SET007 Axioms: SET007+81.ax
%------------------------------------------------------------------------------
% File : SET007+81 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Real Sequences and Basic Operations on Them
% 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 : seq_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 130 ( 8 unt; 0 def)
% Number of atoms : 906 ( 109 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 778 ( 2 ~; 1 |; 514 &)
% ( 21 <=>; 240 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 53 ( 53 usr; 11 con; 0-4 aty)
% Number of variables : 302 ( 296 !; 6 ?)
% 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(cc1_seq_1,axiom,
! [A,B] :
( v2_membered(B)
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( v1_funct_1(C)
=> ( v1_funct_1(C)
& v1_seq_1(C) ) ) ) ) ).
fof(rc1_seq_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) ) ).
fof(fc1_seq_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_xcmplx_0(k1_funct_1(A,B))
& v1_xreal_0(k1_funct_1(A,B)) ) ) ).
fof(fc2_seq_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v1_seq_1(k7_relat_1(A,B)) ) ) ).
fof(fc3_seq_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(k3_seq_1(A,B))
& v1_funct_1(k3_seq_1(A,B))
& v1_seq_1(k3_seq_1(A,B)) ) ) ).
fof(fc4_seq_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(k4_seq_1(A,B))
& v1_funct_1(k4_seq_1(A,B))
& v1_seq_1(k4_seq_1(A,B)) ) ) ).
fof(fc5_seq_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(k5_seq_1(A,B))
& v1_funct_1(k5_seq_1(A,B))
& v1_seq_1(k5_seq_1(A,B)) ) ) ).
fof(fc6_seq_1,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B) )
=> ( v1_relat_1(k12_seq_1(B,A))
& v1_funct_1(k12_seq_1(B,A))
& v1_seq_1(k12_seq_1(B,A)) ) ) ).
fof(fc7_seq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k15_seq_1(A))
& v1_funct_1(k15_seq_1(A))
& v1_seq_1(k15_seq_1(A)) ) ) ).
fof(fc8_seq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k20_seq_1(A))
& v1_funct_1(k20_seq_1(A))
& v1_seq_1(k20_seq_1(A)) ) ) ).
fof(t1_seq_1,axiom,
$true ).
fof(t2_seq_1,axiom,
$true ).
fof(t3_seq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
<=> ( k1_relat_1(A) = k5_numbers
& ! [B] :
( r2_hidden(B,k5_numbers)
=> v1_xreal_0(k1_funct_1(A,B)) ) ) ) ) ).
fof(t4_seq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
<=> ( k1_relat_1(A) = k5_numbers
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> v1_xreal_0(k1_funct_1(A,B)) ) ) ) ) ).
fof(d1_seq_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_seq_1(A)
<=> r1_tarski(k2_relat_1(A),k1_numbers) ) ) ).
fof(d2_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_relat_1(A)
<=> r1_tarski(k5_relset_1(k5_numbers,k1_numbers,A),k4_xboole_0(k1_numbers,k1_tarski(np__0))) ) ) ).
fof(t5_seq_1,axiom,
$true ).
fof(t6_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_relat_1(A)
<=> ! [B] :
~ ( r2_hidden(B,k5_numbers)
& k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 ) ) ) ).
fof(t7_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_relat_1(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) != np__0 ) ) ) ).
fof(t8_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ! [C] :
( r2_hidden(C,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,B,C) )
=> A = B ) ) ) ).
fof(t9_seq_1,axiom,
$true ).
fof(t10_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers)
& k5_relset_1(k5_numbers,k1_numbers,B) = k1_tarski(A) ) ) ).
fof(d3_seq_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 = k3_seq_1(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k3_real_1(k1_seq_1(A,D),k1_seq_1(B,D)) ) ) ) ) ) ) ).
fof(d4_seq_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 = k4_seq_1(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k5_real_1(k1_seq_1(A,D),k1_seq_1(B,D)) ) ) ) ) ) ) ).
fof(d5_seq_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 = k5_seq_1(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k4_real_1(k1_seq_1(A,D),k1_seq_1(B,D)) ) ) ) ) ) ) ).
fof(t11_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( A = k6_seq_1(k5_numbers,k1_numbers,B,C)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,D) = k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ).
fof(t12_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( A = k8_seq_1(k5_numbers,k1_numbers,B,C)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,D) = k4_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ).
fof(d6_seq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k12_seq_1(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(A)
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k3_xcmplx_0(B,k1_seq_1(A,D)) ) ) ) ) ) ) ).
fof(t13_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( B = k14_seq_1(C,A)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,D) = k3_xcmplx_0(A,k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ).
fof(d7_seq_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 = k15_seq_1(A)
<=> ( k1_relat_1(B) = k1_relat_1(A)
& ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> k1_funct_1(B,C) = k1_real_1(k1_seq_1(A,C)) ) ) ) ) ) ).
fof(t14_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( A = k17_seq_1(B)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k1_real_1(k2_seq_1(k5_numbers,k1_numbers,B,C)) ) ) ) ) ).
fof(t15_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k10_seq_1(A,B) = k9_seq_1(A,k17_seq_1(B)) ) ) ).
fof(d8_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k18_seq_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_real_1(k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ).
fof(d9_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k19_seq_1(A,B) = k11_seq_1(A,k18_seq_1(B)) ) ) ).
fof(d10_seq_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 = k20_seq_1(A)
<=> ( k1_relat_1(B) = k1_relat_1(A)
& ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> k1_funct_1(B,C) = k18_complex1(k1_seq_1(A,C)) ) ) ) ) ) ).
fof(t16_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( A = k22_seq_1(B)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k18_complex1(k2_seq_1(k5_numbers,k1_numbers,B,C)) ) ) ) ) ).
fof(t17_seq_1,axiom,
$true ).
fof(t18_seq_1,axiom,
$true ).
fof(t19_seq_1,axiom,
$true ).
fof(t20_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k9_seq_1(k9_seq_1(A,B),C) = k9_seq_1(A,k9_seq_1(B,C)) ) ) ) ).
fof(t21_seq_1,axiom,
$true ).
fof(t22_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k11_seq_1(k11_seq_1(A,B),C) = k11_seq_1(A,k11_seq_1(B,C)) ) ) ) ).
fof(t23_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k11_seq_1(k9_seq_1(A,B),C) = k9_seq_1(k11_seq_1(A,C),k11_seq_1(B,C)) ) ) ) ).
fof(t24_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k11_seq_1(A,k9_seq_1(B,C)) = k9_seq_1(k11_seq_1(A,B),k11_seq_1(A,C)) ) ) ) ).
fof(t25_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k17_seq_1(A) = k14_seq_1(A,k1_real_1(np__1)) ) ).
fof(t26_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k14_seq_1(k11_seq_1(B,C),A) = k11_seq_1(k14_seq_1(B,A),C) ) ) ) ).
fof(t27_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k14_seq_1(k11_seq_1(B,C),A) = k11_seq_1(B,k14_seq_1(C,A)) ) ) ) ).
fof(t28_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k11_seq_1(k10_seq_1(A,B),C) = k10_seq_1(k11_seq_1(A,C),k11_seq_1(B,C)) ) ) ) ).
fof(t29_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k10_seq_1(k11_seq_1(A,B),k11_seq_1(A,C)) = k11_seq_1(A,k10_seq_1(B,C)) ) ) ) ).
fof(t30_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k14_seq_1(k9_seq_1(B,C),A) = k9_seq_1(k14_seq_1(B,A),k14_seq_1(C,A)) ) ) ) ).
fof(t31_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k14_seq_1(C,k3_xcmplx_0(A,B)) = k14_seq_1(k14_seq_1(C,B),A) ) ) ) ).
fof(t32_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k14_seq_1(k10_seq_1(B,C),A) = k10_seq_1(k14_seq_1(B,A),k14_seq_1(C,A)) ) ) ) ).
fof(t33_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k14_seq_1(k19_seq_1(B,C),A) = k19_seq_1(k14_seq_1(B,A),C) ) ) ) ).
fof(t34_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k10_seq_1(A,k9_seq_1(B,C)) = k10_seq_1(k10_seq_1(A,B),C) ) ) ) ).
fof(t35_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k14_seq_1(A,np__1) = A ) ).
fof(t36_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k17_seq_1(k17_seq_1(A)) = A ) ).
fof(t37_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k10_seq_1(A,k17_seq_1(B)) = k9_seq_1(A,B) ) ) ).
fof(t38_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k10_seq_1(A,k10_seq_1(B,C)) = k9_seq_1(k10_seq_1(A,B),C) ) ) ) ).
fof(t39_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k9_seq_1(A,k10_seq_1(B,C)) = k10_seq_1(k9_seq_1(A,B),C) ) ) ) ).
fof(t40_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( k11_seq_1(k17_seq_1(A),B) = k17_seq_1(k11_seq_1(A,B))
& k11_seq_1(A,k17_seq_1(B)) = k17_seq_1(k11_seq_1(A,B)) ) ) ) ).
fof(t41_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_relat_1(A)
=> v2_relat_1(k18_seq_1(A)) ) ) ).
fof(t42_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k18_seq_1(k18_seq_1(A)) = A ) ).
fof(t43_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v2_relat_1(B) )
<=> v2_relat_1(k11_seq_1(A,B)) ) ) ) ).
fof(t44_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k11_seq_1(k18_seq_1(A),k18_seq_1(B)) = k18_seq_1(k11_seq_1(A,B)) ) ) ).
fof(t45_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v2_relat_1(A)
=> k11_seq_1(k19_seq_1(B,A),A) = B ) ) ) ).
fof(t46_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> k11_seq_1(k19_seq_1(A,B),k19_seq_1(C,D)) = k19_seq_1(k11_seq_1(A,C),k11_seq_1(B,D)) ) ) ) ) ).
fof(t47_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v2_relat_1(B) )
=> v2_relat_1(k19_seq_1(A,B)) ) ) ) ).
fof(t48_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k18_seq_1(k19_seq_1(A,B)) = k19_seq_1(B,A) ) ) ).
fof(t49_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k11_seq_1(A,k19_seq_1(B,C)) = k19_seq_1(k11_seq_1(A,B),C) ) ) ) ).
fof(t50_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> k19_seq_1(A,k19_seq_1(B,C)) = k19_seq_1(k11_seq_1(A,C),B) ) ) ) ).
fof(t51_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( v2_relat_1(A)
=> k19_seq_1(B,C) = k19_seq_1(k11_seq_1(B,A),k11_seq_1(C,A)) ) ) ) ) ).
fof(t52_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v2_relat_1(B)
=> ( A = np__0
| v2_relat_1(k14_seq_1(B,A)) ) ) ) ) ).
fof(t53_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_relat_1(A)
=> v2_relat_1(k17_seq_1(A)) ) ) ).
fof(t54_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k18_seq_1(k14_seq_1(B,A)) = k14_seq_1(k18_seq_1(B),k5_xcmplx_0(A)) ) ) ).
fof(t55_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k18_seq_1(k17_seq_1(A)) = k14_seq_1(k18_seq_1(A),k1_real_1(np__1)) ) ).
fof(t56_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( k17_seq_1(k19_seq_1(A,B)) = k19_seq_1(k17_seq_1(A),B)
& k19_seq_1(A,k17_seq_1(B)) = k17_seq_1(k19_seq_1(A,B)) ) ) ) ).
fof(t57_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( k9_seq_1(k19_seq_1(A,B),k19_seq_1(C,B)) = k19_seq_1(k9_seq_1(A,C),B)
& k10_seq_1(k19_seq_1(A,B),k19_seq_1(C,B)) = k19_seq_1(k10_seq_1(A,C),B) ) ) ) ) ).
fof(t58_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v2_relat_1(B) )
=> ( k9_seq_1(k19_seq_1(C,A),k19_seq_1(D,B)) = k19_seq_1(k9_seq_1(k11_seq_1(C,B),k11_seq_1(D,A)),k11_seq_1(A,B))
& k10_seq_1(k19_seq_1(C,A),k19_seq_1(D,B)) = k19_seq_1(k10_seq_1(k11_seq_1(C,B),k11_seq_1(D,A)),k11_seq_1(A,B)) ) ) ) ) ) ) ).
fof(t59_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> k19_seq_1(k19_seq_1(A,B),k19_seq_1(C,D)) = k19_seq_1(k11_seq_1(A,D),k11_seq_1(B,C)) ) ) ) ) ).
fof(t60_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k22_seq_1(k11_seq_1(A,B)) = k11_seq_1(k22_seq_1(A),k22_seq_1(B)) ) ) ).
fof(t61_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_relat_1(A)
=> v2_relat_1(k22_seq_1(A)) ) ) ).
fof(t62_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k18_seq_1(k22_seq_1(A)) = k22_seq_1(k18_seq_1(A)) ) ).
fof(t63_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k22_seq_1(k19_seq_1(A,B)) = k19_seq_1(k22_seq_1(A),k22_seq_1(B)) ) ) ).
fof(t64_seq_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k22_seq_1(k14_seq_1(B,A)) = k14_seq_1(k22_seq_1(B),k18_complex1(A)) ) ) ).
fof(s1_seq_1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = f1_s1_seq_1(B) ) ) ).
fof(s2_seq_1,axiom,
? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s2_seq_1,f2_s2_seq_1)
& ! [B] :
( m1_subset_1(B,f1_s2_seq_1)
=> ( r2_hidden(B,k4_relset_1(f1_s2_seq_1,f2_s2_seq_1,A))
<=> ? [C] :
( m1_subset_1(C,f2_s2_seq_1)
& p1_s2_seq_1(B,C) ) ) )
& ! [B] :
( m1_subset_1(B,f1_s2_seq_1)
=> ( r2_hidden(B,k4_relset_1(f1_s2_seq_1,f2_s2_seq_1,A))
=> p1_s2_seq_1(B,k1_funct_1(A,B)) ) ) ) ).
fof(s3_seq_1,axiom,
? [A] :
( v1_funct_1(A)
& m2_relset_1(A,f1_s3_seq_1,f2_s3_seq_1)
& ! [B] :
( m1_subset_1(B,f1_s3_seq_1)
=> ( r2_hidden(B,k4_relset_1(f1_s3_seq_1,f2_s3_seq_1,A))
<=> p1_s3_seq_1(B) ) )
& ! [B] :
( m1_subset_1(B,f1_s3_seq_1)
=> ( r2_hidden(B,k4_relset_1(f1_s3_seq_1,f2_s3_seq_1,A))
=> k1_funct_1(A,B) = f3_s3_seq_1(B) ) ) ) ).
fof(s4_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,f1_s4_seq_1,f2_s4_seq_1) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,f1_s4_seq_1,f2_s4_seq_1) )
=> ( ( k4_relset_1(f1_s4_seq_1,f2_s4_seq_1,A) = f3_s4_seq_1
& ! [C] :
( m1_subset_1(C,f1_s4_seq_1)
=> ( r2_hidden(C,k4_relset_1(f1_s4_seq_1,f2_s4_seq_1,A))
=> k1_funct_1(A,C) = f4_s4_seq_1(C) ) )
& k4_relset_1(f1_s4_seq_1,f2_s4_seq_1,B) = f3_s4_seq_1
& ! [C] :
( m1_subset_1(C,f1_s4_seq_1)
=> ( r2_hidden(C,k4_relset_1(f1_s4_seq_1,f2_s4_seq_1,B))
=> k1_funct_1(B,C) = f4_s4_seq_1(C) ) ) )
=> A = B ) ) ) ).
fof(dt_k1_seq_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> m1_subset_1(k1_seq_1(A,B),k1_numbers) ) ).
fof(redefinition_k1_seq_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> k1_seq_1(A,B) = k1_funct_1(A,B) ) ).
fof(dt_k2_seq_1,axiom,
! [A,B,C,D] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ).
fof(redefinition_k2_seq_1,axiom,
! [A,B,C,D] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k3_seq_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(k3_seq_1(A,B))
& v1_funct_1(k3_seq_1(A,B)) ) ) ).
fof(commutativity_k3_seq_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) )
=> k3_seq_1(A,B) = k3_seq_1(B,A) ) ).
fof(dt_k4_seq_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(k4_seq_1(A,B))
& v1_funct_1(k4_seq_1(A,B)) ) ) ).
fof(dt_k5_seq_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(k5_seq_1(A,B))
& v1_funct_1(k5_seq_1(A,B)) ) ) ).
fof(commutativity_k5_seq_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) )
=> k5_seq_1(A,B) = k5_seq_1(B,A) ) ).
fof(dt_k6_seq_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(k6_seq_1(A,B,C,D))
& m2_relset_1(k6_seq_1(A,B,C,D),A,k1_numbers) ) ) ).
fof(commutativity_k6_seq_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) )
=> k6_seq_1(A,B,C,D) = k6_seq_1(A,B,D,C) ) ).
fof(redefinition_k6_seq_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) )
=> k6_seq_1(A,B,C,D) = k3_seq_1(C,D) ) ).
fof(dt_k7_seq_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(k7_seq_1(A,B,C,D))
& m2_relset_1(k7_seq_1(A,B,C,D),A,k1_numbers) ) ) ).
fof(redefinition_k7_seq_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) )
=> k7_seq_1(A,B,C,D) = k4_seq_1(C,D) ) ).
fof(dt_k8_seq_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(k8_seq_1(A,B,C,D))
& m2_relset_1(k8_seq_1(A,B,C,D),A,k1_numbers) ) ) ).
fof(commutativity_k8_seq_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) )
=> k8_seq_1(A,B,C,D) = k8_seq_1(A,B,D,C) ) ).
fof(redefinition_k8_seq_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) )
=> k8_seq_1(A,B,C,D) = k5_seq_1(C,D) ) ).
fof(dt_k9_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k9_seq_1(A,B))
& v1_funct_2(k9_seq_1(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k9_seq_1(A,B),k5_numbers,k1_numbers) ) ) ).
fof(commutativity_k9_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> k9_seq_1(A,B) = k9_seq_1(B,A) ) ).
fof(redefinition_k9_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> k9_seq_1(A,B) = k3_seq_1(A,B) ) ).
fof(dt_k10_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k10_seq_1(A,B))
& v1_funct_2(k10_seq_1(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k10_seq_1(A,B),k5_numbers,k1_numbers) ) ) ).
fof(redefinition_k10_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> k10_seq_1(A,B) = k4_seq_1(A,B) ) ).
fof(dt_k11_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k11_seq_1(A,B))
& v1_funct_2(k11_seq_1(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k11_seq_1(A,B),k5_numbers,k1_numbers) ) ) ).
fof(commutativity_k11_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> k11_seq_1(A,B) = k11_seq_1(B,A) ) ).
fof(redefinition_k11_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> k11_seq_1(A,B) = k5_seq_1(A,B) ) ).
fof(dt_k12_seq_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v1_xreal_0(B) )
=> ( v1_relat_1(k12_seq_1(A,B))
& v1_funct_1(k12_seq_1(A,B)) ) ) ).
fof(dt_k13_seq_1,axiom,
! [A,B,C,D] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B)
& v1_xreal_0(D) )
=> ( v1_funct_1(k13_seq_1(A,B,C,D))
& m2_relset_1(k13_seq_1(A,B,C,D),A,k1_numbers) ) ) ).
fof(redefinition_k13_seq_1,axiom,
! [A,B,C,D] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B)
& v1_xreal_0(D) )
=> k13_seq_1(A,B,C,D) = k12_seq_1(C,D) ) ).
fof(dt_k14_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_xreal_0(B) )
=> ( v1_funct_1(k14_seq_1(A,B))
& v1_funct_2(k14_seq_1(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k14_seq_1(A,B),k5_numbers,k1_numbers) ) ) ).
fof(redefinition_k14_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_xreal_0(B) )
=> k14_seq_1(A,B) = k12_seq_1(A,B) ) ).
fof(dt_k15_seq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k15_seq_1(A))
& v1_funct_1(k15_seq_1(A)) ) ) ).
fof(dt_k16_seq_1,axiom,
! [A,B,C] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> ( v1_funct_1(k16_seq_1(A,B,C))
& m2_relset_1(k16_seq_1(A,B,C),A,k1_numbers) ) ) ).
fof(redefinition_k16_seq_1,axiom,
! [A,B,C] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> k16_seq_1(A,B,C) = k15_seq_1(C) ) ).
fof(dt_k17_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k17_seq_1(A))
& v1_funct_2(k17_seq_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k17_seq_1(A),k5_numbers,k1_numbers) ) ) ).
fof(redefinition_k17_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> k17_seq_1(A) = k15_seq_1(A) ) ).
fof(dt_k18_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k18_seq_1(A))
& v1_funct_2(k18_seq_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k18_seq_1(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k19_seq_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k19_seq_1(A,B))
& v1_funct_2(k19_seq_1(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k19_seq_1(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k20_seq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_relat_1(k20_seq_1(A))
& v1_funct_1(k20_seq_1(A)) ) ) ).
fof(dt_k21_seq_1,axiom,
! [A,B,C] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> ( v1_funct_1(k21_seq_1(A,B,C))
& m2_relset_1(k21_seq_1(A,B,C),A,k1_numbers) ) ) ).
fof(redefinition_k21_seq_1,axiom,
! [A,B,C] :
( ( v2_membered(B)
& v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> k21_seq_1(A,B,C) = k20_seq_1(C) ) ).
fof(dt_k22_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k22_seq_1(A))
& v1_funct_2(k22_seq_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k22_seq_1(A),k5_numbers,k1_numbers) ) ) ).
fof(redefinition_k22_seq_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> k22_seq_1(A) = k20_seq_1(A) ) ).
%------------------------------------------------------------------------------