SET007 Axioms: SET007+861.ax

```%------------------------------------------------------------------------------
% File     : SET007+861 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theor
% Axioms   : Substitution in First-Order Formulas. Part II.
% Version  : [Urb08] axioms.
% English  : Substitution in First-Order Formulas. Part II. The Construction of
%            First-Order Formulas

% 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    : substut2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   56 (   1 unit)
%            Number of atoms       :  336 (  77 equality)
%            Maximal formula depth :   24 (   8 average)
%            Number of connectives :  294 (  14 ~  ;   0  |;  85  &)
%                                         (   1 <=>; 194 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   0 propositional; 1-3 arity)
%            Number of functors    :   56 (  15 constant; 0-4 arity)
%            Number of variables   :  185 (   0 singleton; 173 !;  12 ?)
%            Maximal term depth    :    6 (   1 average)
% 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_substut2,axiom,(
! [A] :
( m1_subset_1(A,k1_substut1)
=> ? [B] :
( m2_subset_1(B,k16_substut1,k38_substut1)
& k2_sublemma(B) = k9_cqc_lang
& k19_substut1(B) = A ) ) )).

fof(t2_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( ( v1_cqc_lang(C,A)
& m1_qc_lang1(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_substut1)
=> ? [E] :
( m2_subset_1(E,k16_substut1,k38_substut1)
& k2_sublemma(E) = k8_cqc_lang(A,B,C)
& k19_substut1(E) = D ) ) ) ) ) )).

fof(t3_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(C))
& m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(D)) )
=> C = D ) ) ) ) ) )).

fof(t4_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ( ! [B] :
( m1_subset_1(B,k1_substut1)
=> ? [C] :
( m2_subset_1(C,k16_substut1,k38_substut1)
& k2_sublemma(C) = A
& k19_substut1(C) = B ) )
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> ? [C] :
( m2_subset_1(C,k16_substut1,k38_substut1)
& k2_sublemma(C) = k10_cqc_lang(A)
& k19_substut1(C) = B ) ) ) ) )).

fof(t5_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( ( ! [C] :
( m1_subset_1(C,k1_substut1)
=> ? [D] :
( m2_subset_1(D,k16_substut1,k38_substut1)
& k2_sublemma(D) = A
& k19_substut1(D) = C ) )
& ! [C] :
( m1_subset_1(C,k1_substut1)
=> ? [D] :
( m2_subset_1(D,k16_substut1,k38_substut1)
& k2_sublemma(D) = B
& k19_substut1(D) = C ) ) )
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ? [D] :
( m2_subset_1(D,k16_substut1,k38_substut1)
& k2_sublemma(D) = k11_cqc_lang(A,B)
& k19_substut1(D) = C ) ) ) ) ) )).

fof(t6_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> r1_xboole_0(k1_relat_1(k7_substut1(B,k15_cqc_lang(B,A),C)),k6_domain_1(k2_qc_lang1,B)) ) ) ) )).

fof(t7_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ( r2_hidden(B,k2_relat_1(k7_substut1(B,k15_cqc_lang(B,A),C)))
=> k35_substut1(k1_substut2(k15_cqc_lang(B,A),C)) = k2_qc_lang3(k13_substut1(k7_substut1(B,k15_cqc_lang(B,A),C),A)) ) ) ) ) )).

fof(t8_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ( ~ r2_hidden(B,k2_relat_1(k7_substut1(B,k15_cqc_lang(B,A),C)))
=> k35_substut1(k1_substut2(k15_cqc_lang(B,A),C)) = B ) ) ) ) )).

fof(t9_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> k14_substut1(B,A,k7_substut1(B,k15_cqc_lang(B,A),C)) = k15_sublemma(k2_qc_lang1,k2_substut1(k7_substut1(B,k15_cqc_lang(B,A),C)),k13_sublemma(k2_qc_lang1,k35_substut1(k1_substut2(k15_cqc_lang(B,A),C)),B)) ) ) ) )).

fof(t10_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ! [D] :
( m2_subset_1(D,k16_substut1,k38_substut1)
=> ( ( k19_substut1(D) = k15_sublemma(k2_qc_lang1,k2_substut1(k7_substut1(B,k15_cqc_lang(B,A),C)),k13_sublemma(k2_qc_lang1,k35_substut1(k1_substut2(k15_cqc_lang(B,A),C)),B))
& k2_sublemma(D) = A )
=> ( v3_substut1(k8_sublemma(D,B))
& ? [E] :
( m2_subset_1(E,k16_substut1,k38_substut1)
& E = k1_substut2(k15_cqc_lang(B,A),C) ) ) ) ) ) ) ) )).

fof(t11_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ( ! [C] :
( m1_subset_1(C,k1_substut1)
=> ? [D] :
( m2_subset_1(D,k16_substut1,k38_substut1)
& k2_sublemma(D) = A
& k19_substut1(D) = C ) )
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ? [D] :
( m2_subset_1(D,k16_substut1,k38_substut1)
& k2_sublemma(D) = k15_cqc_lang(B,A)
& k19_substut1(D) = C ) ) ) ) ) )).

fof(t12_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> ? [C] :
( m2_subset_1(C,k16_substut1,k38_substut1)
& k2_sublemma(C) = A
& k19_substut1(C) = B ) ) ) )).

fof(d1_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> k3_substut2(A,B) = k3_cqc_lang(A,B) ) ) )).

fof(d2_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> k4_substut2(A,B,C) = k39_substut1(k2_substut2(A,k3_substut2(B,C))) ) ) ) )).

fof(t13_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> k4_substut2(k9_cqc_lang,A,B) = k9_cqc_lang ) ) )).

fof(t14_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ! [D] :
( m2_subset_1(D,k5_qc_lang1,k7_qc_lang1(A))
=> ! [E] :
( ( v1_cqc_lang(E,A)
& m1_qc_lang1(E,A) )
=> ( k4_substut2(k8_cqc_lang(A,D,E),B,C) = k8_cqc_lang(A,D,k5_sublemma(A,E,k3_substut2(B,C)))
& k6_cqc_sim1(k8_cqc_lang(A,D,E)) = k6_cqc_sim1(k4_substut2(k8_cqc_lang(A,D,E),B,C)) ) ) ) ) ) ) )).

fof(t15_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( ( v1_cqc_lang(C,A)
& m1_qc_lang1(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_substut1)
=> k6_cqc_sim1(k8_cqc_lang(A,B,C)) = k6_cqc_sim1(k39_substut1(k2_substut2(k8_cqc_lang(A,B,C),D))) ) ) ) ) )).

fof(t16_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> k2_substut2(k10_cqc_lang(A),B) = k6_sublemma(k2_substut2(A,B)) ) ) )).

fof(t17_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ( k4_substut2(k10_cqc_lang(A),B,C) = k10_cqc_lang(k4_substut2(A,B,C))
& ( k6_cqc_sim1(A) = k6_cqc_sim1(k4_substut2(A,B,C))
=> k6_cqc_sim1(k10_cqc_lang(A)) = k6_cqc_sim1(k4_substut2(k10_cqc_lang(A),B,C)) ) ) ) ) ) )).

fof(t18_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ( ! [B] :
( m1_subset_1(B,k1_substut1)
=> k6_cqc_sim1(A) = k6_cqc_sim1(k39_substut1(k2_substut2(A,B))) )
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> k6_cqc_sim1(k10_cqc_lang(A)) = k6_cqc_sim1(k39_substut1(k2_substut2(k10_cqc_lang(A),B))) ) ) ) )).

fof(t19_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> k2_substut2(k11_cqc_lang(A,B),C) = k7_sublemma(k2_substut2(A,C),k2_substut2(B,C)) ) ) ) )).

fof(t20_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ! [D] :
( m2_subset_1(D,k1_qc_lang1,k2_qc_lang1)
=> ( k4_substut2(k11_cqc_lang(A,B),C,D) = k11_cqc_lang(k4_substut2(A,C,D),k4_substut2(B,C,D))
& ( ( k6_cqc_sim1(A) = k6_cqc_sim1(k4_substut2(A,C,D))
& k6_cqc_sim1(B) = k6_cqc_sim1(k4_substut2(B,C,D)) )
=> k6_cqc_sim1(k11_cqc_lang(A,B)) = k6_cqc_sim1(k4_substut2(k11_cqc_lang(A,B),C,D)) ) ) ) ) ) ) )).

fof(t21_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( ( ! [C] :
( m1_subset_1(C,k1_substut1)
=> k6_cqc_sim1(A) = k6_cqc_sim1(k39_substut1(k2_substut2(A,C))) )
& ! [C] :
( m1_subset_1(C,k1_substut1)
=> k6_cqc_sim1(B) = k6_cqc_sim1(k39_substut1(k2_substut2(B,C))) ) )
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> k6_cqc_sim1(k11_cqc_lang(A,B)) = k6_cqc_sim1(k39_substut1(k2_substut2(k11_cqc_lang(A,B),C))) ) ) ) ) )).

fof(d3_substut2,axiom,(
k6_substut2 = k7_relat_1(k15_substut1,k38_substut1) )).

fof(d4_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> k7_substut2(A,B,C) = k8_sublemma(k2_substut2(A,k8_funct_2(k38_substut1,k1_substut1,k6_substut2,k2_substut2(k15_cqc_lang(B,A),C))),B) ) ) ) )).

fof(d5_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> k8_substut2(A,B,C) = C ) ) ) )).

fof(t22_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ( k2_substut2(k15_cqc_lang(B,A),C) = k10_sublemma(k7_substut2(A,B,C),k8_substut2(A,B,C))
& v3_substut1(k7_substut2(A,B,C)) ) ) ) ) )).

fof(t23_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ( ! [C] :
( m1_subset_1(C,k1_substut1)
=> k6_cqc_sim1(A) = k6_cqc_sim1(k39_substut1(k2_substut2(A,C))) )
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> k6_cqc_sim1(k15_cqc_lang(B,A)) = k6_cqc_sim1(k39_substut1(k2_substut2(k15_cqc_lang(B,A),C))) ) ) ) ) )).

fof(t24_substut2,axiom,(
! [A] :
( m1_subset_1(A,k1_substut1)
=> k6_cqc_sim1(k9_cqc_lang) = k6_cqc_sim1(k39_substut1(k2_substut2(k9_cqc_lang,A))) ) )).

fof(t25_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> k6_cqc_sim1(A) = k6_cqc_sim1(k39_substut1(k2_substut2(A,B))) ) ) )).

fof(t26_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ~ ( v2_qc_lang1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
=> ! [D] :
( ( v1_cqc_lang(D,B)
& m1_qc_lang1(D,B) )
=> A != k8_cqc_lang(B,C,D) ) ) ) ) ) )).

fof(d6_substut2,axiom,(
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( r2_qc_lang2(A,B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( m1_substut2(C,A,B)
<=> ( r1_xreal_0(np__1,k3_finseq_1(C))
& k1_funct_1(C,np__1) = A
& k1_funct_1(C,k3_finseq_1(C)) = B
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& ~ r1_xreal_0(k3_finseq_1(C),D)
& ! [E] :
( m1_subset_1(E,k8_qc_lang1)
=> ! [F] :
( m1_subset_1(F,k8_qc_lang1)
=> ~ ( k1_funct_1(C,D) = E
& k1_funct_1(C,k1_nat_1(D,np__1)) = F
& r1_qc_lang2(E,F) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t27_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ! [D] :
( m1_substut2(D,B,C)
=> ~ ( r2_qc_lang2(B,C)
& r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(D))
& ! [E] :
( m1_subset_1(E,k8_qc_lang1)
=> ~ ( E = k1_funct_1(D,A)
& r2_qc_lang2(E,C) ) ) ) ) ) ) ) )).

fof(t28_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m1_subset_1(C,k8_qc_lang1)
=> ! [D] :
( m1_substut2(D,C,B)
=> ( ( r2_qc_lang2(C,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(D)) )
=> m2_subset_1(k1_funct_1(D,A),k8_qc_lang1,k7_cqc_lang) ) ) ) ) ) )).

fof(t29_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ! [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
=> ! [E] :
( m1_substut2(E,C,D)
=> ~ ( r1_xreal_0(k6_cqc_sim1(D),A)
& r2_qc_lang2(C,D)
& r1_xreal_0(np__1,B)
& r1_xreal_0(B,k3_finseq_1(E))
& ! [F] :
( m2_subset_1(F,k8_qc_lang1,k7_cqc_lang)
=> ~ ( F = k1_funct_1(E,B)
& r1_xreal_0(k6_cqc_sim1(F),A) ) ) ) ) ) ) ) ) )).

fof(t30_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ( ( k6_cqc_sim1(B) = A
& r2_qc_lang2(C,B) )
=> r1_xreal_0(k6_cqc_sim1(C),A) ) ) ) ) )).

fof(t31_substut2,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ( r2_qc_lang2(C,B)
=> k6_cqc_sim1(C) = A ) )
=> A = np__0 ) ) ) )).

fof(t32_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ( ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( r2_qc_lang2(B,A)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ! [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
=> B != k15_cqc_lang(C,D) ) ) ) )
=> k6_cqc_sim1(A) = np__0 ) ) )).

fof(t33_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ( ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ~ ( r2_qc_lang2(B,A)
& k6_cqc_sim1(B) = np__1 ) )
=> k6_cqc_sim1(A) = np__0 ) ) )).

fof(t34_substut2,axiom,(
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ~ ( r1_xreal_0(np__1,k6_cqc_sim1(A))
& ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ~ ( r2_qc_lang2(B,A)
& k6_cqc_sim1(B) = np__1 ) ) ) ) )).

fof(s1_substut2,axiom,
( ( ! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ( k6_cqc_sim1(A) = np__0
=> p1_s1_substut2(A) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( k6_cqc_sim1(B) = A
=> p1_s1_substut2(B) ) )
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( k6_cqc_sim1(B) = k1_nat_1(A,np__1)
=> p1_s1_substut2(B) ) ) ) ) )
=> ! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> p1_s1_substut2(A) ) )).

fof(s2_substut2,axiom,
( ( ! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ( r1_xreal_0(k6_cqc_sim1(A),np__0)
=> p1_s2_substut2(A) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( r1_xreal_0(k6_cqc_sim1(B),A)
=> p1_s2_substut2(B) ) )
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( r1_xreal_0(k6_cqc_sim1(B),k1_nat_1(A,np__1))
=> p1_s2_substut2(B) ) ) ) ) )
=> ! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> p1_s2_substut2(A) ) )).

fof(s3_substut2,axiom,
( ! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v1_cqc_lang(E,D)
& m1_qc_lang1(E,D) )
=> ! [F] :
( m2_subset_1(F,k5_qc_lang1,k7_qc_lang1(D))
=> ( p1_s3_substut2(k9_cqc_lang)
& p1_s3_substut2(k8_cqc_lang(D,F,E))
& ( p1_s3_substut2(A)
=> p1_s3_substut2(k10_cqc_lang(A)) )
& ( ( p1_s3_substut2(A)
& p1_s3_substut2(B) )
=> p1_s3_substut2(k11_cqc_lang(A,B)) ) ) ) ) ) ) ) )
=> ! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ( k6_cqc_sim1(A) = np__0
=> p1_s3_substut2(A) ) ) )).

fof(dt_m1_substut2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k8_qc_lang1)
& m1_subset_1(B,k8_qc_lang1) )
=> ! [C] :
( m1_substut2(C,A,B)
=> ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) ) ) ) )).

fof(existence_m1_substut2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k8_qc_lang1)
& m1_subset_1(B,k8_qc_lang1) )
=> ? [C] : m1_substut2(C,A,B) ) )).

fof(dt_k1_substut2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k7_cqc_lang)
& m1_subset_1(B,k1_substut1) )
=> m1_subset_1(k1_substut2(A,B),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) )).

fof(redefinition_k1_substut2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k7_cqc_lang)
& m1_subset_1(B,k1_substut1) )
=> k1_substut2(A,B) = k4_tarski(A,B) ) )).

fof(dt_k2_substut2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k7_cqc_lang)
& m1_subset_1(B,k1_substut1) )
=> m2_subset_1(k2_substut2(A,B),k16_substut1,k38_substut1) ) )).

fof(redefinition_k2_substut2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k7_cqc_lang)
& m1_subset_1(B,k1_substut1) )
=> k2_substut2(A,B) = k4_tarski(A,B) ) )).

fof(dt_k3_substut2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k2_qc_lang1)
& m1_subset_1(B,k2_qc_lang1) )
=> m1_subset_1(k3_substut2(A,B),k1_substut1) ) )).

fof(dt_k4_substut2,axiom,(
! [A,B,C] :
( ( m1_subset_1(A,k7_cqc_lang)
& m1_subset_1(B,k2_qc_lang1)
& m1_subset_1(C,k2_qc_lang1) )
=> m2_subset_1(k4_substut2(A,B,C),k8_qc_lang1,k7_cqc_lang) ) )).

fof(dt_k5_substut2,axiom,(
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k5_substut2(A),k1_substut1) ) )).

fof(redefinition_k5_substut2,axiom,(
! [A] :
( m1_subset_1(A,k16_substut1)
=> k5_substut2(A) = k2_mcart_1(A) ) )).

fof(dt_k6_substut2,axiom,
( v1_funct_1(k6_substut2)
& v1_funct_2(k6_substut2,k38_substut1,k1_substut1)
& m2_relset_1(k6_substut2,k38_substut1,k1_substut1) )).

fof(dt_k7_substut2,axiom,(
! [A,B,C] :
( ( m1_subset_1(A,k7_cqc_lang)
& m1_subset_1(B,k2_qc_lang1)
& m1_subset_1(C,k1_substut1) )
=> ( v1_sublemma(k7_substut2(A,B,C))
& m1_subset_1(k7_substut2(A,B,C),k2_zfmisc_1(k16_substut1,k2_qc_lang1)) ) ) )).

fof(dt_k8_substut2,axiom,(
! [A,B,C] :
( ( m1_subset_1(A,k7_cqc_lang)
& m1_subset_1(B,k2_qc_lang1)
& m1_subset_1(C,k1_substut1) )
=> m1_substut1(k8_substut2(A,B,C),k7_substut2(A,B,C)) ) )).
%------------------------------------------------------------------------------
```