SET007 Axioms: SET007+135.ax
%------------------------------------------------------------------------------
% File : SET007+135 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Real Function Differentiability - Part II
% 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 : fdiff_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 0 unt; 0 def)
% Number of atoms : 597 ( 50 equ)
% Maximal formula atoms : 36 ( 11 avg)
% Number of connectives : 562 ( 17 ~; 12 |; 301 &)
% ( 1 <=>; 231 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 11 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 28 ( 27 usr; 0 prp; 1-4 aty)
% Number of functors : 48 ( 48 usr; 6 con; 0-6 aty)
% Number of variables : 184 ( 163 !; 21 ?)
% 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_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B)
& k2_seq_2(A) = k2_seq_2(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k2_seq_1(k5_numbers,k1_numbers,C,k2_nat_1(np__2,D)) = k2_seq_1(k5_numbers,k1_numbers,A,D)
& k2_seq_1(k5_numbers,k1_numbers,C,k1_nat_1(k2_nat_1(np__2,D),np__1)) = k2_seq_1(k5_numbers,k1_numbers,B,D) ) ) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k2_seq_2(A) ) ) ) ) ) ).
fof(t2_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k2_nat_1(np__2,B) )
=> ( v1_seqm_3(A)
& v7_seqm_3(A) ) ) ) ).
fof(t3_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k1_nat_1(k2_nat_1(np__2,B),np__1) )
=> ( v1_seqm_3(A)
& v7_seqm_3(A) ) ) ) ).
fof(t4_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_fdiff_1(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v5_seqm_3(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( k1_rfunct_2(C) = k1_tarski(A)
=> ( v4_seq_2(C)
& k2_seq_2(C) = A
& v4_seq_2(k9_seq_1(B,C))
& k2_seq_2(k9_seq_1(B,C)) = A ) ) ) ) ) ).
fof(t5_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(B) = k1_tarski(A)
& k1_rfunct_2(C) = k1_tarski(A) )
=> B = C ) ) ) ) ).
fof(t6_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_fdiff_1(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( m1_seqm_3(A,B)
=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_fdiff_1(A)
& m2_relset_1(A,k5_numbers,k1_numbers) ) ) ) ) ).
fof(t7_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(D) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,B))
& r1_tarski(k1_tarski(A),k4_relset_1(k1_numbers,k1_numbers,B)) )
=> v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(B,k9_seq_1(C,D)),k2_rfunct_2(B,D)))) ) ) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,E)),k4_relset_1(k1_numbers,k1_numbers,B))
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,B))
& r1_tarski(k1_tarski(A),k4_relset_1(k1_numbers,k1_numbers,B)) )
=> k2_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(B,k9_seq_1(C,E)),k2_rfunct_2(B,E)))) = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,E)),k2_rfunct_2(B,E)))) ) ) ) ) ) ) ) ).
fof(t8_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ~ ( ? [C] :
( m1_rcomp_1(C,A)
& r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,B)) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ~ ( k1_rfunct_2(D) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,B))
& r1_tarski(k1_tarski(A),k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) ) ) ) ).
fof(t9_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,B)))
=> ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,C))
& r1_tarski(k1_rfunct_2(k2_rfunct_2(C,A)),k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) ) ) ).
fof(t10_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ~ ( k2_seq_1(k1_numbers,k1_numbers,C,A) != B
& r1_fdiff_1(C,A)
& ! [D] :
( m1_rcomp_1(D,A)
=> ~ ( r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,D)
& k2_seq_1(k1_numbers,k1_numbers,C,E) = B ) ) ) ) ) ) ) ) ).
fof(t11_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_fdiff_1(B,A)
<=> ( ? [C] :
( m1_rcomp_1(C,A)
& r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,B)) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(D) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,B)) )
=> v4_seq_2(k11_seq_1(k18_seq_1(C),k10_seq_1(k2_rfunct_2(B,k9_seq_1(C,D)),k2_rfunct_2(B,D)))) ) ) ) ) ) ) ) ).
fof(t12_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( ( r1_fdiff_1(C,A)
& k1_fdiff_1(C,A) = B )
=> ( ? [D] :
( m1_rcomp_1(D,A)
& r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C)) )
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,C)) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(C,k9_seq_1(D,E)),k2_rfunct_2(C,E))))
& k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(C,k9_seq_1(D,E)),k2_rfunct_2(C,E)))) = B ) ) ) ) ) )
& ( ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_fdiff_1(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v5_seqm_3(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(E) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),k4_relset_1(k1_numbers,k1_numbers,C)) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(C,k9_seq_1(D,E)),k2_rfunct_2(C,E))))
& k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(C,k9_seq_1(D,E)),k2_rfunct_2(C,E)))) = B ) ) ) )
=> ( ! [D] :
( m1_rcomp_1(D,A)
=> ~ r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,C)) )
| ( r1_fdiff_1(C,A)
& k1_fdiff_1(C,A) = B ) ) ) ) ) ) ) ).
fof(t13_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r1_fdiff_1(B,A)
& r1_fdiff_1(C,k2_seq_1(k1_numbers,k1_numbers,B,A)) )
=> ( r1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A)
& k1_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A) = k4_real_1(k1_fdiff_1(C,k2_seq_1(k1_numbers,k1_numbers,B,A)),k1_fdiff_1(B,A)) ) ) ) ) ) ).
fof(t14_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r1_fdiff_1(C,A)
& r1_fdiff_1(B,A) )
=> ( k2_seq_1(k1_numbers,k1_numbers,B,A) = np__0
| ( r1_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,C,B),A)
& k1_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,C,B),A) = k6_real_1(k5_real_1(k4_real_1(k1_fdiff_1(C,A),k2_seq_1(k1_numbers,k1_numbers,B,A)),k4_real_1(k1_fdiff_1(B,A),k2_seq_1(k1_numbers,k1_numbers,C,A))),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,B,A))) ) ) ) ) ) ) ).
fof(t15_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_fdiff_1(B,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,A) = np__0
| ( r1_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k1_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k1_real_1(k6_real_1(k1_fdiff_1(B,A),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,B,A)))) ) ) ) ) ) ).
fof(t16_fdiff_2,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(B,A)
=> ( r2_fdiff_1(k2_partfun1(k1_numbers,k1_numbers,B,A),A)
& k2_fdiff_1(B,A) = k2_fdiff_1(k2_partfun1(k1_numbers,k1_numbers,B,A),A) ) ) ) ) ).
fof(t17_fdiff_2,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,A)
& r2_fdiff_1(C,A) )
=> ( r2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& k2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k6_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),k2_fdiff_1(C,A)) ) ) ) ) ) ).
fof(t18_fdiff_2,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,A)
& r2_fdiff_1(C,A) )
=> ( r2_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
& k2_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k7_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),k2_fdiff_1(C,A)) ) ) ) ) ) ).
fof(t19_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v3_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(C,B)
=> ( r2_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,C,A),B)
& k2_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,C,A),B) = k13_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(C,B),A) ) ) ) ) ) ).
fof(t20_fdiff_2,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,A)
& r2_fdiff_1(C,A) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
& k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k6_seq_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),C),k8_seq_1(k1_numbers,k1_numbers,B,k2_fdiff_1(C,A))) ) ) ) ) ) ).
fof(t21_fdiff_2,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,A)
& r2_fdiff_1(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(D,A)
& k2_seq_1(k1_numbers,k1_numbers,C,D) = np__0 ) ) )
=> ( r2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A)
& k2_fdiff_1(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A) = k2_rfunct_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),C),k8_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(C,A),B)),k8_seq_1(k1_numbers,k1_numbers,C,C)) ) ) ) ) ) ).
fof(t22_fdiff_2,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_hidden(C,A)
& k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0 ) ) )
=> ( r2_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k2_fdiff_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k16_seq_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),k8_seq_1(k1_numbers,k1_numbers,B,B))) ) ) ) ) ).
fof(t23_fdiff_2,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,A)
& v3_rcomp_1(k10_relset_1(k1_numbers,k1_numbers,B,A))
& m1_subset_1(k10_relset_1(k1_numbers,k1_numbers,B,A),k1_zfmisc_1(k1_numbers))
& r2_fdiff_1(C,k10_relset_1(k1_numbers,k1_numbers,B,A)) )
=> ( r2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A)
& k2_fdiff_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A) = k8_seq_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,k2_fdiff_1(C,k10_relset_1(k1_numbers,k1_numbers,B,A))),k2_fdiff_1(B,A)) ) ) ) ) ) ).
fof(t24_fdiff_2,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r2_hidden(C,A)
& r2_hidden(D,A) )
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,C),k2_seq_1(k1_numbers,k1_numbers,B,D))),k7_square_1(k5_real_1(C,D))) ) ) ) )
=> ( r2_fdiff_1(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k1_fdiff_1(B,C) = np__0 ) ) ) ) ) ) ).
fof(t25_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(D,k2_rcomp_1(A,B))
& r2_hidden(E,k2_rcomp_1(A,B)) )
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,D),k2_seq_1(k1_numbers,k1_numbers,C,E))),k7_square_1(k5_real_1(D,E))) ) ) )
& r1_tarski(k2_rcomp_1(A,B),k4_relset_1(k1_numbers,k1_numbers,C)) )
=> ( r1_xreal_0(B,A)
| ( r2_fdiff_1(C,k2_rcomp_1(A,B))
& r1_partfun2(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)) ) ) ) ) ) ) ).
fof(t26_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(k12_prob_1(A),k4_relset_1(k1_numbers,k1_numbers,B))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r2_hidden(C,k12_prob_1(A))
& r2_hidden(D,k12_prob_1(A)) )
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,C),k2_seq_1(k1_numbers,k1_numbers,B,D))),k7_square_1(k5_real_1(C,D))) ) ) ) )
=> ( r2_fdiff_1(B,k12_prob_1(A))
& r1_partfun2(k1_numbers,k1_numbers,B,k12_prob_1(A)) ) ) ) ) ).
fof(t27_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(k4_limfunc1(A),k4_relset_1(k1_numbers,k1_numbers,B))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r2_hidden(C,k4_limfunc1(A))
& r2_hidden(D,k4_limfunc1(A)) )
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,C),k2_seq_1(k1_numbers,k1_numbers,B,D))),k7_square_1(k5_real_1(C,D))) ) ) ) )
=> ( r2_fdiff_1(B,k4_limfunc1(A))
& r1_partfun2(k1_numbers,k1_numbers,B,k4_limfunc1(A)) ) ) ) ) ).
fof(t28_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v1_partfun1(A,k1_numbers,k1_numbers)
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,B),k2_seq_1(k1_numbers,k1_numbers,A,C))),k7_square_1(k5_real_1(B,C))) ) ) )
=> ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
& r1_partfun2(k1_numbers,k1_numbers,A,k2_subset_1(k1_numbers)) ) ) ) ).
fof(t29_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,k12_prob_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_hidden(C,k12_prob_1(A))
& r1_xreal_0(k1_fdiff_1(B,C),np__0) ) ) )
=> ( r1_rfunct_2(B,k12_prob_1(A))
& v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))) ) ) ) ) ).
fof(t30_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,k12_prob_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_hidden(C,k12_prob_1(A))
& r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) ) )
=> ( r2_rfunct_2(B,k12_prob_1(A))
& v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))) ) ) ) ) ).
fof(t31_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,k12_prob_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,k12_prob_1(A))
=> r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) ) )
=> r3_rfunct_2(B,k12_prob_1(A)) ) ) ) ).
fof(t32_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,k12_prob_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,k12_prob_1(A))
=> r1_xreal_0(k1_fdiff_1(B,C),np__0) ) ) )
=> r4_rfunct_2(B,k12_prob_1(A)) ) ) ) ).
fof(t33_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,k4_limfunc1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_hidden(C,k4_limfunc1(A))
& r1_xreal_0(k1_fdiff_1(B,C),np__0) ) ) )
=> ( r1_rfunct_2(B,k4_limfunc1(A))
& v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))) ) ) ) ) ).
fof(t34_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,k4_limfunc1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_hidden(C,k4_limfunc1(A))
& r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) ) )
=> ( r2_rfunct_2(B,k4_limfunc1(A))
& v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))) ) ) ) ) ).
fof(t35_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,k4_limfunc1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,k4_limfunc1(A))
=> r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) ) )
=> r3_rfunct_2(B,k4_limfunc1(A)) ) ) ) ).
fof(t36_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(B,k4_limfunc1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,k4_limfunc1(A))
=> r1_xreal_0(k1_fdiff_1(B,C),np__0) ) ) )
=> r4_rfunct_2(B,k4_limfunc1(A)) ) ) ) ).
fof(t37_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ r1_xreal_0(k1_fdiff_1(A,B),np__0) ) )
=> ( r1_rfunct_2(A,k2_subset_1(k1_numbers))
& v2_funct_1(A) ) ) ) ).
fof(t38_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ r1_xreal_0(np__0,k1_fdiff_1(A,B)) ) )
=> ( r2_rfunct_2(A,k2_subset_1(k1_numbers))
& v2_funct_1(A) ) ) ) ).
fof(t39_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> r1_xreal_0(np__0,k1_fdiff_1(A,B)) ) )
=> r3_rfunct_2(A,k2_subset_1(k1_numbers)) ) ) ).
fof(t40_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> r1_xreal_0(k1_fdiff_1(A,B),np__0) ) )
=> r4_rfunct_2(A,k2_subset_1(k1_numbers)) ) ) ).
fof(t41_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(C,k2_rcomp_1(A,B))
=> ( ( ? [D] :
( m1_subset_1(D,k1_numbers)
& r2_hidden(D,k2_rcomp_1(A,B))
& r1_xreal_0(k1_fdiff_1(C,D),np__0) )
& ? [D] :
( m1_subset_1(D,k1_numbers)
& r2_hidden(D,k2_rcomp_1(A,B))
& r1_xreal_0(np__0,k1_fdiff_1(C,D)) ) )
| v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))) ) ) ) ) ) ).
fof(t42_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(B,k12_prob_1(A))
=> ( ( ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k12_prob_1(A))
& r1_xreal_0(k1_fdiff_1(B,C),np__0) )
& ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k12_prob_1(A))
& r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) )
| v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))) ) ) ) ) ).
fof(t43_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(B,k4_limfunc1(A))
=> ( ( ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k4_limfunc1(A))
& r1_xreal_0(k1_fdiff_1(B,C),np__0) )
& ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k4_limfunc1(A))
& r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) )
| v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))) ) ) ) ) ).
fof(t44_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
=> ( ( ? [B] :
( m1_subset_1(B,k1_numbers)
& r1_xreal_0(k1_fdiff_1(A,B),np__0) )
& ? [B] :
( m1_subset_1(B,k1_numbers)
& r1_xreal_0(np__0,k1_fdiff_1(A,B)) ) )
| v3_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t45_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v2_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(A,k2_subset_1(k1_numbers))
=> ( ( ? [B] :
( m1_subset_1(B,k1_numbers)
& r1_xreal_0(k1_fdiff_1(A,B),np__0) )
& ? [B] :
( m1_subset_1(B,k1_numbers)
& r1_xreal_0(np__0,k1_fdiff_1(A,B)) ) )
| ( v2_funct_1(A)
& r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,A),k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,A)))
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,A)))
=> k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,A),B) = k6_real_1(np__1,k1_fdiff_1(A,k2_seq_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,A),B))) ) ) ) ) ) ) ).
fof(t46_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v2_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(B,k12_prob_1(A))
=> ( ( ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k12_prob_1(A))
& r1_xreal_0(k1_fdiff_1(B,C),np__0) )
& ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k12_prob_1(A))
& r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) )
| ( v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))
& r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))),k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A)))))
=> k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))),C) = k6_real_1(np__1,k1_fdiff_1(B,k2_seq_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k12_prob_1(A))),C))) ) ) ) ) ) ) ) ).
fof(t47_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v2_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(B,k4_limfunc1(A))
=> ( ( ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k4_limfunc1(A))
& r1_xreal_0(k1_fdiff_1(B,C),np__0) )
& ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k4_limfunc1(A))
& r1_xreal_0(np__0,k1_fdiff_1(B,C)) ) )
| ( v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))
& r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))),k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A)))))
=> k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))),C) = k6_real_1(np__1,k1_fdiff_1(B,k2_seq_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,k4_limfunc1(A))),C))) ) ) ) ) ) ) ) ).
fof(t48_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v2_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(C,k2_rcomp_1(A,B))
=> ( ( ? [D] :
( m1_subset_1(D,k1_numbers)
& r2_hidden(D,k2_rcomp_1(A,B))
& r1_xreal_0(k1_fdiff_1(C,D),np__0) )
& ? [D] :
( m1_subset_1(D,k1_numbers)
& r2_hidden(D,k2_rcomp_1(A,B))
& r1_xreal_0(np__0,k1_fdiff_1(C,D)) ) )
| ( v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))
& r2_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))),k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)))))
=> k1_fdiff_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))),D) = k6_real_1(np__1,k1_fdiff_1(C,k2_seq_1(k1_numbers,k1_numbers,k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B))),D))) ) ) ) ) ) ) ) ) ).
fof(t49_fdiff_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_fdiff_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v1_fdiff_1(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v5_seqm_3(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( k1_rfunct_2(D) = k1_tarski(A)
& r1_tarski(k1_rfunct_2(k9_seq_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,B))
& r1_tarski(k1_rfunct_2(k9_seq_1(k1_fdiff_2(C),D)),k4_relset_1(k1_numbers,k1_numbers,B)) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(k14_seq_1(C,np__2)),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,C)),k2_rfunct_2(B,k10_seq_1(D,C)))))
& k2_seq_2(k11_seq_1(k18_seq_1(k14_seq_1(C,np__2)),k10_seq_1(k2_rfunct_2(B,k9_seq_1(D,C)),k2_rfunct_2(B,k10_seq_1(D,C))))) = k1_fdiff_1(B,A) ) ) ) ) ) ) ) ).
fof(s1_fdiff_2,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& r1_xreal_0(A,B)
& p1_s1_fdiff_2(k2_seq_1(k5_numbers,k1_numbers,f1_s1_fdiff_2,B)) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seqm_3(A)
& v7_seqm_3(A)
& m2_relset_1(A,k5_numbers,k1_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> p1_s1_fdiff_2(k2_seq_1(k5_numbers,k1_numbers,k3_seqm_3(A,f1_s1_fdiff_2),B)) )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k2_seq_1(k5_numbers,k1_numbers,f1_s1_fdiff_2,B)
=> p1_s1_fdiff_2(C) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> B != k2_seqm_3(A,C) ) ) ) ) ) ).
fof(dt_k1_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_fdiff_1(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k1_fdiff_2(A))
& v1_funct_2(k1_fdiff_2(A),k5_numbers,k1_numbers)
& v1_fdiff_1(k1_fdiff_2(A))
& m2_relset_1(k1_fdiff_2(A),k5_numbers,k1_numbers) ) ) ).
fof(redefinition_k1_fdiff_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_fdiff_1(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> k1_fdiff_2(A) = k15_seq_1(A) ) ).
%------------------------------------------------------------------------------