SET007 Axioms: SET007+645.ax

```%------------------------------------------------------------------------------
% File     : SET007+645 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Introduction to Several Concepts of Convexity and Semicontinuity
% Version  : [Urb08] axioms.
% English  : Introduction to Several Concepts of Convexity and Semicontinuity
%            for Function from R to R

% 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    : rfunct_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   34 (   0 unit)
%            Number of atoms       :  297 (  12 equality)
%            Maximal formula depth :   22 (  12 average)
%            Number of connectives :  308 (  45 ~  ;   8  |; 121  &)
%                                         (  18 <=>; 116 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   0 propositional; 1-3 arity)
%            Number of functors    :   30 (   5 constant; 0-4 arity)
%            Number of variables   :  125 (   0 singleton; 115 !;  10 ?)
%            Maximal term depth    :    5 (   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_rfunct_4,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_xreal_0(k1_square_1(A,B),k2_square_1(A,B)) ) ) )).

fof(t2_rfunct_4,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_supinf_1,k4_finseq_2(A,k6_supinf_1))
=> ! [C] :
( m2_finseq_2(C,k6_supinf_1,k4_finseq_2(A,k6_supinf_1))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> k13_rvsum_1(k8_finseq_1(k1_numbers,B,k12_finseq_1(k1_numbers,D)),k8_finseq_1(k1_numbers,C,k12_finseq_1(k1_numbers,E))) = k8_finseq_1(k1_numbers,k14_rvsum_1(A,B,C),k12_finseq_1(k1_numbers,k11_binop_2(D,E))) ) ) ) ) ) )).

fof(t3_rfunct_4,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_supinf_1,k4_finseq_2(A,k6_supinf_1))
=> ( ( k15_rvsum_1(B) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k6_supinf_1,B,C)) ) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> k2_seq_1(k5_numbers,k6_supinf_1,B,C) = np__0 ) ) ) ) ) )).

fof(t4_rfunct_4,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_supinf_1,k4_finseq_2(A,k6_supinf_1))
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> np__0 = k2_seq_1(k5_numbers,k6_supinf_1,B,C) ) )
=> B = k4_finseqop(k1_numbers,A,np__0) ) ) ) )).

fof(t5_rfunct_4,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_supinf_1,k4_finseq_2(A,k6_supinf_1))
=> k14_rvsum_1(A,k4_finseqop(k1_numbers,A,np__0),B) = k4_finseqop(k1_numbers,A,np__0) ) ) )).

fof(d1_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r1_rfunct_4(A,B)
<=> ( r1_tarski(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& r2_hidden(D,B)
& r2_hidden(E,B)
& r2_hidden(k9_binop_2(k11_binop_2(C,D),k11_binop_2(k10_binop_2(np__1,C),E)),B)
& D != E
& r1_xreal_0(k9_binop_2(k11_binop_2(C,k2_seq_1(k6_supinf_1,k6_supinf_1,A,D)),k11_binop_2(k10_binop_2(np__1,C),k2_seq_1(k6_supinf_1,k6_supinf_1,A,E))),k2_seq_1(k6_supinf_1,k6_supinf_1,A,k9_binop_2(k11_binop_2(C,D),k11_binop_2(k10_binop_2(np__1,C),E)))) ) ) ) ) ) ) ) )).

fof(t6_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r1_rfunct_4(A,B)
=> r2_rfunct_3(A,B) ) ) )).

fof(t7_rfunct_4,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k6_supinf_1,k6_supinf_1) )
=> ( r1_rfunct_4(C,k1_rcomp_1(A,B))
<=> ( r1_tarski(k1_rcomp_1(A,B),k1_relat_1(C))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(np__1,D)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m1_subset_1(F,k1_numbers)
& r2_hidden(E,k1_rcomp_1(A,B))
& r2_hidden(F,k1_rcomp_1(A,B))
& E != F
& r1_xreal_0(k9_binop_2(k11_binop_2(D,k2_seq_1(k6_supinf_1,k6_supinf_1,C,E)),k11_binop_2(k10_binop_2(np__1,D),k2_seq_1(k6_supinf_1,k6_supinf_1,C,F))),k2_seq_1(k6_supinf_1,k6_supinf_1,C,k9_binop_2(k11_binop_2(D,E),k11_binop_2(k10_binop_2(np__1,D),F)))) ) ) ) ) ) ) ) ) ) )).

fof(t8_rfunct_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( r2_rfunct_3(B,A)
<=> ( r1_tarski(A,k1_relat_1(B))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(C,A)
& r2_hidden(D,A)
& r2_hidden(E,A) )
=> ( r1_xreal_0(D,C)
| r1_xreal_0(E,D)
| r1_xreal_0(k2_seq_1(k6_supinf_1,k6_supinf_1,B,D),k9_binop_2(k11_binop_2(k12_binop_2(k10_binop_2(E,D),k10_binop_2(E,C)),k2_seq_1(k6_supinf_1,k6_supinf_1,B,C)),k11_binop_2(k12_binop_2(k10_binop_2(D,C),k10_binop_2(E,C)),k2_seq_1(k6_supinf_1,k6_supinf_1,B,E)))) ) ) ) ) ) ) ) ) )).

fof(t9_rfunct_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( r1_rfunct_4(B,A)
<=> ( r1_tarski(A,k1_relat_1(B))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(C,A)
& r2_hidden(D,A)
& r2_hidden(E,A)
& ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(E,D)
& r1_xreal_0(k9_binop_2(k11_binop_2(k12_binop_2(k10_binop_2(E,D),k10_binop_2(E,C)),k2_seq_1(k6_supinf_1,k6_supinf_1,B,C)),k11_binop_2(k12_binop_2(k10_binop_2(D,C),k10_binop_2(E,C)),k2_seq_1(k6_supinf_1,k6_supinf_1,B,E))),k2_seq_1(k6_supinf_1,k6_supinf_1,B,D)) ) ) ) ) ) ) ) )).

fof(t10_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B,C] :
( ( r1_rfunct_4(A,B)
& r1_tarski(C,B) )
=> r1_rfunct_4(A,C) ) ) )).

fof(t11_rfunct_4,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ! [C] :
( r1_rfunct_4(B,C)
<=> r1_rfunct_4(k21_rfunct_3(k6_supinf_1,B,A),C) ) ) ) )).

fof(t12_rfunct_4,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ! [C] :
( ~ r1_xreal_0(A,np__0)
=> ( r1_rfunct_4(B,C)
<=> r1_rfunct_4(k13_seq_1(k6_supinf_1,k6_supinf_1,B,A),C) ) ) ) ) )).

fof(t13_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ! [C] :
( ( r1_rfunct_4(A,C)
& r1_rfunct_4(B,C) )
=> r1_rfunct_4(k6_seq_1(k6_supinf_1,k6_supinf_1,A,B),C) ) ) ) )).

fof(t14_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ! [C] :
( ( r1_rfunct_4(A,C)
& r2_rfunct_3(B,C) )
=> r1_rfunct_4(k6_seq_1(k6_supinf_1,k6_supinf_1,A,B),C) ) ) ) )).

fof(t15_rfunct_4,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k6_supinf_1,k6_supinf_1) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k6_supinf_1,k6_supinf_1) )
=> ! [E] :
( ( r1_rfunct_4(C,E)
& r1_rfunct_4(D,E) )
=> ( ( ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(np__0,B) )
& ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,np__0) ) )
| r1_rfunct_4(k6_seq_1(k6_supinf_1,k1_numbers,k13_seq_1(k6_supinf_1,k6_supinf_1,C,A),k13_seq_1(k6_supinf_1,k6_supinf_1,D,B)),E) ) ) ) ) ) ) )).

fof(t16_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r2_rfunct_3(A,B)
<=> ( r1_tarski(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B)
& r2_hidden(E,B) )
=> ( r1_xreal_0(E,C)
| r1_xreal_0(D,E)
| ( r1_xreal_0(k12_binop_2(k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,E),k2_seq_1(k6_supinf_1,k6_supinf_1,A,C)),k10_binop_2(E,C)),k12_binop_2(k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,C)),k10_binop_2(D,C)))
& r1_xreal_0(k12_binop_2(k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,C)),k10_binop_2(D,C)),k12_binop_2(k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,E)),k10_binop_2(D,E))) ) ) ) ) ) ) ) ) ) )).

fof(t17_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r1_rfunct_4(A,B)
<=> ( r1_tarski(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B)
& r2_hidden(E,B) )
=> ( r1_xreal_0(E,C)
| r1_xreal_0(D,E)
| ( ~ r1_xreal_0(k12_binop_2(k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,C)),k10_binop_2(D,C)),k12_binop_2(k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,E),k2_seq_1(k6_supinf_1,k6_supinf_1,A,C)),k10_binop_2(E,C)))
& ~ r1_xreal_0(k12_binop_2(k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,E)),k10_binop_2(D,E)),k12_binop_2(k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,C)),k10_binop_2(D,C))) ) ) ) ) ) ) ) ) ) )).

fof(t18_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ( v1_partfun1(A,k6_supinf_1,k6_supinf_1)
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k6_supinf_1,k4_finseq_2(B,k6_supinf_1))
=> ! [D] :
( m2_finseq_2(D,k6_supinf_1,k4_finseq_2(B,k6_supinf_1))
=> ! [E] :
( m2_finseq_2(E,k6_supinf_1,k4_finseq_2(B,k6_supinf_1))
=> ( ( k15_rvsum_1(C) = np__1
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k4_finseq_1(C))
=> ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k6_supinf_1,C,F))
& k2_seq_1(k5_numbers,k6_supinf_1,E,F) = k2_seq_1(k6_supinf_1,k6_supinf_1,A,k2_seq_1(k5_numbers,k6_supinf_1,D,F)) ) ) ) )
=> r1_xreal_0(k2_seq_1(k6_supinf_1,k6_supinf_1,A,k15_rvsum_1(k14_rvsum_1(B,C,D))),k15_rvsum_1(k14_rvsum_1(B,C,E))) ) ) ) ) )
<=> r2_rfunct_3(A,k6_supinf_1) ) ) ) )).

fof(t19_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( ( v5_measure5(B)
& m1_subset_1(B,k1_zfmisc_1(k6_supinf_1)) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_rfunct_3(A,B)
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(D,B)
& r2_hidden(E,B)
& ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(E,C) ) ) )
| r1_fcont_1(A,C) ) ) ) ) ) )).

fof(d2_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r2_rfunct_4(A,B)
<=> ( r1_tarski(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& r2_hidden(D,B)
& r2_hidden(E,B)
& r2_hidden(k9_binop_2(k11_binop_2(C,D),k11_binop_2(k10_binop_2(np__1,C),E)),B)
& ~ r1_xreal_0(k2_seq_1(k6_supinf_1,k6_supinf_1,A,k9_binop_2(k11_binop_2(C,D),k11_binop_2(k10_binop_2(np__1,C),E))),k4_square_1(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,E))) ) ) ) ) ) ) ) )).

fof(d3_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r3_rfunct_4(A,B)
<=> ( r1_tarski(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& r2_hidden(D,B)
& r2_hidden(E,B)
& r2_hidden(k9_binop_2(k11_binop_2(C,D),k11_binop_2(k10_binop_2(np__1,C),E)),B)
& k2_seq_1(k6_supinf_1,k6_supinf_1,A,D) != k2_seq_1(k6_supinf_1,k6_supinf_1,A,E)
& r1_xreal_0(k4_square_1(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,E)),k2_seq_1(k6_supinf_1,k6_supinf_1,A,k9_binop_2(k11_binop_2(C,D),k11_binop_2(k10_binop_2(np__1,C),E)))) ) ) ) ) ) ) ) )).

fof(d4_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r4_rfunct_4(A,B)
<=> ( r1_tarski(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& r2_hidden(D,B)
& r2_hidden(E,B)
& r2_hidden(k9_binop_2(k11_binop_2(C,D),k11_binop_2(k10_binop_2(np__1,C),E)),B)
& D != E
& r1_xreal_0(k4_square_1(k2_seq_1(k6_supinf_1,k6_supinf_1,A,D),k2_seq_1(k6_supinf_1,k6_supinf_1,A,E)),k2_seq_1(k6_supinf_1,k6_supinf_1,A,k9_binop_2(k11_binop_2(C,D),k11_binop_2(k10_binop_2(np__1,C),E)))) ) ) ) ) ) ) ) )).

fof(d5_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r5_rfunct_4(A,B)
<=> ( r2_hidden(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,k1_relat_1(A))
& ~ r1_xreal_0(D,k18_complex1(k6_xcmplx_0(E,B)))
& r1_xreal_0(C,k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,B),k2_seq_1(k6_supinf_1,k6_supinf_1,A,E))) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r6_rfunct_4(A,B)
<=> ( r1_tarski(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,B)
=> r5_rfunct_4(k2_partfun1(k6_supinf_1,k6_supinf_1,A,B),C) ) ) ) ) ) )).

fof(d7_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r7_rfunct_4(A,B)
<=> ( r2_hidden(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,k1_relat_1(A))
& ~ r1_xreal_0(D,k18_complex1(k6_xcmplx_0(E,B)))
& r1_xreal_0(C,k10_binop_2(k2_seq_1(k6_supinf_1,k6_supinf_1,A,E),k2_seq_1(k6_supinf_1,k6_supinf_1,A,B))) ) ) ) ) ) ) ) ) ) ) )).

fof(d8_rfunct_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k6_supinf_1,k6_supinf_1) )
=> ! [B] :
( r8_rfunct_4(A,B)
<=> ( r1_tarski(B,k1_relat_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,B)
=> r7_rfunct_4(k2_partfun1(k6_supinf_1,k6_supinf_1,A,B),C) ) ) ) ) ) )).

fof(t20_rfunct_4,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( ( r5_rfunct_4(B,A)
& r7_rfunct_4(B,A) )
<=> r1_fcont_1(B,A) ) ) ) )).

fof(t21_rfunct_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( ( r6_rfunct_4(B,A)
& r8_rfunct_4(B,A) )
<=> r2_fcont_1(B,A) ) ) )).

fof(t22_rfunct_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( r1_rfunct_4(B,A)
=> r4_rfunct_4(B,A) ) ) )).

fof(t23_rfunct_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( r4_rfunct_4(B,A)
=> r2_rfunct_4(B,A) ) ) )).

fof(t24_rfunct_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( r2_rfunct_3(B,A)
=> r3_rfunct_4(B,A) ) ) )).

fof(t25_rfunct_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( r4_rfunct_4(B,A)
=> r3_rfunct_4(B,A) ) ) )).

fof(t26_rfunct_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k6_supinf_1,k6_supinf_1) )
=> ( ( r3_rfunct_4(B,A)
& v2_funct_1(B) )
=> r4_rfunct_4(B,A) ) ) )).
%------------------------------------------------------------------------------
```