SET007 Axioms: SET007+78.ax
%------------------------------------------------------------------------------
% File : SET007+78 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Variables in Formulae of the First Order Language
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : qc_lang3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 110 ( 18 unt; 0 def)
% Number of atoms : 600 ( 211 equ)
% Maximal formula atoms : 67 ( 5 avg)
% Number of connectives : 520 ( 30 ~; 0 |; 196 &)
% ( 18 <=>; 276 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 111 ( 111 usr; 35 con; 0-4 aty)
% Number of variables : 225 ( 208 !; 17 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_qc_lang3,axiom,
$true ).
fof(t2_qc_lang3,axiom,
$true ).
fof(t3_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k5_qc_lang1)
=> m2_subset_1(A,k5_qc_lang1,k7_qc_lang1(k6_qc_lang1(A))) ) ).
fof(d1_qc_lang3,axiom,
$true ).
fof(t4_qc_lang3,axiom,
$true ).
fof(t5_qc_lang3,axiom,
$true ).
fof(t6_qc_lang3,axiom,
! [A] :
( m2_finseq_1(A,k1_qc_lang1)
=> k22_qc_lang1(A) = k1_qc_lang3(A,k2_qc_lang1) ) ).
fof(t7_qc_lang3,axiom,
k24_qc_lang1(k11_qc_lang1) = k1_xboole_0 ).
fof(t8_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_qc_lang1(A)
=> k24_qc_lang1(A) = k22_qc_lang1(k16_qc_lang1(A)) ) ) ).
fof(t9_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( m1_qc_lang1(C,A)
=> k24_qc_lang1(k9_qc_lang1(B,C)) = k22_qc_lang1(C) ) ) ) ).
fof(t10_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v3_qc_lang1(A)
=> k24_qc_lang1(A) = k24_qc_lang1(k17_qc_lang1(A)) ) ) ).
fof(t11_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k24_qc_lang1(k12_qc_lang1(A)) = k24_qc_lang1(A) ) ).
fof(t12_qc_lang3,axiom,
k24_qc_lang1(k1_qc_lang2) = k1_xboole_0 ).
fof(t13_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v4_qc_lang1(A)
=> k24_qc_lang1(A) = k4_subset_1(k2_qc_lang1,k24_qc_lang1(k18_qc_lang1(A)),k24_qc_lang1(k19_qc_lang1(A))) ) ) ).
fof(t14_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k24_qc_lang1(k13_qc_lang1(A,B)) = k4_subset_1(k2_qc_lang1,k24_qc_lang1(A),k24_qc_lang1(B)) ) ) ).
fof(t15_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v5_qc_lang1(A)
=> k24_qc_lang1(A) = k6_subset_1(k2_qc_lang1,k24_qc_lang1(k21_qc_lang1(A)),k23_qc_lang1(k20_qc_lang1(A))) ) ) ).
fof(t16_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k24_qc_lang1(k14_qc_lang1(A,B)) = k6_subset_1(k2_qc_lang1,k24_qc_lang1(B),k23_qc_lang1(A)) ) ) ).
fof(t17_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v1_qc_lang2(A)
=> k24_qc_lang1(A) = k4_subset_1(k2_qc_lang1,k24_qc_lang1(k10_qc_lang2(A)),k24_qc_lang1(k11_qc_lang2(A))) ) ) ).
fof(t18_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k24_qc_lang1(k3_qc_lang2(A,B)) = k4_subset_1(k2_qc_lang1,k24_qc_lang1(A),k24_qc_lang1(B)) ) ) ).
fof(t19_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_qc_lang2(A)
=> k24_qc_lang1(A) = k4_subset_1(k2_qc_lang1,k24_qc_lang1(k12_qc_lang2(A)),k24_qc_lang1(k11_qc_lang2(A))) ) ) ).
fof(t20_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k24_qc_lang1(k2_qc_lang2(A,B)) = k4_subset_1(k2_qc_lang1,k24_qc_lang1(A),k24_qc_lang1(B)) ) ) ).
fof(t21_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v3_qc_lang2(A)
=> k24_qc_lang1(A) = k4_subset_1(k2_qc_lang1,k24_qc_lang1(k13_qc_lang2(A)),k24_qc_lang1(k14_qc_lang2(A))) ) ) ).
fof(t22_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k24_qc_lang1(k4_qc_lang2(A,B)) = k4_subset_1(k2_qc_lang1,k24_qc_lang1(A),k24_qc_lang1(B)) ) ) ).
fof(t23_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k24_qc_lang1(k5_qc_lang2(A,B)) = k6_subset_1(k2_qc_lang1,k24_qc_lang1(B),k23_qc_lang1(A)) ) ) ).
fof(t24_qc_lang3,axiom,
( v6_qc_lang1(k11_qc_lang1)
& v6_qc_lang1(k1_qc_lang2) ) ).
fof(t25_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v6_qc_lang1(A)
<=> v6_qc_lang1(k12_qc_lang1(A)) ) ) ).
fof(t26_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( ( v6_qc_lang1(A)
& v6_qc_lang1(B) )
<=> v6_qc_lang1(k13_qc_lang1(A,B)) ) ) ) ).
fof(t27_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( v6_qc_lang1(k14_qc_lang1(A,B))
<=> r1_tarski(k24_qc_lang1(B),k23_qc_lang1(A)) ) ) ) ).
fof(t28_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( v6_qc_lang1(B)
=> v6_qc_lang1(k14_qc_lang1(A,B)) ) ) ) ).
fof(t29_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( ( v6_qc_lang1(A)
& v6_qc_lang1(B) )
<=> v6_qc_lang1(k3_qc_lang2(A,B)) ) ) ) ).
fof(t30_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( ( v6_qc_lang1(A)
& v6_qc_lang1(B) )
<=> v6_qc_lang1(k2_qc_lang2(A,B)) ) ) ) ).
fof(t31_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( ( v6_qc_lang1(A)
& v6_qc_lang1(B) )
<=> v6_qc_lang1(k4_qc_lang2(A,B)) ) ) ) ).
fof(t32_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( v6_qc_lang1(k5_qc_lang2(A,B))
<=> r1_tarski(k24_qc_lang1(B),k23_qc_lang1(A)) ) ) ) ).
fof(t33_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( v6_qc_lang1(B)
=> v6_qc_lang1(k5_qc_lang2(A,B)) ) ) ) ).
fof(d3_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_qc_lang3(A) = k4_tarski(np__4,A) ) ).
fof(t34_qc_lang3,axiom,
$true ).
fof(t35_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_qc_lang3(A) = k2_qc_lang3(B)
=> A = B ) ) ) ).
fof(t36_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& k2_qc_lang3(B) = A ) ) ).
fof(d4_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_qc_lang3(A) = k4_tarski(np__6,A) ) ).
fof(t37_qc_lang3,axiom,
$true ).
fof(t38_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k3_qc_lang3(A) = k3_qc_lang3(B)
=> A = B ) ) ) ).
fof(t39_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k4_qc_lang1)
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& k3_qc_lang3(B) = A ) ) ).
fof(t40_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k3_qc_lang1)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k4_qc_lang1)
=> A != B ) ) ).
fof(t41_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k3_qc_lang1)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> A != B ) ) ).
fof(t42_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k4_qc_lang1)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> A != B ) ) ).
fof(d5_qc_lang3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_qc_lang1)) )
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( C = k4_qc_lang3(A,B)
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k8_qc_lang1,k1_zfmisc_1(A))
& m2_relset_1(D,k8_qc_lang1,k1_zfmisc_1(A))
& C = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,B)
& ! [E] :
( m1_subset_1(E,k8_qc_lang1)
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(A))
=> ( ( E = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,E) = k1_subset_1(A) )
& ( v2_qc_lang1(E)
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,E) = k1_qc_lang3(k16_qc_lang1(E),A) )
& ( ( v3_qc_lang1(E)
& F = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,k17_qc_lang1(E)) )
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,E) = F )
& ( ( v4_qc_lang1(E)
& F = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,k18_qc_lang1(E))
& G = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,k19_qc_lang1(E)) )
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,E) = k4_subset_1(A,F,G) )
& ( ( v5_qc_lang1(E)
& F = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,k21_qc_lang1(E)) )
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(A),D,E) = F ) ) ) ) ) ) ) ) ) ) ).
fof(t43_qc_lang3,axiom,
$true ).
fof(t44_qc_lang3,axiom,
$true ).
fof(t45_qc_lang3,axiom,
$true ).
fof(t46_qc_lang3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(A,k11_qc_lang1) = k1_xboole_0 ) ).
fof(t49_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ( v3_qc_lang1(A)
=> k4_qc_lang3(B,A) = k4_qc_lang3(B,k17_qc_lang1(A)) ) ) ) ).
fof(t50_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(B,k12_qc_lang1(A)) = k4_qc_lang3(B,A) ) ) ).
fof(t51_qc_lang3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(A,k1_qc_lang2) = k1_xboole_0 ) ).
fof(t52_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ( v4_qc_lang1(A)
=> k4_qc_lang3(B,A) = k4_subset_1(B,k4_qc_lang3(B,k18_qc_lang1(A)),k4_qc_lang3(B,k19_qc_lang1(A))) ) ) ) ).
fof(t53_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(C,k13_qc_lang1(A,B)) = k4_subset_1(C,k4_qc_lang3(C,A),k4_qc_lang3(C,B)) ) ) ) ).
fof(t54_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ( v5_qc_lang1(A)
=> k4_qc_lang3(B,A) = k4_qc_lang3(B,k21_qc_lang1(A)) ) ) ) ).
fof(t55_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(C,k14_qc_lang1(A,B)) = k4_qc_lang3(C,B) ) ) ) ).
fof(t56_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ( v1_qc_lang2(A)
=> k4_qc_lang3(B,A) = k4_subset_1(B,k4_qc_lang3(B,k10_qc_lang2(A)),k4_qc_lang3(B,k11_qc_lang2(A))) ) ) ) ).
fof(t57_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(C,k3_qc_lang2(A,B)) = k4_subset_1(C,k4_qc_lang3(C,A),k4_qc_lang3(C,B)) ) ) ) ).
fof(t58_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ( v2_qc_lang2(A)
=> k4_qc_lang3(B,A) = k4_subset_1(B,k4_qc_lang3(B,k12_qc_lang2(A)),k4_qc_lang3(B,k11_qc_lang2(A))) ) ) ) ).
fof(t59_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(C,k2_qc_lang2(A,B)) = k4_subset_1(C,k4_qc_lang3(C,A),k4_qc_lang3(C,B)) ) ) ) ).
fof(t60_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ( v3_qc_lang2(A)
=> k4_qc_lang3(B,A) = k4_subset_1(B,k4_qc_lang3(B,k13_qc_lang2(A)),k4_qc_lang3(B,k14_qc_lang2(A))) ) ) ) ).
fof(t61_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(C,k4_qc_lang2(A,B)) = k4_subset_1(C,k4_qc_lang3(C,A),k4_qc_lang3(C,B)) ) ) ) ).
fof(t62_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ( v4_qc_lang2(A)
=> k4_qc_lang3(B,A) = k4_qc_lang3(B,k17_qc_lang1(k21_qc_lang1(k17_qc_lang1(A)))) ) ) ) ).
fof(t63_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1)) )
=> k4_qc_lang3(C,k5_qc_lang2(A,B)) = k4_qc_lang3(C,B) ) ) ) ).
fof(d6_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k5_qc_lang3(A) = k4_qc_lang3(k4_qc_lang1,A) ) ).
fof(t64_qc_lang3,axiom,
$true ).
fof(t65_qc_lang3,axiom,
k5_qc_lang3(k11_qc_lang1) = k1_xboole_0 ).
fof(t67_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k5_qc_lang3(k12_qc_lang1(A)) = k5_qc_lang3(A) ) ).
fof(t68_qc_lang3,axiom,
k5_qc_lang3(k1_qc_lang2) = k1_xboole_0 ).
fof(t69_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k5_qc_lang3(k13_qc_lang1(A,B)) = k4_subset_1(k4_qc_lang1,k5_qc_lang3(A),k5_qc_lang3(B)) ) ) ).
fof(t70_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k5_qc_lang3(k14_qc_lang1(A,B)) = k5_qc_lang3(B) ) ) ).
fof(t71_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k5_qc_lang3(k3_qc_lang2(A,B)) = k4_subset_1(k4_qc_lang1,k5_qc_lang3(A),k5_qc_lang3(B)) ) ) ).
fof(t72_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k5_qc_lang3(k2_qc_lang2(A,B)) = k4_subset_1(k4_qc_lang1,k5_qc_lang3(A),k5_qc_lang3(B)) ) ) ).
fof(t73_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k5_qc_lang3(k4_qc_lang2(A,B)) = k4_subset_1(k4_qc_lang1,k5_qc_lang3(A),k5_qc_lang3(B)) ) ) ).
fof(t74_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k5_qc_lang3(k5_qc_lang2(A,B)) = k5_qc_lang3(B) ) ) ).
fof(d7_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k6_qc_lang3(A) = k4_qc_lang3(k3_qc_lang1,A) ) ).
fof(t75_qc_lang3,axiom,
$true ).
fof(t76_qc_lang3,axiom,
k6_qc_lang3(k11_qc_lang1) = k1_xboole_0 ).
fof(t78_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k6_qc_lang3(k12_qc_lang1(A)) = k6_qc_lang3(A) ) ).
fof(t79_qc_lang3,axiom,
k6_qc_lang3(k1_qc_lang2) = k1_xboole_0 ).
fof(t80_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k6_qc_lang3(k13_qc_lang1(A,B)) = k4_subset_1(k3_qc_lang1,k6_qc_lang3(A),k6_qc_lang3(B)) ) ) ).
fof(t81_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k6_qc_lang3(k14_qc_lang1(A,B)) = k6_qc_lang3(B) ) ) ).
fof(t82_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k6_qc_lang3(k3_qc_lang2(A,B)) = k4_subset_1(k3_qc_lang1,k6_qc_lang3(A),k6_qc_lang3(B)) ) ) ).
fof(t83_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k6_qc_lang3(k2_qc_lang2(A,B)) = k4_subset_1(k3_qc_lang1,k6_qc_lang3(A),k6_qc_lang3(B)) ) ) ).
fof(t84_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k6_qc_lang3(k4_qc_lang2(A,B)) = k4_subset_1(k3_qc_lang1,k6_qc_lang3(A),k6_qc_lang3(B)) ) ) ).
fof(t85_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k6_qc_lang3(k5_qc_lang2(A,B)) = k6_qc_lang3(B) ) ) ).
fof(s1_qc_lang3,axiom,
( ( ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,f1_s1_qc_lang3)
=> ! [C] :
( m1_subset_1(C,f1_s1_qc_lang3)
=> ( ( A = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,A) = f4_s1_qc_lang3 )
& ( v2_qc_lang1(A)
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,A) = f5_s1_qc_lang3(A) )
& ( ( v3_qc_lang1(A)
& B = k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,k17_qc_lang1(A)) )
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,A) = f6_s1_qc_lang3(B) )
& ( ( v4_qc_lang1(A)
& B = k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,k18_qc_lang1(A))
& C = k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,k19_qc_lang1(A)) )
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,A) = f7_s1_qc_lang3(B,C) )
& ( ( v5_qc_lang1(A)
& B = k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,k21_qc_lang1(A)) )
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f2_s1_qc_lang3,A) = f8_s1_qc_lang3(A,B) ) ) ) ) )
& ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,f1_s1_qc_lang3)
=> ! [C] :
( m1_subset_1(C,f1_s1_qc_lang3)
=> ( ( A = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,A) = f4_s1_qc_lang3 )
& ( v2_qc_lang1(A)
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,A) = f5_s1_qc_lang3(A) )
& ( ( v3_qc_lang1(A)
& B = k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,k17_qc_lang1(A)) )
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,A) = f6_s1_qc_lang3(B) )
& ( ( v4_qc_lang1(A)
& B = k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,k18_qc_lang1(A))
& C = k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,k19_qc_lang1(A)) )
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,A) = f7_s1_qc_lang3(B,C) )
& ( ( v5_qc_lang1(A)
& B = k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,k21_qc_lang1(A)) )
=> k8_funct_2(k8_qc_lang1,f1_s1_qc_lang3,f3_s1_qc_lang3,A) = f8_s1_qc_lang3(A,B) ) ) ) ) ) )
=> f2_s1_qc_lang3 = f3_s1_qc_lang3 ) ).
fof(s2_qc_lang3,axiom,
( ? [A] :
( m1_subset_1(A,f1_s2_qc_lang3)
& ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k8_qc_lang1,f1_s2_qc_lang3)
& m2_relset_1(B,k8_qc_lang1,f1_s2_qc_lang3)
& A = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,f3_s2_qc_lang3)
& ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ! [D] :
( m1_subset_1(D,f1_s2_qc_lang3)
=> ! [E] :
( m1_subset_1(E,f1_s2_qc_lang3)
=> ( ( C = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,C) = f2_s2_qc_lang3 )
& ( v2_qc_lang1(C)
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,C) = f4_s2_qc_lang3(C) )
& ( ( v3_qc_lang1(C)
& D = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,k17_qc_lang1(C)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,C) = f5_s2_qc_lang3(D) )
& ( ( v4_qc_lang1(C)
& D = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,k18_qc_lang1(C))
& E = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,k19_qc_lang1(C)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,C) = f6_s2_qc_lang3(D,E) )
& ( ( v5_qc_lang1(C)
& D = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,k21_qc_lang1(C)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,B,C) = f7_s2_qc_lang3(C,D) ) ) ) ) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s2_qc_lang3)
=> ! [B] :
( m1_subset_1(B,f1_s2_qc_lang3)
=> ~ ( ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,f1_s2_qc_lang3)
& m2_relset_1(C,k8_qc_lang1,f1_s2_qc_lang3)
& A = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,f3_s2_qc_lang3)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ! [E] :
( m1_subset_1(E,f1_s2_qc_lang3)
=> ! [F] :
( m1_subset_1(F,f1_s2_qc_lang3)
=> ( ( D = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f2_s2_qc_lang3 )
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f4_s2_qc_lang3(D) )
& ( ( v3_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,k17_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f5_s2_qc_lang3(E) )
& ( ( v4_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,k18_qc_lang1(D))
& F = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,k19_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f6_s2_qc_lang3(E,F) )
& ( ( v5_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,k21_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f7_s2_qc_lang3(D,E) ) ) ) ) ) )
& ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,f1_s2_qc_lang3)
& m2_relset_1(C,k8_qc_lang1,f1_s2_qc_lang3)
& B = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,f3_s2_qc_lang3)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ! [E] :
( m1_subset_1(E,f1_s2_qc_lang3)
=> ! [F] :
( m1_subset_1(F,f1_s2_qc_lang3)
=> ( ( D = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f2_s2_qc_lang3 )
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f4_s2_qc_lang3(D) )
& ( ( v3_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,k17_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f5_s2_qc_lang3(E) )
& ( ( v4_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,k18_qc_lang1(D))
& F = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,k19_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f6_s2_qc_lang3(E,F) )
& ( ( v5_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,k21_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s2_qc_lang3,C,D) = f7_s2_qc_lang3(D,E) ) ) ) ) ) )
& A != B ) ) ) ) ).
fof(s3_qc_lang3,axiom,
( ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,f1_s3_qc_lang3)
=> ( B = f2_s3_qc_lang3(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,f1_s3_qc_lang3)
& m2_relset_1(C,k8_qc_lang1,f1_s3_qc_lang3)
& B = k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,A)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ! [E] :
( m1_subset_1(E,f1_s3_qc_lang3)
=> ! [F] :
( m1_subset_1(F,f1_s3_qc_lang3)
=> ( ( D = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,D) = f3_s3_qc_lang3 )
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,D) = f4_s3_qc_lang3(D) )
& ( ( v3_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,k17_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,D) = f5_s3_qc_lang3(E) )
& ( ( v4_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,k18_qc_lang1(D))
& F = k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,k19_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,D) = f6_s3_qc_lang3(E,F) )
& ( ( v5_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,k21_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s3_qc_lang3,C,D) = f7_s3_qc_lang3(D,E) ) ) ) ) ) ) ) ) )
=> f2_s3_qc_lang3(k11_qc_lang1) = f3_s3_qc_lang3 ) ).
fof(s4_qc_lang3,axiom,
( ( ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,f1_s4_qc_lang3)
=> ( B = f3_s4_qc_lang3(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,f1_s4_qc_lang3)
& m2_relset_1(C,k8_qc_lang1,f1_s4_qc_lang3)
& B = k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,A)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ! [E] :
( m1_subset_1(E,f1_s4_qc_lang3)
=> ! [F] :
( m1_subset_1(F,f1_s4_qc_lang3)
=> ( ( D = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,D) = f2_s4_qc_lang3 )
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,D) = f5_s4_qc_lang3(D) )
& ( ( v3_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,k17_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,D) = f6_s4_qc_lang3(E) )
& ( ( v4_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,k18_qc_lang1(D))
& F = k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,k19_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,D) = f7_s4_qc_lang3(E,F) )
& ( ( v5_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,k21_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s4_qc_lang3,C,D) = f8_s4_qc_lang3(D,E) ) ) ) ) ) ) ) ) )
& v2_qc_lang1(f4_s4_qc_lang3) )
=> f3_s4_qc_lang3(f4_s4_qc_lang3) = f5_s4_qc_lang3(f4_s4_qc_lang3) ) ).
fof(s5_qc_lang3,axiom,
( ( ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,f1_s5_qc_lang3)
=> ( B = f8_s5_qc_lang3(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,f1_s5_qc_lang3)
& m2_relset_1(C,k8_qc_lang1,f1_s5_qc_lang3)
& B = k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,A)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ! [E] :
( m1_subset_1(E,f1_s5_qc_lang3)
=> ! [F] :
( m1_subset_1(F,f1_s5_qc_lang3)
=> ( ( D = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,D) = f2_s5_qc_lang3 )
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,D) = f4_s5_qc_lang3(D) )
& ( ( v3_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,k17_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,D) = f5_s5_qc_lang3(E) )
& ( ( v4_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,k18_qc_lang1(D))
& F = k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,k19_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,D) = f6_s5_qc_lang3(E,F) )
& ( ( v5_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,k21_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s5_qc_lang3,C,D) = f7_s5_qc_lang3(D,E) ) ) ) ) ) ) ) ) )
& v3_qc_lang1(f3_s5_qc_lang3) )
=> f8_s5_qc_lang3(f3_s5_qc_lang3) = f5_s5_qc_lang3(f8_s5_qc_lang3(k17_qc_lang1(f3_s5_qc_lang3))) ) ).
fof(s6_qc_lang3,axiom,
( ( ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,f1_s6_qc_lang3)
=> ( B = f7_s6_qc_lang3(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,f1_s6_qc_lang3)
& m2_relset_1(C,k8_qc_lang1,f1_s6_qc_lang3)
& B = k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,A)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ! [E] :
( m1_subset_1(E,f1_s6_qc_lang3)
=> ! [F] :
( m1_subset_1(F,f1_s6_qc_lang3)
=> ( ( D = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,D) = f2_s6_qc_lang3 )
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,D) = f3_s6_qc_lang3(D) )
& ( ( v3_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,k17_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,D) = f4_s6_qc_lang3(E) )
& ( ( v4_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,k18_qc_lang1(D))
& F = k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,k19_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,D) = f5_s6_qc_lang3(E,F) )
& ( ( v5_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,k21_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s6_qc_lang3,C,D) = f6_s6_qc_lang3(D,E) ) ) ) ) ) ) ) ) )
& v4_qc_lang1(f8_s6_qc_lang3) )
=> ! [A] :
( m1_subset_1(A,f1_s6_qc_lang3)
=> ! [B] :
( m1_subset_1(B,f1_s6_qc_lang3)
=> ( ( A = f7_s6_qc_lang3(k18_qc_lang1(f8_s6_qc_lang3))
& B = f7_s6_qc_lang3(k19_qc_lang1(f8_s6_qc_lang3)) )
=> f7_s6_qc_lang3(f8_s6_qc_lang3) = f5_s6_qc_lang3(A,B) ) ) ) ) ).
fof(s7_qc_lang3,axiom,
( ( ! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,f1_s7_qc_lang3)
=> ( B = f8_s7_qc_lang3(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,f1_s7_qc_lang3)
& m2_relset_1(C,k8_qc_lang1,f1_s7_qc_lang3)
& B = k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,A)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ! [E] :
( m1_subset_1(E,f1_s7_qc_lang3)
=> ! [F] :
( m1_subset_1(F,f1_s7_qc_lang3)
=> ( ( D = k11_qc_lang1
=> k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,D) = f2_s7_qc_lang3 )
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,D) = f4_s7_qc_lang3(D) )
& ( ( v3_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,k17_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,D) = f5_s7_qc_lang3(E) )
& ( ( v4_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,k18_qc_lang1(D))
& F = k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,k19_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,D) = f6_s7_qc_lang3(E,F) )
& ( ( v5_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,k21_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,f1_s7_qc_lang3,C,D) = f7_s7_qc_lang3(D,E) ) ) ) ) ) ) ) ) )
& v5_qc_lang1(f3_s7_qc_lang3) )
=> f8_s7_qc_lang3(f3_s7_qc_lang3) = f7_s7_qc_lang3(f3_s7_qc_lang3,f8_s7_qc_lang3(k21_qc_lang1(f3_s7_qc_lang3))) ) ).
fof(dt_k1_qc_lang3,axiom,
! [A,B] :
( ( m1_finseq_1(A,k1_qc_lang1)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> m1_subset_1(k1_qc_lang3(A,B),k1_zfmisc_1(B)) ) ).
fof(dt_k2_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k2_qc_lang3(A),k1_qc_lang1,k2_qc_lang1) ) ).
fof(dt_k3_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k3_qc_lang3(A),k1_qc_lang1,k4_qc_lang1) ) ).
fof(dt_k4_qc_lang3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_qc_lang1))
& m1_subset_1(B,k8_qc_lang1) )
=> m1_subset_1(k4_qc_lang3(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k5_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k5_qc_lang3(A),k1_zfmisc_1(k4_qc_lang1)) ) ).
fof(dt_k6_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k6_qc_lang3(A),k1_zfmisc_1(k3_qc_lang1)) ) ).
fof(d2_qc_lang3,axiom,
! [A] :
( m2_finseq_1(A,k1_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> k1_qc_lang3(A,B) = a_2_0_qc_lang3(A,B) ) ) ).
fof(t47_qc_lang3,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ( v2_qc_lang1(A)
=> ( k4_qc_lang3(B,A) = k1_qc_lang3(k16_qc_lang1(A),B)
& k4_qc_lang3(B,A) = a_2_1_qc_lang3(A,B) ) ) ) ) ).
fof(t48_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_qc_lang1)) )
=> ! [C] :
( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(A))
=> ! [D] :
( m1_qc_lang1(D,A)
=> ( k4_qc_lang3(B,k9_qc_lang1(C,D)) = k1_qc_lang3(D,B)
& k4_qc_lang3(B,k9_qc_lang1(C,D)) = a_3_0_qc_lang3(A,B,D) ) ) ) ) ) ).
fof(t66_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( m1_qc_lang1(C,A)
=> k5_qc_lang3(k9_qc_lang1(B,C)) = a_2_2_qc_lang3(A,C) ) ) ) ).
fof(t77_qc_lang3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( m1_qc_lang1(C,A)
=> k6_qc_lang3(k9_qc_lang1(B,C)) = a_2_3_qc_lang3(A,C) ) ) ) ).
fof(fraenkel_a_2_0_qc_lang3,axiom,
! [A,B,C] :
( ( m2_finseq_1(B,k1_qc_lang1)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1)) )
=> ( r2_hidden(A,a_2_0_qc_lang3(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k1_funct_1(B,D)
& r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(B))
& r2_hidden(k1_funct_1(B,D),C) ) ) ) ).
fof(fraenkel_a_2_1_qc_lang3,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k8_qc_lang1)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1)) )
=> ( r2_hidden(A,a_2_1_qc_lang3(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k1_funct_1(k16_qc_lang1(B),D)
& r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(k16_qc_lang1(B)))
& r2_hidden(k1_funct_1(k16_qc_lang1(B),D),C) ) ) ) ).
fof(fraenkel_a_3_0_qc_lang3,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_qc_lang1))
& m1_qc_lang1(D,B) )
=> ( r2_hidden(A,a_3_0_qc_lang3(B,C,D))
<=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& A = k1_funct_1(D,E)
& r1_xreal_0(np__1,E)
& r1_xreal_0(E,k3_finseq_1(D))
& r2_hidden(k1_funct_1(D,E),C) ) ) ) ).
fof(fraenkel_a_2_2_qc_lang3,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_qc_lang1(C,B) )
=> ( r2_hidden(A,a_2_2_qc_lang3(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k1_funct_1(C,D)
& r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(C))
& r2_hidden(k1_funct_1(C,D),k4_qc_lang1) ) ) ) ).
fof(fraenkel_a_2_3_qc_lang3,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m1_qc_lang1(C,B) )
=> ( r2_hidden(A,a_2_3_qc_lang3(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k1_funct_1(C,D)
& r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(C))
& r2_hidden(k1_funct_1(C,D),k3_qc_lang1) ) ) ) ).
%------------------------------------------------------------------------------