fof(t14_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ( r5_alg_1(A,B)
           => r5_alg_1(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_alg_1,d4_alg_1,d4_alg_1,d2_alg_1,t55_funct_1,d3_alg_1,t55_funct_1,d1_funct_2,d1_funct_2,t11_relset_1,d5_alg_1,t11_alg_1]),
    [file(alg_1,t14_alg_1),interesting(1.00)]).

fof(t13_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => r5_alg_1(A,A) ) ),
    inference(mizar_proof,[status(thm)],[t6_alg_1,t71_relat_1,t8_alg_1,d5_alg_1]),
    [file(alg_1,t13_alg_1),interesting(0.94)]).

fof(t21_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( m1_alg_1(B,A)
         => r1_alg_1(A,k5_alg_1(A,B),k6_alg_1(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d13_alg_1,d11_unialg_1,d3_finseq_1,t6_unialg_2,d11_unialg_1,d13_alg_1,d12_alg_1,t1_unialg_1,d1_xboole_0,d4_finseq_2,d10_unialg_1,d11_unialg_1,t10_finseq_2,d2_unialg_2,d1_alg_1,d3_finseq_1,d3_finseq_1,d11_finseq_1,t2_unialg_2,d4_finseq_2,d5_funct_1,t12_relset_1,d15_alg_1,d13_alg_1,d11_finseq_1,d12_alg_1,d4_finseq_2,t1_alg_1,t1_alg_1,d4_finseq_1,d5_funct_1,d15_alg_1,t1_alg_1,d11_alg_1,d12_alg_1]),
    [file(alg_1,t21_alg_1),interesting(0.81)]).

fof(t6_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => r1_alg_1(A,A,k6_partfun1(u1_struct_0(A))) ) ),
    inference(mizar_proof,[status(thm)],[d1_alg_1,d5_funct_1,t12_relset_1,t4_alg_1,t35_funct_1]),
    [file(alg_1,t6_alg_1),interesting(0.80)]).

fof(t22_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( m1_alg_1(B,A)
         => r3_alg_1(A,k5_alg_1(A,B),k6_alg_1(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t21_alg_1,d3_alg_1,t12_relset_1,d10_xboole_0,d3_tarski,d5_eqrel_1,d1_funct_2,d15_alg_1,d5_funct_1]),
    [file(alg_1,t22_alg_1),interesting(0.74)]).

fof(t18_alg_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => k2_alg_1(A,k6_partfun1(A)) = k6_partfun1(k3_finseq_2(A)) ) ),
    inference(mizar_proof,[status(thm)],[t106_zfmisc_1,d9_alg_1,d3_finseq_1,d9_alg_1,d10_relat_1,t10_finseq_2,d10_relat_1,t106_zfmisc_1,d10_relat_1,t13_finseq_2,d10_relat_1,d9_alg_1,d2_relat_1]),
    [file(alg_1,t18_alg_1),interesting(0.72)]).

fof(t19_alg_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] : 
              ( m2_finseq_1(C,k8_eqrel_1(A,B))
             => ? [D] : 
                  ( m2_finseq_1(D,A)
                  & r6_alg_1(A,B,C,D) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_funct_1,d4_finseq_1,t45_eqrel_1,s1_wellord2,d3_finseq_1,d2_finseq_1,d4_finseq_1,t31_finseq_3,d11_alg_1,d5_funct_1]),
    [file(alg_1,t19_alg_1),interesting(0.63)]).

fof(t4_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( m2_finseq_1(B,u1_struct_0(A))
         => k5_finseqop(u1_struct_0(A),u1_struct_0(A),B,k6_partfun1(u1_struct_0(A))) = B ) ) ),
    inference(mizar_proof,[status(thm)],[t1_alg_1,t1_alg_1,d3_finseq_1,t13_finseq_2,t35_funct_1,t1_alg_1,t10_finseq_2]),
    [file(alg_1,t4_alg_1),interesting(0.59)]).

fof(t11_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
                 => ( ( r4_alg_1(A,B,C)
                      & D = k2_funct_1(C) )
                   => r4_alg_1(B,A,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t8_alg_1,t62_funct_1,t55_funct_1,d1_funct_2,t10_alg_1,t8_alg_1]),
    [file(alg_1,t11_alg_1),interesting(0.57)]).

fof(t10_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
                 => ( ( r4_alg_1(A,B,C)
                      & D = k2_funct_1(C) )
                   => r1_alg_1(B,A,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t8_alg_1,d1_alg_1,d2_unialg_2,d11_unialg_1,d3_finseq_1,t2_unialg_2,d4_finseq_2,d3_finseq_1,t13_finseq_2,d5_funct_1,s5_finseq_1,t61_funct_1,d1_funct_2,t1_alg_1,d3_finseq_1,t1_alg_1,t17_finseq_1,t5_alg_1,t4_alg_1,d11_finseq_1,d11_unialg_1,d4_finseq_2,t2_unialg_2,d1_alg_1,t27_partfun1,d1_funct_2,t23_funct_1,t34_funct_1,d1_alg_1]),
    [file(alg_1,t10_alg_1),interesting(0.47)]).

fof(t24_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r3_alg_1(A,B,C)
               => r4_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_alg_1,t23_alg_1,d2_alg_1,d3_alg_1,d1_funct_2,t12_relset_1,d10_xboole_0,d3_tarski,d5_funct_1,d1_funct_2,d5_eqrel_1,d17_alg_1,d5_funct_1,t8_alg_1]),
    [file(alg_1,t24_alg_1),interesting(0.44)]).

fof(t25_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r3_alg_1(A,B,C)
               => r5_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_alg_1,t24_alg_1]),
    [file(alg_1,t25_alg_1),interesting(0.39)]).

fof(t2_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( B = u1_struct_0(A)
           => k3_unialg_2(A,B) = u1_unialg_1(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_unialg_2,t6_unialg_2,d7_unialg_2,t7_unialg_2,t17_finseq_1]),
    [file(alg_1,t2_alg_1),interesting(0.38)]).

fof(t12_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & v6_unialg_1(C)
                & v7_unialg_1(C)
                & v8_unialg_1(C)
                & l1_unialg_1(C) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] : 
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ( ( r4_alg_1(A,B,D)
                          & r4_alg_1(B,C,E) )
                       => r4_alg_1(A,C,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t8_alg_1,d1_funct_2,t9_alg_1,t47_relat_1,t9_alg_1,t46_funct_1,t8_alg_1,t7_alg_1,t8_alg_1]),
    [file(alg_1,t12_alg_1),interesting(0.37)]).

fof(t23_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ( r1_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C))
                  & r2_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t12_relset_1,d1_alg_1,t21_alg_1,d1_alg_1,d2_unialg_2,d2_unialg_2,d2_unialg_2,d1_alg_1,d11_finseq_1,d3_finseq_1,d11_unialg_1,t6_unialg_2,t19_alg_1,d11_finseq_1,d13_alg_1,d12_alg_1,d4_finseq_2,d11_alg_1,t2_unialg_2,d4_finseq_2,d5_funct_1,t12_relset_1,d12_alg_1,d17_alg_1,t1_alg_1,t1_alg_1,d3_finseq_1,d3_finseq_1,d11_alg_1,t13_finseq_2,t1_alg_1,d17_alg_1,t1_alg_1,t10_finseq_2,d1_alg_1,d8_funct_1,d5_eqrel_1,d5_eqrel_1,d17_alg_1,d16_alg_1,t44_eqrel_1,d2_alg_1]),
    [file(alg_1,t23_alg_1),interesting(0.36)]).

fof(t9_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r4_alg_1(A,B,C)
               => ( k4_relset_1(u1_struct_0(A),u1_struct_0(B),C) = u1_struct_0(A)
                  & k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_alg_1,d3_alg_1,d1_funct_2]),
    [file(alg_1,t9_alg_1),interesting(0.20)]).

fof(t8_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r4_alg_1(A,B,C)
              <=> ( r1_alg_1(A,B,C)
                  & k2_relat_1(C) = u1_struct_0(B)
                  & v2_funct_1(C) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_alg_1,d2_alg_1,d3_alg_1,d2_alg_1,d3_alg_1,d4_alg_1]),
    [file(alg_1,t8_alg_1),interesting(0.17)]).

fof(t1_alg_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( m2_finseq_1(C,A)
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,B)
                    & m2_relset_1(D,A,B) )
                 => ( k4_finseq_1(k5_finseqop(A,B,C,D)) = k4_finseq_1(C)
                    & k3_finseq_1(k5_finseqop(A,B,C,D)) = k3_finseq_1(C)
                    & ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k4_finseq_1(k5_finseqop(A,B,C,D)))
                         => k1_funct_1(k5_finseqop(A,B,C,D),E) = k1_funct_1(D,k1_funct_1(C,E)) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_finseq_1,d1_funct_2,t12_relset_1,t45_relat_1,t46_relat_1,t46_relat_1,t31_finseq_3,t22_funct_1]),
    [file(alg_1,t1_alg_1),interesting(0.12)]).

fof(t5_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & v6_unialg_1(C)
                & v7_unialg_1(C)
                & v8_unialg_1(C)
                & l1_unialg_1(C) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] : 
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ! [F] : 
                          ( m2_finseq_1(F,u1_struct_0(A))
                         => k5_finseqop(u1_struct_0(B),u1_struct_0(C),k5_finseqop(u1_struct_0(A),u1_struct_0(B),F,D),E) = k5_finseqop(u1_struct_0(A),u1_struct_0(C),F,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t1_alg_1,t1_alg_1,t1_alg_1,t1_alg_1,t1_alg_1,t1_alg_1,d3_finseq_1,t13_finseq_2,t1_alg_1,t1_alg_1,t21_funct_2,t1_alg_1,t10_finseq_2]),
    [file(alg_1,t5_alg_1),interesting(0.11)]).

fof(t3_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => k5_finseqop(u1_struct_0(A),u1_struct_0(B),k6_finseq_1(u1_struct_0(A)),C) = k6_finseq_1(u1_struct_0(B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_finseq_1,t1_alg_1,t4_finseq_1,d3_finseq_1,t26_finseq_1]),
    [file(alg_1,t3_alg_1),interesting(0.11)]).

fof(t20_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( m1_alg_1(B,A)
         => ( ~ v3_struct_0(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & v3_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & v6_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & v7_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & v8_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & l1_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_finseq_1,d13_alg_1,d3_finseq_1,t6_unialg_2,d13_alg_1,d4_unialg_1,d3_finseq_1,d13_alg_1,d3_finseq_1,t6_unialg_2,d13_alg_1,d5_unialg_1,d3_finseq_1,d13_alg_1,d3_finseq_1,t6_unialg_2,d13_alg_1,d15_funct_1,t25_finseq_1,d13_alg_1,t25_finseq_1,d7_unialg_1,d8_unialg_1,d9_unialg_1]),
    [file(alg_1,t20_alg_1),interesting(0.02)]).

fof(t16_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => k2_relat_1(C) = u1_struct_0(k1_alg_1(A,B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t146_relat_1,d6_alg_1]),
    [file(alg_1,t16_alg_1),interesting(0.01)]).

fof(d1_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
              <=> ( r1_unialg_2(A,B)
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(u1_unialg_1(A)))
                       => ! [E] : 
                            ( m2_unialg_2(E,u1_struct_0(A),k1_unialg_2(A))
                           => ! [F] : 
                                ( m2_unialg_2(F,u1_struct_0(B),k1_unialg_2(B))
                               => ( ( E = k1_funct_1(u1_unialg_1(A),D)
                                    & F = k1_funct_1(u1_unialg_1(B),D) )
                                 => ! [G] : 
                                      ( m2_finseq_1(G,u1_struct_0(A))
                                     => ( r2_hidden(G,k4_relset_1(k3_finseq_2(u1_struct_0(A)),u1_struct_0(A),E))
                                       => k1_funct_1(C,k1_funct_1(E,G)) = k1_funct_1(F,k5_finseqop(u1_struct_0(A),u1_struct_0(B),G,C)) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file(alg_1,d1_alg_1),
    [interesting(0.00)]).

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(0.00)]).

fof(t12_relset_1,theorem,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( r1_tarski(k1_relat_1(C),A)
        & r1_tarski(k2_relat_1(C),B) ) ) ),
    file(relset_1,t12_relset_1),
    [interesting(0.00)]).

fof(d4_finseq_1,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( m1_finseq_1(B,A)
      <=> r1_tarski(k2_relat_1(B),A) ) ) ),
    file(finseq_1,d4_finseq_1),
    [interesting(0.00)]).

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(0.00)]).

fof(t45_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => r1_tarski(k2_relat_1(k5_relat_1(A,B)),k2_relat_1(B)) ) ) ),
    file(relat_1,t45_relat_1),
    [interesting(0.00)]).

fof(t46_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(k2_relat_1(A),k1_relat_1(B))
           => k1_relat_1(k5_relat_1(A,B)) = k1_relat_1(A) ) ) ) ),
    file(relat_1,t46_relat_1),
    [interesting(0.00)]).

fof(t31_finseq_3,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( k3_finseq_1(A) = k3_finseq_1(B)
          <=> k4_finseq_1(A) = k4_finseq_1(B) ) ) ) ),
    file(finseq_3,t31_finseq_3),
    [interesting(0.00)]).

fof(t22_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(k5_relat_1(C,B)))
           => k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) ),
    file(funct_1,t22_funct_1),
    [interesting(0.00)]).

fof(d3_finseq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k3_finseq_1(A)
          <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ),
    file(finseq_1,d3_finseq_1),
    [interesting(0.00)]).

fof(t13_finseq_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] : 
          ( m2_finseq_1(C,B)
         => ( r2_hidden(A,k4_finseq_1(C))
           => r2_hidden(k1_funct_1(C,A),B) ) ) ) ),
    file(finseq_2,t13_finseq_2),
    [interesting(0.00)]).

fof(t35_funct_1,theorem,(
    ! [A,B] : 
      ( r2_hidden(B,A)
     => k1_funct_1(k6_relat_1(A),B) = B ) ),
    file(funct_1,t35_funct_1),
    [interesting(0.00)]).

fof(t10_finseq_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( k3_finseq_1(B) = A
                  & k3_finseq_1(C) = A
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k2_finseq_1(A))
                       => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) )
               => B = C ) ) ) ) ),
    file(finseq_2,t10_finseq_2),
    [interesting(0.00)]).

fof(t71_relat_1,theorem,(
    ! [A] : 
      ( k1_relat_1(k6_relat_1(A)) = A
      & k2_relat_1(k6_relat_1(A)) = A ) ),
    file(relat_1,t71_relat_1),
    [interesting(0.00)]).

fof(d4_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r4_alg_1(A,B,C)
              <=> ( r2_alg_1(A,B,C)
                  & r3_alg_1(A,B,C) ) ) ) ) ) ),
    file(alg_1,d4_alg_1),
    [interesting(0.00)]).

fof(d2_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r2_alg_1(A,B,C)
              <=> ( r1_alg_1(A,B,C)
                  & v2_funct_1(C) ) ) ) ) ) ),
    file(alg_1,d2_alg_1),
    [interesting(0.00)]).

fof(d3_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r3_alg_1(A,B,C)
              <=> ( r1_alg_1(A,B,C)
                  & k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ),
    file(alg_1,d3_alg_1),
    [interesting(0.00)]).

fof(d5_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ( r5_alg_1(A,B)
          <=> ? [C] : 
                ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
                & r4_alg_1(A,B,C) ) ) ) ) ),
    file(alg_1,d5_alg_1),
    [interesting(0.00)]).

fof(t55_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ( k2_relat_1(A) = k1_relat_1(k2_funct_1(A))
          & k1_relat_1(A) = k2_relat_1(k2_funct_1(A)) ) ) ) ),
    file(funct_1,t55_funct_1),
    [interesting(0.00)]).

fof(t11_relset_1,theorem,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => ( ( r1_tarski(k1_relat_1(C),A)
          & r1_tarski(k2_relat_1(C),B) )
       => m2_relset_1(C,A,B) ) ) ),
    file(relset_1,t11_relset_1),
    [interesting(0.00)]).

fof(t62_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => v2_funct_1(k2_funct_1(A)) ) ) ),
    file(funct_1,t62_funct_1),
    [interesting(0.00)]).

fof(d2_unialg_2,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ( r1_unialg_2(A,B)
          <=> k3_unialg_1(A) = k3_unialg_1(B) ) ) ) ),
    file(unialg_2,d2_unialg_2),
    [interesting(0.00)]).

fof(d11_unialg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( m2_finseq_1(B,k5_numbers)
         => ( B = k3_unialg_1(A)
          <=> ( k3_finseq_1(B) = k3_finseq_1(u1_unialg_1(A))
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k4_finseq_1(B))
                   => ! [D] : 
                        ( ( ~ v1_xboole_0(D)
                          & v1_funct_1(D)
                          & v1_unialg_1(D,u1_struct_0(A))
                          & m2_relset_1(D,k13_finseq_1(u1_struct_0(A)),u1_struct_0(A)) )
                       => ( D = k1_funct_1(u1_unialg_1(A),C)
                         => k1_funct_1(B,C) = k2_unialg_1(u1_struct_0(A),D) ) ) ) ) ) ) ) ) ),
    file(unialg_1,d11_unialg_1),
    [interesting(0.00)]).

fof(t2_unialg_2,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & v1_funct_1(B)
            & v1_unialg_1(B,A)
            & v2_unialg_1(B,A)
            & m2_relset_1(B,k3_finseq_2(A),A) )
         => k1_relat_1(B) = k4_finseq_2(k2_unialg_1(A,B),A) ) ) ),
    file(unialg_2,t2_unialg_2),
    [interesting(0.00)]).

fof(d4_finseq_2,definition,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : k4_finseq_2(A,B) = a_2_0_finseq_2(A,B) ) ),
    file(finseq_2,d4_finseq_2),
    [interesting(0.00)]).

fof(s5_finseq_1,theorem,
    ( ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ~ ( r2_hidden(A,k2_finseq_1(f2_s5_finseq_1))
            & ! [B] : 
                ( m1_subset_1(B,f1_s5_finseq_1)
               => ~ p1_s5_finseq_1(A,B) ) ) )
   => ? [A] : 
        ( m2_finseq_1(A,f1_s5_finseq_1)
        & k4_finseq_1(A) = k2_finseq_1(f2_s5_finseq_1)
        & ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,k2_finseq_1(f2_s5_finseq_1))
             => p1_s5_finseq_1(B,k1_funct_1(A,B)) ) ) ) ),
    file(finseq_1,s5_finseq_1),
    [interesting(0.00)]).

fof(t61_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ( k5_relat_1(A,k2_funct_1(A)) = k6_relat_1(k1_relat_1(A))
          & k5_relat_1(k2_funct_1(A),A) = k6_relat_1(k2_relat_1(A)) ) ) ) ),
    file(funct_1,t61_funct_1),
    [interesting(0.00)]).

fof(t17_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ( k4_finseq_1(A) = k4_finseq_1(B)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k4_finseq_1(A))
                   => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
           => A = B ) ) ) ),
    file(finseq_1,t17_finseq_1),
    [interesting(0.00)]).

fof(t21_funct_2,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m2_relset_1(D,A,B) )
     => ! [E] : 
          ( ( v1_relat_1(E)
            & v1_funct_1(E) )
         => ( r2_hidden(C,A)
           => ( B = k1_xboole_0
              | k1_funct_1(k5_relat_1(D,E),C) = k1_funct_1(E,k1_funct_1(D,C)) ) ) ) ) ),
    file(funct_2,t21_funct_2),
    [interesting(0.00)]).

fof(d11_finseq_1,definition,(
    ! [A,B] : 
      ( B = k13_finseq_1(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> m2_finseq_1(C,A) ) ) ),
    file(finseq_1,d11_finseq_1),
    [interesting(0.00)]).

fof(t27_partfun1,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & m2_relset_1(D,A,B) )
     => ( r2_hidden(C,k4_relset_1(A,B,D))
       => r2_hidden(k1_funct_1(D,C),B) ) ) ),
    file(partfun1,t27_partfun1),
    [interesting(0.00)]).

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(0.00)]).

fof(t34_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( B = k6_relat_1(A)
      <=> ( k1_relat_1(B) = A
          & ! [C] : 
              ( r2_hidden(C,A)
             => k1_funct_1(B,C) = C ) ) ) ) ),
    file(funct_1,t34_funct_1),
    [interesting(0.00)]).

fof(t47_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(k1_relat_1(A),k2_relat_1(B))
           => k2_relat_1(k5_relat_1(B,A)) = k2_relat_1(A) ) ) ) ),
    file(relat_1,t47_relat_1),
    [interesting(0.00)]).

fof(t46_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(A)
              & v2_funct_1(B) )
           => v2_funct_1(k5_relat_1(A,B)) ) ) ) ),
    file(funct_1,t46_funct_1),
    [interesting(0.00)]).

fof(t6_unialg_2,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( r2_hidden(B,k4_finseq_1(u1_unialg_1(A)))
         => m2_unialg_2(k1_funct_1(u1_unialg_1(A),B),u1_struct_0(A),k1_unialg_2(A)) ) ) ),
    file(unialg_2,t6_unialg_2),
    [interesting(0.00)]).

fof(t7_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & v6_unialg_1(C)
                & v7_unialg_1(C)
                & v8_unialg_1(C)
                & l1_unialg_1(C) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] : 
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ( ( r1_alg_1(A,B,D)
                          & r1_alg_1(B,C,E) )
                       => r1_alg_1(A,C,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_alg_1,d2_unialg_2,d2_unialg_2,d1_alg_1,d11_unialg_1,d3_finseq_1,t6_unialg_2,d5_funct_1,t12_relset_1,d11_unialg_1,t2_unialg_2,d4_finseq_2,d11_finseq_1,t1_alg_1,d4_finseq_2,d1_alg_1,t21_funct_2,t5_alg_1]),
    [file(alg_1,t7_alg_1),interesting(0.00)]).

fof(t15_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & v6_unialg_1(C)
                & v7_unialg_1(C)
                & v8_unialg_1(C)
                & l1_unialg_1(C) )
             => ( ( r5_alg_1(A,B)
                  & r5_alg_1(B,C) )
               => r5_alg_1(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_alg_1,d5_alg_1,t12_alg_1,d5_alg_1]),
    [file(alg_1,t15_alg_1),interesting(0.00)]).

fof(t146_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => k9_relat_1(A,k1_relat_1(A)) = k2_relat_1(A) ) ),
    file(relat_1,t146_relat_1),
    [interesting(0.00)]).

fof(d6_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ! [D] : 
                    ( ( v3_unialg_1(D)
                      & m3_unialg_2(D,B) )
                   => ( D = k1_alg_1(A,B,C)
                    <=> u1_struct_0(D) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(A)) ) ) ) ) ) ) ),
    file(alg_1,d6_alg_1),
    [interesting(0.00)]).

fof(d8_unialg_2,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ( m3_unialg_2(B,A)
          <=> ( m1_subset_1(u1_struct_0(B),k1_zfmisc_1(u1_struct_0(A)))
              & ! [C] : 
                  ( ( ~ v1_xboole_0(C)
                    & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
                 => ( C = u1_struct_0(B)
                   => ( u1_unialg_1(B) = k3_unialg_2(A,C)
                      & v1_unialg_2(C,A) ) ) ) ) ) ) ) ),
    file(unialg_2,d8_unialg_2),
    [interesting(0.00)]).

fof(d7_unialg_2,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] : 
              ( m2_finseq_1(C,k4_partfun1(k13_finseq_1(B),B))
             => ( C = k3_unialg_2(A,B)
              <=> ( k4_finseq_1(C) = k4_finseq_1(u1_unialg_1(A))
                  & ! [D,E] : 
                      ( m2_unialg_2(E,u1_struct_0(A),k1_unialg_2(A))
                     => ( ( r2_hidden(D,k4_finseq_1(C))
                          & E = k1_funct_1(u1_unialg_1(A),D) )
                       => k1_funct_1(C,D) = k2_unialg_2(A,B,E) ) ) ) ) ) ) ) ),
    file(unialg_2,d7_unialg_2),
    [interesting(0.00)]).

fof(t7_unialg_2,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( B = u1_struct_0(A)
           => ( v1_unialg_2(B,A)
              & ! [C] : 
                  ( m2_unialg_2(C,u1_struct_0(A),k1_unialg_2(A))
                 => k2_unialg_2(A,B,C) = C ) ) ) ) ) ),
    file(unialg_2,t7_unialg_2),
    [interesting(0.00)]).

fof(t17_alg_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_unialg_1(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ( r3_alg_1(A,B,C)
                <=> k1_alg_1(A,B,C) = B ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_alg_1,t146_relat_1,d1_funct_2,d6_alg_1,d8_unialg_2,d8_unialg_2,t2_alg_1,d6_alg_1,d1_funct_2,t146_relat_1,d3_alg_1]),
    [file(alg_1,t17_alg_1),interesting(0.00)]).

fof(t106_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,t106_zfmisc_1),
    [interesting(0.00)]).

fof(d9_alg_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m2_relset_1(B,A,A)
         => ! [C] : 
              ( m2_relset_1(C,k3_finseq_2(A),k3_finseq_2(A))
             => ( C = k2_alg_1(A,B)
              <=> ! [D] : 
                    ( m2_finseq_1(D,A)
                   => ! [E] : 
                        ( m2_finseq_1(E,A)
                       => ( r2_hidden(k4_tarski(D,E),C)
                        <=> ( k3_finseq_1(D) = k3_finseq_1(E)
                            & ! [F] : 
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ( r2_hidden(F,k4_finseq_1(D))
                                 => r2_hidden(k4_tarski(k1_funct_1(D,F),k1_funct_1(E,F)),B) ) ) ) ) ) ) ) ) ) ) ),
    file(alg_1,d9_alg_1),
    [interesting(0.00)]).

fof(d10_relat_1,definition,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( B = k6_relat_1(A)
      <=> ! [C,D] : 
            ( r2_hidden(k4_tarski(C,D),B)
          <=> ( r2_hidden(C,A)
              & C = D ) ) ) ) ),
    file(relat_1,d10_relat_1),
    [interesting(0.00)]).

fof(d2_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( A = B
          <=> ! [C,D] : 
                ( r2_hidden(k4_tarski(C,D),A)
              <=> r2_hidden(k4_tarski(C,D),B) ) ) ) ) ),
    file(relat_1,d2_relat_1),
    [interesting(0.00)]).

fof(d13_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( m1_alg_1(B,A)
         => ! [C] : 
              ( m2_finseq_1(C,k4_partfun1(k13_finseq_1(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B)))
             => ( C = k4_alg_1(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(u1_unialg_1(A))
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(C))
                       => ! [E] : 
                            ( m2_unialg_2(E,u1_struct_0(A),k1_unialg_2(A))
                           => ( k1_funct_1(u1_unialg_1(A),D) = E
                             => k1_funct_1(C,D) = k3_alg_1(A,B,E) ) ) ) ) ) ) ) ) ) ),
    file(alg_1,d13_alg_1),
    [interesting(0.00)]).

fof(d4_unialg_1,definition,(
    ! [A,B] : 
      ( m2_finseq_1(B,k4_partfun1(k13_finseq_1(A),A))
     => ( v4_unialg_1(B,A)
      <=> ! [C] : 
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ! [D] : 
                ( ( v1_funct_1(D)
                  & m2_relset_1(D,k13_finseq_1(A),A) )
               => ( ( r2_hidden(C,k4_finseq_1(B))
                    & D = k1_funct_1(B,C) )
                 => v1_unialg_1(D,A) ) ) ) ) ) ),
    file(unialg_1,d4_unialg_1),
    [interesting(0.00)]).

fof(d5_unialg_1,definition,(
    ! [A,B] : 
      ( m2_finseq_1(B,k4_partfun1(k13_finseq_1(A),A))
     => ( v5_unialg_1(B,A)
      <=> ! [C] : 
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ! [D] : 
                ( ( v1_funct_1(D)
                  & m2_relset_1(D,k13_finseq_1(A),A) )
               => ( ( r2_hidden(C,k4_finseq_1(B))
                    & D = k1_funct_1(B,C) )
                 => v2_unialg_1(D,A) ) ) ) ) ) ),
    file(unialg_1,d5_unialg_1),
    [interesting(0.00)]).

fof(d15_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_relat_1(A)
      <=> ! [B] : ~ ( r2_hidden(B,k1_relat_1(A))
            & v1_xboole_0(k1_funct_1(A,B)) ) ) ) ),
    file(funct_1,d15_funct_1),
    [interesting(0.00)]).

fof(t25_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( k3_finseq_1(A) = 0
      <=> A = k1_xboole_0 ) ) ),
    file(finseq_1,t25_finseq_1),
    [interesting(0.00)]).

fof(d7_unialg_1,definition,(
    ! [A] : 
      ( l1_unialg_1(A)
     => ( v6_unialg_1(A)
      <=> v4_unialg_1(u1_unialg_1(A),u1_struct_0(A)) ) ) ),
    file(unialg_1,d7_unialg_1),
    [interesting(0.00)]).

fof(d8_unialg_1,definition,(
    ! [A] : 
      ( l1_unialg_1(A)
     => ( v7_unialg_1(A)
      <=> v5_unialg_1(u1_unialg_1(A),u1_struct_0(A)) ) ) ),
    file(unialg_1,d8_unialg_1),
    [interesting(0.00)]).

fof(d9_unialg_1,definition,(
    ! [A] : 
      ( l1_unialg_1(A)
     => ( v8_unialg_1(A)
      <=> ( u1_unialg_1(A) != k1_xboole_0
          & v2_relat_1(u1_unialg_1(A)) ) ) ) ),
    file(unialg_1,d9_unialg_1),
    [interesting(0.00)]).

fof(d12_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( m1_alg_1(B,A)
         => ! [C] : 
              ( m2_unialg_2(C,u1_struct_0(A),k1_unialg_2(A))
             => ! [D] : 
                  ( ( ~ v1_xboole_0(D)
                    & v1_funct_1(D)
                    & v1_unialg_1(D,k8_eqrel_1(u1_struct_0(A),B))
                    & v2_unialg_1(D,k8_eqrel_1(u1_struct_0(A),B))
                    & m2_relset_1(D,k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B)) )
                 => ( D = k3_alg_1(A,B,C)
                  <=> ( k4_relset_1(k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B),D) = k4_finseq_2(k2_unialg_1(u1_struct_0(A),C),k8_eqrel_1(u1_struct_0(A),B))
                      & ! [E] : 
                          ( m2_finseq_1(E,k8_eqrel_1(u1_struct_0(A),B))
                         => ( r2_hidden(E,k4_relset_1(k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B),D))
                           => ! [F] : 
                                ( m2_finseq_1(F,u1_struct_0(A))
                               => ( r6_alg_1(u1_struct_0(A),B,E,F)
                                 => k1_funct_1(D,E) = k6_eqrel_1(u1_struct_0(A),B,k1_funct_1(C,F)) ) ) ) ) ) ) ) ) ) ) ),
    file(alg_1,d12_alg_1),
    [interesting(0.00)]).

fof(t1_unialg_1,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k13_finseq_1(A),A) )
     => ( ~ ( ~ v1_xboole_0(B)
            & k1_relat_1(B) = k1_xboole_0 )
        & ~ ( k1_relat_1(B) != k1_xboole_0
            & v1_xboole_0(B) ) ) ) ),
    file(unialg_1,t1_unialg_1),
    [interesting(0.00)]).

fof(d1_xboole_0,definition,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    [interesting(0.00)]).

fof(d10_unialg_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & v1_funct_1(B)
            & v1_unialg_1(B,A)
            & m2_relset_1(B,k13_finseq_1(A),A) )
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k2_unialg_1(A,B)
              <=> ! [D] : 
                    ( m2_finseq_1(D,A)
                   => ( r2_hidden(D,k1_relat_1(B))
                     => C = k3_finseq_1(D) ) ) ) ) ) ) ),
    file(unialg_1,d10_unialg_1),
    [interesting(0.00)]).

fof(d15_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( m1_alg_1(B,A)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(k5_alg_1(A,B)))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(k5_alg_1(A,B))) )
             => ( C = k6_alg_1(A,B)
              <=> ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => k8_funct_2(u1_struct_0(A),u1_struct_0(k5_alg_1(A,B)),C,D) = k6_eqrel_1(u1_struct_0(A),B,D) ) ) ) ) ) ),
    file(alg_1,d15_alg_1),
    [interesting(0.00)]).

fof(d11_alg_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] : 
              ( m2_finseq_1(C,k8_eqrel_1(A,B))
             => ! [D] : 
                  ( m2_finseq_1(D,A)
                 => ( r6_alg_1(A,B,C,D)
                  <=> ( k3_finseq_1(D) = k3_finseq_1(C)
                      & ! [E] : 
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k4_finseq_1(D))
                           => k6_eqrel_1(A,B,k1_funct_1(D,E)) = k1_funct_1(C,E) ) ) ) ) ) ) ) ) ),
    file(alg_1,d11_alg_1),
    [interesting(0.00)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.00)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.00)]).

fof(d5_eqrel_1,definition,(
    ! [A,B] : 
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] : 
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( C = k7_eqrel_1(A,B)
          <=> ! [D] : 
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,C)
                <=> ? [E] : 
                      ( r2_hidden(E,A)
                      & D = k6_eqrel_1(A,B,E) ) ) ) ) ) ) ),
    file(eqrel_1,d5_eqrel_1),
    [interesting(0.00)]).

fof(t45_eqrel_1,theorem,(
    ! [A,B,C] : 
      ( ( v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,B,B)
        & m2_relset_1(C,B,B) )
     => ~ ( r2_hidden(A,k7_eqrel_1(B,C))
          & ! [D] : 
              ( m1_subset_1(D,B)
             => A != k6_eqrel_1(B,C,D) ) ) ) ),
    file(eqrel_1,t45_eqrel_1),
    [interesting(0.00)]).

fof(s1_wellord2,theorem,
    ( ! [A] : ~ ( r2_hidden(A,f1_s1_wellord2)
        & ! [B] : ~ ( r2_hidden(B,f2_s1_wellord2)
            & p1_s1_wellord2(A,B) ) )
   => ? [A] : 
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s1_wellord2
        & r1_tarski(k2_relat_1(A),f2_s1_wellord2)
        & ! [B] : 
            ( r2_hidden(B,f1_s1_wellord2)
           => p1_s1_wellord2(B,k1_funct_1(A,B)) ) ) ),
    file(wellord2,s1_wellord2),
    [interesting(0.00)]).

fof(d2_finseq_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( v1_finseq_1(A)
      <=> ? [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & k1_relat_1(A) = k2_finseq_1(B) ) ) ) ),
    file(finseq_1,d2_finseq_1),
    [interesting(0.00)]).

fof(d17_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ! [D] : 
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B))
                      & m2_relset_1(D,u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B)) )
                   => ( D = k8_alg_1(A,B,C)
                    <=> ! [E] : 
                          ( m1_subset_1(E,u1_struct_0(A))
                         => k1_funct_1(D,k6_eqrel_1(u1_struct_0(A),k7_alg_1(A,B,C),E)) = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) ) ) ) ) ) ) ) ),
    file(alg_1,d17_alg_1),
    [interesting(0.00)]).

fof(d8_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B,C] : 
            ( ( r2_hidden(B,k1_relat_1(A))
              & r2_hidden(C,k1_relat_1(A))
              & k1_funct_1(A,B) = k1_funct_1(A,C) )
           => B = C ) ) ) ),
    file(funct_1,d8_funct_1),
    [interesting(0.00)]).

fof(d16_alg_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ! [D] : 
                    ( m1_alg_1(D,A)
                   => ( D = k7_alg_1(A,B,C)
                    <=> ! [E] : 
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ! [F] : 
                              ( m1_subset_1(F,u1_struct_0(A))
                             => ( r2_hidden(k4_tarski(E,F),D)
                              <=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,F) ) ) ) ) ) ) ) ) ) ),
    file(alg_1,d16_alg_1),
    [interesting(0.00)]).

fof(t44_eqrel_1,theorem,(
    ! [A,B,C] : 
      ( ( v3_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ! [D] : 
          ( r2_hidden(D,A)
         => ( r2_hidden(k4_tarski(D,B),C)
          <=> k6_eqrel_1(A,C,D) = k6_eqrel_1(A,C,B) ) ) ) ),
    file(eqrel_1,t44_eqrel_1),
    [interesting(0.00)]).

fof(t4_finseq_1,theorem,
    ( k2_finseq_1(0) = k1_xboole_0
    & k2_finseq_1(1) = k1_tarski(1)
    & k2_finseq_1(2) = k2_tarski(1,2) ),
    file(finseq_1,t4_finseq_1),
    [interesting(0.00)]).

fof(t26_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( A = k1_xboole_0
      <=> k4_finseq_1(A) = k1_xboole_0 ) ) ),
    file(finseq_1,t26_finseq_1),
    [interesting(0.00)]).