% Mizar ND problem: t99_orders_1,orders_1,356,16 
fof(dh_c1_18__orders_1,definition,
    ( ! [A] : 
        ( ( v1_relat_2(A)
          & v4_relat_2(A)
          & v8_relat_2(A)
          & v1_partfun1(A,c1_18__orders_1,c1_18__orders_1)
          & m2_relset_1(A,c1_18__orders_1,c1_18__orders_1) )
       => ( k4_relset_1(c1_18__orders_1,c1_18__orders_1,A) = c1_18__orders_1
          & k5_relset_1(c1_18__orders_1,c1_18__orders_1,A) = c1_18__orders_1 ) )
   => ! [B,C] : 
        ( ( v1_relat_2(C)
          & v4_relat_2(C)
          & v8_relat_2(C)
          & v1_partfun1(C,B,B)
          & m2_relset_1(C,B,B) )
       => ( k4_relset_1(B,B,C) = B
          & k5_relset_1(B,B,C) = B ) ) ),
    introduced(definition,[new_symbol(c1_18__orders_1),file(orders_1,c1_18__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,c1_18__orders_1)]).

fof(dh_c2_18__orders_1,definition,
    ( ( ( v1_relat_2(c2_18__orders_1)
        & v4_relat_2(c2_18__orders_1)
        & v8_relat_2(c2_18__orders_1)
        & v1_partfun1(c2_18__orders_1,c1_18__orders_1,c1_18__orders_1)
        & m2_relset_1(c2_18__orders_1,c1_18__orders_1,c1_18__orders_1) )
     => ( k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1) = c1_18__orders_1
        & k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1) = c1_18__orders_1 ) )
   => ! [A] : 
        ( ( v1_relat_2(A)
          & v4_relat_2(A)
          & v8_relat_2(A)
          & v1_partfun1(A,c1_18__orders_1,c1_18__orders_1)
          & m2_relset_1(A,c1_18__orders_1,c1_18__orders_1) )
       => ( k4_relset_1(c1_18__orders_1,c1_18__orders_1,A) = c1_18__orders_1
          & k5_relset_1(c1_18__orders_1,c1_18__orders_1,A) = c1_18__orders_1 ) ) ),
    introduced(definition,[new_symbol(c2_18__orders_1),file(orders_1,c2_18__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,c2_18__orders_1)]).

fof(cc1_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A) ) ) ),
    file(ordinal1,cc1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc1_ordinal1)]).

fof(cc2_finset_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_finset_1(B) ) ) ),
    file(finset_1,cc2_finset_1),
    [interesting(0.9),axiom,file(finset_1,cc2_finset_1)]).

fof(cc2_ordinal1,theorem,(
    ! [A] : 
      ( ( v1_ordinal1(A)
        & v2_ordinal1(A) )
     => v3_ordinal1(A) ) ),
    file(ordinal1,cc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc2_ordinal1)]).

fof(fc14_finset_1,theorem,(
    ! [A,B] : 
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_zfmisc_1(A,B)) ) ),
    file(finset_1,fc14_finset_1),
    [interesting(0.9),axiom,file(finset_1,fc14_finset_1)]).

fof(rc1_finset_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A) ) ),
    file(finset_1,rc1_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]).

fof(rc1_ordinal1,theorem,(
    ? [A] : 
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc1_ordinal1)]).

fof(rc3_finset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(finset_1,rc3_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]).

fof(rc3_ordinal1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc3_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc3_ordinal1)]).

fof(rc4_finset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(finset_1,rc4_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc4_finset_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(cc1_finset_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_finset_1(A) ) ),
    file(finset_1,cc1_finset_1),
    [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]).

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(cc3_ordinal1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) ),
    file(ordinal1,cc3_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc3_ordinal1)]).

fof(fc4_subset_1,theorem,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc4_subset_1)]).

fof(rc1_subset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]).

fof(rc2_subset_1,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]).

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_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    [interesting(0.9),axiom,file(relat_1,k1_relat_1)]).

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_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    [interesting(0.9),axiom,file(relset_1,m1_relset_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(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_subset_1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc1_subset_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(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_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(dt_c1_18__orders_1,assumption,(
    $true ),
    introduced(assumption,[file(orders_1,c1_18__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,c1_18__orders_1)]).

fof(dt_c2_18__orders_1,assumption,
    ( v1_relat_2(c2_18__orders_1)
    & v4_relat_2(c2_18__orders_1)
    & v8_relat_2(c2_18__orders_1)
    & v1_partfun1(c2_18__orders_1,c1_18__orders_1,c1_18__orders_1)
    & m2_relset_1(c2_18__orders_1,c1_18__orders_1,c1_18__orders_1) ),
    introduced(assumption,[file(orders_1,c2_18__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,c2_18__orders_1)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]).

fof(rc1_partfun1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A) ) ),
    file(partfun1,rc1_partfun1),
    [interesting(0.9),axiom,file(partfun1,rc1_partfun1)]).

fof(rc2_ordinal1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc2_ordinal1)]).

fof(rc2_partfun1,theorem,(
    ! [A,B] : 
    ? [C] : 
      ( m1_relset_1(C,A,B)
      & v1_relat_1(C)
      & v1_funct_1(C) ) ),
    file(partfun1,rc2_partfun1),
    [interesting(0.9),axiom,file(partfun1,rc2_partfun1)]).

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

fof(fc2_ordinal1,theorem,
    ( v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0)
    & v1_xboole_0(k1_xboole_0)
    & v1_ordinal1(k1_xboole_0)
    & v2_ordinal1(k1_xboole_0)
    & v3_ordinal1(k1_xboole_0) ),
    file(ordinal1,fc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,fc2_ordinal1)]).

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(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(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(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(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(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_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(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(e1_18_1__orders_1,plain,(
    r1_tarski(k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1),c1_18__orders_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1])],[rc1_partfun1,rc2_ordinal1,rc2_partfun1,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_ordinal1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,fc2_ordinal1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,t1_subset,t4_subset,t5_subset,dt_k2_zfmisc_1,cc1_finset_1,cc1_relset_1,cc3_ordinal1,fc4_subset_1,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_subset_1,reflexivity_r1_tarski,redefinition_k4_relset_1,dt_k4_relset_1,dt_c1_18__orders_1,dt_c2_18__orders_1,t3_subset]),
    [interesting(0.65),file(orders_1,e1_18_1__orders_1),[file(orders_1,e1_18_1__orders_1)]]).

fof(dt_c1_18_1__orders_1,assumption,(
    $true ),
    introduced(assumption,[file(orders_1,c1_18_1__orders_1)]),
    [interesting(0.65),axiom,file(orders_1,c1_18_1__orders_1)]).

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_c1_18_1__orders_1,definition,
    ( ~ ( r2_hidden(c1_18_1__orders_1,c1_18__orders_1)
        & ~ r2_hidden(c1_18_1__orders_1,k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) )
   => ! [A] : 
        ~ ( r2_hidden(A,c1_18__orders_1)
          & ~ r2_hidden(A,k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ) ),
    introduced(definition,[new_symbol(c1_18_1__orders_1),file(orders_1,c1_18_1__orders_1)]),
    [interesting(0.65),axiom,file(orders_1,c1_18_1__orders_1)]).

fof(e2_18_1__orders_1,assumption,(
    r2_hidden(c1_18_1__orders_1,c1_18__orders_1) ),
    introduced(assumption,[file(orders_1,e2_18_1__orders_1)]),
    [interesting(0.65),axiom,file(orders_1,e2_18_1__orders_1)]).

fof(commutativity_k2_tarski,theorem,(
    ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ),
    file(tarski,k2_tarski),
    [interesting(0.9),axiom,file(tarski,k2_tarski)]).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    [interesting(0.9),axiom,file(tarski,k1_tarski)]).

fof(dt_k2_tarski,axiom,(
    $true ),
    file(tarski,k2_tarski),
    [interesting(0.9),axiom,file(tarski,k2_tarski)]).

fof(fc1_finset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_tarski(A))
      & v1_finset_1(k1_tarski(A)) ) ),
    file(finset_1,fc1_finset_1),
    [interesting(0.9),axiom,file(finset_1,fc1_finset_1)]).

fof(fc2_finset_1,theorem,(
    ! [A,B] : 
      ( ~ v1_xboole_0(k2_tarski(A,B))
      & v1_finset_1(k2_tarski(A,B)) ) ),
    file(finset_1,fc2_finset_1),
    [interesting(0.9),axiom,file(finset_1,fc2_finset_1)]).

fof(fc2_subset_1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc2_subset_1)]).

fof(fc3_subset_1,theorem,(
    ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ),
    file(subset_1,fc3_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc3_subset_1)]).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    [interesting(0.9),axiom,file(tarski,k4_tarski)]).

fof(d5_tarski,definition,(
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ),
    file(tarski,d5_tarski),
    [interesting(0.9),axiom,file(tarski,d5_tarski)]).

fof(t12_orders_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_2(C)
        & v4_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ( r2_hidden(B,A)
       => r2_hidden(k4_tarski(B,B),C) ) ) ),
    file(orders_1,t12_orders_1),
    [interesting(0.9),axiom,file(orders_1,t12_orders_1)]).

fof(e3_18_1__orders_1,plain,(
    r2_hidden(k4_tarski(c1_18_1__orders_1,c1_18_1__orders_1),c2_18__orders_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_18__orders_1,dt_c1_18_1__orders_1,dt_c2_18__orders_1,e2_18_1__orders_1])],[rc1_partfun1,rc2_ordinal1,rc2_partfun1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_ordinal1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,fc2_ordinal1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,commutativity_k2_tarski,existence_m1_relset_1,existence_m1_subset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_tarski,dt_k2_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,cc1_finset_1,cc1_relset_1,cc3_ordinal1,fc1_finset_1,fc1_subset_1,fc2_finset_1,fc2_subset_1,fc3_subset_1,fc4_subset_1,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m2_relset_1,redefinition_m2_relset_1,dt_k4_tarski,dt_m2_relset_1,dt_c1_18__orders_1,dt_c1_18_1__orders_1,dt_c2_18__orders_1,t1_subset,t7_boole,d5_tarski,e2_18_1__orders_1,t12_orders_1]),
    [interesting(0.65),file(orders_1,e3_18_1__orders_1),[file(orders_1,e3_18_1__orders_1)]]).

fof(d4_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( B = k1_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : r2_hidden(k4_tarski(C,D),A) ) ) ) ),
    file(relat_1,d4_relat_1),
    [interesting(0.9),axiom,file(relat_1,d4_relat_1)]).

fof(e4_18_1__orders_1,plain,(
    r2_hidden(c1_18_1__orders_1,k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_18__orders_1,dt_c1_18_1__orders_1,dt_c2_18__orders_1,e2_18_1__orders_1])],[rc1_partfun1,rc2_ordinal1,rc2_partfun1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc1_ordinal1,cc1_relset_1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,fc2_ordinal1,fc4_subset_1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,commutativity_k2_tarski,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_tarski,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc1_finset_1,cc3_ordinal1,fc1_finset_1,fc1_subset_1,fc2_finset_1,fc2_subset_1,fc3_subset_1,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k4_relset_1,dt_k1_relat_1,dt_k4_relset_1,dt_k4_tarski,dt_c1_18__orders_1,dt_c1_18_1__orders_1,dt_c2_18__orders_1,t1_subset,t7_boole,d5_tarski,e3_18_1__orders_1,d4_relat_1]),
    [interesting(0.65),file(orders_1,e4_18_1__orders_1),[file(orders_1,e4_18_1__orders_1)]]).

fof(i4_18_1__orders_1,theorem,(
    $true ),
    introduced(tautology,[file(orders_1,i4_18_1__orders_1)]),
    [interesting(0.65),trivial,file(orders_1,i4_18_1__orders_1)]).

fof(i3_18_1__orders_1,plain,(
    r2_hidden(c1_18_1__orders_1,k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_18__orders_1,dt_c1_18_1__orders_1,dt_c2_18__orders_1,e2_18_1__orders_1])],[e4_18_1__orders_1,i4_18_1__orders_1]),
    [interesting(0.65),file(orders_1,i3_18_1__orders_1),[file(orders_1,i3_18_1__orders_1)]]).

fof(i2_18_1__orders_1,plain,(
    ~ ( r2_hidden(c1_18_1__orders_1,c1_18__orders_1)
      & ~ r2_hidden(c1_18_1__orders_1,k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_18__orders_1,dt_c1_18_1__orders_1,dt_c2_18__orders_1]),discharge_asm(discharge,[e2_18_1__orders_1])],[e2_18_1__orders_1,i3_18_1__orders_1]),
    [interesting(0.65),file(orders_1,i2_18_1__orders_1),[file(orders_1,i2_18_1__orders_1)]]).

fof(i2_18_1_tmp__orders_1,plain,(
    ~ ( r2_hidden(c1_18_1__orders_1,c1_18__orders_1)
      & ~ r2_hidden(c1_18_1__orders_1,k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1]),discharge_asm(discharge,[dt_c1_18_1__orders_1])],[dt_c1_18_1__orders_1,i2_18_1__orders_1]),
    [interesting(0.65),i1_18_1__orders_1]).

fof(i1_18_1__orders_1,plain,(
    r1_tarski(c1_18__orders_1,k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ),
    inference(let,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1])],[i2_18_1_tmp__orders_1,cc1_ordinal1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,dt_k2_zfmisc_1,cc1_finset_1,cc1_relset_1,cc3_ordinal1,fc4_subset_1,rc1_subset_1,rc2_subset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_subset_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k4_relset_1,dt_k4_relset_1,dt_c1_18__orders_1,dt_c2_18__orders_1,d3_tarski,dh_c1_18_1__orders_1]),
    [interesting(0.65),file(orders_1,i1_18_1__orders_1),[file(orders_1,i1_18_1__orders_1)]]).

fof(e1_18__orders_1,plain,(
    k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1) = c1_18__orders_1 ),
    inference(conclusion,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1])],[cc1_ordinal1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,dt_k2_zfmisc_1,cc1_finset_1,cc1_relset_1,cc3_ordinal1,fc4_subset_1,rc1_subset_1,rc2_subset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_subset_1,reflexivity_r1_tarski,redefinition_k4_relset_1,dt_k4_relset_1,dt_c1_18__orders_1,dt_c2_18__orders_1,d10_xboole_0,e1_18_1__orders_1,i1_18_1__orders_1]),
    [interesting(0.8),file(orders_1,e1_18__orders_1),[file(orders_1,e1_18__orders_1)]]).

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

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_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(e2_18__orders_1,plain,(
    r1_tarski(k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1),c1_18__orders_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1])],[rc1_partfun1,rc2_ordinal1,rc2_partfun1,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_ordinal1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,fc2_ordinal1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,t1_subset,t4_subset,t5_subset,dt_k2_zfmisc_1,cc1_finset_1,cc1_relset_1,cc3_ordinal1,fc4_subset_1,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k2_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_subset_1,reflexivity_r1_tarski,redefinition_k5_relset_1,dt_k5_relset_1,dt_c1_18__orders_1,dt_c2_18__orders_1,t3_subset]),
    [interesting(0.8),file(orders_1,e2_18__orders_1),[file(orders_1,e2_18__orders_1)]]).

fof(dt_c3_18__orders_1,assumption,(
    $true ),
    introduced(assumption,[file(orders_1,c3_18__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,c3_18__orders_1)]).

fof(dh_c3_18__orders_1,definition,
    ( ~ ( r2_hidden(c3_18__orders_1,c1_18__orders_1)
        & ~ r2_hidden(c3_18__orders_1,k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) )
   => ! [A] : 
        ~ ( r2_hidden(A,c1_18__orders_1)
          & ~ r2_hidden(A,k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ) ),
    introduced(definition,[new_symbol(c3_18__orders_1),file(orders_1,c3_18__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,c3_18__orders_1)]).

fof(e3_18__orders_1,assumption,(
    r2_hidden(c3_18__orders_1,c1_18__orders_1) ),
    introduced(assumption,[file(orders_1,e3_18__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,e3_18__orders_1)]).

fof(e4_18__orders_1,plain,(
    r2_hidden(k4_tarski(c3_18__orders_1,c3_18__orders_1),c2_18__orders_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1,dt_c3_18__orders_1,e3_18__orders_1])],[rc1_partfun1,rc2_ordinal1,rc2_partfun1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_ordinal1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,fc2_ordinal1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,commutativity_k2_tarski,existence_m1_relset_1,existence_m1_subset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_tarski,dt_k2_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,cc1_finset_1,cc1_relset_1,cc3_ordinal1,fc1_finset_1,fc1_subset_1,fc2_finset_1,fc2_subset_1,fc3_subset_1,fc4_subset_1,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m2_relset_1,redefinition_m2_relset_1,dt_k4_tarski,dt_m2_relset_1,dt_c1_18__orders_1,dt_c2_18__orders_1,dt_c3_18__orders_1,t1_subset,t7_boole,d5_tarski,e3_18__orders_1,t12_orders_1]),
    [interesting(0.8),file(orders_1,e4_18__orders_1),[file(orders_1,e4_18__orders_1)]]).

fof(d5_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( B = k2_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : r2_hidden(k4_tarski(D,C),A) ) ) ) ),
    file(relat_1,d5_relat_1),
    [interesting(0.9),axiom,file(relat_1,d5_relat_1)]).

fof(e5_18__orders_1,plain,(
    r2_hidden(c3_18__orders_1,k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1,dt_c3_18__orders_1,e3_18__orders_1])],[rc1_partfun1,rc2_ordinal1,rc2_partfun1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc1_ordinal1,cc1_relset_1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,fc2_ordinal1,fc4_subset_1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,commutativity_k2_tarski,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_tarski,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc1_finset_1,cc3_ordinal1,fc1_finset_1,fc1_subset_1,fc2_finset_1,fc2_subset_1,fc3_subset_1,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k5_relset_1,dt_k2_relat_1,dt_k4_tarski,dt_k5_relset_1,dt_c1_18__orders_1,dt_c2_18__orders_1,dt_c3_18__orders_1,t1_subset,t7_boole,d5_tarski,e4_18__orders_1,d5_relat_1]),
    [interesting(0.8),file(orders_1,e5_18__orders_1),[file(orders_1,e5_18__orders_1)]]).

fof(i7_18__orders_1,theorem,(
    $true ),
    introduced(tautology,[file(orders_1,i7_18__orders_1)]),
    [interesting(0.8),trivial,file(orders_1,i7_18__orders_1)]).

fof(i6_18__orders_1,plain,(
    r2_hidden(c3_18__orders_1,k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1,dt_c3_18__orders_1,e3_18__orders_1])],[e5_18__orders_1,i7_18__orders_1]),
    [interesting(0.8),file(orders_1,i6_18__orders_1),[file(orders_1,i6_18__orders_1)]]).

fof(i5_18__orders_1,plain,(
    ~ ( r2_hidden(c3_18__orders_1,c1_18__orders_1)
      & ~ r2_hidden(c3_18__orders_1,k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1,dt_c3_18__orders_1]),discharge_asm(discharge,[e3_18__orders_1])],[e3_18__orders_1,i6_18__orders_1]),
    [interesting(0.8),file(orders_1,i5_18__orders_1),[file(orders_1,i5_18__orders_1)]]).

fof(i5_18_tmp__orders_1,plain,(
    ~ ( r2_hidden(c3_18__orders_1,c1_18__orders_1)
      & ~ r2_hidden(c3_18__orders_1,k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1]),discharge_asm(discharge,[dt_c3_18__orders_1])],[dt_c3_18__orders_1,i5_18__orders_1]),
    [interesting(0.8),i4_18__orders_1]).

fof(i4_18__orders_1,plain,(
    r1_tarski(c1_18__orders_1,k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1)) ),
    inference(let,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1])],[i5_18_tmp__orders_1,cc1_ordinal1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,dt_k2_zfmisc_1,cc1_finset_1,cc1_relset_1,cc3_ordinal1,fc4_subset_1,rc1_subset_1,rc2_subset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k2_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_subset_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k5_relset_1,dt_k5_relset_1,dt_c1_18__orders_1,dt_c2_18__orders_1,d3_tarski,dh_c3_18__orders_1]),
    [interesting(0.8),file(orders_1,i4_18__orders_1),[file(orders_1,i4_18__orders_1)]]).

fof(i3_18__orders_1,plain,(
    k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1) = c1_18__orders_1 ),
    inference(conclusion,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1])],[cc1_ordinal1,cc2_finset_1,cc2_ordinal1,fc14_finset_1,rc1_finset_1,rc1_ordinal1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,dt_k2_zfmisc_1,cc1_finset_1,cc1_relset_1,cc3_ordinal1,fc4_subset_1,rc1_subset_1,rc2_subset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k2_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_subset_1,reflexivity_r1_tarski,redefinition_k5_relset_1,dt_k5_relset_1,dt_c1_18__orders_1,dt_c2_18__orders_1,d10_xboole_0,e2_18__orders_1,i4_18__orders_1]),
    [interesting(0.8),file(orders_1,i3_18__orders_1),[file(orders_1,i3_18__orders_1)]]).

fof(i2_18__orders_1,plain,
    ( k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1) = c1_18__orders_1
    & k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1) = c1_18__orders_1 ),
    inference(conclusion,[status(thm),assumptions([dt_c1_18__orders_1,dt_c2_18__orders_1])],[e1_18__orders_1,i3_18__orders_1]),
    [interesting(0.8),file(orders_1,i2_18__orders_1),[file(orders_1,i2_18__orders_1)]]).

fof(i2_18_tmp__orders_1,plain,
    ( ( v1_relat_2(c2_18__orders_1)
      & v4_relat_2(c2_18__orders_1)
      & v8_relat_2(c2_18__orders_1)
      & v1_partfun1(c2_18__orders_1,c1_18__orders_1,c1_18__orders_1)
      & m2_relset_1(c2_18__orders_1,c1_18__orders_1,c1_18__orders_1) )
   => ( k4_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1) = c1_18__orders_1
      & k5_relset_1(c1_18__orders_1,c1_18__orders_1,c2_18__orders_1) = c1_18__orders_1 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_18__orders_1]),discharge_asm(discharge,[dt_c2_18__orders_1])],[dt_c2_18__orders_1,i2_18__orders_1]),
    [interesting(0.8),i1_18__orders_1]).

fof(i1_18__orders_1,plain,(
    ! [A] : 
      ( ( v1_relat_2(A)
        & v4_relat_2(A)
        & v8_relat_2(A)
        & v1_partfun1(A,c1_18__orders_1,c1_18__orders_1)
        & m2_relset_1(A,c1_18__orders_1,c1_18__orders_1) )
     => ( k4_relset_1(c1_18__orders_1,c1_18__orders_1,A) = c1_18__orders_1
        & k5_relset_1(c1_18__orders_1,c1_18__orders_1,A) = c1_18__orders_1 ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_18__orders_1])],[i2_18_tmp__orders_1,dh_c2_18__orders_1]),
    [interesting(0.8),file(orders_1,i1_18__orders_1),[file(orders_1,i1_18__orders_1)]]).

fof(i1_18_tmp__orders_1,plain,(
    ! [A] : 
      ( ( v1_relat_2(A)
        & v4_relat_2(A)
        & v8_relat_2(A)
        & v1_partfun1(A,c1_18__orders_1,c1_18__orders_1)
        & m2_relset_1(A,c1_18__orders_1,c1_18__orders_1) )
     => ( k4_relset_1(c1_18__orders_1,c1_18__orders_1,A) = c1_18__orders_1
        & k5_relset_1(c1_18__orders_1,c1_18__orders_1,A) = c1_18__orders_1 ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_18__orders_1])],[dt_c1_18__orders_1,i1_18__orders_1]),
    [interesting(1),t99_orders_1]).

fof(t99_orders_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_2(B)
        & v4_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ( k4_relset_1(A,A,B) = A
        & k5_relset_1(A,A,B) = A ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_18_tmp__orders_1,dh_c1_18__orders_1]),
    [interesting(1),file(orders_1,t99_orders_1),[file(orders_1,t99_orders_1)]]).