SET007 Axioms: SET007+106.ax
%------------------------------------------------------------------------------
% File : SET007+106 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Real Function Continuity
% 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 : fcont_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 3 unt; 0 def)
% Number of atoms : 407 ( 29 equ)
% Maximal formula atoms : 15 ( 7 avg)
% Number of connectives : 375 ( 25 ~; 6 |; 160 &)
% ( 10 <=>; 174 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-4 aty)
% Number of functors : 38 ( 38 usr; 4 con; 0-6 aty)
% Number of variables : 162 ( 158 !; 4 ?)
% 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(d1_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_fcont_1(A,B)
<=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( r1_tarski(k1_rfunct_2(C),k4_relset_1(k1_numbers,k1_numbers,A))
& v4_seq_2(C)
& k2_seq_2(C) = B )
=> ( v4_seq_2(k2_rfunct_2(A,C))
& k2_seq_1(k1_numbers,k1_numbers,A,B) = k2_seq_2(k2_rfunct_2(A,C)) ) ) ) ) ) ) ) ).
fof(t1_fcont_1,axiom,
$true ).
fof(t2_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_fcont_1(B,A)
<=> ( r2_hidden(A,k4_relset_1(k1_numbers,k1_numbers,B))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( r1_tarski(k1_rfunct_2(C),k4_relset_1(k1_numbers,k1_numbers,B))
& v4_seq_2(C)
& k2_seq_2(C) = A
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) != A ) )
=> ( v4_seq_2(k2_rfunct_2(B,C))
& k2_seq_1(k1_numbers,k1_numbers,B,A) = k2_seq_2(k2_rfunct_2(B,C)) ) ) ) ) ) ) ) ).
fof(t3_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_fcont_1(B,A)
<=> ( r2_hidden(A,k4_relset_1(k1_numbers,k1_numbers,B))
& ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( v1_xreal_0(E)
=> ~ ( r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
& ~ r1_xreal_0(D,k18_complex1(k6_xcmplx_0(E,A)))
& r1_xreal_0(C,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,B,A)))) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_fcont_1(A,B)
<=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [C] :
( m1_rcomp_1(C,k2_seq_1(k1_numbers,k1_numbers,A,B))
=> ? [D] :
( m1_rcomp_1(D,B)
& ! [E] :
( v1_xreal_0(E)
=> ( ( r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
& r2_hidden(E,D) )
=> r2_hidden(k2_seq_1(k1_numbers,k1_numbers,A,E),C) ) ) ) ) ) ) ) ) ).
fof(t5_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_fcont_1(A,B)
<=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [C] :
( m1_rcomp_1(C,k2_seq_1(k1_numbers,k1_numbers,A,B))
=> ? [D] :
( m1_rcomp_1(D,B)
& r1_tarski(k10_relset_1(k1_numbers,k1_numbers,A,D),C) ) ) ) ) ) ) ).
fof(t6_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_hidden(A,k4_relset_1(k1_numbers,k1_numbers,B))
=> ( ! [C] :
( m1_rcomp_1(C,A)
=> k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),C) != k1_seq_4(A) )
| r1_fcont_1(B,A) ) ) ) ) ).
fof(t7_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [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_fcont_1(B,A)
& r1_fcont_1(C,A) )
=> ( r1_fcont_1(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& r1_fcont_1(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
& r1_fcont_1(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t8_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r1_fcont_1(C,A)
=> r1_fcont_1(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ).
fof(t9_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_fcont_1(B,A)
=> ( r1_fcont_1(k21_seq_1(k1_numbers,k1_numbers,B),A)
& r1_fcont_1(k16_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t10_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_fcont_1(B,A)
=> ( k2_seq_1(k1_numbers,k1_numbers,B,A) = np__0
| r1_fcont_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t11_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [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_fcont_1(B,A)
& r1_fcont_1(C,A) )
=> ( k2_seq_1(k1_numbers,k1_numbers,B,A) = np__0
| r1_fcont_1(k2_rfunct_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) ).
fof(t12_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [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_fcont_1(B,A)
& r1_fcont_1(C,k2_seq_1(k1_numbers,k1_numbers,B,A)) )
=> r1_fcont_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A) ) ) ) ) ).
fof(d2_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( r2_fcont_1(A,B)
<=> ( r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,B)
=> r1_fcont_1(k2_partfun1(k1_numbers,k1_numbers,A,B),C) ) ) ) ) ) ).
fof(t13_fcont_1,axiom,
$true ).
fof(t14_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(B,A)
<=> ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( r1_tarski(k1_rfunct_2(C),A)
& v4_seq_2(C)
& r2_hidden(k2_seq_2(C),A) )
=> ( v4_seq_2(k2_rfunct_2(B,C))
& k2_seq_1(k1_numbers,k1_numbers,B,k2_seq_2(C)) = k2_seq_2(k2_rfunct_2(B,C)) ) ) ) ) ) ) ).
fof(t15_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(B,A)
<=> ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( r2_hidden(C,A)
& ~ r1_xreal_0(D,np__0)
& ! [E] :
( v1_xreal_0(E)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ! [F] :
( v1_xreal_0(F)
=> ~ ( r2_hidden(F,A)
& ~ r1_xreal_0(E,k18_complex1(k6_xcmplx_0(F,C)))
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,B,C)))) ) ) ) ) ) ) ) ) ) ) ).
fof(t16_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(B,A)
<=> r2_fcont_1(k2_partfun1(k1_numbers,k1_numbers,B,A),A) ) ) ).
fof(t17_fcont_1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r2_fcont_1(C,A)
& r1_tarski(B,A) )
=> r2_fcont_1(C,B) ) ) ).
fof(t18_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_hidden(A,k4_relset_1(k1_numbers,k1_numbers,B))
=> r2_fcont_1(B,k1_seq_4(A)) ) ) ) ).
fof(t19_fcont_1,axiom,
! [A,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_fcont_1(B,A)
& r2_fcont_1(C,A) )
=> ( r2_fcont_1(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& r2_fcont_1(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
& r2_fcont_1(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ).
fof(t20_fcont_1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r2_fcont_1(C,A)
& r2_fcont_1(D,B) )
=> ( r2_fcont_1(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B))
& r2_fcont_1(k7_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B))
& r2_fcont_1(k8_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) ) ).
fof(t21_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(C,B)
=> r2_fcont_1(k13_seq_1(k1_numbers,k1_numbers,C,A),B) ) ) ) ).
fof(t22_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(B,A)
=> ( r2_fcont_1(k21_seq_1(k1_numbers,k1_numbers,B),A)
& r2_fcont_1(k16_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) ).
fof(t23_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fcont_1(B,A)
& k3_funct_2(k1_numbers,k1_numbers,B,k1_seq_4(np__0)) = k1_xboole_0 )
=> r2_fcont_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ).
fof(t24_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_fcont_1(B,A)
& k3_funct_2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,A),k1_seq_4(np__0)) = k1_xboole_0 )
=> r2_fcont_1(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ).
fof(t25_fcont_1,axiom,
! [A,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_fcont_1(B,A)
& k3_funct_2(k1_numbers,k1_numbers,B,k1_seq_4(np__0)) = k1_xboole_0
& r2_fcont_1(C,A) )
=> r2_fcont_1(k2_rfunct_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ).
fof(t26_fcont_1,axiom,
! [A,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_fcont_1(B,A)
& r2_fcont_1(C,k10_relset_1(k1_numbers,k1_numbers,B,A)) )
=> r2_fcont_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,B,C),A) ) ) ) ).
fof(t27_fcont_1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r2_fcont_1(C,A)
& r2_fcont_1(D,B) )
=> r2_fcont_1(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,D),k3_xboole_0(A,k3_funct_2(k1_numbers,k1_numbers,C,B))) ) ) ) ).
fof(t28_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v1_partfun1(A,k1_numbers,k1_numbers)
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_seq_1(k1_numbers,k1_numbers,A,k2_xcmplx_0(B,C)) = k3_real_1(k2_seq_1(k1_numbers,k1_numbers,A,B),k2_seq_1(k1_numbers,k1_numbers,A,C)) ) ) )
=> ( ! [B] :
( v1_xreal_0(B)
=> ~ r1_fcont_1(A,B) )
| r2_fcont_1(A,k1_numbers) ) ) ) ).
fof(t29_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v1_rcomp_1(k4_relset_1(k1_numbers,k1_numbers,A))
& r2_fcont_1(A,k4_relset_1(k1_numbers,k1_numbers,A)) )
=> v1_rcomp_1(k5_relset_1(k1_numbers,k1_numbers,A)) ) ) ).
fof(t30_fcont_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) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& v1_rcomp_1(A)
& r2_fcont_1(B,A) )
=> v1_rcomp_1(k10_relset_1(k1_numbers,k1_numbers,B,A)) ) ) ) ).
fof(t31_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ~ ( k4_relset_1(k1_numbers,k1_numbers,A) != k1_xboole_0
& v1_rcomp_1(k4_relset_1(k1_numbers,k1_numbers,A))
& r2_fcont_1(A,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,B) = k4_seq_4(k5_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) = k5_seq_4(k5_relset_1(k1_numbers,k1_numbers,A)) ) ) ) ) ) ).
fof(t32_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ~ ( B != k1_xboole_0
& r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
& v1_rcomp_1(B)
& r2_fcont_1(A,B)
& ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& k2_seq_1(k1_numbers,k1_numbers,A,C) = k4_seq_4(k10_relset_1(k1_numbers,k1_numbers,A,B))
& k2_seq_1(k1_numbers,k1_numbers,A,D) = k5_seq_4(k10_relset_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ) ) ).
fof(d3_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( r3_fcont_1(A,B)
<=> ( r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ? [C] :
( v1_xreal_0(C)
& ~ r1_xreal_0(C,np__0)
& ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ( ( r2_hidden(D,B)
& r2_hidden(E,B) )
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,A,E))),k3_xcmplx_0(C,k18_complex1(k6_xcmplx_0(D,E)))) ) ) ) ) ) ) ) ).
fof(t33_fcont_1,axiom,
$true ).
fof(t34_fcont_1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r3_fcont_1(C,A)
& r1_tarski(B,A) )
=> r3_fcont_1(C,B) ) ) ).
fof(t35_fcont_1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r3_fcont_1(C,A)
& r3_fcont_1(D,B) )
=> r3_fcont_1(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) ).
fof(t36_fcont_1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r3_fcont_1(C,A)
& r3_fcont_1(D,B) )
=> r3_fcont_1(k7_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) ).
fof(t37_fcont_1,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(E)
& m2_relset_1(E,k1_numbers,k1_numbers) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,k1_numbers,k1_numbers) )
=> ( ( r3_fcont_1(E,A)
& r3_fcont_1(F,B)
& r3_rfunct_1(E,C)
& r3_rfunct_1(F,D) )
=> r3_fcont_1(k8_seq_1(k1_numbers,k1_numbers,E,F),k3_xboole_0(k3_xboole_0(k3_xboole_0(A,C),B),D)) ) ) ) ).
fof(t38_fcont_1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r3_fcont_1(C,A)
=> r3_fcont_1(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ).
fof(t39_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r3_fcont_1(B,A)
=> ( r3_fcont_1(k16_seq_1(k1_numbers,k1_numbers,B),A)
& r3_fcont_1(k21_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) ).
fof(t40_fcont_1,axiom,
! [A,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) )
=> r3_fcont_1(B,A) ) ) ).
fof(t41_fcont_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> r3_fcont_1(k1_partfun2(k1_numbers,A),A) ) ).
fof(t42_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r3_fcont_1(B,A)
=> r2_fcont_1(B,A) ) ) ).
fof(t43_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ? [B] :
( v1_xreal_0(B)
& k5_relset_1(k1_numbers,k1_numbers,A) = k1_seq_4(B) )
=> r2_fcont_1(A,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ).
fof(t44_fcont_1,axiom,
! [A,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_fcont_1(B,A) ) ) ).
fof(t45_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = B ) )
=> r2_fcont_1(A,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ).
fof(t46_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( A = k1_partfun2(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A))
=> r2_fcont_1(A,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ).
fof(t47_fcont_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) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_partfun1(k1_numbers,k1_numbers,B,A) = k1_partfun2(k1_numbers,A) )
=> r2_fcont_1(B,A) ) ) ) ).
fof(t48_fcont_1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,D))
& ! [E] :
( v1_xreal_0(E)
=> ( r2_hidden(E,A)
=> k2_seq_1(k1_numbers,k1_numbers,D,E) = k2_xcmplx_0(k3_xcmplx_0(B,E),C) ) ) )
=> r2_fcont_1(D,A) ) ) ) ) ).
fof(t49_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = k5_square_1(B) ) )
=> r2_fcont_1(A,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ).
fof(t50_fcont_1,axiom,
! [A,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] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,B,C) = k5_square_1(C) ) ) )
=> r2_fcont_1(B,A) ) ) ).
fof(t51_fcont_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(B,k4_relset_1(k1_numbers,k1_numbers,A))
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = k18_complex1(B) ) )
=> r2_fcont_1(A,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ).
fof(t52_fcont_1,axiom,
! [A,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] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> k2_seq_1(k1_numbers,k1_numbers,B,C) = k18_complex1(C) ) ) )
=> r2_fcont_1(B,A) ) ) ).
fof(t53_fcont_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
& r5_rfunct_2(B,A) )
=> ( ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( r1_xreal_0(C,D)
& k10_relset_1(k1_numbers,k1_numbers,B,A) = k1_rcomp_1(C,D) ) ) )
| r2_fcont_1(B,A) ) ) ) ).
fof(t54_fcont_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v2_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r1_xreal_0(A,B)
& r1_tarski(k1_rcomp_1(A,B),k4_relset_1(k1_numbers,k1_numbers,C)) )
=> ( ( ~ r1_rfunct_2(C,k1_rcomp_1(A,B))
& ~ r2_rfunct_2(C,k1_rcomp_1(A,B)) )
| r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))),k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))) ) ) ) ) ) ).
%------------------------------------------------------------------------------