TPTP Problem File: ALG229+4.p
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%------------------------------------------------------------------------------
% File : ALG229+4 : TPTP v8.2.0. Released v3.4.0.
% Domain : General Algebra
% Problem : Algebraic Operation on Subsets of Many Sorted Sets T10
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Mar97] Marasik (1997), Algebraic Operation on Subsets of Many
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t10_closure3 [Urb08]
% Status : Unknown
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 42816 (6323 unt; 0 def)
% Number of atoms : 313714 (35224 equ)
% Maximal formula atoms : 208 ( 7 avg)
% Number of connectives : 315401 (44503 ~;3334 |;153677 &)
% (7374 <=>;106513 =>; 0 <=; 0 <~>)
% Maximal formula depth : 150 ( 8 avg)
% Maximal term depth : 12 ( 1 avg)
% Number of predicates : 2353 (2351 usr; 3 prp; 0-8 aty)
% Number of functors : 5944 (5944 usr;1283 con; 0-10 aty)
% Number of variables : 120495 (114672 !;5823 ?)
% SPC : FOF_UNK_RFO_SEQ
% Comments : Chainy large version: includes all preceding MML articles.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
include('Axioms/SET007/SET007+0.ax').
include('Axioms/SET007/SET007+1.ax').
include('Axioms/SET007/SET007+2.ax').
include('Axioms/SET007/SET007+3.ax').
include('Axioms/SET007/SET007+4.ax').
include('Axioms/SET007/SET007+5.ax').
include('Axioms/SET007/SET007+6.ax').
include('Axioms/SET007/SET007+7.ax').
include('Axioms/SET007/SET007+8.ax').
include('Axioms/SET007/SET007+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+12.ax').
include('Axioms/SET007/SET007+13.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+15.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+17.ax').
include('Axioms/SET007/SET007+18.ax').
include('Axioms/SET007/SET007+19.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+21.ax').
include('Axioms/SET007/SET007+22.ax').
include('Axioms/SET007/SET007+23.ax').
include('Axioms/SET007/SET007+24.ax').
include('Axioms/SET007/SET007+25.ax').
include('Axioms/SET007/SET007+26.ax').
include('Axioms/SET007/SET007+27.ax').
include('Axioms/SET007/SET007+28.ax').
include('Axioms/SET007/SET007+29.ax').
include('Axioms/SET007/SET007+30.ax').
include('Axioms/SET007/SET007+31.ax').
include('Axioms/SET007/SET007+32.ax').
include('Axioms/SET007/SET007+33.ax').
include('Axioms/SET007/SET007+34.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+36.ax').
include('Axioms/SET007/SET007+37.ax').
include('Axioms/SET007/SET007+38.ax').
include('Axioms/SET007/SET007+39.ax').
include('Axioms/SET007/SET007+40.ax').
include('Axioms/SET007/SET007+41.ax').
include('Axioms/SET007/SET007+42.ax').
include('Axioms/SET007/SET007+43.ax').
include('Axioms/SET007/SET007+44.ax').
include('Axioms/SET007/SET007+45.ax').
include('Axioms/SET007/SET007+46.ax').
include('Axioms/SET007/SET007+47.ax').
include('Axioms/SET007/SET007+48.ax').
include('Axioms/SET007/SET007+49.ax').
include('Axioms/SET007/SET007+50.ax').
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include('Axioms/SET007/SET007+81.ax').
include('Axioms/SET007/SET007+82.ax').
include('Axioms/SET007/SET007+83.ax').
include('Axioms/SET007/SET007+84.ax').
include('Axioms/SET007/SET007+85.ax').
include('Axioms/SET007/SET007+86.ax').
include('Axioms/SET007/SET007+87.ax').
include('Axioms/SET007/SET007+88.ax').
include('Axioms/SET007/SET007+89.ax').
include('Axioms/SET007/SET007+90.ax').
include('Axioms/SET007/SET007+91.ax').
include('Axioms/SET007/SET007+92.ax').
include('Axioms/SET007/SET007+93.ax').
include('Axioms/SET007/SET007+94.ax').
include('Axioms/SET007/SET007+95.ax').
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include('Axioms/SET007/SET007+97.ax').
include('Axioms/SET007/SET007+98.ax').
include('Axioms/SET007/SET007+99.ax').
include('Axioms/SET007/SET007+100.ax').
include('Axioms/SET007/SET007+101.ax').
include('Axioms/SET007/SET007+102.ax').
include('Axioms/SET007/SET007+103.ax').
include('Axioms/SET007/SET007+104.ax').
include('Axioms/SET007/SET007+105.ax').
include('Axioms/SET007/SET007+106.ax').
include('Axioms/SET007/SET007+107.ax').
include('Axioms/SET007/SET007+108.ax').
include('Axioms/SET007/SET007+109.ax').
include('Axioms/SET007/SET007+110.ax').
include('Axioms/SET007/SET007+111.ax').
include('Axioms/SET007/SET007+112.ax').
include('Axioms/SET007/SET007+113.ax').
include('Axioms/SET007/SET007+114.ax').
include('Axioms/SET007/SET007+115.ax').
include('Axioms/SET007/SET007+116.ax').
include('Axioms/SET007/SET007+117.ax').
include('Axioms/SET007/SET007+118.ax').
include('Axioms/SET007/SET007+119.ax').
include('Axioms/SET007/SET007+120.ax').
include('Axioms/SET007/SET007+121.ax').
include('Axioms/SET007/SET007+122.ax').
include('Axioms/SET007/SET007+123.ax').
include('Axioms/SET007/SET007+124.ax').
include('Axioms/SET007/SET007+125.ax').
include('Axioms/SET007/SET007+126.ax').
include('Axioms/SET007/SET007+127.ax').
include('Axioms/SET007/SET007+128.ax').
include('Axioms/SET007/SET007+129.ax').
include('Axioms/SET007/SET007+130.ax').
include('Axioms/SET007/SET007+131.ax').
include('Axioms/SET007/SET007+132.ax').
include('Axioms/SET007/SET007+133.ax').
include('Axioms/SET007/SET007+134.ax').
include('Axioms/SET007/SET007+135.ax').
include('Axioms/SET007/SET007+136.ax').
include('Axioms/SET007/SET007+137.ax').
include('Axioms/SET007/SET007+138.ax').
include('Axioms/SET007/SET007+139.ax').
include('Axioms/SET007/SET007+140.ax').
include('Axioms/SET007/SET007+141.ax').
include('Axioms/SET007/SET007+142.ax').
include('Axioms/SET007/SET007+143.ax').
include('Axioms/SET007/SET007+144.ax').
include('Axioms/SET007/SET007+145.ax').
include('Axioms/SET007/SET007+146.ax').
include('Axioms/SET007/SET007+147.ax').
include('Axioms/SET007/SET007+148.ax').
include('Axioms/SET007/SET007+149.ax').
include('Axioms/SET007/SET007+150.ax').
include('Axioms/SET007/SET007+151.ax').
include('Axioms/SET007/SET007+152.ax').
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include('Axioms/SET007/SET007+157.ax').
include('Axioms/SET007/SET007+158.ax').
include('Axioms/SET007/SET007+159.ax').
include('Axioms/SET007/SET007+160.ax').
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include('Axioms/SET007/SET007+166.ax').
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include('Axioms/SET007/SET007+172.ax').
include('Axioms/SET007/SET007+173.ax').
include('Axioms/SET007/SET007+174.ax').
include('Axioms/SET007/SET007+175.ax').
include('Axioms/SET007/SET007+176.ax').
include('Axioms/SET007/SET007+177.ax').
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include('Axioms/SET007/SET007+179.ax').
include('Axioms/SET007/SET007+180.ax').
include('Axioms/SET007/SET007+181.ax').
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include('Axioms/SET007/SET007+186.ax').
include('Axioms/SET007/SET007+187.ax').
include('Axioms/SET007/SET007+188.ax').
include('Axioms/SET007/SET007+189.ax').
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include('Axioms/SET007/SET007+220.ax').
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include('Axioms/SET007/SET007+426.ax').
include('Axioms/SET007/SET007+427.ax').
include('Axioms/SET007/SET007+428.ax').
include('Axioms/SET007/SET007+429.ax').
include('Axioms/SET007/SET007+430.ax').
include('Axioms/SET007/SET007+431.ax').
include('Axioms/SET007/SET007+432.ax').
include('Axioms/SET007/SET007+433.ax').
include('Axioms/SET007/SET007+434.ax').
include('Axioms/SET007/SET007+435.ax').
include('Axioms/SET007/SET007+436.ax').
include('Axioms/SET007/SET007+437.ax').
include('Axioms/SET007/SET007+438.ax').
include('Axioms/SET007/SET007+439.ax').
include('Axioms/SET007/SET007+440.ax').
include('Axioms/SET007/SET007+441.ax').
include('Axioms/SET007/SET007+442.ax').
include('Axioms/SET007/SET007+443.ax').
include('Axioms/SET007/SET007+444.ax').
include('Axioms/SET007/SET007+445.ax').
include('Axioms/SET007/SET007+446.ax').
include('Axioms/SET007/SET007+447.ax').
include('Axioms/SET007/SET007+448.ax').
include('Axioms/SET007/SET007+449.ax').
include('Axioms/SET007/SET007+450.ax').
include('Axioms/SET007/SET007+451.ax').
include('Axioms/SET007/SET007+452.ax').
include('Axioms/SET007/SET007+453.ax').
include('Axioms/SET007/SET007+454.ax').
include('Axioms/SET007/SET007+455.ax').
include('Axioms/SET007/SET007+456.ax').
include('Axioms/SET007/SET007+457.ax').
include('Axioms/SET007/SET007+458.ax').
include('Axioms/SET007/SET007+459.ax').
include('Axioms/SET007/SET007+460.ax').
include('Axioms/SET007/SET007+461.ax').
include('Axioms/SET007/SET007+462.ax').
include('Axioms/SET007/SET007+463.ax').
include('Axioms/SET007/SET007+464.ax').
include('Axioms/SET007/SET007+465.ax').
include('Axioms/SET007/SET007+466.ax').
include('Axioms/SET007/SET007+467.ax').
include('Axioms/SET007/SET007+468.ax').
include('Axioms/SET007/SET007+469.ax').
include('Axioms/SET007/SET007+470.ax').
include('Axioms/SET007/SET007+471.ax').
include('Axioms/SET007/SET007+472.ax').
include('Axioms/SET007/SET007+473.ax').
include('Axioms/SET007/SET007+474.ax').
include('Axioms/SET007/SET007+475.ax').
include('Axioms/SET007/SET007+476.ax').
include('Axioms/SET007/SET007+477.ax').
include('Axioms/SET007/SET007+478.ax').
include('Axioms/SET007/SET007+479.ax').
include('Axioms/SET007/SET007+480.ax').
include('Axioms/SET007/SET007+481.ax').
include('Axioms/SET007/SET007+482.ax').
include('Axioms/SET007/SET007+483.ax').
include('Axioms/SET007/SET007+484.ax').
include('Axioms/SET007/SET007+485.ax').
include('Axioms/SET007/SET007+486.ax').
include('Axioms/SET007/SET007+487.ax').
include('Axioms/SET007/SET007+488.ax').
include('Axioms/SET007/SET007+489.ax').
include('Axioms/SET007/SET007+490.ax').
include('Axioms/SET007/SET007+491.ax').
include('Axioms/SET007/SET007+492.ax').
include('Axioms/SET007/SET007+493.ax').
include('Axioms/SET007/SET007+494.ax').
include('Axioms/SET007/SET007+495.ax').
include('Axioms/SET007/SET007+496.ax').
include('Axioms/SET007/SET007+497.ax').
include('Axioms/SET007/SET007+498.ax').
include('Axioms/SET007/SET007+499.ax').
include('Axioms/SET007/SET007+500.ax').
include('Axioms/SET007/SET007+501.ax').
include('Axioms/SET007/SET007+502.ax').
include('Axioms/SET007/SET007+503.ax').
include('Axioms/SET007/SET007+504.ax').
include('Axioms/SET007/SET007+505.ax').
include('Axioms/SET007/SET007+506.ax').
include('Axioms/SET007/SET007+507.ax').
include('Axioms/SET007/SET007+508.ax').
include('Axioms/SET007/SET007+509.ax').
include('Axioms/SET007/SET007+510.ax').
include('Axioms/SET007/SET007+511.ax').
include('Axioms/SET007/SET007+512.ax').
include('Axioms/SET007/SET007+513.ax').
include('Axioms/SET007/SET007+514.ax').
include('Axioms/SET007/SET007+515.ax').
include('Axioms/SET007/SET007+516.ax').
%------------------------------------------------------------------------------
fof(fraenkel_a_4_0_closure3,axiom,
! [A,B,C,D,E] :
( ( m1_pboole(C,B)
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(B,C))) )
=> ( r2_hidden(A,a_4_0_closure3(B,C,D,E))
<=> ? [F] :
( m1_closure2(F,B,C,k6_closure2(B,C))
& A = k1_funct_1(F,E)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_4_4_closure3,axiom,
! [A,B,C,D,E] :
( ( m1_pboole(C,B)
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(B,C))) )
=> ( r2_hidden(A,a_4_4_closure3(B,C,D,E))
<=> ? [F] :
( m1_closure3(F,B,C,k6_closure2(B,C))
& A = k1_funct_1(F,E)
& r2_hidden(F,D) ) ) ) ).
fof(dt_m1_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ! [D] :
( m1_closure3(D,A,B,C)
=> m1_pboole(D,A) ) ) ).
fof(existence_m1_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ? [D] : m1_closure3(D,A,B,C) ) ).
fof(redefinition_m1_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ! [D] :
( m1_closure3(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(reflexivity_r1_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> r1_closure3(A,B,D,D) ) ).
fof(reflexivity_r2_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> r2_closure3(A,B,C,C) ) ).
fof(dt_k1_closure3,axiom,
$true ).
fof(dt_k2_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> m4_pboole(k2_closure3(A,B,C),A,B) ) ).
fof(dt_k3_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> m1_subset_1(k3_closure3(A,B,C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ).
fof(commutativity_k3_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k3_closure3(A,B,C,D) = k3_closure3(A,B,D,C) ) ).
fof(idempotence_k3_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k3_closure3(A,B,C,C) = C ) ).
fof(redefinition_k3_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k3_closure3(A,B,C,D) = k2_xboole_0(C,D) ) ).
fof(dt_k4_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> m1_subset_1(k4_closure3(A,B,C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ).
fof(commutativity_k4_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k4_closure3(A,B,C,D) = k4_closure3(A,B,D,C) ) ).
fof(idempotence_k4_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k4_closure3(A,B,C,C) = C ) ).
fof(redefinition_k4_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k4_closure3(A,B,C,D) = k3_xboole_0(C,D) ) ).
fof(dt_k5_closure3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v11_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
& l1_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A))) ) ) ).
fof(fc1_closure3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ( v1_struct_0(g1_struct_0(u1_struct_0(A)))
& ~ v3_struct_0(g1_struct_0(u1_struct_0(A))) ) ) ).
fof(t1_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> k1_funct_4(B,C) = C ) ) ) ).
fof(t2_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r2_hidden(C,D)
=> r2_pboole(A,k6_mssubfam(A,B,k5_closure2(A,B,D)),C) ) ) ) ) ).
fof(t3_closure3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B))))
=> ( r1_tarski(C,k6_msualg_2(A,B))
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
=> ( r6_pboole(u1_struct_0(A),D,k6_mssubfam(u1_struct_0(A),u4_msualg_1(A,B),k5_closure2(u1_struct_0(A),u4_msualg_1(A,B),C)))
=> v3_msualg_2(D,A,B) ) ) ) ) ) ) ).
fof(d1_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r1_closure3(A,B,C,D)
<=> ! [E] :
~ ( r2_hidden(E,D)
& ! [F] :
~ ( r2_hidden(F,C)
& r1_tarski(E,F) ) ) ) ) ) ) ).
fof(d2_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r2_closure3(A,B,C,D)
<=> ! [E] :
~ ( r2_hidden(E,C)
& ! [F] :
~ ( r2_hidden(F,D)
& r1_tarski(F,E) ) ) ) ) ) ) ).
fof(t4_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
=> ( ( r1_closure3(A,B,D,C)
& r1_closure3(A,B,E,D) )
=> r1_closure3(A,B,E,C) ) ) ) ) ) ).
fof(t5_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
=> ( ( r2_closure3(A,B,C,D)
& r2_closure3(A,B,D,E) )
=> r2_closure3(A,B,C,E) ) ) ) ) ) ).
fof(d3_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( C = k1_closure3(A,B)
<=> C = a_2_0_closure3(A,B) ) ) ) ).
fof(t6_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> B = k1_funct_4(k1_pboole(A),k7_relat_1(B,k1_closure3(A,B))) ) ) ).
fof(t7_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ( ( k1_closure3(A,B) = k1_closure3(A,C)
& k7_relat_1(B,k1_closure3(A,B)) = k7_relat_1(C,k1_closure3(A,C)) )
=> r6_pboole(A,B,C) ) ) ) ) ).
fof(t8_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( ~ r2_hidden(C,k1_closure3(A,B))
=> k1_funct_1(B,C) = k1_xboole_0 ) ) ) ) ).
fof(t9_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k6_closure2(A,B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
~ ( r2_hidden(E,k1_funct_1(C,D))
& ! [F] :
( m1_closure2(F,A,B,k6_closure2(A,B))
=> ~ ( r2_hidden(E,k1_funct_1(F,D))
& v1_pre_circ(F,A)
& v1_finset_1(k1_closure3(A,F))
& r2_pboole(A,F,C) ) ) ) ) ) ) ) ).
fof(d4_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m4_pboole(D,A,B)
=> ( D = k2_closure3(A,B,C)
<=> ! [E] :
( r2_hidden(E,A)
=> k1_funct_1(D,E) = k3_tarski(a_4_0_closure3(A,B,C,E)) ) ) ) ) ) ).
fof(fc2_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ( v1_relat_1(k2_closure3(A,B,C))
& v3_relat_1(k2_closure3(A,B,C))
& v1_funct_1(k2_closure3(A,B,C))
& v1_pre_circ(k2_closure3(A,B,C),A) ) ) ).
fof(t10_closure3,conjecture,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> r6_pboole(A,k2_closure3(A,B,C),k2_mboolean(A,k5_closure2(A,B,C))) ) ) ).
%------------------------------------------------------------------------------