    [interesting(0.9),axiom,file(pboole,r6_pboole)]).

fof(reflexivity_r6_pboole,theorem,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => r6_pboole(A,B,B) ) ),
    file(pboole,r6_pboole),
    [interesting(0.9),axiom,file(pboole,r6_pboole)]).

fof(redefinition_k2_extens_1,definition,(
    ! [A,B] : 
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A) )
     => k2_extens_1(A,B) = k3_funct_6(B) ) ),
    file(extens_1,k2_extens_1),
    [interesting(0.9),axiom,file(extens_1,k2_extens_1)]).

fof(redefinition_r6_pboole,definition,(
    ! [A,B,C] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A) )
     => ( r6_pboole(A,B,C)
      <=> B = C ) ) ),
    file(pboole,r6_pboole),
    [interesting(0.9),axiom,file(pboole,r6_pboole)]).

fof(dt_k2_extens_1,axiom,(
    ! [A,B] : 
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A) )
     => m1_pboole(k2_extens_1(A,B),A) ) ),
    file(extens_1,k2_extens_1),
    [interesting(0.9),axiom,file(extens_1,k2_extens_1)]).

fof(t14_extens_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] : 
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => r6_pboole(u1_struct_0(A),k2_extens_1(u1_struct_0(A),k15_msafree(A,B)),B) ) ) ),
    file(extens_1,t14_extens_1),
    [interesting(0.9),axiom,file(extens_1,t14_extens_1)]).

fof(e7_2__birkhoff,plain,(
    r6_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k15_msafree(c1_2__birkhoff,c3_2__birkhoff)),c3_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[reflexivity_r1_tarski,dt_k1_zfmisc_1,dt_k2_zfmisc_1,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_relset_1,cc20_membered,rc5_struct_0,t3_subset,t4_subset,t5_subset,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,existence_l2_msualg_1,existence_m1_subset_1,dt_g3_msualg_1,dt_l2_msualg_1,dt_m1_subset_1,dt_u5_msualg_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,fc1_msualg_1,fc6_msualg_2,t2_subset,antisymmetry_r2_hidden,abstractness_v4_msualg_1,existence_l3_msualg_1,existence_m4_pboole,dt_k1_xboole_0,dt_l3_msualg_1,dt_m4_pboole,dt_u4_msualg_1,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,rc1_extens_1,rc1_membered,rc3_funct_1,rc5_msualg_1,rc5_pboole,rc6_msualg_1,t1_subset,existence_m1_msafree,dt_k11_msafree,dt_m1_msafree,cc15_membered,cc1_closure2,cc1_funct_1,cc1_pboole,cc2_funct_1,cc2_pboole,fc2_msualg_9,rc1_pboole,rc2_funct_1,rc2_pboole,rc4_funct_1,t6_boole,t7_boole,t8_boole,existence_l1_struct_0,existence_m3_pboole,dt_k13_msafree,dt_k3_funct_6,dt_l1_struct_0,dt_m3_pboole,cc3_pboole,fc1_msualg_9,fc1_struct_0,rc1_funcop_1,rc1_funct_1,rc3_pboole,rc3_struct_0,rc4_pboole,rc5_funct_1,symmetry_r6_pboole,reflexivity_r6_pboole,existence_l1_msualg_1,existence_m1_pboole,redefinition_k2_extens_1,redefinition_k2_mssubfam,redefinition_r6_pboole,dt_k15_msafree,dt_k2_extens_1,dt_k2_mssubfam,dt_l1_msualg_1,dt_m1_pboole,dt_u1_struct_0,dt_c1_2__birkhoff,dt_c3_2__birkhoff,cc1_msafree2,t14_extens_1]),
    [interesting(0.8),file(birkhoff,e7_2__birkhoff),[file(birkhoff,e7_2__birkhoff)]]).

fof(t13_extens_1,theorem,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ! [D] : 
              ( m3_pboole(D,A,B,C)
             => ( v2_msualg_3(D,A,B,C)
              <=> r6_pboole(A,k2_extens_1(A,D),C) ) ) ) ) ),
    file(extens_1,t13_extens_1),
    [interesting(0.9),axiom,file(extens_1,t13_extens_1)]).

fof(e8_2__birkhoff,plain,(
    v2_msualg_3(k15_msafree(c1_2__birkhoff,c3_2__birkhoff),u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[reflexivity_r1_tarski,dt_k1_zfmisc_1,dt_k2_zfmisc_1,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_relset_1,cc20_membered,rc5_struct_0,t3_subset,t4_subset,t5_subset,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,fc1_funct_1,fc8_funcop_1,existence_m1_subset_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_m1_subset_1,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,fc1_pboole,t2_subset,free_g3_msualg_1,antisymmetry_r2_hidden,existence_l2_msualg_1,dt_g3_msualg_1,dt_k1_xboole_0,dt_l2_msualg_1,dt_u5_msualg_1,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc1_msualg_1,fc6_membered,fc6_msualg_2,rc1_membered,rc3_funct_1,t1_subset,abstractness_v4_msualg_1,existence_l3_msualg_1,existence_m4_pboole,dt_l3_msualg_1,dt_m4_pboole,dt_u4_msualg_1,cc15_membered,cc1_closure2,cc1_funct_1,cc1_pboole,cc2_funct_1,cc2_pboole,rc1_extens_1,rc1_pboole,rc2_funct_1,rc2_pboole,rc4_funct_1,rc5_msualg_1,rc5_pboole,rc6_msualg_1,t6_boole,t7_boole,t8_boole,existence_l1_msualg_1,existence_l1_struct_0,existence_m1_msafree,dt_k11_msafree,dt_k3_funct_6,dt_l1_msualg_1,dt_l1_struct_0,dt_m1_msafree,cc1_msafree2,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,rc1_funcop_1,rc1_funct_1,rc3_pboole,rc3_struct_0,rc4_pboole,rc5_funct_1,symmetry_r6_pboole,reflexivity_r6_pboole,existence_m1_pboole,existence_m3_pboole,redefinition_k2_extens_1,redefinition_k2_mssubfam,redefinition_r6_pboole,dt_k13_msafree,dt_k15_msafree,dt_k2_extens_1,dt_k2_mssubfam,dt_m1_pboole,dt_m3_pboole,dt_u1_struct_0,dt_c1_2__birkhoff,dt_c3_2__birkhoff,cc3_pboole,e7_2__birkhoff,t13_extens_1]),
    [interesting(0.8),file(birkhoff,e8_2__birkhoff),[file(birkhoff,e8_2__birkhoff)]]).

fof(d3_msualg_3,definition,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ! [D] : 
              ( m3_pboole(D,A,B,C)
             => ( v2_msualg_3(D,A,B,C)
              <=> ! [E] : 
                    ( r2_hidden(E,A)
                   => k2_relat_1(k1_funct_1(D,E)) = k1_funct_1(C,E) ) ) ) ) ) ),
    file(msualg_3,d3_msualg_3),
    [interesting(0.9),axiom,file(msualg_3,d3_msualg_3)]).

fof(e9_2__birkhoff,plain,(
    k5_relset_1(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff)) = k1_funct_1(c3_2__birkhoff,c7_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c6_2__birkhoff,e1_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,cc2_msualg_1,fc1_pboole,free_g3_msualg_1,existence_l2_msualg_1,dt_g3_msualg_1,dt_l2_msualg_1,dt_u5_msualg_1,fc1_msualg_1,fc6_msualg_2,rc1_closure2,rc5_msafree2,rc6_msafree2,reflexivity_r1_tarski,abstractness_v4_msualg_1,existence_l3_msualg_1,existence_m4_pboole,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l3_msualg_1,dt_m4_pboole,dt_u4_msualg_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_msualg_1,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc3_msafree2,cc4_membered,cc4_msafree2,fc6_membered,rc1_extens_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc5_msualg_1,rc5_pboole,rc6_msualg_1,existence_l1_msualg_1,existence_l1_struct_0,existence_m1_msafree,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k11_msafree,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_m1_msafree,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc2_funct_1,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,fc6_funct_1,fc7_funcop_1,rc1_funcop_1,rc1_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc4_pboole,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_pboole,existence_m3_pboole,redefinition_k1_msualg_3,redefinition_k5_relset_1,dt_k13_msafree,dt_k15_msafree,dt_k1_funct_1,dt_k1_msualg_3,dt_k2_relat_1,dt_k5_relset_1,dt_m1_pboole,dt_m3_pboole,dt_u1_struct_0,dt_c1_2__birkhoff,dt_c3_2__birkhoff,dt_c7_2__birkhoff,de_c7_2__birkhoff,cc3_pboole,t1_subset,t7_boole,e8_2__birkhoff,d3_msualg_3]),
    [interesting(0.8),file(birkhoff,e9_2__birkhoff),[file(birkhoff,e9_2__birkhoff)]]).

fof(e6_2__birkhoff,plain,
    ( r2_hidden(c9_2__birkhoff,k4_relset_1(k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff)))
    & c8_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff),c9_2__birkhoff) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[dt_k2_zfmisc_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc7_funcop_1,rc1_funcop_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc2_funct_1,cc3_pboole,fc1_msualg_1,fc1_struct_0,fc2_pboole,rc1_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,antisymmetry_r2_hidden,redefinition_k1_msualg_3,redefinition_k4_relset_1,dt_k1_funct_1,dt_k1_msualg_3,dt_k4_relset_1,dt_u1_struct_0,dt_u4_msualg_1,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,dt_c8_2__birkhoff,dt_c9_2__birkhoff,dh_c9_2__birkhoff,e5_2__birkhoff]),
    [interesting(0.8),file(birkhoff,e6_2__birkhoff),[file(birkhoff,e6_2__birkhoff)]]).

fof(e10_2__birkhoff,plain,(
    ? [A] : 
      ( r2_hidden(A,k4_relset_1(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff)))
      & c9_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),A) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,cc2_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,reflexivity_r1_tarski,abstractness_v4_msualg_1,existence_m4_pboole,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m4_pboole,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,fc7_funcop_1,rc1_closure2,rc1_extens_1,rc1_funcop_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,rc5_msafree2,rc5_msualg_1,rc5_pboole,rc6_msafree2,rc6_msualg_1,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_msafree,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m3_pboole,redefinition_m2_relset_1,dt_k11_msafree,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_msualg_1,cc1_pboole,cc2_funct_1,cc3_msafree2,cc3_pboole,cc4_msafree2,fc1_msualg_1,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,fc6_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k1_msualg_3,redefinition_k4_relset_1,redefinition_k5_relset_1,dt_k13_msafree,dt_k15_msafree,dt_k1_funct_1,dt_k1_msualg_3,dt_k1_relat_1,dt_k2_relat_1,dt_k4_relset_1,dt_k5_relset_1,dt_u1_struct_0,dt_u4_msualg_1,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,dt_c8_2__birkhoff,dt_c9_2__birkhoff,de_c7_2__birkhoff,rc1_funct_1,t1_subset,t7_boole,e9_2__birkhoff,e6_2__birkhoff,d5_funct_1]),
    [interesting(0.8),file(birkhoff,e10_2__birkhoff),[file(birkhoff,e10_2__birkhoff)]]).

fof(dt_c10_2__birkhoff,plain,(
    $true ),
    inference(consider,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[dh_c10_2__birkhoff,e10_2__birkhoff]),
    [interesting(0.8),file(birkhoff,c10_2__birkhoff),[file(birkhoff,c10_2__birkhoff)]]).

fof(dt_k13_pboole,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_funcop_1(B) )
     => ( v1_relat_1(k13_pboole(A,B))
        & v1_funct_1(k13_pboole(A,B))
        & v1_funcop_1(k13_pboole(A,B)) ) ) ),
    file(pboole,k13_pboole),
    [interesting(0.9),axiom,file(pboole,k13_pboole)]).

fof(dt_k7_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(A)
     => v1_relat_1(k7_relat_1(A,B)) ) ),
    file(relat_1,k7_relat_1),
    [interesting(0.9),axiom,file(relat_1,k7_relat_1)]).

fof(fc4_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B)) ) ) ),
    file(funct_1,fc4_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc4_funct_1)]).

fof(redefinition_k3_msualg_3,definition,(
    ! [A,B,C,D,E,F] : 
      ( ( m1_pboole(B,A)
        & v2_relat_1(C)
        & m1_pboole(C,A)
        & v2_relat_1(D)
        & m1_pboole(D,A)
        & m3_pboole(E,A,B,C)
        & m3_pboole(F,A,C,D) )
     => k3_msualg_3(A,B,C,D,E,F) = k13_pboole(E,F) ) ),
    file(msualg_3,k3_msualg_3),
    [interesting(0.9),axiom,file(msualg_3,k3_msualg_3)]).

fof(redefinition_k7_funct_2,definition,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(B)
        & v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m1_relset_1(D,A,B)
        & v1_funct_1(E)
        & v1_funct_2(E,B,C)
        & m1_relset_1(E,B,C) )
     => k7_funct_2(A,B,C,D,E) = k5_relat_1(D,E) ) ),
    file(funct_2,k7_funct_2),
    [interesting(0.9),axiom,file(funct_2,k7_funct_2)]).

fof(redefinition_k8_relset_1,definition,(
    ! [A,B,C,D] : 
      ( m1_relset_1(C,A,B)
     => k8_relset_1(A,B,C,D) = k7_relat_1(C,D) ) ),
    file(relset_1,k8_relset_1),
    [interesting(0.9),axiom,file(relset_1,k8_relset_1)]).

fof(dt_k1_msafree,axiom,(
    ! [A,B,C,D,E] : 
      ( ( m1_pboole(B,A)
        & m1_pboole(C,A)
        & m4_pboole(D,A,B)
        & m3_pboole(E,A,B,C) )
     => m3_pboole(k1_msafree(A,B,C,D,E),A,D,C) ) ),
    file(msafree,k1_msafree),
    [interesting(0.9),axiom,file(msafree,k1_msafree)]).

fof(dt_k3_msualg_3,axiom,(
    ! [A,B,C,D,E,F] : 
      ( ( m1_pboole(B,A)
        & v2_relat_1(C)
        & m1_pboole(C,A)
        & v2_relat_1(D)
        & m1_pboole(D,A)
        & m3_pboole(E,A,B,C)
        & m3_pboole(F,A,C,D) )
     => m3_pboole(k3_msualg_3(A,B,C,D,E,F),A,B,D) ) ),
    file(msualg_3,k3_msualg_3),
    [interesting(0.9),axiom,file(msualg_3,k3_msualg_3)]).

fof(dt_k7_funct_2,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(B)
        & v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m1_relset_1(D,A,B)
        & v1_funct_1(E)
        & v1_funct_2(E,B,C)
        & m1_relset_1(E,B,C) )
     => ( v1_funct_1(k7_funct_2(A,B,C,D,E))
        & v1_funct_2(k7_funct_2(A,B,C,D,E),A,C)
        & m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ),
    file(funct_2,k7_funct_2),
    [interesting(0.9),axiom,file(funct_2,k7_funct_2)]).

fof(dt_k8_relset_1,axiom,(
    ! [A,B,C,D] : 
      ( m1_relset_1(C,A,B)
     => m2_relset_1(k8_relset_1(A,B,C,D),A,B) ) ),
    file(relset_1,k8_relset_1),
    [interesting(0.9),axiom,file(relset_1,k8_relset_1)]).

fof(e11_2__birkhoff,plain,
    ( r2_hidden(c10_2__birkhoff,k4_relset_1(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff)))
    & c9_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),c10_2__birkhoff) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_l2_msualg_1,dt_u5_msualg_1,fc1_msualg_1,fc6_msualg_2,rc1_closure2,rc5_msafree2,rc6_msafree2,abstractness_v4_msualg_1,dt_k2_zfmisc_1,dt_l3_msualg_1,dt_m4_pboole,dt_u4_msualg_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc3_msafree2,cc4_membered,cc4_msafree2,fc7_funcop_1,rc1_extens_1,rc1_funcop_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,rc5_msualg_1,rc5_pboole,rc6_msualg_1,redefinition_m2_relset_1,dt_k11_msafree,dt_k1_relat_1,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_m1_msafree,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc2_funct_1,cc3_pboole,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,rc1_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,antisymmetry_r2_hidden,redefinition_k1_msualg_3,redefinition_k4_relset_1,dt_k13_msafree,dt_k15_msafree,dt_k1_funct_1,dt_k1_msualg_3,dt_k4_relset_1,dt_u1_struct_0,dt_c10_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff,dt_c7_2__birkhoff,dt_c9_2__birkhoff,dh_c10_2__birkhoff,e10_2__birkhoff]),
    [interesting(0.8),file(birkhoff,e11_2__birkhoff),[file(birkhoff,e11_2__birkhoff)]]).

fof(t23_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(B))
           => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ),
    file(funct_1,t23_funct_1),
    [interesting(0.9),axiom,file(funct_1,t23_funct_1)]).

fof(e1_2_1__birkhoff,plain,(
    c8_2__birkhoff = k1_funct_1(k7_funct_2(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff)),c10_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_m1_finseq_2,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,reflexivity_r1_tarski,abstractness_v4_msualg_1,existence_m4_pboole,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m4_pboole,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,fc7_funcop_1,fc8_funcop_1,rc1_closure2,rc1_extens_1,rc1_funcop_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,rc5_msafree2,rc5_msualg_1,rc5_pboole,rc6_msafree2,rc6_msualg_1,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_msafree,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m3_pboole,redefinition_m2_relset_1,dt_k11_msafree,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc2_funct_1,cc3_msafree2,cc3_pboole,cc4_msafree2,fc1_msualg_1,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k1_msualg_3,redefinition_k4_relset_1,redefinition_k7_funct_2,dt_k13_msafree,dt_k15_msafree,dt_k1_funct_1,dt_k1_msualg_3,dt_k1_relat_1,dt_k4_relset_1,dt_k5_relat_1,dt_k7_funct_2,dt_u1_struct_0,dt_u4_msualg_1,dt_c10_2__birkhoff,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,dt_c8_2__birkhoff,dt_c9_2__birkhoff,de_c7_2__birkhoff,fc1_funct_1,rc1_funct_1,t1_subset,t7_boole,e6_2__birkhoff,e11_2__birkhoff,t23_funct_1]),
    [interesting(0.65),file(birkhoff,e1_2_1__birkhoff),[file(birkhoff,e1_2_1__birkhoff)]]).

fof(t2_msualg_3,theorem,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ! [D] : 
              ( m1_pboole(D,A)
             => ! [E] : 
                  ( m3_pboole(E,A,B,C)
                 => ! [F] : 
                      ( m3_pboole(F,A,C,D)
                     => ( k1_relat_1(k13_pboole(E,F)) = A
                        & ! [G] : 
                            ( r2_hidden(G,A)
                           => k1_funct_1(k13_pboole(E,F),G) = k5_relat_1(k1_funct_1(E,G),k1_funct_1(F,G)) ) ) ) ) ) ) ) ),
    file(msualg_3,t2_msualg_3),
    [interesting(0.9),axiom,file(msualg_3,t2_msualg_3)]).

fof(e2_2_1__birkhoff,plain,(
    k1_funct_1(k7_funct_2(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c4_2__birkhoff,c7_2__birkhoff)),c10_2__birkhoff) = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k3_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c4_2__birkhoff),c7_2__birkhoff),c10_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c8_2__birkhoff,e3_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_m1_finseq_2,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,reflexivity_r1_tarski,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,abstractness_v4_msualg_1,existence_m4_pboole,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m4_pboole,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,rc1_closure2,rc1_extens_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc5_msafree2,rc5_msualg_1,rc5_pboole,rc5_struct_0,rc6_msafree2,rc6_msualg_1,t3_subset,t4_subset,t5_subset,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_msafree,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k11_msafree,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc2_funct_1,cc3_msafree2,cc4_msafree2,fc1_funct_1,fc1_msualg_1,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,fc7_funcop_1,fc8_funcop_1,rc1_funcop_1,rc1_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc4_pboole,rc5_funct_1,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_pboole,existence_m3_pboole,redefinition_k1_msualg_3,redefinition_k3_msualg_3,redefinition_k7_funct_2,dt_k13_msafree,dt_k13_pboole,dt_k15_msafree,dt_k1_funct_1,dt_k1_msualg_3,dt_k1_relat_1,dt_k3_msualg_3,dt_k5_relat_1,dt_k7_funct_2,dt_m1_pboole,dt_m3_pboole,dt_u1_struct_0,dt_u4_msualg_1,dt_c10_2__birkhoff,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,de_c7_2__birkhoff,cc3_pboole,t1_subset,t7_boole,t2_msualg_3]),
    [interesting(0.65),file(birkhoff,e2_2_1__birkhoff),[file(birkhoff,e2_2_1__birkhoff)]]).

fof(d1_birkhoff,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] : 
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] : 
              ( ( v5_msualg_1(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] : 
                  ( m3_pboole(D,u1_struct_0(A),B,u4_msualg_1(A,C))
                 => ! [E] : 
                      ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,C))
                     => ( E = k1_birkhoff(A,B,C,D)
                      <=> ( r1_msualg_3(A,k11_msafree(A,B),C,E)
                          & r6_pboole(u1_struct_0(A),k1_msafree(u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,C),k13_msafree(A,B),E),k3_msualg_3(u1_struct_0(A),k13_msafree(A,B),B,u4_msualg_1(A,C),k15_msafree(A,B),D)) ) ) ) ) ) ) ) ),
    file(birkhoff,d1_birkhoff),
    [interesting(0.9),axiom,file(birkhoff,d1_birkhoff)]).

fof(e3_2_1__birkhoff,plain,(
    k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k3_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c4_2__birkhoff),c7_2__birkhoff),c10_2__birkhoff) = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k1_msafree(u1_struct_0(c1_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c7_2__birkhoff),c10_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c8_2__birkhoff,e3_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,reflexivity_r1_tarski,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,antisymmetry_r2_hidden,existence_m1_relset_1,dt_g3_msualg_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m1_relset_1,dt_u5_msualg_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,fc6_msualg_2,rc1_closure2,rc1_membered,rc3_funct_1,rc5_msafree2,rc5_struct_0,rc6_msafree2,t1_subset,t3_subset,t4_subset,t5_subset,abstractness_v4_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_m1_msafree,existence_m1_subset_1,existence_m2_relset_1,existence_m4_pboole,redefinition_m2_relset_1,dt_k13_pboole,dt_l1_struct_0,dt_l2_msualg_1,dt_m1_msafree,dt_m1_subset_1,dt_m2_relset_1,dt_m4_pboole,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_pboole,cc2_funct_1,cc2_pboole,cc3_msafree2,cc4_msafree2,fc1_msualg_1,fc1_struct_0,fc2_pboole,fc7_funcop_1,rc1_extens_1,rc1_funcop_1,rc1_funct_1,rc1_pboole,rc2_funct_1,rc2_pboole,rc3_pboole,rc3_struct_0,rc4_funct_1,rc4_pboole,rc5_funct_1,rc5_msualg_1,rc5_pboole,rc6_msualg_1,t2_subset,t6_boole,t7_boole,t8_boole,symmetry_r6_pboole,reflexivity_r6_pboole,existence_l1_msualg_1,existence_l3_msualg_1,existence_m1_pboole,existence_m3_pboole,redefinition_k1_msualg_3,redefinition_k3_msualg_3,redefinition_r6_pboole,dt_k11_msafree,dt_k13_msafree,dt_k15_msafree,dt_k1_birkhoff,dt_k1_funct_1,dt_k1_msafree,dt_k1_msualg_3,dt_k3_msualg_3,dt_l1_msualg_1,dt_l3_msualg_1,dt_m1_pboole,dt_m3_pboole,dt_u1_struct_0,dt_u4_msualg_1,dt_c10_2__birkhoff,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,de_c7_2__birkhoff,cc1_msafree2,cc3_pboole,fc1_msualg_9,fc2_msualg_9,d1_birkhoff]),
    [interesting(0.65),file(birkhoff,e3_2_1__birkhoff),[file(birkhoff,e3_2_1__birkhoff)]]).

fof(d1_msafree,definition,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ! [D] : 
              ( m4_pboole(D,A,B)
             => ! [E] : 
                  ( m3_pboole(E,A,B,C)
                 => ! [F] : 
                      ( m3_pboole(F,A,D,C)
                     => ( F = k1_msafree(A,B,C,D,E)
                      <=> ! [G] : 
                            ( r2_hidden(G,A)
                           => ! [H] : 
                                ( ( v1_funct_1(H)
                                  & v1_funct_2(H,k1_funct_1(B,G),k1_funct_1(C,G))
                                  & m2_relset_1(H,k1_funct_1(B,G),k1_funct_1(C,G)) )
                               => ( H = k1_funct_1(E,G)
                                 => k1_funct_1(F,G) = k8_relset_1(k1_funct_1(B,G),k1_funct_1(C,G),H,k1_funct_1(D,G)) ) ) ) ) ) ) ) ) ) ),
    file(msafree,d1_msafree),
    [interesting(0.9),axiom,file(msafree,d1_msafree)]).

fof(e4_2_1__birkhoff,plain,(
    k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k1_msafree(u1_struct_0(c1_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c7_2__birkhoff),c10_2__birkhoff) = k1_funct_1(k8_relset_1(k1_funct_1(u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff),c7_2__birkhoff),k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff)),c10_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c8_2__birkhoff,e3_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,reflexivity_r1_tarski,abstractness_v4_msualg_1,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,rc1_closure2,rc1_extens_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc5_msafree2,rc5_msualg_1,rc6_msafree2,rc6_msualg_1,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_msafree,existence_m1_relset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k7_relat_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,dt_m1_relset_1,dt_m1_subset_1,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc1_relset_1,cc2_funct_1,cc3_msafree2,cc4_msafree2,fc1_msualg_1,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,fc4_funct_1,fc7_funcop_1,rc1_funcop_1,rc1_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc4_pboole,rc5_funct_1,rc5_pboole,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_pboole,existence_m2_relset_1,existence_m3_pboole,existence_m4_pboole,redefinition_k1_msualg_3,redefinition_k8_relset_1,redefinition_m2_relset_1,dt_k11_msafree,dt_k13_msafree,dt_k1_birkhoff,dt_k1_funct_1,dt_k1_msafree,dt_k1_msualg_3,dt_k8_relset_1,dt_m1_pboole,dt_m2_relset_1,dt_m3_pboole,dt_m4_pboole,dt_u1_struct_0,dt_u4_msualg_1,dt_c10_2__birkhoff,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,de_c7_2__birkhoff,cc3_pboole,t1_subset,t7_boole,d1_msafree]),
    [interesting(0.65),file(birkhoff,e4_2_1__birkhoff),[file(birkhoff,e4_2_1__birkhoff)]]).

fof(t72_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,A)
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ),
    file(funct_1,t72_funct_1),
    [interesting(0.9),axiom,file(funct_1,t72_funct_1)]).

fof(e5_2_1__birkhoff,plain,(
    k1_funct_1(k8_relset_1(k1_funct_1(u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff),c7_2__birkhoff),k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff)),c10_2__birkhoff) = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff),c7_2__birkhoff),c10_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,reflexivity_r1_tarski,abstractness_v4_msualg_1,existence_m4_pboole,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m4_pboole,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,fc7_funcop_1,rc1_closure2,rc1_extens_1,rc1_funcop_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,rc5_msafree2,rc5_msualg_1,rc5_pboole,rc6_msafree2,rc6_msualg_1,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_msafree,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m3_pboole,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc2_funct_1,cc3_msafree2,cc3_pboole,cc4_msafree2,fc1_msualg_1,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k1_msualg_3,redefinition_k4_relset_1,redefinition_k8_relset_1,dt_k11_msafree,dt_k13_msafree,dt_k15_msafree,dt_k1_birkhoff,dt_k1_funct_1,dt_k1_msualg_3,dt_k4_relset_1,dt_k7_relat_1,dt_k8_relset_1,dt_u1_struct_0,dt_u4_msualg_1,dt_c10_2__birkhoff,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,dt_c9_2__birkhoff,de_c7_2__birkhoff,fc4_funct_1,rc1_funct_1,t1_subset,t7_boole,e11_2__birkhoff,t72_funct_1]),
    [interesting(0.65),file(birkhoff,e5_2_1__birkhoff),[file(birkhoff,e5_2_1__birkhoff)]]).

fof(e16_2__birkhoff,plain,(
    c8_2__birkhoff = k1_funct_1(k1_msualg_3(u1_struct_0(c1_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff),c7_2__birkhoff),c10_2__birkhoff) ),
    inference(iterative_eq,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c8_2__birkhoff,e3_2__birkhoff])],[dt_k13_finseq_1,dt_m1_finseq_2,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,abstractness_v4_msualg_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc7_funcop_1,fc8_funcop_1,rc1_closure2,rc1_extens_1,rc1_funcop_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,rc5_msafree2,rc5_msualg_1,rc5_struct_0,rc6_msafree2,rc6_msualg_1,redefinition_m2_relset_1,dt_k13_pboole,dt_k5_relat_1,dt_k7_relat_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,dt_m4_pboole,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc2_funct_1,cc3_msafree2,cc3_pboole,cc4_msafree2,fc1_funct_1,fc1_msualg_1,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,fc4_funct_1,rc1_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_pboole,redefinition_k1_msualg_3,redefinition_k3_msualg_3,redefinition_k7_funct_2,redefinition_k8_relset_1,dt_k11_msafree,dt_k13_msafree,dt_k15_msafree,dt_k1_birkhoff,dt_k1_funct_1,dt_k1_msafree,dt_k1_msualg_3,dt_k3_msualg_3,dt_k7_funct_2,dt_k8_relset_1,dt_u1_struct_0,dt_u4_msualg_1,dt_c10_2__birkhoff,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,dt_c8_2__birkhoff,e1_2_1__birkhoff,e2_2_1__birkhoff,e3_2_1__birkhoff,e4_2_1__birkhoff,e5_2_1__birkhoff]),
    [interesting(0.8),file(birkhoff,e16_2__birkhoff),[file(birkhoff,e16_2__birkhoff)]]).

fof(d1_funct_2,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( ( ( B = k1_xboole_0
           => A = k1_xboole_0 )
         => ( v1_funct_2(C,A,B)
          <=> A = k4_relset_1(A,B,C) ) )
        & ( B = k1_xboole_0
         => ( A = k1_xboole_0
            | ( v1_funct_2(C,A,B)
            <=> C = k1_xboole_0 ) ) ) ) ) ),
    file(funct_2,d1_funct_2),
    [interesting(0.9),axiom,file(funct_2,d1_funct_2)]).

fof(e12_2__birkhoff,plain,(
    k4_relset_1(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(c3_2__birkhoff,c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c3_2__birkhoff,k15_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff)) = k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c3_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,existence_l2_msualg_1,dt_g3_msualg_1,dt_l2_msualg_1,dt_u5_msualg_1,fc1_msualg_1,fc6_msualg_2,rc1_closure2,rc5_msafree2,rc6_msafree2,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v4_msualg_1,existence_l3_msualg_1,existence_m4_pboole,dt_l3_msualg_1,dt_m4_pboole,dt_u4_msualg_1,cc2_pboole,cc3_msafree2,cc4_msafree2,fc7_funcop_1,rc1_extens_1,rc1_funcop_1,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,rc5_msualg_1,rc5_pboole,rc6_msualg_1,t1_subset,t4_subset,t5_subset,existence_l1_msualg_1,existence_l1_struct_0,existence_m1_msafree,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m3_pboole,dt_k11_msafree,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_m1_msafree,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m3_pboole,dt_c6_2__birkhoff,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_closure2,cc1_funct_1,cc1_membered,cc1_msafree2,cc1_pboole,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc3_pboole,cc4_membered,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,rc1_funct_1,rc1_membered,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k1_msualg_3,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k13_msafree,dt_k15_msafree,dt_k1_funct_1,dt_k1_msualg_3,dt_k1_xboole_0,dt_k4_relset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_2__birkhoff,dt_c3_2__birkhoff,dt_c7_2__birkhoff,de_c7_2__birkhoff,fc6_membered,t6_boole,d1_funct_2]),
    [interesting(0.8),file(birkhoff,e12_2__birkhoff),[file(birkhoff,e12_2__birkhoff)]]).

fof(e13_2__birkhoff,plain,(
    k4_relset_1(k1_funct_1(u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff),c7_2__birkhoff)) = k1_funct_1(u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),c7_2__birkhoff) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v4_msualg_1,cc2_pboole,fc7_funcop_1,rc1_closure2,rc1_funcop_1,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,rc5_msafree2,rc5_msualg_1,rc6_msafree2,rc6_msualg_1,t1_subset,t4_subset,t5_subset,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m3_pboole,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m3_pboole,dt_c6_2__birkhoff,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_closure2,cc1_funct_1,cc1_membered,cc1_msafree2,cc1_pboole,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc3_msafree2,cc3_pboole,cc4_membered,cc4_msafree2,fc1_msualg_1,fc1_struct_0,fc2_msualg_9,fc2_pboole,rc1_funct_1,rc1_membered,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k1_msualg_3,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k11_msafree,dt_k1_birkhoff,dt_k1_funct_1,dt_k1_msualg_3,dt_k1_xboole_0,dt_k4_relset_1,dt_m2_relset_1,dt_u1_struct_0,dt_u4_msualg_1,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,de_c7_2__birkhoff,fc6_membered,t6_boole,d1_funct_2]),
    [interesting(0.8),file(birkhoff,e13_2__birkhoff),[file(birkhoff,e13_2__birkhoff)]]).

fof(d23_pboole,definition,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ! [C] : 
          ( m1_pboole(C,A)
         => ( m4_pboole(C,A,B)
          <=> r2_pboole(A,C,B) ) ) ) ),
    file(pboole,d23_pboole),
    [interesting(0.9),axiom,file(pboole,d23_pboole)]).

fof(e14_2__birkhoff,plain,(
    r2_pboole(u1_struct_0(c1_2__birkhoff),k13_msafree(c1_2__birkhoff,c3_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[reflexivity_r1_tarski,dt_k1_zfmisc_1,dt_k2_zfmisc_1,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_relset_1,cc20_membered,rc5_struct_0,t3_subset,t4_subset,t5_subset,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,fc1_funct_1,fc8_funcop_1,rc1_funcop_1,rc4_pboole,existence_m1_subset_1,existence_m3_pboole,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_m1_subset_1,dt_m3_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc3_pboole,fc1_pboole,t2_subset,free_g3_msualg_1,antisymmetry_r2_hidden,dt_g3_msualg_1,dt_k1_xboole_0,dt_u5_msualg_1,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,fc6_msualg_2,rc1_membered,rc3_funct_1,t1_subset,t8_boole,abstractness_v4_msualg_1,cc15_membered,cc1_closure2,cc1_funct_1,cc1_pboole,cc2_funct_1,cc2_pboole,fc1_msualg_1,rc1_extens_1,rc1_pboole,rc2_funct_1,rc2_pboole,rc4_funct_1,rc5_msualg_1,rc6_msualg_1,t6_boole,t7_boole,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_msafree,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,cc1_msafree2,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,rc1_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_pboole,reflexivity_r2_pboole,existence_m1_pboole,existence_m4_pboole,dt_k11_msafree,dt_k13_msafree,dt_m1_pboole,dt_m4_pboole,dt_u1_struct_0,dt_u4_msualg_1,dt_c1_2__birkhoff,dt_c3_2__birkhoff,d23_pboole]),
    [interesting(0.8),file(birkhoff,e14_2__birkhoff),[file(birkhoff,e14_2__birkhoff)]]).

fof(e15_2__birkhoff,plain,(
    r1_tarski(k1_funct_1(k13_msafree(c1_2__birkhoff,c3_2__birkhoff),c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),c7_2__birkhoff)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c6_2__birkhoff,e1_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[dt_k2_zfmisc_1,cc1_relset_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,fc1_funct_1,fc7_funcop_1,fc8_funcop_1,rc1_funcop_1,rc4_pboole,existence_m3_pboole,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_m3_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,cc3_pboole,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,abstractness_v4_msualg_1,existence_m4_pboole,dt_k1_xboole_0,dt_m4_pboole,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc1_msualg_1,fc6_membered,rc1_closure2,rc1_extens_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc5_msafree2,rc5_msualg_1,rc5_pboole,rc6_msafree2,rc6_msualg_1,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_msafree,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,dt_m1_subset_1,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_pboole,cc2_funct_1,cc3_msafree2,cc4_msafree2,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,rc1_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,reflexivity_r1_tarski,antisymmetry_r2_hidden,reflexivity_r2_pboole,existence_m1_pboole,dt_k11_msafree,dt_k13_msafree,dt_k1_funct_1,dt_m1_pboole,dt_u1_struct_0,dt_u4_msualg_1,dt_c1_2__birkhoff,dt_c3_2__birkhoff,dt_c7_2__birkhoff,de_c7_2__birkhoff,t1_subset,t3_subset,t7_boole,e14_2__birkhoff,d5_pboole]),
    [interesting(0.8),file(birkhoff,e15_2__birkhoff),[file(birkhoff,e15_2__birkhoff)]]).

fof(e17_2__birkhoff,plain,(
    r2_hidden(c8_2__birkhoff,k5_relset_1(k1_funct_1(u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),c7_2__birkhoff),k1_funct_1(u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),c7_2__birkhoff),k1_msualg_3(u1_struct_0(c1_2__birkhoff),u4_msualg_1(c1_2__birkhoff,k11_msafree(c1_2__birkhoff,c3_2__birkhoff)),u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff),c7_2__birkhoff))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c8_2__birkhoff,e3_2__birkhoff,dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,cc2_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,abstractness_v4_msualg_1,existence_m4_pboole,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m4_pboole,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,fc7_funcop_1,rc1_closure2,rc1_extens_1,rc1_funcop_1,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc4_pboole,rc5_msafree2,rc5_msualg_1,rc5_pboole,rc6_msafree2,rc6_msualg_1,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_msafree,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m3_pboole,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_msafree,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,dt_c6_2__birkhoff,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_msualg_1,cc1_pboole,cc2_funct_1,cc3_msafree2,cc3_pboole,cc4_msafree2,fc1_msualg_1,fc1_msualg_9,fc1_struct_0,fc2_msualg_9,fc2_pboole,fc6_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k1_msualg_3,redefinition_k4_relset_1,redefinition_k5_relset_1,dt_k11_msafree,dt_k13_msafree,dt_k15_msafree,dt_k1_birkhoff,dt_k1_funct_1,dt_k1_msualg_3,dt_k1_relat_1,dt_k2_relat_1,dt_k4_relset_1,dt_k5_relset_1,dt_u1_struct_0,dt_u4_msualg_1,dt_c10_2__birkhoff,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c7_2__birkhoff,dt_c8_2__birkhoff,dt_c9_2__birkhoff,de_c7_2__birkhoff,rc1_funct_1,t1_subset,t3_subset,t7_boole,e16_2__birkhoff,e11_2__birkhoff,e12_2__birkhoff,e13_2__birkhoff,e15_2__birkhoff,d5_funct_1]),
    [interesting(0.8),file(birkhoff,e17_2__birkhoff),[file(birkhoff,e17_2__birkhoff)]]).

fof(e18_2__birkhoff,plain,(
    r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c8_2__birkhoff,e3_2__birkhoff,dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[existence_m1_finseq_2,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,cc2_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,fc6_msualg_2,reflexivity_r1_tarski,abstractness_v4_msualg_1,dt_k1_xboole_0,dt_k2_zfmisc_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_membered,rc1_closure2,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,rc5_msafree2,rc5_msualg_1,rc6_msafree2,rc6_msualg_1,existence_l1_msualg_1,existence_l1_struct_0,existence_l2_msualg_1,existence_l3_msualg_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m3_pboole,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k3_funct_6,dt_l1_msualg_1,dt_l1_struct_0,dt_l2_msualg_1,dt_l3_msualg_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m3_pboole,cc15_membered,cc1_closure2,cc1_funct_1,cc1_msafree2,cc1_msualg_1,cc1_pboole,cc2_funct_1,cc3_msafree2,cc3_pboole,cc4_msafree2,fc1_msualg_1,fc1_struct_0,fc2_msualg_9,fc2_pboole,fc6_funct_1,rc2_funct_1,rc3_pboole,rc3_struct_0,rc5_funct_1,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_pboole,redefinition_k1_msualg_3,redefinition_k2_mssubfam,redefinition_k5_relset_1,dt_k11_msafree,dt_k1_birkhoff,dt_k1_funct_1,dt_k1_msualg_3,dt_k2_mssubfam,dt_k2_relat_1,dt_k5_relset_1,dt_m1_pboole,dt_u1_struct_0,dt_u4_msualg_1,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,dt_c7_2__birkhoff,dt_c8_2__birkhoff,de_c7_2__birkhoff,fc7_funcop_1,rc1_funcop_1,rc1_funct_1,rc4_pboole,t1_subset,t7_boole,e17_2__birkhoff,t13_mssubfam]),
    [interesting(0.8),file(birkhoff,e18_2__birkhoff),[file(birkhoff,e18_2__birkhoff)]]).

fof(i9_2__birkhoff,theorem,(
    $true ),
    introduced(tautology,[file(birkhoff,i9_2__birkhoff)]),
    [interesting(0.8),trivial,file(birkhoff,i9_2__birkhoff)]).

fof(i8_2__birkhoff,plain,(
    r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) ),
    inference(conclusion,[status(thm),assumptions([dt_c8_2__birkhoff,e3_2__birkhoff,dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[e18_2__birkhoff,i9_2__birkhoff]),
    [interesting(0.8),file(birkhoff,i8_2__birkhoff),[file(birkhoff,i8_2__birkhoff)]]).

fof(i7_2__birkhoff,plain,(
    ~ ( r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff))
      & ~ r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c8_2__birkhoff,dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff]),discharge_asm(discharge,[e3_2__birkhoff])],[e3_2__birkhoff,i8_2__birkhoff]),
    [interesting(0.8),file(birkhoff,i7_2__birkhoff),[file(birkhoff,i7_2__birkhoff)]]).

fof(i7_2_tmp__birkhoff,plain,(
    ~ ( r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff))
      & ~ r2_hidden(c8_2__birkhoff,k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff]),discharge_asm(discharge,[dt_c8_2__birkhoff])],[dt_c8_2__birkhoff,i7_2__birkhoff]),
    [interesting(0.8),i6_2__birkhoff]).

fof(i6_2__birkhoff,plain,(
    r1_tarski(k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff),k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) ),
    inference(let,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,e1_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[i7_2_tmp__birkhoff,rc1_closure2,rc5_msafree2,rc6_msafree2,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_relset_1,cc20_membered,cc3_msafree2,cc4_msafree2,fc2_pboole,rc5_struct_0,redefinition_m2_relset_1,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,cc1_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_msualg_2,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,abstractness_v4_msualg_1,dt_l2_msualg_1,cc15_membered,cc1_closure2,cc1_funct_1,cc1_pboole,cc2_funct_1,fc1_msualg_1,rc2_funct_1,rc5_msualg_1,rc6_msualg_1,dt_k11_msafree,dt_k3_funct_6,dt_l1_msualg_1,dt_l1_struct_0,dt_l3_msualg_1,dt_m1_pboole,dt_m3_pboole,dt_u4_msualg_1,cc1_msafree2,cc3_pboole,fc1_struct_0,fc2_msualg_9,fc7_funcop_1,rc1_funcop_1,rc1_funct_1,rc3_pboole,rc3_struct_0,rc4_pboole,rc5_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k2_mssubfam,dt_k1_birkhoff,dt_k1_funct_1,dt_k2_mssubfam,dt_u1_struct_0,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,d3_tarski,dh_c8_2__birkhoff]),
    [interesting(0.8),file(birkhoff,i6_2__birkhoff),[file(birkhoff,i6_2__birkhoff)]]).

fof(i5_2__birkhoff,plain,(
    ~ ( r2_hidden(c6_2__birkhoff,u1_struct_0(c1_2__birkhoff))
      & ~ r1_tarski(k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff),k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c6_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff]),discharge_asm(discharge,[e1_2__birkhoff])],[e1_2__birkhoff,i6_2__birkhoff]),
    [interesting(0.8),file(birkhoff,i5_2__birkhoff),[file(birkhoff,i5_2__birkhoff)]]).

fof(i5_2_tmp__birkhoff,plain,(
    ~ ( r2_hidden(c6_2__birkhoff,u1_struct_0(c1_2__birkhoff))
      & ~ r1_tarski(k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),c6_2__birkhoff),k1_funct_1(k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff)),c6_2__birkhoff)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff]),discharge_asm(discharge,[dt_c6_2__birkhoff])],[dt_c6_2__birkhoff,i5_2__birkhoff]),
    [interesting(0.8),i4_2__birkhoff]).

fof(i4_2__birkhoff,plain,(
    r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff))) ),
    inference(let,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c4_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[i5_2_tmp__birkhoff,rc1_closure2,rc5_msafree2,rc6_msafree2,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_relset_1,cc20_membered,cc3_msafree2,cc4_msafree2,fc2_pboole,rc5_struct_0,redefinition_m2_relset_1,dt_k13_finseq_1,dt_k5_relat_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,fc1_funct_1,fc8_funcop_1,redefinition_k3_finseq_2,redefinition_k7_pboole,dt_k3_finseq_2,dt_k6_pboole,dt_k7_pboole,dt_u1_msualg_1,dt_u2_msualg_1,dt_u3_msualg_1,fc1_pboole,free_g3_msualg_1,dt_g3_msualg_1,dt_u5_msualg_1,cc1_membered,cc2_membered,cc2_pboole,cc3_membered,cc4_membered,fc6_msualg_2,rc1_membered,rc1_pboole,rc2_pboole,rc3_funct_1,rc4_funct_1,abstractness_v4_msualg_1,dt_l2_msualg_1,cc15_membered,cc1_closure2,cc1_funct_1,cc1_pboole,cc2_funct_1,fc1_msualg_1,rc2_funct_1,rc5_msualg_1,rc6_msualg_1,dt_k11_msafree,dt_k3_funct_6,dt_l1_msualg_1,dt_l1_struct_0,dt_l3_msualg_1,dt_m3_pboole,dt_u4_msualg_1,cc1_msafree2,cc3_pboole,fc1_struct_0,fc2_msualg_9,fc7_funcop_1,rc1_funcop_1,rc1_funct_1,rc3_pboole,rc3_struct_0,rc4_pboole,rc5_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,reflexivity_r2_pboole,redefinition_k2_mssubfam,dt_k1_birkhoff,dt_k1_funct_1,dt_k2_mssubfam,dt_m1_pboole,dt_u1_struct_0,dt_c1_2__birkhoff,dt_c2_2__birkhoff,dt_c3_2__birkhoff,dt_c4_2__birkhoff,d5_pboole,dh_c6_2__birkhoff]),
    [interesting(0.8),file(birkhoff,i4_2__birkhoff),[file(birkhoff,i4_2__birkhoff)]]).

fof(i4_2_tmp__birkhoff,plain,
    ( m3_pboole(c4_2__birkhoff,u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
   => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),c4_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,c4_2__birkhoff))) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff]),discharge_asm(discharge,[dt_c4_2__birkhoff])],[dt_c4_2__birkhoff,i4_2__birkhoff]),
    [interesting(0.8),i3_2__birkhoff]).

fof(i3_2__birkhoff,plain,(
    ! [A] : 
      ( m3_pboole(A,u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
     => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),A),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,A))) ) ),
    inference(let,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c1_2__birkhoff,dt_c3_2__birkhoff])],[i4_2_tmp__birkhoff,dh_c4_2__birkhoff]),
    [interesting(0.8),file(birkhoff,i3_2__birkhoff),[file(birkhoff,i3_2__birkhoff)]]).

fof(i3_2_tmp__birkhoff,plain,
    ( ( v2_relat_1(c3_2__birkhoff)
      & m1_pboole(c3_2__birkhoff,u1_struct_0(c1_2__birkhoff)) )
   => ! [A] : 
        ( m3_pboole(A,u1_struct_0(c1_2__birkhoff),c3_2__birkhoff,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
       => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),A),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,c3_2__birkhoff,c2_2__birkhoff,A))) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c1_2__birkhoff]),discharge_asm(discharge,[dt_c3_2__birkhoff])],[dt_c3_2__birkhoff,i3_2__birkhoff]),
    [interesting(0.8),i2_2__birkhoff]).

fof(i2_2__birkhoff,plain,(
    ! [A] : 
      ( ( v2_relat_1(A)
        & m1_pboole(A,u1_struct_0(c1_2__birkhoff)) )
     => ! [B] : 
          ( m3_pboole(B,u1_struct_0(c1_2__birkhoff),A,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
         => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),B),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,A,c2_2__birkhoff,B))) ) ) ),
    inference(let,[status(thm),assumptions([dt_c2_2__birkhoff,dt_c1_2__birkhoff])],[i3_2_tmp__birkhoff,dh_c3_2__birkhoff]),
    [interesting(0.8),file(birkhoff,i2_2__birkhoff),[file(birkhoff,i2_2__birkhoff)]]).

fof(i2_2_tmp__birkhoff,plain,
    ( ( v5_msualg_1(c2_2__birkhoff,c1_2__birkhoff)
      & l3_msualg_1(c2_2__birkhoff,c1_2__birkhoff) )
   => ! [A] : 
        ( ( v2_relat_1(A)
          & m1_pboole(A,u1_struct_0(c1_2__birkhoff)) )
       => ! [B] : 
            ( m3_pboole(B,u1_struct_0(c1_2__birkhoff),A,u4_msualg_1(c1_2__birkhoff,c2_2__birkhoff))
           => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),B),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,A,c2_2__birkhoff,B))) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__birkhoff]),discharge_asm(discharge,[dt_c2_2__birkhoff])],[dt_c2_2__birkhoff,i2_2__birkhoff]),
    [interesting(0.8),i1_2__birkhoff]).

fof(i1_2__birkhoff,plain,(
    ! [A] : 
      ( ( v5_msualg_1(A,c1_2__birkhoff)
        & l3_msualg_1(A,c1_2__birkhoff) )
     => ! [B] : 
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(c1_2__birkhoff)) )
         => ! [C] : 
              ( m3_pboole(C,u1_struct_0(c1_2__birkhoff),B,u4_msualg_1(c1_2__birkhoff,A))
             => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),C),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,B,A,C))) ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_2__birkhoff])],[i2_2_tmp__birkhoff,dh_c2_2__birkhoff]),
    [interesting(0.8),file(birkhoff,i1_2__birkhoff),[file(birkhoff,i1_2__birkhoff)]]).

fof(i1_2_tmp__birkhoff,plain,
    ( ( ~ v3_struct_0(c1_2__birkhoff)
      & ~ v2_msualg_1(c1_2__birkhoff)
      & l1_msualg_1(c1_2__birkhoff) )
   => ! [A] : 
        ( ( v5_msualg_1(A,c1_2__birkhoff)
          & l3_msualg_1(A,c1_2__birkhoff) )
       => ! [B] : 
            ( ( v2_relat_1(B)
              & m1_pboole(B,u1_struct_0(c1_2__birkhoff)) )
           => ! [C] : 
                ( m3_pboole(C,u1_struct_0(c1_2__birkhoff),B,u4_msualg_1(c1_2__birkhoff,A))
               => r2_pboole(u1_struct_0(c1_2__birkhoff),k2_mssubfam(u1_struct_0(c1_2__birkhoff),C),k2_mssubfam(u1_struct_0(c1_2__birkhoff),k1_birkhoff(c1_2__birkhoff,B,A,C))) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__birkhoff])],[dt_c1_2__birkhoff,i1_2__birkhoff]),
    [interesting(1),t1_birkhoff]).

fof(t1_birkhoff,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] : 
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] : 
              ( ( v2_relat_1(C)
                & m1_pboole(C,u1_struct_0(A)) )
             => ! [D] : 
                  ( m3_pboole(D,u1_struct_0(A),C,u4_msualg_1(A,B))
                 => r2_pboole(u1_struct_0(A),k2_mssubfam(u1_struct_0(A),D),k2_mssubfam(u1_struct_0(A),k1_birkhoff(A,C,B,D))) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_2_tmp__birkhoff,dh_c1_2__birkhoff]),
    [interesting(1),file(birkhoff,t1_birkhoff),[file(birkhoff,t1_birkhoff)]]).