SET007 Axioms: SET007+103.ax
%------------------------------------------------------------------------------
% File : SET007+103 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Partial Functions from a Domain to the Set of Real Numbers
% 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 : rfunct_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 132 ( 23 unt; 0 def)
% Number of atoms : 836 ( 100 equ)
% Maximal formula atoms : 23 ( 6 avg)
% Number of connectives : 738 ( 34 ~; 0 |; 410 &)
% ( 16 <=>; 278 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-4 aty)
% Number of functors : 42 ( 42 usr; 4 con; 0-4 aty)
% Number of variables : 302 ( 299 !; 3 ?)
% 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(fc1_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ( v1_relat_1(k1_rfunct_1(A,B))
& v1_funct_1(k1_rfunct_1(A,B))
& v1_seq_1(k1_rfunct_1(A,B)) ) ) ).
fof(fc2_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k3_rfunct_1(A))
& v1_funct_1(k3_rfunct_1(A))
& v1_seq_1(k3_rfunct_1(A)) ) ) ).
fof(d1_rfunct_1,axiom,
$true ).
fof(d2_rfunct_1,axiom,
$true ).
fof(d3_rfunct_1,axiom,
$true ).
fof(d4_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k1_rfunct_1(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k4_xboole_0(k1_relat_1(B),k10_relat_1(B,k1_tarski(np__0))))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k4_real_1(k1_seq_1(A,D),k2_real_1(k1_seq_1(B,D))) ) ) ) ) ) ) ).
fof(d5_rfunct_1,axiom,
$true ).
fof(d6_rfunct_1,axiom,
$true ).
fof(d7_rfunct_1,axiom,
$true ).
fof(d8_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k3_rfunct_1(A)
<=> ( k1_relat_1(B) = k4_xboole_0(k1_relat_1(A),k10_relat_1(A,k1_tarski(np__0)))
& ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> k1_funct_1(B,C) = k2_real_1(k1_seq_1(A,C)) ) ) ) ) ) ).
fof(t1_rfunct_1,axiom,
$true ).
fof(t2_rfunct_1,axiom,
$true ).
fof(t3_rfunct_1,axiom,
$true ).
fof(t4_rfunct_1,axiom,
$true ).
fof(t5_rfunct_1,axiom,
$true ).
fof(t6_rfunct_1,axiom,
$true ).
fof(t7_rfunct_1,axiom,
$true ).
fof(t8_rfunct_1,axiom,
$true ).
fof(t9_rfunct_1,axiom,
$true ).
fof(t10_rfunct_1,axiom,
$true ).
fof(t11_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( r1_tarski(k1_relat_1(k3_rfunct_1(A)),k1_relat_1(A))
& k3_xboole_0(k1_relat_1(A),k4_xboole_0(k1_relat_1(A),k10_relat_1(A,k1_tarski(np__0)))) = k4_xboole_0(k1_relat_1(A),k10_relat_1(A,k1_tarski(np__0))) ) ) ).
fof(t12_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k4_xboole_0(k1_relat_1(k5_seq_1(A,B)),k10_relat_1(k5_seq_1(A,B),k1_tarski(np__0))) = k3_xboole_0(k4_xboole_0(k1_relat_1(A),k10_relat_1(A,k1_tarski(np__0))),k4_xboole_0(k1_relat_1(B),k10_relat_1(B,k1_tarski(np__0)))) ) ) ).
fof(t13_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ~ ( r2_hidden(B,k1_relat_1(k3_rfunct_1(C)))
& k1_seq_1(C,B) = np__0 ) ) ) ) ).
fof(t14_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k10_relat_1(k3_rfunct_1(A),k1_tarski(np__0)) = k1_xboole_0 ) ).
fof(t15_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( k10_relat_1(k20_seq_1(A),k1_tarski(np__0)) = k10_relat_1(A,k1_tarski(np__0))
& k10_relat_1(k15_seq_1(A),k1_tarski(np__0)) = k10_relat_1(A,k1_tarski(np__0)) ) ) ).
fof(t16_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k1_relat_1(k3_rfunct_1(k3_rfunct_1(A))) = k1_relat_1(k7_relat_1(A,k1_relat_1(k3_rfunct_1(A)))) ) ).
fof(t17_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ( B != np__0
=> k10_relat_1(k12_seq_1(A,B),k1_tarski(np__0)) = k10_relat_1(A,k1_tarski(np__0)) ) ) ) ).
fof(t18_rfunct_1,axiom,
$true ).
fof(t19_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k3_seq_1(k3_seq_1(A,B),C) = k3_seq_1(A,k3_seq_1(B,C)) ) ) ) ).
fof(t20_rfunct_1,axiom,
$true ).
fof(t21_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k5_seq_1(k5_seq_1(A,B),C) = k5_seq_1(A,k5_seq_1(B,C)) ) ) ) ).
fof(t22_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k5_seq_1(k3_seq_1(A,B),C) = k3_seq_1(k5_seq_1(A,C),k5_seq_1(B,C)) ) ) ) ).
fof(t23_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k5_seq_1(A,k3_seq_1(B,C)) = k3_seq_1(k5_seq_1(A,B),k5_seq_1(A,C)) ) ) ) ).
fof(t24_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( v1_xreal_0(C)
=> k12_seq_1(k5_seq_1(A,B),C) = k5_seq_1(k12_seq_1(A,C),B) ) ) ) ).
fof(t25_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( v1_xreal_0(C)
=> k12_seq_1(k5_seq_1(A,B),C) = k5_seq_1(A,k12_seq_1(B,C)) ) ) ) ).
fof(t26_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k5_seq_1(k4_seq_1(A,B),C) = k4_seq_1(k5_seq_1(A,C),k5_seq_1(B,C)) ) ) ) ).
fof(t27_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k4_seq_1(k5_seq_1(A,B),k5_seq_1(A,C)) = k5_seq_1(A,k4_seq_1(B,C)) ) ) ) ).
fof(t28_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( v1_xreal_0(C)
=> k12_seq_1(k3_seq_1(A,B),C) = k3_seq_1(k12_seq_1(A,C),k12_seq_1(B,C)) ) ) ) ).
fof(t29_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k12_seq_1(A,k3_xcmplx_0(B,C)) = k12_seq_1(k12_seq_1(A,C),B) ) ) ) ).
fof(t30_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( v1_xreal_0(C)
=> k12_seq_1(k4_seq_1(A,B),C) = k4_seq_1(k12_seq_1(A,C),k12_seq_1(B,C)) ) ) ) ).
fof(t31_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k4_seq_1(A,B) = k12_seq_1(k4_seq_1(B,A),k1_real_1(np__1)) ) ) ).
fof(t32_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k4_seq_1(A,k3_seq_1(B,C)) = k4_seq_1(k4_seq_1(A,B),C) ) ) ) ).
fof(t33_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k12_seq_1(A,np__1) = A ) ).
fof(t34_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k4_seq_1(A,k4_seq_1(B,C)) = k3_seq_1(k4_seq_1(A,B),C) ) ) ) ).
fof(t35_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k3_seq_1(A,k4_seq_1(B,C)) = k4_seq_1(k3_seq_1(A,B),C) ) ) ) ).
fof(t36_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k20_seq_1(k5_seq_1(A,B)) = k5_seq_1(k20_seq_1(A),k20_seq_1(B)) ) ) ).
fof(t37_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( v1_xreal_0(B)
=> k20_seq_1(k12_seq_1(A,B)) = k12_seq_1(k20_seq_1(A),k18_complex1(B)) ) ) ).
fof(t38_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k15_seq_1(A) = k12_seq_1(A,k1_real_1(np__1)) ) ).
fof(t39_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k15_seq_1(k15_seq_1(A)) = A ) ).
fof(t40_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k4_seq_1(A,B) = k3_seq_1(A,k15_seq_1(B)) ) ) ).
fof(t41_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k4_seq_1(A,k15_seq_1(B)) = k3_seq_1(A,B) ) ) ).
fof(t42_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k3_rfunct_1(k3_rfunct_1(A)) = k7_relat_1(A,k1_relat_1(k3_rfunct_1(A))) ) ).
fof(t43_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k3_rfunct_1(k5_seq_1(A,B)) = k5_seq_1(k3_rfunct_1(A),k3_rfunct_1(B)) ) ) ).
fof(t44_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ( B != np__0
=> k3_rfunct_1(k12_seq_1(A,B)) = k12_seq_1(k3_rfunct_1(A),k5_xcmplx_0(B)) ) ) ) ).
fof(t45_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k3_rfunct_1(k15_seq_1(A)) = k12_seq_1(k3_rfunct_1(A),k1_real_1(np__1)) ) ).
fof(t46_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k3_rfunct_1(k20_seq_1(A)) = k20_seq_1(k3_rfunct_1(A)) ) ).
fof(t47_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k1_rfunct_1(A,B) = k5_seq_1(A,k3_rfunct_1(B)) ) ) ).
fof(t48_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( v1_xreal_0(C)
=> k12_seq_1(k1_rfunct_1(A,B),C) = k1_rfunct_1(k12_seq_1(A,C),B) ) ) ) ).
fof(t49_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k5_seq_1(k1_rfunct_1(A,B),B) = k7_relat_1(A,k1_relat_1(k3_rfunct_1(B))) ) ) ).
fof(t50_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> k5_seq_1(k1_rfunct_1(A,B),k1_rfunct_1(C,D)) = k1_rfunct_1(k5_seq_1(A,C),k5_seq_1(B,D)) ) ) ) ) ).
fof(t51_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k3_rfunct_1(k1_rfunct_1(A,B)) = k1_rfunct_1(k7_relat_1(B,k1_relat_1(k3_rfunct_1(B))),A) ) ) ).
fof(t52_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k5_seq_1(A,k1_rfunct_1(B,C)) = k1_rfunct_1(k5_seq_1(A,B),C) ) ) ) ).
fof(t53_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> k1_rfunct_1(A,k1_rfunct_1(B,C)) = k1_rfunct_1(k5_seq_1(A,k7_relat_1(C,k1_relat_1(k3_rfunct_1(C)))),B) ) ) ) ).
fof(t54_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ( k15_seq_1(k1_rfunct_1(A,B)) = k1_rfunct_1(k15_seq_1(A),B)
& k1_rfunct_1(A,k15_seq_1(B)) = k15_seq_1(k1_rfunct_1(A,B)) ) ) ) ).
fof(t55_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( k3_seq_1(k1_rfunct_1(A,B),k1_rfunct_1(C,B)) = k1_rfunct_1(k3_seq_1(A,C),B)
& k4_seq_1(k1_rfunct_1(A,B),k1_rfunct_1(C,B)) = k1_rfunct_1(k4_seq_1(A,C),B) ) ) ) ) ).
fof(t56_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> k3_seq_1(k1_rfunct_1(A,B),k1_rfunct_1(C,D)) = k1_rfunct_1(k3_seq_1(k5_seq_1(A,D),k5_seq_1(C,B)),k5_seq_1(B,D)) ) ) ) ) ).
fof(t57_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> k1_rfunct_1(k1_rfunct_1(A,B),k1_rfunct_1(C,D)) = k1_rfunct_1(k5_seq_1(A,k7_relat_1(D,k1_relat_1(k3_rfunct_1(D)))),k5_seq_1(B,C)) ) ) ) ) ).
fof(t58_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> k4_seq_1(k1_rfunct_1(A,B),k1_rfunct_1(C,D)) = k1_rfunct_1(k4_seq_1(k5_seq_1(A,D),k5_seq_1(C,B)),k5_seq_1(B,D)) ) ) ) ) ).
fof(t59_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> k20_seq_1(k1_rfunct_1(A,B)) = k1_rfunct_1(k20_seq_1(A),k20_seq_1(B)) ) ) ).
fof(t60_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( k7_relat_1(k3_seq_1(B,C),A) = k3_seq_1(k7_relat_1(B,A),k7_relat_1(C,A))
& k7_relat_1(k3_seq_1(B,C),A) = k3_seq_1(k7_relat_1(B,A),C)
& k7_relat_1(k3_seq_1(B,C),A) = k3_seq_1(B,k7_relat_1(C,A)) ) ) ) ).
fof(t61_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( k7_relat_1(k5_seq_1(B,C),A) = k5_seq_1(k7_relat_1(B,A),k7_relat_1(C,A))
& k7_relat_1(k5_seq_1(B,C),A) = k5_seq_1(k7_relat_1(B,A),C)
& k7_relat_1(k5_seq_1(B,C),A) = k5_seq_1(B,k7_relat_1(C,A)) ) ) ) ).
fof(t62_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ( k7_relat_1(k15_seq_1(B),A) = k15_seq_1(k7_relat_1(B,A))
& k7_relat_1(k3_rfunct_1(B),A) = k3_rfunct_1(k7_relat_1(B,A))
& k7_relat_1(k20_seq_1(B),A) = k20_seq_1(k7_relat_1(B,A)) ) ) ).
fof(t63_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( k7_relat_1(k4_seq_1(B,C),A) = k4_seq_1(k7_relat_1(B,A),k7_relat_1(C,A))
& k7_relat_1(k4_seq_1(B,C),A) = k4_seq_1(k7_relat_1(B,A),C)
& k7_relat_1(k4_seq_1(B,C),A) = k4_seq_1(B,k7_relat_1(C,A)) ) ) ) ).
fof(t64_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( k7_relat_1(k1_rfunct_1(B,C),A) = k1_rfunct_1(k7_relat_1(B,A),k7_relat_1(C,A))
& k7_relat_1(k1_rfunct_1(B,C),A) = k1_rfunct_1(k7_relat_1(B,A),C)
& k7_relat_1(k1_rfunct_1(B,C),A) = k1_rfunct_1(B,k7_relat_1(C,A)) ) ) ) ).
fof(t65_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ! [C] :
( v1_xreal_0(C)
=> k7_relat_1(k12_seq_1(B,C),A) = k12_seq_1(k7_relat_1(B,A),C) ) ) ).
fof(t66_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( ( ( v1_partfun1(B,A,k1_numbers)
& v1_partfun1(C,A,k1_numbers) )
=> v1_partfun1(k6_seq_1(A,k1_numbers,B,C),A,k1_numbers) )
& ( v1_partfun1(k6_seq_1(A,k1_numbers,B,C),A,k1_numbers)
=> ( v1_partfun1(B,A,k1_numbers)
& v1_partfun1(C,A,k1_numbers) ) )
& ( ( v1_partfun1(B,A,k1_numbers)
& v1_partfun1(C,A,k1_numbers) )
=> v1_partfun1(k7_seq_1(A,k1_numbers,B,C),A,k1_numbers) )
& ( v1_partfun1(k7_seq_1(A,k1_numbers,B,C),A,k1_numbers)
=> ( v1_partfun1(B,A,k1_numbers)
& v1_partfun1(C,A,k1_numbers) ) )
& ( ( v1_partfun1(B,A,k1_numbers)
& v1_partfun1(C,A,k1_numbers) )
=> v1_partfun1(k8_seq_1(A,k1_numbers,B,C),A,k1_numbers) )
& ( v1_partfun1(k8_seq_1(A,k1_numbers,B,C),A,k1_numbers)
=> ( v1_partfun1(B,A,k1_numbers)
& v1_partfun1(C,A,k1_numbers) ) ) ) ) ) ) ).
fof(t67_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( v1_partfun1(C,A,k1_numbers)
<=> v1_partfun1(k13_seq_1(A,k1_numbers,C,B),A,k1_numbers) ) ) ) ) ).
fof(t68_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ( v1_partfun1(B,A,k1_numbers)
<=> v1_partfun1(k16_seq_1(A,k1_numbers,B),A,k1_numbers) ) ) ) ).
fof(t69_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ( v1_partfun1(B,A,k1_numbers)
<=> v1_partfun1(k21_seq_1(A,k1_numbers,B),A,k1_numbers) ) ) ) ).
fof(t70_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ( v1_partfun1(k4_rfunct_1(A,k1_numbers,B),A,k1_numbers)
<=> ( k11_relset_1(A,k1_numbers,B,k1_tarski(np__0)) = k1_xboole_0
& v1_partfun1(B,A,k1_numbers) ) ) ) ) ).
fof(t71_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( ( v1_partfun1(B,A,k1_numbers)
& k11_relset_1(A,k1_numbers,C,k1_tarski(np__0)) = k1_xboole_0
& v1_partfun1(C,A,k1_numbers) )
<=> v1_partfun1(k2_rfunct_1(A,k1_numbers,B,C),A,k1_numbers) ) ) ) ) ).
fof(t72_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers) )
=> ( ( v1_partfun1(C,A,k1_numbers)
& v1_partfun1(D,A,k1_numbers) )
=> ( k2_seq_1(A,k1_numbers,k6_seq_1(A,k1_numbers,C,D),B) = k3_real_1(k2_seq_1(A,k1_numbers,C,B),k2_seq_1(A,k1_numbers,D,B))
& k2_seq_1(A,k1_numbers,k7_seq_1(A,k1_numbers,C,D),B) = k5_real_1(k2_seq_1(A,k1_numbers,C,B),k2_seq_1(A,k1_numbers,D,B))
& k2_seq_1(A,k1_numbers,k8_seq_1(A,k1_numbers,C,D),B) = k4_real_1(k2_seq_1(A,k1_numbers,C,B),k2_seq_1(A,k1_numbers,D,B)) ) ) ) ) ) ) ).
fof(t73_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers) )
=> ( v1_partfun1(D,A,k1_numbers)
=> k2_seq_1(A,k1_numbers,k13_seq_1(A,k1_numbers,D,C),B) = k3_xcmplx_0(C,k2_seq_1(A,k1_numbers,D,B)) ) ) ) ) ) ).
fof(t74_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( v1_partfun1(C,A,k1_numbers)
=> ( k2_seq_1(A,k1_numbers,k16_seq_1(A,k1_numbers,C),B) = k1_real_1(k2_seq_1(A,k1_numbers,C,B))
& k2_seq_1(A,k1_numbers,k21_seq_1(A,k1_numbers,C),B) = k18_complex1(k2_seq_1(A,k1_numbers,C,B)) ) ) ) ) ) ).
fof(t75_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( v1_partfun1(k4_rfunct_1(A,k1_numbers,C),A,k1_numbers)
=> k2_seq_1(A,k1_numbers,k4_rfunct_1(A,k1_numbers,C),B) = k2_real_1(k2_seq_1(A,k1_numbers,C,B)) ) ) ) ) ).
fof(t76_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers) )
=> ( ( v1_partfun1(C,A,k1_numbers)
& v1_partfun1(k4_rfunct_1(A,k1_numbers,D),A,k1_numbers) )
=> k2_seq_1(A,k1_numbers,k2_rfunct_1(A,k1_numbers,C,D),B) = k4_real_1(k2_seq_1(A,k1_numbers,C,B),k2_real_1(k2_seq_1(A,k1_numbers,D,B))) ) ) ) ) ) ).
fof(t77_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,k1_numbers) )
=> ( C = k5_rfunct_1(A,B)
<=> ( k4_relset_1(B,k1_numbers,C) = B
& ! [D] :
( m1_subset_1(D,B)
=> ( ( r2_hidden(D,A)
=> k2_seq_1(B,k1_numbers,C,D) = np__1 )
& ( ~ r2_hidden(D,A)
=> k2_seq_1(B,k1_numbers,C,D) = np__0 ) ) ) ) ) ) ) ).
fof(t78_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> v1_partfun1(k5_rfunct_1(A,B),B,k1_numbers) ) ).
fof(t79_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ( r2_hidden(C,A)
<=> k2_seq_1(B,k1_numbers,k5_rfunct_1(A,B),C) = np__1 ) ) ) ).
fof(t80_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ( ~ r2_hidden(C,A)
<=> k2_seq_1(B,k1_numbers,k5_rfunct_1(A,B),C) = np__0 ) ) ) ).
fof(t81_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ( r2_hidden(C,k4_xboole_0(B,A))
<=> k2_seq_1(B,k1_numbers,k5_rfunct_1(A,B),C) = np__0 ) ) ) ).
fof(t82_rfunct_1,axiom,
$true ).
fof(t83_rfunct_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k2_seq_1(A,k1_numbers,k5_rfunct_1(A,A),B) = np__1 ) ) ).
fof(t84_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ( k2_seq_1(B,k1_numbers,k5_rfunct_1(A,B),C) != np__1
<=> k2_seq_1(B,k1_numbers,k5_rfunct_1(A,B),C) = np__0 ) ) ) ).
fof(t85_rfunct_1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ( r1_xboole_0(A,B)
=> k6_seq_1(C,k1_numbers,k5_rfunct_1(A,C),k5_rfunct_1(B,C)) = k5_rfunct_1(k2_xboole_0(A,B),C) ) ) ).
fof(t86_rfunct_1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> k8_seq_1(C,k1_numbers,k5_rfunct_1(A,C),k5_rfunct_1(B,C)) = k5_rfunct_1(k3_xboole_0(A,B),C) ) ).
fof(d9_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( r1_rfunct_1(A,B)
<=> ? [C] :
( v1_xreal_0(C)
& ! [D] :
( r2_hidden(D,k3_xboole_0(B,k1_relat_1(A)))
=> r1_xreal_0(k1_seq_1(A,D),C) ) ) ) ) ).
fof(d10_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( r2_rfunct_1(A,B)
<=> ? [C] :
( v1_xreal_0(C)
& ! [D] :
( r2_hidden(D,k3_xboole_0(B,k1_relat_1(A)))
=> r1_xreal_0(C,k1_seq_1(A,D)) ) ) ) ) ).
fof(d11_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( r3_rfunct_1(A,B)
<=> ( r1_rfunct_1(A,B)
& r2_rfunct_1(A,B) ) ) ) ).
fof(t87_rfunct_1,axiom,
$true ).
fof(t88_rfunct_1,axiom,
$true ).
fof(t89_rfunct_1,axiom,
$true ).
fof(t90_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ( r3_rfunct_1(B,A)
<=> ? [C] :
( v1_xreal_0(C)
& ! [D] :
( r2_hidden(D,k3_xboole_0(A,k1_relat_1(B)))
=> r1_xreal_0(k18_complex1(k1_seq_1(B,D)),C) ) ) ) ) ).
fof(t91_rfunct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( ( ( r1_tarski(A,B)
& r1_rfunct_1(C,B) )
=> r1_rfunct_1(C,A) )
& ( ( r1_tarski(A,B)
& r2_rfunct_1(C,B) )
=> r2_rfunct_1(C,A) )
& ( ( r1_tarski(A,B)
& r3_rfunct_1(C,B) )
=> r3_rfunct_1(C,A) ) ) ) ).
fof(t92_rfunct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( ( r1_rfunct_1(C,A)
& r2_rfunct_1(C,B) )
=> r3_rfunct_1(C,k3_xboole_0(A,B)) ) ) ).
fof(t93_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ( r1_xboole_0(A,k1_relat_1(B))
=> r3_rfunct_1(B,A) ) ) ).
fof(t94_rfunct_1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( np__0 = B
=> r3_rfunct_1(k12_seq_1(C,B),A) ) ) ) ).
fof(t95_rfunct_1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( ( ( r1_rfunct_1(C,A)
& r1_xreal_0(np__0,B) )
=> r1_rfunct_1(k12_seq_1(C,B),A) )
& ( ( r1_rfunct_1(C,A)
& r1_xreal_0(B,np__0) )
=> r2_rfunct_1(k12_seq_1(C,B),A) ) ) ) ) ).
fof(t96_rfunct_1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( ( ( r2_rfunct_1(C,A)
& r1_xreal_0(np__0,B) )
=> r2_rfunct_1(k12_seq_1(C,B),A) )
& ( ( r2_rfunct_1(C,A)
& r1_xreal_0(B,np__0) )
=> r1_rfunct_1(k12_seq_1(C,B),A) ) ) ) ) ).
fof(t97_rfunct_1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( r3_rfunct_1(C,A)
=> r3_rfunct_1(k12_seq_1(C,B),A) ) ) ) ).
fof(t98_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> r2_rfunct_1(k20_seq_1(B),A) ) ).
fof(t99_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ( r3_rfunct_1(B,A)
=> ( r3_rfunct_1(k20_seq_1(B),A)
& r3_rfunct_1(k15_seq_1(B),A) ) ) ) ).
fof(t100_rfunct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> ( ( ( r1_rfunct_1(C,A)
& r1_rfunct_1(D,B) )
=> r1_rfunct_1(k3_seq_1(C,D),k3_xboole_0(A,B)) )
& ( ( r2_rfunct_1(C,A)
& r2_rfunct_1(D,B) )
=> r2_rfunct_1(k3_seq_1(C,D),k3_xboole_0(A,B)) )
& ( ( r3_rfunct_1(C,A)
& r3_rfunct_1(D,B) )
=> r3_rfunct_1(k3_seq_1(C,D),k3_xboole_0(A,B)) ) ) ) ) ).
fof(t101_rfunct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_seq_1(D) )
=> ( ( r3_rfunct_1(C,A)
& r3_rfunct_1(D,B) )
=> ( r3_rfunct_1(k5_seq_1(C,D),k3_xboole_0(A,B))
& r3_rfunct_1(k4_seq_1(C,D),k3_xboole_0(A,B)) ) ) ) ) ).
fof(t102_rfunct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( ( r1_rfunct_1(C,A)
& r1_rfunct_1(C,B) )
=> r1_rfunct_1(C,k2_xboole_0(A,B)) ) ) ).
fof(t103_rfunct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( ( r2_rfunct_1(C,A)
& r2_rfunct_1(C,B) )
=> r2_rfunct_1(C,k2_xboole_0(A,B)) ) ) ).
fof(t104_rfunct_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_seq_1(C) )
=> ( ( r3_rfunct_1(C,A)
& r3_rfunct_1(C,B) )
=> r3_rfunct_1(C,k2_xboole_0(A,B)) ) ) ).
fof(t105_rfunct_1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,k1_numbers) )
=> ( ( r1_partfun2(C,k1_numbers,D,A)
& r1_partfun2(C,k1_numbers,E,B) )
=> ( r1_partfun2(C,k1_numbers,k6_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B))
& r1_partfun2(C,k1_numbers,k7_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B))
& r1_partfun2(C,k1_numbers,k8_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B)) ) ) ) ) ) ).
fof(t106_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> ( r1_partfun2(B,k1_numbers,D,A)
=> r1_partfun2(B,k1_numbers,k13_seq_1(B,k1_numbers,D,C),A) ) ) ) ) ).
fof(t107_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,k1_numbers) )
=> ( r1_partfun2(B,k1_numbers,C,A)
=> ( r1_partfun2(B,k1_numbers,k21_seq_1(B,k1_numbers,C),A)
& r1_partfun2(B,k1_numbers,k16_seq_1(B,k1_numbers,C),A) ) ) ) ) ).
fof(t108_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,k1_numbers) )
=> ( r1_partfun2(B,k1_numbers,C,A)
=> r3_rfunct_1(C,A) ) ) ) ).
fof(t109_rfunct_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,B,k1_numbers) )
=> ( r1_partfun2(B,k1_numbers,C,A)
=> ( ! [D] :
( v1_xreal_0(D)
=> r3_rfunct_1(k13_seq_1(B,k1_numbers,C,D),A) )
& r3_rfunct_1(k16_seq_1(B,k1_numbers,C),A)
& r3_rfunct_1(k21_seq_1(B,k1_numbers,C),A) ) ) ) ) ).
fof(t110_rfunct_1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,k1_numbers) )
=> ( ( ( r1_rfunct_1(D,A)
& r1_partfun2(C,k1_numbers,E,B) )
=> r1_rfunct_1(k6_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B)) )
& ( ( r2_rfunct_1(D,A)
& r1_partfun2(C,k1_numbers,E,B) )
=> r2_rfunct_1(k6_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B)) )
& ( ( r3_rfunct_1(D,A)
& r1_partfun2(C,k1_numbers,E,B) )
=> r3_rfunct_1(k6_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B)) ) ) ) ) ) ).
fof(t111_rfunct_1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,C,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,C,k1_numbers) )
=> ( ( ( r1_rfunct_1(D,A)
& r1_partfun2(C,k1_numbers,E,B) )
=> r1_rfunct_1(k7_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B)) )
& ( ( r2_rfunct_1(D,A)
& r1_partfun2(C,k1_numbers,E,B) )
=> r2_rfunct_1(k7_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B)) )
& ( ( r3_rfunct_1(D,A)
& r1_partfun2(C,k1_numbers,E,B) )
=> ( r3_rfunct_1(k7_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B))
& r3_rfunct_1(k7_seq_1(C,k1_numbers,E,D),k3_xboole_0(A,B))
& r3_rfunct_1(k8_seq_1(C,k1_numbers,D,E),k3_xboole_0(A,B)) ) ) ) ) ) ) ).
fof(dt_k1_rfunct_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ( v1_relat_1(k1_rfunct_1(A,B))
& v1_funct_1(k1_rfunct_1(A,B)) ) ) ).
fof(dt_k2_rfunct_1,axiom,
! [A,B,C,D] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B)
& v1_funct_1(D)
& m1_relset_1(D,A,B) )
=> ( v1_funct_1(k2_rfunct_1(A,B,C,D))
& m2_relset_1(k2_rfunct_1(A,B,C,D),A,k1_numbers) ) ) ).
fof(redefinition_k2_rfunct_1,axiom,
! [A,B,C,D] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B)
& v1_funct_1(D)
& m1_relset_1(D,A,B) )
=> k2_rfunct_1(A,B,C,D) = k1_rfunct_1(C,D) ) ).
fof(dt_k3_rfunct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k3_rfunct_1(A))
& v1_funct_1(k3_rfunct_1(A)) ) ) ).
fof(dt_k4_rfunct_1,axiom,
! [A,B,C] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> ( v1_funct_1(k4_rfunct_1(A,B,C))
& m2_relset_1(k4_rfunct_1(A,B,C),A,k1_numbers) ) ) ).
fof(redefinition_k4_rfunct_1,axiom,
! [A,B,C] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> k4_rfunct_1(A,B,C) = k3_rfunct_1(C) ) ).
fof(dt_k5_rfunct_1,axiom,
! [A,B] :
( v1_funct_1(k5_rfunct_1(A,B))
& m2_relset_1(k5_rfunct_1(A,B),B,k1_numbers) ) ).
fof(redefinition_k5_rfunct_1,axiom,
! [A,B] : k5_rfunct_1(A,B) = k4_funct_3(A,B) ).
%------------------------------------------------------------------------------