SET007 Axioms: SET007+884.ax
%------------------------------------------------------------------------------
% File : SET007+884 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Real Valued Functions
% 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 : partfun3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 130 ( 0 unt; 0 def)
% Number of atoms : 1158 ( 29 equ)
% Maximal formula atoms : 16 ( 8 avg)
% Number of connectives : 1086 ( 58 ~; 0 |; 885 &)
% ( 5 <=>; 138 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 30 ( 29 usr; 0 prp; 1-3 aty)
% Number of functors : 49 ( 49 usr; 2 con; 0-4 aty)
% Number of variables : 314 ( 309 !; 5 ?)
% 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_partfun3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k7_xcmplx_0(A,A))
& v1_xreal_0(k7_xcmplx_0(A,A))
& ~ v3_xreal_0(k7_xcmplx_0(A,A)) ) ) ).
fof(fc2_partfun3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k3_xcmplx_0(A,A))
& v1_xreal_0(k3_xcmplx_0(A,A))
& ~ v3_xreal_0(k3_xcmplx_0(A,A)) ) ) ).
fof(fc3_partfun3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k3_xcmplx_0(A,k5_xcmplx_0(A)))
& v1_xreal_0(k3_xcmplx_0(A,k5_xcmplx_0(A)))
& ~ v3_xreal_0(k3_xcmplx_0(A,k5_xcmplx_0(A))) ) ) ).
fof(fc4_partfun3,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ( v1_xcmplx_0(k8_square_1(A))
& v1_xreal_0(k8_square_1(A))
& ~ v3_xreal_0(k8_square_1(A)) ) ) ).
fof(fc5_partfun3,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ( ~ v1_xboole_0(k8_square_1(A))
& v1_xcmplx_0(k8_square_1(A))
& v1_xreal_0(k8_square_1(A))
& v2_xreal_0(k8_square_1(A))
& ~ v3_xreal_0(k8_square_1(A)) ) ) ).
fof(fc6_partfun3,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) )
=> ( v1_xboole_0(k10_relat_1(A,k1_tarski(np__0)))
& v1_funct_1(k10_relat_1(A,k1_tarski(np__0)))
& v1_membered(k10_relat_1(A,k1_tarski(np__0)))
& v2_membered(k10_relat_1(A,k1_tarski(np__0)))
& v3_membered(k10_relat_1(A,k1_tarski(np__0)))
& v4_membered(k10_relat_1(A,k1_tarski(np__0)))
& v5_membered(k10_relat_1(A,k1_tarski(np__0))) ) ) ).
fof(fc7_partfun3,axiom,
! [A,B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ( v1_relat_1(k2_funcop_1(A,B))
& v2_relat_1(k2_funcop_1(A,B))
& v1_funct_1(k2_funcop_1(A,B))
& v1_partfun3(k2_funcop_1(A,B)) ) ) ).
fof(fc8_partfun3,axiom,
! [A,B] :
( ( v1_xreal_0(B)
& v3_xreal_0(B) )
=> ( v1_relat_1(k2_funcop_1(A,B))
& v2_relat_1(k2_funcop_1(A,B))
& v1_funct_1(k2_funcop_1(A,B))
& v2_partfun3(k2_funcop_1(A,B)) ) ) ).
fof(fc9_partfun3,axiom,
! [A,B] :
( ( v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( v1_relat_1(k2_funcop_1(A,B))
& v1_funct_1(k2_funcop_1(A,B))
& v3_partfun3(k2_funcop_1(A,B)) ) ) ).
fof(fc10_partfun3,axiom,
! [A,B] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( v1_relat_1(k2_funcop_1(A,B))
& v1_funct_1(k2_funcop_1(A,B))
& v4_partfun3(k2_funcop_1(A,B)) ) ) ).
fof(fc11_partfun3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_relat_1(k2_funcop_1(A,np__0))
& ~ v2_relat_1(k2_funcop_1(A,np__0))
& v1_funct_1(k2_funcop_1(A,np__0))
& v3_partfun3(k2_funcop_1(A,np__0))
& v4_partfun3(k2_funcop_1(A,np__0)) ) ) ).
fof(cc1_partfun3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_partfun3(A) )
=> ( v1_relat_1(A)
& v2_relat_1(A)
& v4_partfun3(A) ) ) ).
fof(cc2_partfun3,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_partfun3(A) )
=> ( v1_relat_1(A)
& v2_relat_1(A)
& v3_partfun3(A) ) ) ).
fof(rc1_partfun3,axiom,
! [A] :
? [B] :
( m1_relset_1(B,A,k1_numbers)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v1_seq_1(B)
& v2_partfun3(B)
& v3_partfun3(B) ) ).
fof(rc2_partfun3,axiom,
! [A] :
? [B] :
( m1_relset_1(B,A,k1_numbers)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v1_seq_1(B)
& v1_partfun3(B)
& v4_partfun3(B) ) ).
fof(rc3_partfun3,axiom,
? [A] :
( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) ) ).
fof(fc12_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k3_seq_1(B,C))
& v1_funct_1(k3_seq_1(B,C))
& v1_seq_1(k3_seq_1(B,C))
& v3_partfun3(k3_seq_1(B,C)) ) ) ).
fof(fc13_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k3_seq_1(B,C))
& v1_funct_1(k3_seq_1(B,C))
& v1_seq_1(k3_seq_1(B,C))
& v4_partfun3(k3_seq_1(B,C)) ) ) ).
fof(fc14_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k3_seq_1(B,C))
& v2_relat_1(k3_seq_1(B,C))
& v1_funct_1(k3_seq_1(B,C))
& v1_seq_1(k3_seq_1(B,C))
& v1_partfun3(k3_seq_1(B,C))
& v4_partfun3(k3_seq_1(B,C)) ) ) ).
fof(fc15_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k3_seq_1(B,C))
& v2_relat_1(k3_seq_1(B,C))
& v1_funct_1(k3_seq_1(B,C))
& v1_seq_1(k3_seq_1(B,C))
& v1_partfun3(k3_seq_1(B,C))
& v4_partfun3(k3_seq_1(B,C)) ) ) ).
fof(fc16_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v2_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k3_seq_1(B,C))
& v2_relat_1(k3_seq_1(B,C))
& v1_funct_1(k3_seq_1(B,C))
& v1_seq_1(k3_seq_1(B,C))
& v2_partfun3(k3_seq_1(B,C))
& v3_partfun3(k3_seq_1(B,C)) ) ) ).
fof(fc17_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k3_seq_1(B,C))
& v2_relat_1(k3_seq_1(B,C))
& v1_funct_1(k3_seq_1(B,C))
& v1_seq_1(k3_seq_1(B,C))
& v2_partfun3(k3_seq_1(B,C))
& v3_partfun3(k3_seq_1(B,C)) ) ) ).
fof(fc18_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k4_seq_1(B,C))
& v1_funct_1(k4_seq_1(B,C))
& v1_seq_1(k4_seq_1(B,C))
& v4_partfun3(k4_seq_1(B,C)) ) ) ).
fof(fc19_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k4_seq_1(B,C))
& v1_funct_1(k4_seq_1(B,C))
& v1_seq_1(k4_seq_1(B,C))
& v3_partfun3(k4_seq_1(B,C)) ) ) ).
fof(fc20_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k4_seq_1(B,C))
& v2_relat_1(k4_seq_1(B,C))
& v1_funct_1(k4_seq_1(B,C))
& v1_seq_1(k4_seq_1(B,C))
& v1_partfun3(k4_seq_1(B,C))
& v4_partfun3(k4_seq_1(B,C)) ) ) ).
fof(fc21_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k4_seq_1(B,C))
& v2_relat_1(k4_seq_1(B,C))
& v1_funct_1(k4_seq_1(B,C))
& v1_seq_1(k4_seq_1(B,C))
& v2_partfun3(k4_seq_1(B,C))
& v3_partfun3(k4_seq_1(B,C)) ) ) ).
fof(fc22_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k4_seq_1(B,C))
& v2_relat_1(k4_seq_1(B,C))
& v1_funct_1(k4_seq_1(B,C))
& v1_seq_1(k4_seq_1(B,C))
& v2_partfun3(k4_seq_1(B,C))
& v3_partfun3(k4_seq_1(B,C)) ) ) ).
fof(fc23_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v2_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k4_seq_1(B,C))
& v2_relat_1(k4_seq_1(B,C))
& v1_funct_1(k4_seq_1(B,C))
& v1_seq_1(k4_seq_1(B,C))
& v1_partfun3(k4_seq_1(B,C))
& v4_partfun3(k4_seq_1(B,C)) ) ) ).
fof(fc24_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C))
& v4_partfun3(k5_seq_1(B,C)) ) ) ).
fof(fc25_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C))
& v4_partfun3(k5_seq_1(B,C)) ) ) ).
fof(fc26_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C))
& v3_partfun3(k5_seq_1(B,C)) ) ) ).
fof(fc27_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C))
& v3_partfun3(k5_seq_1(B,C)) ) ) ).
fof(fc28_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v2_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v2_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C))
& v2_partfun3(k5_seq_1(B,C))
& v3_partfun3(k5_seq_1(B,C)) ) ) ).
fof(fc29_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v2_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C))
& v2_partfun3(k5_seq_1(B,C))
& v3_partfun3(k5_seq_1(B,C)) ) ) ).
fof(fc30_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v2_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C))
& v1_partfun3(k5_seq_1(B,C))
& v4_partfun3(k5_seq_1(B,C)) ) ) ).
fof(fc31_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v2_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v2_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C))
& v1_partfun3(k5_seq_1(B,C))
& v4_partfun3(k5_seq_1(B,C)) ) ) ).
fof(fc32_partfun3,axiom,
! [A,B,C] :
( ( v2_relat_1(B)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers)
& v2_relat_1(C)
& v1_funct_1(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,C))
& v2_relat_1(k5_seq_1(B,C))
& v1_funct_1(k5_seq_1(B,C))
& v1_seq_1(k5_seq_1(B,C)) ) ) ).
fof(fc33_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k5_seq_1(B,B))
& v1_funct_1(k5_seq_1(B,B))
& v1_seq_1(k5_seq_1(B,B))
& v4_partfun3(k5_seq_1(B,B)) ) ) ).
fof(fc34_partfun3,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& ~ v2_xreal_0(B)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B))
& v4_partfun3(k12_seq_1(C,B)) ) ) ).
fof(fc35_partfun3,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B))
& v4_partfun3(k12_seq_1(C,B)) ) ) ).
fof(fc36_partfun3,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& ~ v2_xreal_0(B)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B))
& v3_partfun3(k12_seq_1(C,B)) ) ) ).
fof(fc37_partfun3,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B))
& v3_partfun3(k12_seq_1(C,B)) ) ) ).
fof(fc38_partfun3,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v2_xreal_0(B)
& v1_funct_1(C)
& v2_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v2_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B))
& v2_partfun3(k12_seq_1(C,B))
& v3_partfun3(k12_seq_1(C,B)) ) ) ).
fof(fc39_partfun3,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v3_xreal_0(B)
& v1_funct_1(C)
& v1_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v2_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B))
& v2_partfun3(k12_seq_1(C,B))
& v3_partfun3(k12_seq_1(C,B)) ) ) ).
fof(fc40_partfun3,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v2_xreal_0(B)
& v1_funct_1(C)
& v1_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v2_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B))
& v1_partfun3(k12_seq_1(C,B))
& v4_partfun3(k12_seq_1(C,B)) ) ) ).
fof(fc41_partfun3,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v3_xreal_0(B)
& v1_funct_1(C)
& v2_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v2_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B))
& v1_partfun3(k12_seq_1(C,B))
& v4_partfun3(k12_seq_1(C,B)) ) ) ).
fof(fc42_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_xreal_0(B)
& v2_relat_1(C)
& v1_funct_1(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k12_seq_1(C,B))
& v2_relat_1(k12_seq_1(C,B))
& v1_funct_1(k12_seq_1(C,B))
& v1_seq_1(k12_seq_1(C,B)) ) ) ).
fof(fc43_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C))
& v4_partfun3(k1_rfunct_1(B,C)) ) ) ).
fof(fc44_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C))
& v4_partfun3(k1_rfunct_1(B,C)) ) ) ).
fof(fc45_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v4_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C))
& v3_partfun3(k1_rfunct_1(B,C)) ) ) ).
fof(fc46_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v3_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C))
& v3_partfun3(k1_rfunct_1(B,C)) ) ) ).
fof(fc47_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v2_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v2_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C))
& v2_partfun3(k1_rfunct_1(B,C))
& v3_partfun3(k1_rfunct_1(B,C)) ) ) ).
fof(fc48_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v2_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C))
& v2_partfun3(k1_rfunct_1(B,C))
& v3_partfun3(k1_rfunct_1(B,C)) ) ) ).
fof(fc49_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v2_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C))
& v1_partfun3(k1_rfunct_1(B,C))
& v4_partfun3(k1_rfunct_1(B,C)) ) ) ).
fof(fc50_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v2_partfun3(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v2_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C))
& v1_partfun3(k1_rfunct_1(B,C))
& v4_partfun3(k1_rfunct_1(B,C)) ) ) ).
fof(fc51_partfun3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,B))
& v1_funct_1(k1_rfunct_1(B,B))
& v4_partfun3(k1_rfunct_1(B,B)) ) ) ).
fof(fc52_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers)
& v2_relat_1(C)
& v1_funct_1(C)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_relat_1(k1_rfunct_1(B,C))
& v2_relat_1(k1_rfunct_1(B,C))
& v1_funct_1(k1_rfunct_1(B,C)) ) ) ).
fof(fc53_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k10_pscomp_1(A,B))
& v1_funct_1(k10_pscomp_1(A,B))
& v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
& v1_seq_1(k10_pscomp_1(A,B))
& v3_partfun3(k10_pscomp_1(A,B)) ) ) ).
fof(fc54_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k10_pscomp_1(A,B))
& v1_funct_1(k10_pscomp_1(A,B))
& v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
& v1_seq_1(k10_pscomp_1(A,B))
& v4_partfun3(k10_pscomp_1(A,B)) ) ) ).
fof(fc55_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k10_pscomp_1(A,B))
& v2_relat_1(k10_pscomp_1(A,B))
& v1_funct_1(k10_pscomp_1(A,B))
& v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
& v1_seq_1(k10_pscomp_1(A,B))
& v1_partfun3(k10_pscomp_1(A,B))
& v4_partfun3(k10_pscomp_1(A,B)) ) ) ).
fof(fc56_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k10_pscomp_1(A,B))
& v2_relat_1(k10_pscomp_1(A,B))
& v1_funct_1(k10_pscomp_1(A,B))
& v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
& v1_seq_1(k10_pscomp_1(A,B))
& v2_partfun3(k10_pscomp_1(A,B))
& v3_partfun3(k10_pscomp_1(A,B)) ) ) ).
fof(fc57_partfun3,axiom,
! [A,B] :
( ( v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k10_pscomp_1(A,B))
& v2_relat_1(k10_pscomp_1(A,B))
& v1_funct_1(k10_pscomp_1(A,B))
& v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
& v1_seq_1(k10_pscomp_1(A,B)) ) ) ).
fof(fc58_partfun3,axiom,
! [A,B] :
( ( v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k15_seq_1(B))
& v2_relat_1(k15_seq_1(B))
& v1_funct_1(k15_seq_1(B))
& v1_seq_1(k15_seq_1(B)) ) ) ).
fof(fc59_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k15_seq_1(B))
& v1_funct_1(k15_seq_1(B))
& v1_seq_1(k15_seq_1(B))
& v4_partfun3(k15_seq_1(B)) ) ) ).
fof(fc60_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k15_seq_1(B))
& v1_funct_1(k15_seq_1(B))
& v1_seq_1(k15_seq_1(B))
& v3_partfun3(k15_seq_1(B)) ) ) ).
fof(fc61_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k15_seq_1(B))
& v2_relat_1(k15_seq_1(B))
& v1_funct_1(k15_seq_1(B))
& v1_seq_1(k15_seq_1(B))
& v2_partfun3(k15_seq_1(B))
& v3_partfun3(k15_seq_1(B)) ) ) ).
fof(fc62_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k15_seq_1(B))
& v2_relat_1(k15_seq_1(B))
& v1_funct_1(k15_seq_1(B))
& v1_seq_1(k15_seq_1(B))
& v1_partfun3(k15_seq_1(B))
& v4_partfun3(k15_seq_1(B)) ) ) ).
fof(fc63_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k20_seq_1(B))
& v1_funct_1(k20_seq_1(B))
& v1_seq_1(k20_seq_1(B))
& v4_partfun3(k20_seq_1(B)) ) ) ).
fof(fc64_partfun3,axiom,
! [A,B] :
( ( v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k20_seq_1(B))
& v2_relat_1(k20_seq_1(B))
& v1_funct_1(k20_seq_1(B))
& v1_seq_1(k20_seq_1(B))
& v1_partfun3(k20_seq_1(B))
& v4_partfun3(k20_seq_1(B)) ) ) ).
fof(fc65_partfun3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v3_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k3_rfunct_1(B))
& v1_funct_1(k3_rfunct_1(B))
& v3_partfun3(k3_rfunct_1(B)) ) ) ).
fof(fc66_partfun3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k3_rfunct_1(B))
& v1_funct_1(k3_rfunct_1(B))
& v4_partfun3(k3_rfunct_1(B)) ) ) ).
fof(fc67_partfun3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k3_rfunct_1(B))
& v2_relat_1(k3_rfunct_1(B))
& v1_funct_1(k3_rfunct_1(B))
& v1_partfun3(k3_rfunct_1(B))
& v4_partfun3(k3_rfunct_1(B)) ) ) ).
fof(fc68_partfun3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v2_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k3_rfunct_1(B))
& v2_relat_1(k3_rfunct_1(B))
& v1_funct_1(k3_rfunct_1(B))
& v2_partfun3(k3_rfunct_1(B))
& v3_partfun3(k3_rfunct_1(B)) ) ) ).
fof(fc69_partfun3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k3_rfunct_1(B))
& v2_relat_1(k3_rfunct_1(B))
& v1_funct_1(k3_rfunct_1(B)) ) ) ).
fof(fc70_partfun3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k1_partfun3(A))
& v1_funct_1(k1_partfun3(A))
& v1_seq_1(k1_partfun3(A)) ) ) ).
fof(fc71_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v4_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k1_partfun3(B))
& v1_funct_1(k1_partfun3(B))
& v1_seq_1(k1_partfun3(B))
& v4_partfun3(k1_partfun3(B)) ) ) ).
fof(fc72_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& v1_partfun3(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k1_partfun3(B))
& v2_relat_1(k1_partfun3(B))
& v1_funct_1(k1_partfun3(B))
& v1_seq_1(k1_partfun3(B))
& v1_partfun3(k1_partfun3(B))
& v4_partfun3(k1_partfun3(B)) ) ) ).
fof(rc4_partfun3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v1_seq_1(B)
& v9_pscomp_1(B,A)
& v1_partfun3(B)
& v4_partfun3(B) ) ) ).
fof(rc5_partfun3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v1_seq_1(B)
& v9_pscomp_1(B,A)
& v2_partfun3(B)
& v3_partfun3(B) ) ) ).
fof(t1_partfun3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_funct_1(A)
& r1_tarski(B,k1_relat_1(k2_funct_1(A))) )
=> k9_relat_1(A,k9_relat_1(k2_funct_1(A),B)) = B ) ) ).
fof(d1_partfun3,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_partfun3(A)
<=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r2_hidden(B,k2_relat_1(A))
& r1_xreal_0(B,np__0) ) ) ) ) ).
fof(d2_partfun3,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_partfun3(A)
<=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r2_hidden(B,k2_relat_1(A))
& r1_xreal_0(np__0,B) ) ) ) ) ).
fof(d3_partfun3,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_partfun3(A)
<=> ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(B,k2_relat_1(A))
=> r1_xreal_0(B,np__0) ) ) ) ) ).
fof(d4_partfun3,axiom,
! [A] :
( v1_relat_1(A)
=> ( v4_partfun3(A)
<=> ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(B,k2_relat_1(A))
=> r1_xreal_0(np__0,B) ) ) ) ) ).
fof(t2_partfun3,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k1_relat_1(k3_rfunct_1(A)) = k1_relat_1(A) ) ).
fof(t3_partfun3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v2_relat_1(C)
& v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> k4_relset_1(A,k1_numbers,k2_rfunct_1(A,k1_numbers,B,C)) = k5_subset_1(A,k4_relset_1(A,k1_numbers,B),k4_relset_1(A,k1_numbers,C)) ) ) ) ).
fof(d5_partfun3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k1_partfun3(A)
<=> ( k1_relat_1(B) = k1_relat_1(A)
& ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> k1_funct_1(B,C) = k9_square_1(k1_seq_1(A,C)) ) ) ) ) ) ).
fof(dt_k1_partfun3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k1_partfun3(A))
& v1_funct_1(k1_partfun3(A)) ) ) ).
fof(dt_k2_partfun3,axiom,
! [A,B,C] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> ( v1_funct_1(k2_partfun3(A,B,C))
& m2_relset_1(k2_partfun3(A,B,C),A,k1_numbers) ) ) ).
fof(redefinition_k2_partfun3,axiom,
! [A,B,C] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> k2_partfun3(A,B,C) = k1_partfun3(C) ) ).
fof(dt_k3_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_funct_1(k3_partfun3(A,B,C))
& v1_funct_2(k3_partfun3(A,B,C),A,k1_numbers)
& m2_relset_1(k3_partfun3(A,B,C),A,k1_numbers) ) ) ).
fof(commutativity_k3_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> k3_partfun3(A,B,C) = k3_partfun3(A,C,B) ) ).
fof(redefinition_k3_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> k3_partfun3(A,B,C) = k3_seq_1(B,C) ) ).
fof(dt_k4_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_funct_1(k4_partfun3(A,B,C))
& v1_funct_2(k4_partfun3(A,B,C),A,k1_numbers)
& m2_relset_1(k4_partfun3(A,B,C),A,k1_numbers) ) ) ).
fof(redefinition_k4_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> k4_partfun3(A,B,C) = k4_seq_1(B,C) ) ).
fof(dt_k5_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_funct_1(k5_partfun3(A,B,C))
& v1_funct_2(k5_partfun3(A,B,C),A,k1_numbers)
& m2_relset_1(k5_partfun3(A,B,C),A,k1_numbers) ) ) ).
fof(commutativity_k5_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> k5_partfun3(A,B,C) = k5_partfun3(A,C,B) ) ).
fof(redefinition_k5_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> k5_partfun3(A,B,C) = k5_seq_1(B,C) ) ).
fof(dt_k6_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_funct_1(k6_partfun3(A,B))
& v1_funct_2(k6_partfun3(A,B),A,k1_numbers)
& m2_relset_1(k6_partfun3(A,B),A,k1_numbers) ) ) ).
fof(redefinition_k6_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> k6_partfun3(A,B) = k15_seq_1(B) ) ).
fof(dt_k7_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_funct_1(k7_partfun3(A,B))
& v1_funct_2(k7_partfun3(A,B),A,k1_numbers)
& m2_relset_1(k7_partfun3(A,B),A,k1_numbers) ) ) ).
fof(redefinition_k7_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> k7_partfun3(A,B) = k20_seq_1(B) ) ).
fof(dt_k8_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_funct_1(k8_partfun3(A,B))
& v1_funct_2(k8_partfun3(A,B),A,k1_numbers)
& m2_relset_1(k8_partfun3(A,B),A,k1_numbers) ) ) ).
fof(redefinition_k8_partfun3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> k8_partfun3(A,B) = k1_partfun3(B) ) ).
fof(dt_k9_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_xreal_0(C) )
=> ( v1_funct_1(k9_partfun3(A,B,C))
& v1_funct_2(k9_partfun3(A,B,C),A,k1_numbers)
& m2_relset_1(k9_partfun3(A,B,C),A,k1_numbers) ) ) ).
fof(redefinition_k9_partfun3,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v1_xreal_0(C) )
=> k9_partfun3(A,B,C) = k12_seq_1(B,C) ) ).
fof(dt_k10_partfun3,axiom,
! [A,B] :
( ( v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_funct_1(k10_partfun3(A,B))
& v1_funct_2(k10_partfun3(A,B),A,k1_numbers)
& m2_relset_1(k10_partfun3(A,B),A,k1_numbers) ) ) ).
fof(redefinition_k10_partfun3,axiom,
! [A,B] :
( ( v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers) )
=> k10_partfun3(A,B) = k3_rfunct_1(B) ) ).
fof(dt_k11_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v2_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_funct_1(k11_partfun3(A,B,C))
& v1_funct_2(k11_partfun3(A,B,C),A,k1_numbers)
& m2_relset_1(k11_partfun3(A,B,C),A,k1_numbers) ) ) ).
fof(redefinition_k11_partfun3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m1_relset_1(B,A,k1_numbers)
& v2_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> k11_partfun3(A,B,C) = k1_rfunct_1(B,C) ) ).
fof(dt_k12_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k12_partfun3(A,B,C))
& v1_funct_2(k12_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k12_partfun3(A,B,C),A)
& m2_relset_1(k12_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) ).
fof(commutativity_k12_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> k12_partfun3(A,B,C) = k12_partfun3(A,C,B) ) ).
fof(redefinition_k12_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> k12_partfun3(A,B,C) = k3_seq_1(B,C) ) ).
fof(dt_k13_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k13_partfun3(A,B,C))
& v1_funct_2(k13_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k13_partfun3(A,B,C),A)
& m2_relset_1(k13_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) ).
fof(redefinition_k13_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> k13_partfun3(A,B,C) = k4_seq_1(B,C) ) ).
fof(dt_k14_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k14_partfun3(A,B,C))
& v1_funct_2(k14_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k14_partfun3(A,B,C),A)
& m2_relset_1(k14_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) ).
fof(commutativity_k14_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> k14_partfun3(A,B,C) = k14_partfun3(A,C,B) ) ).
fof(redefinition_k14_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> k14_partfun3(A,B,C) = k5_seq_1(B,C) ) ).
fof(dt_k15_partfun3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k15_partfun3(A,B))
& v1_funct_2(k15_partfun3(A,B),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k15_partfun3(A,B),A)
& m2_relset_1(k15_partfun3(A,B),u1_struct_0(A),k1_numbers) ) ) ).
fof(redefinition_k15_partfun3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers) )
=> k15_partfun3(A,B) = k15_seq_1(B) ) ).
fof(dt_k16_partfun3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k16_partfun3(A,B))
& v1_funct_2(k16_partfun3(A,B),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k16_partfun3(A,B),A)
& m2_relset_1(k16_partfun3(A,B),u1_struct_0(A),k1_numbers) ) ) ).
fof(redefinition_k16_partfun3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers) )
=> k16_partfun3(A,B) = k20_seq_1(B) ) ).
fof(dt_k17_partfun3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& v4_partfun3(B)
& m1_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k17_partfun3(A,B))
& v1_funct_2(k17_partfun3(A,B),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k17_partfun3(A,B),A)
& m2_relset_1(k17_partfun3(A,B),u1_struct_0(A),k1_numbers) ) ) ).
fof(redefinition_k17_partfun3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& v4_partfun3(B)
& m1_relset_1(B,u1_struct_0(A),k1_numbers) )
=> k17_partfun3(A,B) = k1_partfun3(B) ) ).
fof(dt_k18_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_xreal_0(C) )
=> ( v1_funct_1(k18_partfun3(A,B,C))
& v1_funct_2(k18_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k18_partfun3(A,B,C),A)
& m2_relset_1(k18_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) ).
fof(redefinition_k18_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_xreal_0(C) )
=> k18_partfun3(A,B,C) = k12_seq_1(B,C) ) ).
fof(dt_k19_partfun3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k19_partfun3(A,B))
& v1_funct_2(k19_partfun3(A,B),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k19_partfun3(A,B),A)
& m2_relset_1(k19_partfun3(A,B),u1_struct_0(A),k1_numbers) ) ) ).
fof(redefinition_k19_partfun3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers) )
=> k19_partfun3(A,B) = k3_rfunct_1(B) ) ).
fof(dt_k20_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v2_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k20_partfun3(A,B,C))
& v1_funct_2(k20_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k20_partfun3(A,B,C),A)
& m2_relset_1(k20_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) ).
fof(redefinition_k20_partfun3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v2_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(C,A)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> k20_partfun3(A,B,C) = k1_rfunct_1(B,C) ) ).
%------------------------------------------------------------------------------