SET007 Axioms: SET007+902.ax
%------------------------------------------------------------------------------
% File : SET007+902 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Inner Product and Conjugate of Finite Sequences of Complex 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 : complsp2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 101 ( 0 unt; 0 def)
% Number of atoms : 455 ( 182 equ)
% Maximal formula atoms : 17 ( 4 avg)
% Number of connectives : 354 ( 0 ~; 0 |; 67 &)
% ( 1 <=>; 286 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-3 aty)
% Number of functors : 61 ( 61 usr; 12 con; 0-6 aty)
% Number of variables : 250 ( 250 !; 0 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(d1_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( B = k1_complsp2(A)
<=> ( k3_finseq_1(B) = k3_finseq_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(A)) )
=> k9_matrix_5(B,C) = k15_complex1(k9_matrix_5(A,C)) ) ) ) ) ) ) ).
fof(t1_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( r2_hidden(A,k4_finseq_1(k3_complsp1(B,C)))
=> k9_matrix_5(k3_complsp1(B,C),A) = k3_binop_2(k9_matrix_5(B,A),k9_matrix_5(C,A)) ) ) ) ) ).
fof(t2_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( r2_hidden(A,k4_finseq_1(k4_complsp1(B,C)))
=> k9_matrix_5(k4_complsp1(B,C),A) = k4_binop_2(k9_matrix_5(B,A),k9_matrix_5(C,A)) ) ) ) ) ).
fof(t3_complsp2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> k3_finseq_1(k6_complsp1(A,B)) = k3_finseq_1(B) ) ) ).
fof(t4_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k4_finseq_1(A) = k4_finseq_1(k5_complsp1(A)) ) ).
fof(t5_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k3_finseq_1(k5_complsp1(A)) = k3_finseq_1(A) ) ).
fof(t6_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k3_finseq_1(k3_complsp1(A,B)) = k3_finseq_1(A) ) ) ) ).
fof(t7_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k3_finseq_1(k4_complsp1(A,B)) = k3_finseq_1(A) ) ) ) ).
fof(t8_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> m2_finseq_2(A,k2_numbers,k8_complsp1(k3_finseq_1(A))) ) ).
fof(t9_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k2_numbers,k4_finseq_2(A,k2_numbers))
=> ! [C] :
( m2_finseq_2(C,k2_numbers,k4_finseq_2(A,k2_numbers))
=> k2_complsp2(A,B,C) = k3_complsp1(B,k5_complsp1(C)) ) ) ) ).
fof(t10_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k4_complsp1(A,B) = k3_complsp1(A,k5_complsp1(B)) ) ) ) ).
fof(t11_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k2_numbers,k4_finseq_2(A,k2_numbers))
=> k4_complsp2(A,k7_binop_2(np__1),B) = k5_complsp1(B) ) ) ).
fof(t12_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k6_complsp1(k7_binop_2(np__1),A) = k5_complsp1(A) ) ).
fof(t13_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> k9_matrix_5(k5_complsp1(B),A) = k1_binop_2(k9_matrix_5(B,A)) ) ) ).
fof(t14_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ! [D] :
( m2_finseq_2(D,k2_numbers,k4_finseq_2(A,k2_numbers))
=> ( C = k9_matrix_5(D,B)
=> k9_matrix_5(k5_complsp2(A,D),B) = k1_binop_2(C) ) ) ) ) ) ).
fof(t15_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( v1_xcmplx_0(B)
=> k4_finseq_1(k6_complsp1(B,A)) = k4_finseq_1(A) ) ) ).
fof(t16_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( v1_xcmplx_0(C)
=> k9_matrix_5(k6_complsp1(C,B),A) = k3_xcmplx_0(C,k9_matrix_5(B,A)) ) ) ) ).
fof(t17_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( v1_xcmplx_0(B)
=> k1_complsp2(k6_complsp1(B,A)) = k6_complsp1(k15_complex1(B),k1_complsp2(A)) ) ) ).
fof(t18_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k2_numbers,k4_finseq_2(A,k2_numbers))
=> ! [D] :
( m2_finseq_2(D,k2_numbers,k4_finseq_2(A,k2_numbers))
=> k9_matrix_5(k3_complsp2(A,C,D),B) = k3_binop_2(k9_matrix_5(C,B),k9_matrix_5(D,B)) ) ) ) ) ).
fof(t19_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k1_complsp2(k3_complsp1(A,B)) = k3_complsp1(k1_complsp2(A),k1_complsp2(B)) ) ) ) ).
fof(t20_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k2_numbers,k4_finseq_2(A,k2_numbers))
=> ! [D] :
( m2_finseq_2(D,k2_numbers,k4_finseq_2(A,k2_numbers))
=> k9_matrix_5(k2_complsp2(A,C,D),B) = k4_binop_2(k9_matrix_5(C,B),k9_matrix_5(D,B)) ) ) ) ) ).
fof(t21_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k1_complsp2(k4_complsp1(A,B)) = k4_complsp1(k1_complsp2(A),k1_complsp2(B)) ) ) ) ).
fof(t22_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k1_complsp2(k1_complsp2(A)) = A ) ).
fof(t23_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k1_complsp2(k5_complsp1(A)) = k5_complsp1(k1_complsp2(A)) ) ).
fof(t24_complsp2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k2_xcmplx_0(A,k15_complex1(A)) = k11_binop_2(np__2,k3_complex1(A)) ) ).
fof(t25_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( k3_finseq_1(B) = k3_finseq_1(C)
=> k9_matrix_5(k4_complsp1(B,C),A) = k4_binop_2(k9_matrix_5(B,A),k9_matrix_5(C,A)) ) ) ) ) ).
fof(t26_complsp2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( k3_finseq_1(B) = k3_finseq_1(C)
=> k9_matrix_5(k3_complsp1(B,C),A) = k3_binop_2(k9_matrix_5(B,A),k9_matrix_5(C,A)) ) ) ) ) ).
fof(d2_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k6_complsp2(A) = k6_complsp1(k12_binop_2(np__1,np__2),k3_complsp1(A,k1_complsp2(A))) ) ).
fof(t27_complsp2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k6_xcmplx_0(A,k15_complex1(A)) = k3_xcmplx_0(k11_binop_2(np__2,k4_complex1(A)),k7_complex1) ) ).
fof(d3_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k7_complsp2(A) = k6_complsp1(k4_xcmplx_0(k3_xcmplx_0(k12_binop_2(np__1,np__2),k7_complex1)),k4_complsp1(A,k1_complsp2(A))) ) ).
fof(d4_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> k8_complsp2(A,B) = k2_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(k1_euclid_2(k6_complsp2(A),k6_complsp2(B)),k3_xcmplx_0(k7_complex1,k1_euclid_2(k6_complsp2(A),k7_complsp2(B)))),k3_xcmplx_0(k7_complex1,k1_euclid_2(k7_complsp2(A),k6_complsp2(B)))),k1_euclid_2(k7_complsp2(A),k7_complsp2(B))) ) ) ).
fof(t28_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k3_complsp1(A,k3_complsp1(B,C)) = k3_complsp1(k3_complsp1(A,B),C) ) ) ) ) ).
fof(t29_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k3_complsp1(A,B) = k3_complsp1(B,A) ) ) ) ).
fof(t30_complsp2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( k3_finseq_1(B) = k3_finseq_1(C)
=> k6_complsp1(A,k3_complsp1(B,C)) = k3_complsp1(k6_complsp1(A,B),k6_complsp1(A,C)) ) ) ) ) ).
fof(t31_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k4_complsp1(A,B) = k3_complsp1(A,k5_complsp1(B)) ) ) ) ).
fof(t32_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_complsp1(A,B) = k11_complsp1(k3_finseq_1(A)) )
=> ( A = k5_complsp1(B)
& B = k5_complsp1(A) ) ) ) ) ).
fof(t33_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k3_complsp1(A,k11_complsp1(k3_finseq_1(A))) = A ) ).
fof(t34_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k3_complsp1(A,k5_complsp1(A)) = k11_complsp1(k3_finseq_1(A)) ) ).
fof(t35_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k5_complsp1(k3_complsp1(A,B)) = k3_complsp1(k5_complsp1(A),k5_complsp1(B)) ) ) ) ).
fof(t36_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k4_complsp1(k4_complsp1(A,B),C) = k4_complsp1(A,k3_complsp1(B,C)) ) ) ) ) ).
fof(t37_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k3_complsp1(A,k4_complsp1(B,C)) = k4_complsp1(k3_complsp1(A,B),C) ) ) ) ) ).
fof(t38_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k5_complsp1(k5_complsp1(A)) = A ) ).
fof(t39_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k5_complsp1(k4_complsp1(A,B)) = k3_complsp1(k5_complsp1(A),B) ) ) ) ).
fof(t40_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k4_complsp1(A,k4_complsp1(B,C)) = k3_complsp1(k4_complsp1(A,B),C) ) ) ) ) ).
fof(t41_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( v1_xcmplx_0(B)
=> k6_complsp1(B,k11_complsp1(k3_finseq_1(A))) = k11_complsp1(k3_finseq_1(A)) ) ) ).
fof(t42_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( v1_xcmplx_0(B)
=> k5_complsp1(k6_complsp1(B,A)) = k6_complsp1(B,k5_complsp1(A)) ) ) ).
fof(t43_complsp2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( k3_finseq_1(B) = k3_finseq_1(C)
=> k6_complsp1(A,k4_complsp1(B,C)) = k4_complsp1(k6_complsp1(A,B),k6_complsp1(A,C)) ) ) ) ) ).
fof(t44_complsp2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( A = C
& B = D )
=> k2_binop_1(k2_numbers,k2_numbers,k2_numbers,k27_binop_2,A,B) = k2_binop_1(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,C,D) ) ) ) ) ) ).
fof(t45_complsp2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(k2_numbers,k2_numbers),k2_numbers)
& m2_relset_1(A,k2_zfmisc_1(k2_numbers,k2_numbers),k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) )
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ! [D] :
( m2_finseq_1(D,k2_numbers)
=> ! [E] :
( m2_finseq_1(E,k1_numbers)
=> ! [F] :
( m2_finseq_1(F,k1_numbers)
=> ( ( C = E
& D = F
& k3_finseq_1(C) = k3_finseq_1(F)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(G,k4_finseq_1(C))
=> k2_binop_1(k2_numbers,k2_numbers,k2_numbers,A,k9_matrix_5(C,G),k9_matrix_5(D,G)) = k1_binop_1(B,k1_funct_1(E,G),k1_funct_1(F,G)) ) ) )
=> k1_finseqop(k2_numbers,k2_numbers,k2_numbers,A,C,D) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,B,E,F) ) ) ) ) ) ) ) ).
fof(t46_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( A = C
& B = D
& k3_finseq_1(A) = k3_finseq_1(D) )
=> k1_finseqop(k2_numbers,k2_numbers,k2_numbers,k27_binop_2,A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,C,D) ) ) ) ) ) ).
fof(t47_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( A = C
& B = D
& k3_finseq_1(A) = k3_finseq_1(D) )
=> k3_complsp1(A,B) = k3_rvsum_1(C,D) ) ) ) ) ) ).
fof(t48_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ( k3_finseq_1(k6_complsp2(A)) = k3_finseq_1(A)
& k3_finseq_1(k7_complsp2(A)) = k3_finseq_1(A) ) ) ).
fof(t49_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( k6_complsp2(k3_complsp1(A,B)) = k3_rvsum_1(k6_complsp2(A),k6_complsp2(B))
& k7_complsp2(k3_complsp1(A,B)) = k3_rvsum_1(k7_complsp2(A),k7_complsp2(B)) ) ) ) ) ).
fof(t50_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( A = C
& B = D
& k3_finseq_1(A) = k3_finseq_1(D) )
=> k1_finseqop(k2_numbers,k2_numbers,k2_numbers,k28_binop_2,A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k34_binop_2,C,D) ) ) ) ) ) ).
fof(t51_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( A = C
& B = D
& k3_finseq_1(A) = k3_finseq_1(D) )
=> k4_complsp1(A,B) = k7_rvsum_1(C,D) ) ) ) ) ) ).
fof(t52_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( k6_complsp2(k4_complsp1(A,B)) = k7_rvsum_1(k6_complsp2(A),k6_complsp2(B))
& k7_complsp2(k4_complsp1(A,B)) = k7_rvsum_1(k7_complsp2(A),k7_complsp2(B)) ) ) ) ) ).
fof(t53_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( v1_xcmplx_0(B)
=> ! [C] :
( v1_xcmplx_0(C)
=> k6_complsp1(B,k6_complsp1(C,A)) = k6_complsp1(k3_xcmplx_0(B,C),A) ) ) ) ).
fof(t54_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( v1_xcmplx_0(B)
=> k6_complsp1(k4_xcmplx_0(B),A) = k5_complsp1(k6_complsp1(B,A)) ) ) ).
fof(t55_complsp2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k2_numbers,k2_numbers)
& m2_relset_1(A,k2_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_numbers,k1_numbers)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(C))
=> k8_funct_2(k2_numbers,k2_numbers,A,k9_matrix_5(C,E)) = k8_funct_2(k1_numbers,k1_numbers,B,k10_complsp2(D,E)) ) ) )
=> k5_finseqop(k2_numbers,k2_numbers,C,A) = k5_finseqop(k1_numbers,k1_numbers,D,B) ) ) ) ) ) ).
fof(t56_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( ( A = B
& k3_finseq_1(A) = k3_finseq_1(B) )
=> k5_finseqop(k2_numbers,k2_numbers,A,k25_binop_2) = k5_finseqop(k1_numbers,k1_numbers,B,k31_binop_2) ) ) ) ).
fof(t57_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( ( A = B
& k3_finseq_1(A) = k3_finseq_1(B) )
=> k5_complsp1(A) = k5_rvsum_1(B) ) ) ) ).
fof(t58_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ( k6_complsp2(k6_complsp1(k7_complex1,A)) = k5_rvsum_1(k7_complsp2(A))
& k7_complsp2(k6_complsp1(k7_complex1,A)) = k6_complsp2(A) ) ) ).
fof(t59_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k8_complsp2(k6_complsp1(k7_complex1,A),B) = k5_binop_2(k7_complex1,k8_complsp2(A,B)) ) ) ) ).
fof(t60_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k8_complsp2(A,k6_complsp1(k7_complex1,B)) = k1_binop_2(k5_binop_2(k7_complex1,k8_complsp2(A,B))) ) ) ) ).
fof(t61_complsp2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( A = C
& B = D
& k3_finseq_1(B) = k3_finseq_1(D) )
=> k5_finseqop(k2_numbers,k2_numbers,B,k1_complsp1(A)) = k5_finseqop(k1_numbers,k1_numbers,D,k2_rvsum_1(C)) ) ) ) ) ) ).
fof(t62_complsp2,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( A = C
& B = D
& k3_finseq_1(B) = k3_finseq_1(D) )
=> k6_complsp1(A,B) = k9_rvsum_1(C,D) ) ) ) ) ) ).
fof(t63_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( v1_xcmplx_0(B)
=> ! [C] :
( v1_xcmplx_0(C)
=> k6_complsp1(k2_xcmplx_0(B,C),A) = k3_complsp1(k6_complsp1(B,A),k6_complsp1(C,A)) ) ) ) ).
fof(t64_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( v1_xcmplx_0(B)
=> ! [C] :
( v1_xcmplx_0(C)
=> k6_complsp1(k6_xcmplx_0(B,C),A) = k4_complsp1(k6_complsp1(B,A),k6_complsp1(C,A)) ) ) ) ).
fof(t65_complsp2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k6_complsp2(k6_complsp1(A,B)) = k7_rvsum_1(k9_rvsum_1(k3_complex1(A),k6_complsp2(B)),k9_rvsum_1(k4_complex1(A),k7_complsp2(B)))
& k7_complsp2(k6_complsp1(A,B)) = k3_rvsum_1(k9_rvsum_1(k4_complex1(A),k6_complsp2(B)),k9_rvsum_1(k3_complex1(A),k7_complsp2(B))) ) ) ) ).
fof(t66_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k8_complsp2(k3_complsp1(A,B),C) = k3_binop_2(k8_complsp2(A,C),k8_complsp2(B,C)) ) ) ) ) ).
fof(t67_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k8_complsp2(k5_complsp1(A),B) = k1_binop_2(k8_complsp2(A,B)) ) ) ) ).
fof(t68_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k8_complsp2(A,k5_complsp1(B)) = k1_binop_2(k8_complsp2(A,B)) ) ) ) ).
fof(t69_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k8_complsp2(k5_complsp1(A),k5_complsp1(B)) = k8_complsp2(A,B) ) ) ) ).
fof(t70_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k8_complsp2(k4_complsp1(A,B),C) = k4_binop_2(k8_complsp2(A,C),k8_complsp2(B,C)) ) ) ) ) ).
fof(t71_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k8_complsp2(A,k3_complsp1(B,C)) = k3_binop_2(k8_complsp2(A,B),k8_complsp2(A,C)) ) ) ) ) ).
fof(t72_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k8_complsp2(A,k4_complsp1(B,C)) = k4_binop_2(k8_complsp2(A,B),k8_complsp2(A,C)) ) ) ) ) ).
fof(t73_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ! [D] :
( m2_finseq_1(D,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C)
& k3_finseq_1(C) = k3_finseq_1(D) )
=> k8_complsp2(k3_complsp1(A,B),k3_complsp1(C,D)) = k3_binop_2(k3_binop_2(k3_binop_2(k8_complsp2(A,C),k8_complsp2(A,D)),k8_complsp2(B,C)),k8_complsp2(B,D)) ) ) ) ) ) ).
fof(t74_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ! [D] :
( m2_finseq_1(D,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C)
& k3_finseq_1(C) = k3_finseq_1(D) )
=> k8_complsp2(k4_complsp1(A,B),k4_complsp1(C,D)) = k3_binop_2(k4_binop_2(k4_binop_2(k8_complsp2(A,C),k8_complsp2(A,D)),k8_complsp2(B,C)),k8_complsp2(B,D)) ) ) ) ) ) ).
fof(t75_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k8_complsp2(A,B) = k15_complex1(k8_complsp2(B,A)) ) ) ) ).
fof(t76_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k8_complsp2(k3_complsp1(A,B),k3_complsp1(A,B)) = k2_xcmplx_0(k2_xcmplx_0(k8_complsp2(A,A),k11_binop_2(np__2,k3_complex1(k8_complsp2(A,B)))),k8_complsp2(B,B)) ) ) ) ).
fof(t77_complsp2,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k8_complsp2(k4_complsp1(A,B),k4_complsp1(A,B)) = k2_xcmplx_0(k6_xcmplx_0(k8_complsp2(A,A),k11_binop_2(np__2,k3_complex1(k8_complsp2(A,B)))),k8_complsp2(B,B)) ) ) ) ).
fof(t78_complsp2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( k3_finseq_1(B) = k3_finseq_1(C)
=> k8_complsp2(k6_complsp1(A,B),C) = k5_binop_2(A,k8_complsp2(B,C)) ) ) ) ) ).
fof(t79_complsp2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( k3_finseq_1(B) = k3_finseq_1(C)
=> k8_complsp2(B,k6_complsp1(A,C)) = k5_binop_2(k15_complex1(A),k8_complsp2(B,C)) ) ) ) ) ).
fof(t80_complsp2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ! [D] :
( m2_finseq_1(D,k2_numbers)
=> ! [E] :
( m2_finseq_1(E,k2_numbers)
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& k3_finseq_1(D) = k3_finseq_1(E) )
=> k8_complsp2(k3_complsp1(k6_complsp1(A,C),k6_complsp1(B,D)),E) = k3_binop_2(k5_binop_2(A,k8_complsp2(C,E)),k5_binop_2(B,k8_complsp2(D,E))) ) ) ) ) ) ) ).
fof(t81_complsp2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ! [D] :
( m2_finseq_1(D,k2_numbers)
=> ! [E] :
( m2_finseq_1(E,k2_numbers)
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& k3_finseq_1(D) = k3_finseq_1(E) )
=> k8_complsp2(C,k3_complsp1(k6_complsp1(A,D),k6_complsp1(B,E))) = k3_binop_2(k5_binop_2(k15_complex1(A),k8_complsp2(C,D)),k5_binop_2(k15_complex1(B),k8_complsp2(C,E))) ) ) ) ) ) ) ).
fof(dt_k1_complsp2,axiom,
! [A] :
( m1_finseq_1(A,k2_numbers)
=> m2_finseq_1(k1_complsp2(A),k2_numbers) ) ).
fof(dt_k2_complsp2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers))
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> m2_finseq_2(k2_complsp2(A,B,C),k2_numbers,k4_finseq_2(A,k2_numbers)) ) ).
fof(redefinition_k2_complsp2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers))
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> k2_complsp2(A,B,C) = k4_complsp1(B,C) ) ).
fof(dt_k3_complsp2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers))
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> m2_finseq_2(k3_complsp2(A,B,C),k2_numbers,k4_finseq_2(A,k2_numbers)) ) ).
fof(redefinition_k3_complsp2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers))
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> k3_complsp2(A,B,C) = k3_complsp1(B,C) ) ).
fof(dt_k4_complsp2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& v1_xcmplx_0(B)
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> m2_finseq_2(k4_complsp2(A,B,C),k2_numbers,k4_finseq_2(A,k2_numbers)) ) ).
fof(redefinition_k4_complsp2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& v1_xcmplx_0(B)
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> k4_complsp2(A,B,C) = k6_complsp1(B,C) ) ).
fof(dt_k5_complsp2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers)) )
=> m2_finseq_2(k5_complsp2(A,B),k2_numbers,k4_finseq_2(A,k2_numbers)) ) ).
fof(redefinition_k5_complsp2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers)) )
=> k5_complsp2(A,B) = k5_complsp1(B) ) ).
fof(dt_k6_complsp2,axiom,
! [A] :
( m1_finseq_1(A,k2_numbers)
=> m2_finseq_1(k6_complsp2(A),k1_numbers) ) ).
fof(dt_k7_complsp2,axiom,
! [A] :
( m1_finseq_1(A,k2_numbers)
=> m2_finseq_1(k7_complsp2(A),k1_numbers) ) ).
fof(dt_k8_complsp2,axiom,
! [A,B] :
( ( m1_finseq_1(A,k2_numbers)
& m1_finseq_1(B,k2_numbers) )
=> m1_subset_1(k8_complsp2(A,B),k2_numbers) ) ).
fof(dt_k9_complsp2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k2_numbers) )
=> m2_finseq_2(k9_complsp2(A,B),k2_numbers,k4_finseq_2(A,k2_numbers)) ) ).
fof(redefinition_k9_complsp2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k2_numbers) )
=> k9_complsp2(A,B) = k2_finseq_2(A,B) ) ).
fof(dt_k10_complsp2,axiom,
! [A,B] :
( m1_finseq_1(A,k1_numbers)
=> m1_subset_1(k10_complsp2(A,B),k1_numbers) ) ).
fof(redefinition_k10_complsp2,axiom,
! [A,B] :
( m1_finseq_1(A,k1_numbers)
=> k10_complsp2(A,B) = k1_funct_1(A,B) ) ).
%------------------------------------------------------------------------------