% Mizar problem: t19_incsp_1,incsp_1,201,30 
fof(t19_incsp_1,conjecture,(
    ! [A] : 
      ( l2_incsp_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] : 
              ( m1_subset_1(C,u4_incsp_1(A))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_incsp_1(A)))
                 => ( ( r5_incsp_1(A,D,C)
                      & r2_incsp_1(A,B,C) )
                  <=> r5_incsp_1(A,k4_subset_1(u1_incsp_1(A),D,k6_domain_1(u1_incsp_1(A),B)),C) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k4_subset_1,commutativity_k7_domain_1,d1_tarski,d2_xboole_0,d5_incsp_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_tarski,dt_k2_xboole_0,dt_k4_subset_1,dt_k6_domain_1,dt_k7_domain_1,dt_l1_incsp_1,dt_l2_incsp_1,dt_m1_subset_1,dt_u1_incsp_1,dt_u4_incsp_1,existence_l1_incsp_1,existence_l2_incsp_1,existence_m1_subset_1,fc1_subset_1,fc1_xboole_0,fc2_subset_1,fc2_xboole_0,fc3_subset_1,fc3_xboole_0,idempotence_k2_xboole_0,idempotence_k4_subset_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,redefinition_k4_subset_1,redefinition_k6_domain_1,redefinition_k7_domain_1,reflexivity_r1_tarski,t13_incsp_1,t17_incsp_1,t1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t69_enumset1,t6_boole,t7_boole,t7_xboole_1,t8_boole]),
    [file(incsp_1,t19_incsp_1)]).

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

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

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(commutativity_k4_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ),
    file(subset_1,k4_subset_1),
    []).

fof(commutativity_k7_domain_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k7_domain_1(A,B,C) = k7_domain_1(A,C,B) ) ),
    file(domain_1,k7_domain_1),
    []).

fof(d1_tarski,axiom,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    []).

fof(d2_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    []).

fof(d5_incsp_1,axiom,(
    ! [A] : 
      ( l2_incsp_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_incsp_1(A)))
         => ! [C] : 
              ( m1_subset_1(C,u4_incsp_1(A))
             => ( r5_incsp_1(A,B,C)
              <=> ! [D] : 
                    ( m1_subset_1(D,u1_incsp_1(A))
                   => ( r2_hidden(D,B)
                     => r2_incsp_1(A,D,C) ) ) ) ) ) ) ),
    file(incsp_1,d5_incsp_1),
    []).

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

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

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

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

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k4_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(subset_1,k4_subset_1),
    []).

fof(dt_k6_domain_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ),
    file(domain_1,k6_domain_1),
    []).

fof(dt_k7_domain_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k7_domain_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(domain_1,k7_domain_1),
    []).

fof(dt_l1_incsp_1,axiom,(
    $true ),
    file(incsp_1,l1_incsp_1),
    []).

fof(dt_l2_incsp_1,axiom,(
    ! [A] : 
      ( l2_incsp_1(A)
     => l1_incsp_1(A) ) ),
    file(incsp_1,l2_incsp_1),
    []).

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

fof(dt_u1_incsp_1,axiom,(
    ! [A] : 
      ( l1_incsp_1(A)
     => ~ v1_xboole_0(u1_incsp_1(A)) ) ),
    file(incsp_1,u1_incsp_1),
    []).

fof(dt_u4_incsp_1,axiom,(
    ! [A] : 
      ( l2_incsp_1(A)
     => ~ v1_xboole_0(u4_incsp_1(A)) ) ),
    file(incsp_1,u4_incsp_1),
    []).

fof(existence_l1_incsp_1,axiom,(
    ? [A] : l1_incsp_1(A) ),
    file(incsp_1,l1_incsp_1),
    []).

fof(existence_l2_incsp_1,axiom,(
    ? [A] : l2_incsp_1(A) ),
    file(incsp_1,l2_incsp_1),
    []).

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

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

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

fof(fc2_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ),
    file(xboole_0,fc2_xboole_0),
    []).

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

fof(fc3_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ),
    file(xboole_0,fc3_xboole_0),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

fof(idempotence_k4_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,B) = B ) ),
    file(subset_1,k4_subset_1),
    []).

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

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

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

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(redefinition_k4_subset_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ),
    file(subset_1,k4_subset_1),
    []).

fof(redefinition_k6_domain_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => k6_domain_1(A,B) = k1_tarski(B) ) ),
    file(domain_1,k6_domain_1),
    []).

fof(redefinition_k7_domain_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k7_domain_1(A,B,C) = k2_tarski(B,C) ) ),
    file(domain_1,k7_domain_1),
    []).

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

fof(t13_incsp_1,axiom,(
    ! [A] : 
      ( l2_incsp_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u1_incsp_1(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_incsp_1(A))
             => ! [D] : 
                  ( m1_subset_1(D,u4_incsp_1(A))
                 => ( r5_incsp_1(A,k7_domain_1(u1_incsp_1(A),B,C),D)
                  <=> ( r2_incsp_1(A,B,D)
                      & r2_incsp_1(A,C,D) ) ) ) ) ) ) ),
    file(incsp_1,t13_incsp_1),
    []).

fof(t17_incsp_1,axiom,(
    ! [A] : 
      ( l2_incsp_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u4_incsp_1(A))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_incsp_1(A)))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_incsp_1(A)))
                 => ( ( r1_tarski(C,D)
                      & r5_incsp_1(A,D,B) )
                   => r5_incsp_1(A,C,B) ) ) ) ) ) ),
    file(incsp_1,t17_incsp_1),
    []).

fof(t1_boole,axiom,(
    ! [A] : k2_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t1_boole),
    []).

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

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

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

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

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

fof(t69_enumset1,axiom,(
    ! [A] : k2_tarski(A,A) = k1_tarski(A) ),
    file(enumset1,t69_enumset1),
    []).

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

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

fof(t7_xboole_1,axiom,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    []).

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