SET007 Axioms: SET007+132.ax
%------------------------------------------------------------------------------
% File : SET007+132 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Factorial and Newton Coefficients
% 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 : newton [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 139 ( 22 unt; 0 def)
% Number of atoms : 488 ( 98 equ)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 395 ( 46 ~; 2 |; 93 &)
% ( 18 <=>; 236 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-3 aty)
% Number of functors : 51 ( 51 usr; 8 con; 0-3 aty)
% Number of variables : 229 ( 222 !; 7 ?)
% 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_newton,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v4_ordinal2(B) )
=> ( v1_xcmplx_0(k2_newton(A,B))
& v1_xreal_0(k2_newton(A,B)) ) ) ).
fof(fc2_newton,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> ( v4_ordinal2(k2_newton(A,B))
& v1_xcmplx_0(k2_newton(A,B))
& v1_xreal_0(k2_newton(A,B))
& ~ v3_xreal_0(k2_newton(A,B))
& v1_int_1(k2_newton(A,B)) ) ) ).
fof(fc3_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_xcmplx_0(k5_newton(A))
& v1_xreal_0(k5_newton(A)) ) ) ).
fof(fc4_newton,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> ( v1_xcmplx_0(k7_newton(A,B))
& v1_xreal_0(k7_newton(A,B)) ) ) ).
fof(rc1_newton,axiom,
? [A] :
( m1_subset_1(A,k5_numbers)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_int_1(A)
& v1_int_2(A) ) ).
fof(fc5_newton,axiom,
( ~ v1_xboole_0(k12_newton)
& ~ v1_finset_1(k12_newton)
& v1_membered(k12_newton)
& v2_membered(k12_newton)
& v3_membered(k12_newton)
& v4_membered(k12_newton)
& v5_membered(k12_newton) ) ).
fof(t1_newton,axiom,
$true ).
fof(t2_newton,axiom,
$true ).
fof(t3_newton,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(A))
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
=> A = B ) ) ) ).
fof(t4_newton,axiom,
$true ).
fof(t6_newton,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> k3_finseq_1(k9_rvsum_1(A,B)) = k3_finseq_1(B) ) ) ).
fof(t7_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ( r2_hidden(A,k4_finseq_1(C))
<=> r2_hidden(A,k4_finseq_1(k9_rvsum_1(B,C))) ) ) ) ) ).
fof(d1_newton,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> k2_newton(A,B) = k16_rvsum_1(k1_newton(B,A)) ) ) ).
fof(t8_newton,axiom,
$true ).
fof(t9_newton,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_newton(A,np__0) = np__1 ) ).
fof(t10_newton,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_newton(A,np__1) = A ) ).
fof(t11_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_newton(B,k2_xcmplx_0(A,np__1)) = k3_xcmplx_0(k2_newton(B,A),B) ) ) ).
fof(t12_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_newton(k3_xcmplx_0(B,C),A) = k3_xcmplx_0(k2_newton(B,A),k2_newton(C,A)) ) ) ) ).
fof(t13_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_newton(C,k2_xcmplx_0(A,B)) = k3_xcmplx_0(k2_newton(C,A),k2_newton(C,B)) ) ) ) ).
fof(t14_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_newton(k2_newton(C,A),B) = k2_newton(C,k3_xcmplx_0(A,B)) ) ) ) ).
fof(t15_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k3_newton(np__1,A) = np__1 ) ).
fof(t16_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ( r1_xreal_0(np__1,A)
=> k3_newton(np__0,A) = np__0 ) ) ).
fof(d2_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k5_newton(A) = k16_rvsum_1(k4_newton(A)) ) ).
fof(t17_newton,axiom,
$true ).
fof(t18_newton,axiom,
k6_newton(np__0) = np__1 ).
fof(t19_newton,axiom,
k6_newton(np__1) = np__1 ).
fof(t20_newton,axiom,
k6_newton(np__2) = np__2 ).
fof(t21_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k6_newton(k2_xcmplx_0(A,np__1)) = k3_xcmplx_0(k6_newton(A),k2_xcmplx_0(A,np__1)) ) ).
fof(t22_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> m2_subset_1(k6_newton(A),k1_numbers,k5_numbers) ) ).
fof(t23_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ r1_xreal_0(k6_newton(A),np__0) ) ).
fof(t24_newton,axiom,
$true ).
fof(t25_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k3_xcmplx_0(k6_newton(A),k6_newton(B)) != np__0 ) ) ).
fof(d3_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( ( r1_xreal_0(A,B)
=> ( C = k7_newton(A,B)
<=> ! [D] :
( v4_ordinal2(D)
=> ( D = k6_xcmplx_0(B,A)
=> C = k7_xcmplx_0(k6_newton(B),k3_xcmplx_0(k6_newton(A),k6_newton(D))) ) ) ) )
& ( ~ r1_xreal_0(A,B)
=> ( C = k7_newton(A,B)
<=> C = np__0 ) ) ) ) ) ).
fof(t26_newton,axiom,
$true ).
fof(t27_newton,axiom,
k8_newton(np__0,np__0) = np__1 ).
fof(t28_newton,axiom,
$true ).
fof(t29_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k8_newton(np__0,A) = np__1 ) ).
fof(t30_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(B,A)
=> ! [C] :
( v4_ordinal2(C)
=> ( C = k6_xcmplx_0(A,B)
=> k8_newton(B,A) = k8_newton(C,A) ) ) ) ) ) ).
fof(t31_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k8_newton(A,A) = np__1 ) ).
fof(t32_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k8_newton(k2_xcmplx_0(B,np__1),k2_xcmplx_0(A,np__1)) = k2_xcmplx_0(k8_newton(k2_xcmplx_0(B,np__1),A),k8_newton(B,A)) ) ) ).
fof(t33_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ( r1_xreal_0(np__1,A)
=> k8_newton(np__1,A) = A ) ) ).
fof(t34_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ( r1_xreal_0(np__1,A)
& B = k6_xcmplx_0(A,np__1) )
=> k8_newton(B,A) = A ) ) ) ).
fof(t35_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> m2_subset_1(k8_newton(B,A),k1_numbers,k5_numbers) ) ) ).
fof(t36_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ( ( k3_finseq_1(C) = B
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r2_hidden(D,k4_finseq_1(C))
& E = k6_xcmplx_0(k1_nat_1(A,D),np__1) )
=> k1_funct_1(C,D) = k8_newton(A,E) ) ) ) )
=> ( B = np__0
| k15_rvsum_1(C) = k8_newton(k1_nat_1(A,np__1),k1_nat_1(A,B)) ) ) ) ) ) ).
fof(d4_newton,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( D = k9_newton(A,B,C)
<=> ( k3_finseq_1(D) = k2_xcmplx_0(C,np__1)
& ! [E] :
( v4_ordinal2(E)
=> ! [F] :
( v4_ordinal2(F)
=> ! [G] :
( v4_ordinal2(G)
=> ( ( r2_hidden(E,k4_finseq_1(D))
& G = k6_xcmplx_0(E,np__1)
& F = k6_xcmplx_0(C,G) )
=> k1_funct_1(D,E) = k3_xcmplx_0(k3_xcmplx_0(k8_newton(G,C),k2_newton(A,F)),k2_newton(B,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(t37_newton,axiom,
$true ).
fof(t38_newton,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k9_newton(A,B,np__0) = k12_finseq_1(k5_numbers,np__1) ) ) ).
fof(t39_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k1_funct_1(k9_newton(B,C,A),np__1) = k2_newton(B,A) ) ) ) ).
fof(t40_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k1_funct_1(k9_newton(B,C,A),k2_xcmplx_0(A,np__1)) = k2_newton(C,A) ) ) ) ).
fof(t41_newton,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> k2_newton(k2_xcmplx_0(A,B),C) = k15_rvsum_1(k9_newton(A,B,C)) ) ) ) ).
fof(d5_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( B = k10_newton(A)
<=> ( k3_finseq_1(B) = k2_xcmplx_0(A,np__1)
& ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& D = k6_xcmplx_0(C,np__1) )
=> k1_funct_1(B,C) = k8_newton(D,A) ) ) ) ) ) ) ) ).
fof(t42_newton,axiom,
$true ).
fof(t43_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k10_newton(A) = k9_newton(np__1,np__1,A) ) ).
fof(t44_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k3_newton(np__2,A) = k15_rvsum_1(k10_newton(A)) ) ).
fof(t45_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> r1_xreal_0(B,k2_nat_1(B,A)) ) ) ) ).
fof(t46_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(k2_nat_1(C,A),B) )
=> r1_xreal_0(C,B) ) ) ) ) ).
fof(t47_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( A != np__0
=> r1_nat_1(A,k11_newton(A)) ) ) ).
fof(t48_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& r1_xreal_0(k7_xcmplx_0(k1_nat_1(A,np__1),A),np__1) ) ) ).
fof(t49_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__1,k7_xcmplx_0(A,k1_nat_1(A,np__1))) ) ).
fof(t50_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_xreal_0(A,k11_newton(A)) ) ).
fof(t51_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__1
& r1_nat_1(A,B)
& r1_nat_1(A,k1_nat_1(B,np__1)) ) ) ) ).
fof(t52_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(B,A)
& r1_nat_1(B,k1_nat_1(A,np__1)) )
<=> B = np__1 ) ) ) ).
fof(t53_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B != np__0
=> r1_nat_1(B,k11_newton(k1_nat_1(B,A))) ) ) ) ).
fof(t54_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> ( B = np__0
| r1_nat_1(B,k11_newton(A)) ) ) ) ) ).
fof(t55_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( B != np__1
& B != np__0
& r1_nat_1(B,k1_nat_1(k11_newton(A),np__1))
& r1_xreal_0(B,A) ) ) ) ).
fof(t56_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_nat_1(A,k5_nat_1(B,C)) = k5_nat_1(k5_nat_1(A,B),C) ) ) ) ).
fof(t57_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,B)
<=> k5_nat_1(A,B) = B ) ) ) ).
fof(t58_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(C,B) )
<=> r1_nat_1(k5_nat_1(A,C),B) ) ) ) ) ).
fof(t59_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k5_nat_1(A,np__1) = A ) ).
fof(t60_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_nat_1(k5_nat_1(A,B),k2_nat_1(A,B)) ) ) ).
fof(t61_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k6_nat_1(A,k6_nat_1(B,C)) = k6_nat_1(k6_nat_1(A,B),C) ) ) ) ).
fof(t62_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,B)
=> k6_nat_1(A,B) = A ) ) ) ).
fof(t63_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(A,C) )
<=> r1_nat_1(A,k6_nat_1(B,C)) ) ) ) ) ).
fof(t64_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k6_nat_1(A,np__1) = np__1 ) ).
fof(t65_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k6_nat_1(A,np__0) = A ) ).
fof(t66_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k5_nat_1(k6_nat_1(A,B),B) = B ) ) ).
fof(t67_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k6_nat_1(A,k5_nat_1(A,B)) = A ) ) ).
fof(t68_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k6_nat_1(A,k5_nat_1(A,B)) = k5_nat_1(k6_nat_1(B,A),A) ) ) ).
fof(t69_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,B)
=> r1_nat_1(k6_nat_1(A,C),k6_nat_1(B,C)) ) ) ) ) ).
fof(t70_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,B)
=> r1_nat_1(k6_nat_1(C,A),k6_nat_1(C,B)) ) ) ) ) ).
fof(t71_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(k6_nat_1(A,B),np__0) ) ) ) ).
fof(t72_newton,axiom,
$true ).
fof(t73_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& r1_xreal_0(k5_nat_1(A,B),np__0) ) ) ) ).
fof(t74_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_nat_1(k5_nat_1(k6_nat_1(A,B),k6_nat_1(A,C)),k6_nat_1(A,k5_nat_1(B,C))) ) ) ) ).
fof(t75_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,B)
=> r1_nat_1(k5_nat_1(A,k6_nat_1(C,B)),k6_nat_1(k5_nat_1(A,C),B)) ) ) ) ) ).
fof(t76_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_nat_1(k6_nat_1(A,B),k5_nat_1(A,B)) ) ) ).
fof(t77_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k4_nat_1(A,B) = k6_xcmplx_0(A,k2_nat_1(B,k3_nat_1(A,B))) ) ) ) ).
fof(t78_newton,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r1_xreal_0(np__0,A)
=> r1_xreal_0(np__0,k6_int_1(B,A)) ) ) ) ).
fof(t79_newton,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(A,k6_int_1(B,A)) ) ) ) ).
fof(t80_newton,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( A != np__0
=> B = k2_xcmplx_0(k3_xcmplx_0(k5_int_1(B,A),A),k6_int_1(B,A)) ) ) ) ).
fof(t81_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ ( r1_xreal_0(A,np__0)
& r1_xreal_0(B,np__0) )
& ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> k2_xcmplx_0(k3_xcmplx_0(C,A),k3_xcmplx_0(D,B)) != k6_nat_1(A,B) ) ) ) ) ) ).
fof(d6_newton,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
=> ( A = k12_newton
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,A)
<=> v1_int_2(B) ) ) ) ) ).
fof(d7_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k5_numbers))
=> ( B = k13_newton(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,B)
<=> ( ~ r1_xreal_0(A,C)
& v1_int_2(C) ) ) ) ) ) ) ).
fof(t82_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k13_newton(A),k12_newton) ) ).
fof(t83_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,A)
=> r1_tarski(k13_newton(A),k13_newton(B)) ) ) ) ).
fof(t84_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k13_newton(A),k2_finseq_1(A)) ) ).
fof(t85_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v1_finset_1(k13_newton(A)) ) ).
fof(t86_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers)
& v1_int_2(B)
& ~ r1_xreal_0(B,A) ) ) ).
fof(t87_newton,axiom,
k12_newton != k1_xboole_0 ).
fof(t97_newton,axiom,
~ v1_finset_1(k12_newton) ).
fof(t98_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( v1_int_2(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_nat_1(A,k2_nat_1(B,C))
& ~ r1_nat_1(A,B)
& ~ r1_nat_1(A,C) ) ) ) ) ) ).
fof(t99_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> m2_subset_1(k3_newton(B,A),k1_numbers,k5_numbers) ) ) ).
fof(t100_newton,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_newton(A,np__2) = k3_xcmplx_0(A,A)
& k5_square_1(A) = k2_newton(A,np__2) ) ) ).
fof(t101_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( k5_int_1(A,B) = k3_nat_1(A,B)
& k6_int_1(A,B) = k4_nat_1(A,B) ) ) ) ).
fof(dt_k1_newton,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v1_xreal_0(B) )
=> m2_finseq_1(k1_newton(A,B),k1_numbers) ) ).
fof(redefinition_k1_newton,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v1_xreal_0(B) )
=> k1_newton(A,B) = k2_finseq_2(A,B) ) ).
fof(dt_k2_newton,axiom,
$true ).
fof(dt_k3_newton,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& v4_ordinal2(B) )
=> m1_subset_1(k3_newton(A,B),k1_numbers) ) ).
fof(redefinition_k3_newton,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& v4_ordinal2(B) )
=> k3_newton(A,B) = k2_newton(A,B) ) ).
fof(dt_k4_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> m2_finseq_1(k4_newton(A),k1_numbers) ) ).
fof(redefinition_k4_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k4_newton(A) = k1_finseq_2(A) ) ).
fof(dt_k5_newton,axiom,
$true ).
fof(dt_k6_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> m1_subset_1(k6_newton(A),k1_numbers) ) ).
fof(redefinition_k6_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> k6_newton(A) = k5_newton(A) ) ).
fof(dt_k7_newton,axiom,
$true ).
fof(dt_k8_newton,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> m1_subset_1(k8_newton(A,B),k1_numbers) ) ).
fof(redefinition_k8_newton,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> k8_newton(A,B) = k7_newton(A,B) ) ).
fof(dt_k9_newton,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v4_ordinal2(C) )
=> m2_finseq_1(k9_newton(A,B,C),k1_numbers) ) ).
fof(dt_k10_newton,axiom,
! [A] :
( v4_ordinal2(A)
=> m2_finseq_1(k10_newton(A),k1_numbers) ) ).
fof(dt_k11_newton,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k11_newton(A),k1_numbers,k5_numbers) ) ).
fof(redefinition_k11_newton,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> k11_newton(A) = k5_newton(A) ) ).
fof(dt_k12_newton,axiom,
m1_subset_1(k12_newton,k1_zfmisc_1(k5_numbers)) ).
fof(dt_k13_newton,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k13_newton(A),k1_zfmisc_1(k5_numbers)) ) ).
fof(dt_k14_newton,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_int_2(k14_newton(A))
& m2_subset_1(k14_newton(A),k1_numbers,k5_numbers) ) ) ).
fof(t5_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> k2_finseq_1(A) = k2_xboole_0(k2_xboole_0(k1_tarski(np__1),a_1_0_newton(A)),k1_tarski(A)) ) ) ).
fof(t88_newton,axiom,
a_0_0_newton = k1_xboole_0 ).
fof(t89_newton,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> r1_tarski(a_1_1_newton(A),k5_numbers) ) ).
fof(t90_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(a_1_2_newton(A),k2_finseq_1(A)) ) ).
fof(t91_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v1_finset_1(a_1_2_newton(A)) ) ).
fof(t92_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ r2_hidden(A,a_1_2_newton(A)) ) ).
fof(t93_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v1_finset_1(k2_xboole_0(a_1_2_newton(A),k1_tarski(A))) ) ).
fof(t94_newton,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ( ~ r1_xreal_0(B,A)
=> r1_tarski(k2_xboole_0(a_1_1_newton(A),k1_tarski(A)),a_1_1_newton(B)) ) ) ) ).
fof(t95_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(B,A)
& r2_hidden(B,a_1_2_newton(A)) ) ) ) ).
fof(d8_newton,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ( B = k14_newton(A)
<=> ? [C] :
( v1_finset_1(C)
& C = a_1_1_newton(B)
& A = k4_card_1(C) ) ) ) ) ).
fof(t96_newton,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> k13_newton(A) = a_1_1_newton(A) ) ).
fof(fraenkel_a_1_0_newton,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_0_newton(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& ~ r1_xreal_0(C,np__1)
& ~ r1_xreal_0(B,C) ) ) ) ).
fof(fraenkel_a_0_0_newton,axiom,
! [A] :
( r2_hidden(A,a_0_0_newton)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = B
& ~ r1_xreal_0(np__2,B)
& v1_int_2(B) ) ) ).
fof(fraenkel_a_1_1_newton,axiom,
! [A,B] :
( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_1_1_newton(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& ~ r1_xreal_0(B,C)
& v1_int_2(C) ) ) ) ).
fof(fraenkel_a_1_2_newton,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_2_newton(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& ~ r1_xreal_0(B,C)
& v1_int_2(C) ) ) ) ).
%------------------------------------------------------------------------------