SET007 Axioms: SET007+347.ax
%------------------------------------------------------------------------------
% File : SET007+347 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Properties of Connecting Points with Line Segments in E^2_T
% 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 : topreal3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 53 ( 6 unt; 0 def)
% Number of atoms : 347 ( 97 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 314 ( 20 ~; 31 |; 74 &)
% ( 2 <=>; 187 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 9 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 11 ( 9 usr; 1 prp; 0-3 aty)
% Number of functors : 46 ( 46 usr; 7 con; 0-4 aty)
% Number of variables : 164 ( 162 !; 2 ?)
% 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(t1_topreal3,axiom,
$true ).
fof(t2_topreal3,axiom,
$true ).
fof(t3_topreal3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ( ~ r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2),A)
& ~ r1_xreal_0(B,k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)) ) ) ) ) ).
fof(t4_topreal3,axiom,
$true ).
fof(t5_topreal3,axiom,
$true ).
fof(t6_topreal3,axiom,
! [A,B,C] :
( r2_hidden(np__1,k4_finseq_1(k11_finseq_1(A,B,C)))
& r2_hidden(np__2,k4_finseq_1(k11_finseq_1(A,B,C)))
& r2_hidden(np__3,k4_finseq_1(k11_finseq_1(A,B,C))) ) ).
fof(t7_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( k21_euclid(k17_euclid(np__2,A,B)) = k3_real_1(k21_euclid(A),k21_euclid(B))
& k22_euclid(k17_euclid(np__2,A,B)) = k3_real_1(k22_euclid(A),k22_euclid(B)) ) ) ) ).
fof(t8_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( k21_euclid(k20_euclid(np__2,A,B)) = k5_real_1(k21_euclid(A),k21_euclid(B))
& k22_euclid(k20_euclid(np__2,A,B)) = k5_real_1(k22_euclid(A),k22_euclid(B)) ) ) ) ).
fof(t9_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( v1_xreal_0(B)
=> ( k21_euclid(k18_euclid(B,np__2,A)) = k3_xcmplx_0(B,k21_euclid(A))
& k22_euclid(k18_euclid(B,np__2,A)) = k3_xcmplx_0(B,k22_euclid(A)) ) ) ) ).
fof(t10_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ( ( A = k10_finseq_1(C,D)
& B = k10_finseq_1(E,F) )
=> ( k17_euclid(np__2,A,B) = k10_finseq_1(k2_xcmplx_0(C,E),k2_xcmplx_0(D,F))
& k20_euclid(np__2,A,B) = k10_finseq_1(k6_xcmplx_0(C,E),k6_xcmplx_0(D,F)) ) ) ) ) ) ) ) ) ).
fof(t11_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( k21_euclid(A) = k21_euclid(B)
& k22_euclid(A) = k22_euclid(B) )
=> A = B ) ) ) ).
fof(t12_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k14_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
=> ( ( C = A
& D = B )
=> k1_binop_1(k13_euclid(np__2),C,D) = k9_square_1(k3_real_1(k7_square_1(k5_real_1(k21_euclid(A),k21_euclid(B))),k7_square_1(k5_real_1(k22_euclid(A),k22_euclid(B))))) ) ) ) ) ) ).
fof(t13_topreal3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> u1_struct_0(k15_euclid(A)) = u1_struct_0(k14_euclid(A)) ) ).
fof(t14_topreal3,axiom,
$true ).
fof(t17_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(A,k3_topreal1(np__2,k23_euclid(B,C),k23_euclid(B,D)))
=> k21_euclid(A) = B ) ) ) ) ) ).
fof(t18_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(A,k3_topreal1(np__2,k23_euclid(B,C),k23_euclid(D,C)))
=> k22_euclid(A) = C ) ) ) ) ) ).
fof(t19_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( k22_euclid(A) = k22_euclid(B)
=> ( k21_euclid(A) = k21_euclid(B)
| r2_hidden(k23_euclid(k6_real_1(k3_real_1(k21_euclid(A),k21_euclid(B)),np__2),k22_euclid(A)),k3_topreal1(np__2,A,B)) ) ) ) ) ).
fof(t20_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( k21_euclid(A) = k21_euclid(B)
=> ( k22_euclid(A) = k22_euclid(B)
| r2_hidden(k23_euclid(k21_euclid(A),k6_real_1(k3_real_1(k22_euclid(A),k22_euclid(B)),np__2)),k3_topreal1(np__2,A,B)) ) ) ) ) ).
fof(t21_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( D = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,B,C)
=> ( E = np__0
| r1_xreal_0(F,k1_nat_1(E,np__1))
| k4_topreal1(np__2,D,F) = k1_xboole_0 ) ) ) ) ) ) ) ) ).
fof(t22_topreal3,axiom,
$true ).
fof(t23_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( D = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,B,C)
=> k5_topreal1(np__2,D) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,B),k3_topreal1(np__2,B,C)) ) ) ) ) ) ).
fof(t24_topreal3,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k4_finseq_1(A))
& r2_hidden(C,k4_finseq_1(k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,B)))
& r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,B))) )
=> k4_topreal1(np__2,A,C) = k4_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,B),C) ) ) ) ) ).
fof(t25_topreal3,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(A))
& r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(A)) )
=> k4_topreal1(np__2,k8_finseq_1(u1_struct_0(k15_euclid(np__2)),A,B),C) = k4_topreal1(np__2,A,C) ) ) ) ) ).
fof(t26_topreal3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k4_topreal1(A,B,C),k5_topreal1(A,B)) ) ) ) ).
fof(t27_topreal3,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_tarski(k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,B)),k5_topreal1(np__2,A)) ) ) ).
fof(t28_topreal3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(B)))
=> ( ( r2_hidden(C,k9_metric_1(k14_euclid(B),E,A))
& r2_hidden(D,k9_metric_1(k14_euclid(B),E,A)) )
=> r1_tarski(k3_topreal1(B,C,D),k9_metric_1(k14_euclid(B),E,A)) ) ) ) ) ) ) ).
fof(t29_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( m1_subset_1(I,u1_struct_0(k14_euclid(np__2)))
=> ( ( I = A
& A = k23_euclid(D,E)
& B = k23_euclid(F,G)
& C = k23_euclid(F,E)
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),I,H)) )
=> r2_hidden(C,k9_metric_1(k14_euclid(np__2),I,H)) ) ) ) ) ) ) ) ) ) ) ).
fof(t30_topreal3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(k23_euclid(A,B),k9_metric_1(k14_euclid(np__2),E,C))
& r2_hidden(k23_euclid(A,D),k9_metric_1(k14_euclid(np__2),E,C)) )
=> r2_hidden(k23_euclid(A,k7_xcmplx_0(k2_xcmplx_0(B,D),np__2)),k9_metric_1(k14_euclid(np__2),E,C)) ) ) ) ) ) ) ).
fof(t31_topreal3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(k23_euclid(A,B),k9_metric_1(k14_euclid(np__2),E,C))
& r2_hidden(k23_euclid(D,B),k9_metric_1(k14_euclid(np__2),E,C)) )
=> r2_hidden(k23_euclid(k7_xcmplx_0(k2_xcmplx_0(A,D),np__2),B),k9_metric_1(k14_euclid(np__2),E,C)) ) ) ) ) ) ) ).
fof(t32_topreal3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k14_euclid(np__2)))
=> ~ ( A != B
& C != D
& r2_hidden(k23_euclid(A,D),k9_metric_1(k14_euclid(np__2),F,E))
& r2_hidden(k23_euclid(B,C),k9_metric_1(k14_euclid(np__2),F,E))
& ~ r2_hidden(k23_euclid(A,C),k9_metric_1(k14_euclid(np__2),F,E))
& ~ r2_hidden(k23_euclid(B,D),k9_metric_1(k14_euclid(np__2),F,E)) ) ) ) ) ) ) ) ).
fof(t33_topreal3,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k14_euclid(np__2)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k9_metric_1(k14_euclid(np__2),C,B))
& r1_xreal_0(np__1,D)
& r1_xreal_0(D,k5_real_1(k3_finseq_1(A),np__1))
& ~ r1_xboole_0(k4_topreal1(np__2,A,D),k9_metric_1(k14_euclid(np__2),C,B))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,E)
& r1_xreal_0(E,k5_real_1(k3_finseq_1(A),np__1)) )
=> ( k3_xboole_0(k4_topreal1(np__2,A,E),k9_metric_1(k14_euclid(np__2),C,B)) = k1_xboole_0
| r1_xreal_0(D,E) ) ) )
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,D),k9_metric_1(k14_euclid(np__2),C,B)) ) ) ) ) ) ).
fof(t34_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( k22_euclid(A) = k22_euclid(B)
=> ( k22_euclid(C) = k22_euclid(B)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,B,k23_euclid(k21_euclid(B),k22_euclid(C))),k3_topreal1(np__2,k23_euclid(k21_euclid(B),k22_euclid(C)),C)),k3_topreal1(np__2,A,B)) = k1_struct_0(k15_euclid(np__2),B) ) ) ) ) ) ).
fof(t35_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( k21_euclid(A) = k21_euclid(B)
=> ( k21_euclid(C) = k21_euclid(B)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,B,k23_euclid(k21_euclid(C),k22_euclid(B))),k3_topreal1(np__2,k23_euclid(k21_euclid(C),k22_euclid(B)),C)),k3_topreal1(np__2,A,B)) = k1_struct_0(k15_euclid(np__2),B) ) ) ) ) ) ).
fof(t36_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,k23_euclid(k21_euclid(A),k22_euclid(B))),k3_topreal1(np__2,k23_euclid(k21_euclid(A),k22_euclid(B)),B)) = k1_struct_0(k15_euclid(np__2),k23_euclid(k21_euclid(A),k22_euclid(B))) ) ) ).
fof(t37_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,k23_euclid(k21_euclid(B),k22_euclid(A))),k3_topreal1(np__2,k23_euclid(k21_euclid(B),k22_euclid(A)),B)) = k1_struct_0(k15_euclid(np__2),k23_euclid(k21_euclid(B),k22_euclid(A))) ) ) ).
fof(t38_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( k21_euclid(A) = k21_euclid(B)
=> ( k22_euclid(A) = k22_euclid(B)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,k23_euclid(k21_euclid(A),k6_real_1(k3_real_1(k22_euclid(A),k22_euclid(B)),np__2))),k3_topreal1(np__2,k23_euclid(k21_euclid(A),k6_real_1(k3_real_1(k22_euclid(A),k22_euclid(B)),np__2)),B)) = k1_struct_0(k15_euclid(np__2),k23_euclid(k21_euclid(A),k6_real_1(k3_real_1(k22_euclid(A),k22_euclid(B)),np__2))) ) ) ) ) ).
fof(t39_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( k22_euclid(A) = k22_euclid(B)
=> ( k21_euclid(A) = k21_euclid(B)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,k23_euclid(k6_real_1(k3_real_1(k21_euclid(A),k21_euclid(B)),np__2),k22_euclid(A))),k3_topreal1(np__2,k23_euclid(k6_real_1(k3_real_1(k21_euclid(A),k21_euclid(B)),np__2),k22_euclid(A)),B)) = k1_struct_0(k15_euclid(np__2),k23_euclid(k6_real_1(k3_real_1(k21_euclid(A),k21_euclid(B)),np__2),k22_euclid(A))) ) ) ) ) ).
fof(t40_topreal3,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k4_finseq_1(A))
& v4_topreal1(A) )
=> ( r1_xreal_0(B,np__2)
| v4_topreal1(k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,B)) ) ) ) ) ).
fof(t41_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(A),k22_euclid(B)),B)
=> ( k21_euclid(A) = k21_euclid(B)
| k22_euclid(A) = k22_euclid(B)
| ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
& v4_topreal1(C) ) ) ) ) ) ) ).
fof(t42_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(B),k22_euclid(A)),B)
=> ( k21_euclid(A) = k21_euclid(B)
| k22_euclid(A) = k22_euclid(B)
| ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
& v4_topreal1(C) ) ) ) ) ) ) ).
fof(t43_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( k21_euclid(A) = k21_euclid(B)
& C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k21_euclid(A),k6_real_1(k3_real_1(k22_euclid(A),k22_euclid(B)),np__2)),B) )
=> ( k22_euclid(A) = k22_euclid(B)
| ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
& v4_topreal1(C) ) ) ) ) ) ) ).
fof(t44_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( k22_euclid(A) = k22_euclid(B)
& C = k3_finseq_4(u1_struct_0(k15_euclid(np__2)),A,k23_euclid(k6_real_1(k3_real_1(k21_euclid(A),k21_euclid(B)),np__2),k22_euclid(A)),B) )
=> ( k21_euclid(A) = k21_euclid(B)
| ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = A
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = B
& v4_topreal1(C) ) ) ) ) ) ) ).
fof(t45_topreal3,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k4_finseq_1(A))
& r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A)) )
=> k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__1))) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k16_finseq_1(u1_struct_0(k15_euclid(np__2)),A,B)),k3_topreal1(np__2,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(B,np__1)))) ) ) ) ).
fof(t46_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r1_xreal_0(np__2,k3_finseq_1(B))
=> ( r2_hidden(A,k5_topreal1(np__2,B))
| ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C) = A ) ) ) ) ) ) ).
fof(t47_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( A != B
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,B),k5_topreal1(np__2,C)) = k1_struct_0(k15_euclid(np__2),A)
& r2_hidden(B,k5_topreal1(np__2,C)) ) ) ) ) ).
fof(t48_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k14_euclid(np__2)))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( v4_topreal1(B)
& r2_hidden(A,k9_metric_1(k14_euclid(np__2),D,C))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k4_topreal1(np__2,B,E))
& r1_xreal_0(np__1,E)
& r1_xreal_0(k1_nat_1(E,np__1),k3_finseq_1(B)) )
=> ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k9_metric_1(k14_euclid(np__2),D,C))
| r1_xboole_0(k4_topreal1(np__2,B,E),k9_metric_1(k14_euclid(np__2),D,C))
| k1_nat_1(E,np__1) = k3_finseq_1(B) ) ) ) ) ) ) ) ).
fof(t49_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(C,k9_metric_1(k14_euclid(np__2),E,D)) )
=> ( r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
| r2_hidden(C,k3_topreal1(np__2,A,B))
| ( ~ ( k21_euclid(B) = k21_euclid(C)
& k22_euclid(B) != k22_euclid(C) )
& ~ ( k21_euclid(B) != k21_euclid(C)
& k22_euclid(B) = k22_euclid(C) ) )
| ( k21_euclid(A) != k21_euclid(B)
& k22_euclid(A) != k22_euclid(B) )
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,B),k3_topreal1(np__2,B,C)) = k1_struct_0(k15_euclid(np__2),B) ) ) ) ) ) ) ) ).
fof(t50_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(k23_euclid(k21_euclid(B),k22_euclid(C)),k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(C,k9_metric_1(k14_euclid(np__2),E,D))
& k21_euclid(A) = k21_euclid(B) )
=> ( r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
| r2_hidden(k23_euclid(k21_euclid(B),k22_euclid(C)),k3_topreal1(np__2,A,B))
| k21_euclid(B) = k21_euclid(C)
| k22_euclid(B) = k22_euclid(C)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,B,k23_euclid(k21_euclid(B),k22_euclid(C))),k3_topreal1(np__2,k23_euclid(k21_euclid(B),k22_euclid(C)),C)),k3_topreal1(np__2,A,B)) = k1_struct_0(k15_euclid(np__2),B) ) ) ) ) ) ) ) ).
fof(t51_topreal3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(np__2)))
=> ( ( r2_hidden(B,k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(k23_euclid(k21_euclid(C),k22_euclid(B)),k9_metric_1(k14_euclid(np__2),E,D))
& r2_hidden(C,k9_metric_1(k14_euclid(np__2),E,D))
& k22_euclid(A) = k22_euclid(B) )
=> ( r2_hidden(A,k9_metric_1(k14_euclid(np__2),E,D))
| r2_hidden(k23_euclid(k21_euclid(C),k22_euclid(B)),k3_topreal1(np__2,A,B))
| k21_euclid(B) = k21_euclid(C)
| k22_euclid(B) = k22_euclid(C)
| k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,B,k23_euclid(k21_euclid(C),k22_euclid(B))),k3_topreal1(np__2,k23_euclid(k21_euclid(C),k22_euclid(B)),C)),k3_topreal1(np__2,A,B)) = k1_struct_0(k15_euclid(np__2),B) ) ) ) ) ) ) ) ).
fof(t15_topreal3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(A,B)
=> a_3_0_topreal3(A,B,C) = k3_topreal1(np__2,k23_euclid(C,A),k23_euclid(C,B)) ) ) ) ) ).
fof(t16_topreal3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(A,B)
=> a_3_1_topreal3(A,B,C) = k3_topreal1(np__2,k23_euclid(A,C),k23_euclid(B,C)) ) ) ) ) ).
fof(fraenkel_a_3_0_topreal3,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(B)
& v1_xreal_0(C)
& v1_xreal_0(D) )
=> ( r2_hidden(A,a_3_0_topreal3(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
& A = E
& k21_euclid(E) = D
& r1_xreal_0(B,k22_euclid(E))
& r1_xreal_0(k22_euclid(E),C) ) ) ) ).
fof(fraenkel_a_3_1_topreal3,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(B)
& v1_xreal_0(C)
& v1_xreal_0(D) )
=> ( r2_hidden(A,a_3_1_topreal3(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
& A = E
& k22_euclid(E) = D
& r1_xreal_0(B,k21_euclid(E))
& r1_xreal_0(k21_euclid(E),C) ) ) ) ).
%------------------------------------------------------------------------------