SET007 Axioms: SET007+108.ax
%------------------------------------------------------------------------------
% File : SET007+108 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Real Function Differentiability
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 9 unt; 0 def)
% Number of atoms : 399 ( 29 equ)
% Maximal formula atoms : 18 ( 7 avg)
% Number of connectives : 350 ( 3 ~; 1 |; 201 &)
% ( 10 <=>; 135 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 29 ( 27 usr; 1 prp; 0-4 aty)
% Number of functors : 29 ( 29 usr; 4 con; 0-4 aty)
% Number of variables : 120 ( 109 !; 11 ?)
% 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(rc1_fdiff_1,axiom,
? [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v1_fdiff_1(A) ) ).
fof(rc2_fdiff_1,axiom,
? [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v3_seqm_3(A)
& v4_seqm_3(A)
& v5_seqm_3(A) ) ).
fof(cc1_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v5_seqm_3(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( m1_seqm_3(B,A)
=> ( v1_relat_1(B)
& v5_seqm_3(B) ) ) ) ).
fof(rc3_fdiff_1,axiom,
? [A] :
( m1_relset_1(A,k1_numbers,k1_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v2_fdiff_1(A) ) ).
fof(rc4_fdiff_1,axiom,
? [A] :
( m1_relset_1(A,k1_numbers,k1_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v3_fdiff_1(A) ) ).
fof(fc1_fdiff_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_fdiff_1(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k1_seqm_3(A,B))
& v1_funct_1(k1_seqm_3(A,B))
& v1_funct_2(k1_seqm_3(A,B),k5_numbers,k1_numbers)
& v1_seq_1(k1_seqm_3(A,B))
& v1_fdiff_1(k1_seqm_3(A,B)) ) ) ).
fof(fc2_fdiff_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v5_seqm_3(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k1_seqm_3(A,B))
& v1_funct_1(k1_seqm_3(A,B))
& v1_funct_2(k1_seqm_3(A,B),k5_numbers,k1_numbers)
& v1_seq_1(k1_seqm_3(A,B))
& v3_seqm_3(k1_seqm_3(A,B))
& v4_seqm_3(k1_seqm_3(A,B))
& v5_seqm_3(k1_seqm_3(A,B)) ) ) ).
fof(t1_fdiff_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(B,A)
<=> r2_hidden(B,k1_numbers) ) )
<=> A = k1_numbers ) ) ).
fof(d1_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_fdiff_1(A)
<=> ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0 ) ) ) ).
fof(d2_fdiff_1,axiom,
$true ).
fof(d3_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v2_fdiff_1(A)
<=> ( v1_partfun1(A,k1_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) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(B),k2_rfunct_2(A,B)))
& k2_seq_2(k11_seq_1(k18_seq_1(B),k2_rfunct_2(A,B))) = np__0 ) ) ) ) ) ).
fof(d4_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v3_fdiff_1(A)
<=> ( v1_partfun1(A,k1_numbers,k1_numbers)
& ? [B] :
( m1_subset_1(B,k1_numbers)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> k2_seq_1(k1_numbers,k1_numbers,A,C) = k4_real_1(B,C) ) ) ) ) ) ).
fof(t2_fdiff_1,axiom,
$true ).
fof(t3_fdiff_1,axiom,
$true ).
fof(t4_fdiff_1,axiom,
$true ).
fof(t5_fdiff_1,axiom,
$true ).
fof(t6_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v3_fdiff_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v3_fdiff_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v1_funct_1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& v3_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& m2_relset_1(k6_seq_1(k1_numbers,k1_numbers,A,B),k1_numbers,k1_numbers)
& v1_funct_1(k7_seq_1(k1_numbers,k1_numbers,A,B))
& v3_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,A,B))
& m2_relset_1(k7_seq_1(k1_numbers,k1_numbers,A,B),k1_numbers,k1_numbers) ) ) ) ).
fof(t7_fdiff_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v3_fdiff_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v1_funct_1(k13_seq_1(k1_numbers,k1_numbers,B,A))
& v3_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,B,A))
& m2_relset_1(k13_seq_1(k1_numbers,k1_numbers,B,A),k1_numbers,k1_numbers) ) ) ) ).
fof(t8_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v2_fdiff_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v2_fdiff_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v1_funct_1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& v2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& m2_relset_1(k6_seq_1(k1_numbers,k1_numbers,A,B),k1_numbers,k1_numbers)
& v1_funct_1(k7_seq_1(k1_numbers,k1_numbers,A,B))
& v2_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,A,B))
& m2_relset_1(k7_seq_1(k1_numbers,k1_numbers,A,B),k1_numbers,k1_numbers)
& v1_funct_1(k8_seq_1(k1_numbers,k1_numbers,A,B))
& v2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,A,B))
& m2_relset_1(k8_seq_1(k1_numbers,k1_numbers,A,B),k1_numbers,k1_numbers) ) ) ) ).
fof(t9_fdiff_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v2_fdiff_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v1_funct_1(k13_seq_1(k1_numbers,k1_numbers,B,A))
& v2_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,B,A))
& m2_relset_1(k13_seq_1(k1_numbers,k1_numbers,B,A),k1_numbers,k1_numbers) ) ) ) ).
fof(t10_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v3_fdiff_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v3_fdiff_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> v2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ).
fof(t11_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v2_fdiff_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v3_fdiff_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v1_funct_1(k8_seq_1(k1_numbers,k1_numbers,A,B))
& v2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,A,B))
& m2_relset_1(k8_seq_1(k1_numbers,k1_numbers,A,B),k1_numbers,k1_numbers)
& v1_funct_1(k8_seq_1(k1_numbers,k1_numbers,B,A))
& v2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,B,A))
& m2_relset_1(k8_seq_1(k1_numbers,k1_numbers,B,A),k1_numbers,k1_numbers) ) ) ) ).
fof(d5_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_fdiff_1(A,B)
<=> ? [C] :
( m1_rcomp_1(C,B)
& r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,A))
& ? [D] :
( v1_funct_1(D)
& v3_fdiff_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers)
& ? [E] :
( v1_funct_1(E)
& v2_fdiff_1(E)
& m2_relset_1(E,k1_numbers,k1_numbers)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,C)
=> k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,F),k2_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k2_seq_1(k1_numbers,k1_numbers,D,k6_xcmplx_0(F,B)),k2_seq_1(k1_numbers,k1_numbers,E,k6_xcmplx_0(F,B))) ) ) ) ) ) ) ) ) ).
fof(d6_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_fdiff_1(A,B)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k1_fdiff_1(A,B)
<=> ? [D] :
( m1_rcomp_1(D,B)
& r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,A))
& ? [E] :
( v1_funct_1(E)
& v3_fdiff_1(E)
& m2_relset_1(E,k1_numbers,k1_numbers)
& ? [F] :
( v1_funct_1(F)
& v2_fdiff_1(F)
& m2_relset_1(F,k1_numbers,k1_numbers)
& C = k2_seq_1(k1_numbers,k1_numbers,E,np__1)
& ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( r2_hidden(G,D)
=> k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,G),k2_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k2_seq_1(k1_numbers,k1_numbers,E,k6_xcmplx_0(G,B)),k2_seq_1(k1_numbers,k1_numbers,F,k6_xcmplx_0(G,B))) ) ) ) ) ) ) ) ) ) ) ).
fof(d7_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( r2_fdiff_1(A,B)
<=> ( r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,B)
=> r1_fdiff_1(k2_partfun1(k1_numbers,k1_numbers,A,B),C) ) ) ) ) ) ).
fof(t12_fdiff_1,axiom,
$true ).
fof(t13_fdiff_1,axiom,
$true ).
fof(t14_fdiff_1,axiom,
$true ).
fof(t15_fdiff_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(B,A)
=> m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) ) ).
fof(t16_fdiff_1,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)
<=> ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> r1_fdiff_1(B,C) ) ) ) ) ) ) ).
fof(t17_fdiff_1,axiom,
! [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)
=> v3_rcomp_1(A) ) ) ) ).
fof(d8_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( r2_fdiff_1(A,B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( C = k2_fdiff_1(A,B)
<=> ( k4_relset_1(k1_numbers,k1_numbers,C) = B
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,B)
=> k2_seq_1(k1_numbers,k1_numbers,C,D) = k1_fdiff_1(A,D) ) ) ) ) ) ) ) ).
fof(t18_fdiff_1,axiom,
$true ).
fof(t19_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v3_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> k5_relset_1(k1_numbers,k1_numbers,A) != k1_tarski(C) )
| ( r2_fdiff_1(A,B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(A,B),C) = np__0 ) ) ) ) ) ) ) ).
fof(t20_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_rcomp_1(C,B)
=> ( ( r1_fdiff_1(A,B)
& r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,A)) )
=> ! [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(B)
& r1_tarski(k1_rfunct_2(k9_seq_1(D,E)),C) )
=> ( v4_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E))))
& k1_fdiff_1(A,B) = k2_seq_2(k11_seq_1(k18_seq_1(D),k10_seq_1(k2_rfunct_2(A,k9_seq_1(D,E)),k2_rfunct_2(A,E)))) ) ) ) ) ) ) ) ) ).
fof(t21_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r1_fdiff_1(A,C)
& r1_fdiff_1(B,C) )
=> ( r1_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,A,B),C)
& k1_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,A,B),C) = k3_real_1(k1_fdiff_1(A,C),k1_fdiff_1(B,C)) ) ) ) ) ) ).
fof(t22_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r1_fdiff_1(A,C)
& r1_fdiff_1(B,C) )
=> ( r1_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,A,B),C)
& k1_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,A,B),C) = k5_real_1(k1_fdiff_1(A,C),k1_fdiff_1(B,C)) ) ) ) ) ) ).
fof(t23_fdiff_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r1_fdiff_1(B,C)
=> ( r1_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,B,A),C)
& k1_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,B,A),C) = k4_real_1(A,k1_fdiff_1(B,C)) ) ) ) ) ) ).
fof(t24_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r1_fdiff_1(A,C)
& r1_fdiff_1(B,C) )
=> ( r1_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,A,B),C)
& k1_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,A,B),C) = k3_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,B,C),k1_fdiff_1(A,C)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,A,C),k1_fdiff_1(B,C))) ) ) ) ) ) ).
fof(t25_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v3_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_partfun1(k1_numbers,k1_numbers,A,B) = k1_partfun2(k1_numbers,B) )
=> ( r2_fdiff_1(A,B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,B)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(A,B),C) = np__1 ) ) ) ) ) ) ).
fof(t26_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v3_rcomp_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ( ( r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,A,B)))
& r2_fdiff_1(A,C)
& r2_fdiff_1(B,C) )
=> ( r2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,A,B),C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,C)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k6_seq_1(k1_numbers,k1_numbers,A,B),C),D) = k3_real_1(k1_fdiff_1(A,D),k1_fdiff_1(B,D)) ) ) ) ) ) ) ) ).
fof(t27_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v3_rcomp_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ( ( r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,A,B)))
& r2_fdiff_1(A,C)
& r2_fdiff_1(B,C) )
=> ( r2_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,A,B),C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,C)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k7_seq_1(k1_numbers,k1_numbers,A,B),C),D) = k5_real_1(k1_fdiff_1(A,D),k1_fdiff_1(B,D)) ) ) ) ) ) ) ) ).
fof(t28_fdiff_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v3_rcomp_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ( ( r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,k13_seq_1(k1_numbers,k1_numbers,B,A)))
& r2_fdiff_1(B,C) )
=> ( r2_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,B,A),C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,C)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k13_seq_1(k1_numbers,k1_numbers,B,A),C),D) = k4_real_1(A,k1_fdiff_1(B,D)) ) ) ) ) ) ) ) ).
fof(t29_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v3_rcomp_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ( ( r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,A,B)))
& r2_fdiff_1(A,C)
& r2_fdiff_1(B,C) )
=> ( r2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,A,B),C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,C)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(k8_seq_1(k1_numbers,k1_numbers,A,B),C),D) = k3_real_1(k4_real_1(k2_seq_1(k1_numbers,k1_numbers,B,D),k1_fdiff_1(A,D)),k4_real_1(k2_seq_1(k1_numbers,k1_numbers,A,D),k1_fdiff_1(B,D))) ) ) ) ) ) ) ) ).
fof(t30_fdiff_1,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))
& r1_partfun2(k1_numbers,k1_numbers,B,A) )
=> ( r2_fdiff_1(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,A),C) = np__0 ) ) ) ) ) ) ).
fof(t31_fdiff_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v3_rcomp_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(C,k4_relset_1(k1_numbers,k1_numbers,D))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,C)
=> k2_seq_1(k1_numbers,k1_numbers,D,E) = k3_real_1(k4_real_1(A,E),B) ) ) )
=> ( r2_fdiff_1(D,C)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,C)
=> k2_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(D,C),E) = A ) ) ) ) ) ) ) ) ).
fof(t32_fdiff_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_fdiff_1(A,B)
=> r1_fcont_1(A,B) ) ) ) ).
fof(t33_fdiff_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fdiff_1(B,A)
=> r2_fcont_1(B,A) ) ) ).
fof(t34_fdiff_1,axiom,
! [A,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,A)
& r1_tarski(B,A) )
=> r2_fdiff_1(C,B) ) ) ) ).
fof(t35_fdiff_1,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)
& v2_fdiff_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ~ ( k2_seq_1(k1_numbers,k1_numbers,C,np__0) = np__0
& r1_fcont_1(C,np__0) ) ) ) ) ) ).
fof(dt_k1_fdiff_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& m1_relset_1(A,k1_numbers,k1_numbers)
& v1_xreal_0(B) )
=> m1_subset_1(k1_fdiff_1(A,B),k1_numbers) ) ).
fof(dt_k2_fdiff_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& m1_relset_1(A,k1_numbers,k1_numbers) )
=> ( v1_funct_1(k2_fdiff_1(A,B))
& m2_relset_1(k2_fdiff_1(A,B),k1_numbers,k1_numbers) ) ) ).
%------------------------------------------------------------------------------