SET007 Axioms: SET007+294.ax
%------------------------------------------------------------------------------
% File : SET007+294 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Fundamental Types of Metric Affine Spaces
% 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 : euclmetr [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 45 ( 1 unt; 0 def)
% Number of atoms : 462 ( 22 equ)
% Maximal formula atoms : 24 ( 10 avg)
% Number of connectives : 516 ( 99 ~; 9 |; 260 &)
% ( 13 <=>; 135 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 11 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 43 ( 41 usr; 1 prp; 0-5 aty)
% Number of functors : 12 ( 12 usr; 2 con; 0-3 aty)
% Number of variables : 124 ( 122 !; 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(rc1_euclmetr,axiom,
? [A] :
( l1_analmetr(A)
& ~ v3_struct_0(A)
& v2_analmetr(A)
& v3_analmetr(A)
& v1_euclmetr(A)
& v2_euclmetr(A)
& v3_euclmetr(A)
& v4_euclmetr(A)
& v5_euclmetr(A)
& v6_euclmetr(A)
& v7_euclmetr(A) ) ).
fof(rc2_euclmetr,axiom,
? [A] :
( l1_analmetr(A)
& ~ v3_struct_0(A)
& v2_analmetr(A)
& v1_euclmetr(A)
& v2_euclmetr(A)
& v3_euclmetr(A)
& v4_euclmetr(A)
& v5_euclmetr(A)
& v6_euclmetr(A)
& v7_euclmetr(A) ) ).
fof(fc1_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& v2_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v2_diraf(k3_analmetr(A))
& v2_aff_2(k3_analmetr(A)) ) ) ).
fof(fc2_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& v3_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v2_diraf(k3_analmetr(A))
& v4_aff_2(k3_analmetr(A)) ) ) ).
fof(fc3_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& v5_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v2_diraf(k3_analmetr(A))
& v7_aff_2(k3_analmetr(A)) ) ) ).
fof(fc4_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& v6_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v2_diraf(k3_analmetr(A))
& v11_aff_2(k3_analmetr(A)) ) ) ).
fof(fc5_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& v4_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v2_diraf(k3_analmetr(A))
& v1_translac(k3_analmetr(A)) ) ) ).
fof(fc6_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& v7_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v2_diraf(k3_analmetr(A))
& v4_aff_2(k3_analmetr(A)) ) ) ).
fof(fc7_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& v1_euclmetr(A)
& v3_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v2_diraf(k3_analmetr(A))
& v2_aff_2(k3_analmetr(A))
& v4_aff_2(k3_analmetr(A)) ) ) ).
fof(fc8_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& v2_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v2_aff_2(k3_analmetr(A)) ) ) ).
fof(fc9_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& v3_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v4_aff_2(k3_analmetr(A)) ) ) ).
fof(fc10_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& v5_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v7_aff_2(k3_analmetr(A)) ) ) ).
fof(fc11_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& v6_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v11_aff_2(k3_analmetr(A)) ) ) ).
fof(fc12_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& v4_euclmetr(A)
& l1_analmetr(A) )
=> ( ~ v3_struct_0(k3_analmetr(A))
& ~ v3_realset2(k3_analmetr(A))
& v1_analoaf(k3_analmetr(A))
& v1_diraf(k3_analmetr(A))
& v1_translac(k3_analmetr(A)) ) ) ).
fof(d1_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ( v1_euclmetr(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r5_analmetr(A,B,C,D,E)
& r5_analmetr(A,C,D,B,E) )
=> r5_analmetr(A,C,E,B,D) ) ) ) ) ) ) ) ).
fof(d2_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ( v2_euclmetr(A)
<=> v2_aff_2(k3_analmetr(A)) ) ) ).
fof(d3_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ( v3_euclmetr(A)
<=> v4_aff_2(k3_analmetr(A)) ) ) ).
fof(d4_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ( v4_euclmetr(A)
<=> v1_translac(k3_analmetr(A)) ) ) ).
fof(d5_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ( v5_euclmetr(A)
<=> v7_aff_2(k3_analmetr(A)) ) ) ).
fof(d6_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ( v6_euclmetr(A)
<=> v11_aff_2(k3_analmetr(A)) ) ) ).
fof(d7_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ( v7_euclmetr(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r5_analmetr(A,B,C,B,D)
& r5_analmetr(A,B,E,B,F)
& r5_analmetr(A,B,G,B,H)
& r5_analmetr(A,C,E,D,F)
& r5_analmetr(A,C,G,D,H) )
=> ( r4_analmetr(A,B,G,B,C)
| r4_analmetr(A,B,C,B,E)
| r5_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ~ r6_analmetr(A,B,C,D)
=> ( B != C
& C != D
& B != D ) ) ) ) ) ) ).
fof(t2_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r7_analmetr(A,B,C,F)
& r7_analmetr(A,D,E,F) )
=> ( r4_analmetr(A,B,C,D,E)
& r4_analmetr(A,B,C,E,D) ) ) ) ) ) ) ) ) ).
fof(t3_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B != C
& ( r7_analmetr(A,B,C,E)
| r7_analmetr(A,C,B,E) )
& ( r7_analmetr(A,B,C,D)
| r7_analmetr(A,C,B,D) )
& ~ r10_analmetr(A,E,D) ) ) ) ) ) ) ).
fof(t4_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r6_analmetr(A,B,C,D)
=> ( r6_analmetr(A,B,D,C)
& r6_analmetr(A,C,B,D)
& r6_analmetr(A,C,D,B)
& r6_analmetr(A,D,B,C)
& r6_analmetr(A,D,C,B) ) ) ) ) ) ) ).
fof(t5_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( ~ r6_analmetr(A,B,C,D)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r5_analmetr(A,E,B,C,D)
& r5_analmetr(A,E,C,B,D) ) ) ) ) ) ) ) ).
fof(t6_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r5_analmetr(A,E,B,C,D)
& r5_analmetr(A,E,C,B,D)
& r5_analmetr(A,F,B,C,D)
& r5_analmetr(A,F,C,B,D) )
=> ( r6_analmetr(A,B,C,D)
| E = F ) ) ) ) ) ) ) ) ).
fof(t7_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r5_analmetr(A,B,C,D,E)
& r5_analmetr(A,C,D,B,E)
& r6_analmetr(A,B,C,D)
& B != D
& B != C
& C != D ) ) ) ) ) ) ).
fof(t8_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v1_euclmetr(A)
<=> v3_conaffm(A) ) ) ).
fof(t9_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v7_euclmetr(A)
<=> v4_conaffm(A) ) ) ).
fof(t10_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v2_euclmetr(A)
<=> v2_conmetr(A) ) ) ).
fof(t11_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v3_euclmetr(A)
<=> v1_conaffm(A) ) ) ).
fof(t12_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v5_euclmetr(A)
<=> v5_conmetr(A) ) ) ).
fof(t13_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v6_euclmetr(A)
<=> v8_conmetr(A) ) ) ).
fof(t14_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v7_euclmetr(A)
=> v3_euclmetr(A) ) ) ).
fof(t15_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( ( v1_euclmetr(A)
& v3_euclmetr(A) )
=> v2_euclmetr(A) ) ) ).
fof(t16_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( r5_analmetr(A,B,C,B,D)
& r5_analmetr(A,B,E,B,F)
& r5_analmetr(A,B,E,B,G)
& r5_analmetr(A,C,E,D,F)
& r5_analmetr(A,C,E,D,G) )
=> ( r6_analmetr(A,B,C,E)
| B = D
| F = G ) ) ) ) ) ) ) ) ) ).
fof(t17_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( ~ r6_analmetr(A,B,C,E)
& B != D
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( r5_analmetr(A,B,E,B,F)
& r5_analmetr(A,C,E,D,F) ) ) ) ) ) ) ) ) ).
fof(t18_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( r1_analmetr(A,B,C)
& k1_rlvect_1(A) != D
& k1_rlvect_1(A) != E
& r2_analmetr(A,D,E,B,C)
& D = k4_rlvect_1(A,k3_rlvect_1(A,B,F),k3_rlvect_1(A,C,G))
& ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( H != np__0
& E = k4_rlvect_1(A,k3_rlvect_1(A,B,k4_real_1(H,G)),k3_rlvect_1(A,C,k1_real_1(k4_real_1(H,F)))) ) ) ) ) ) ) ) ) ) ) ).
fof(t19_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r1_analmetr(A,B,C)
& k1_rlvect_1(A) != D
& k1_rlvect_1(A) != E
& r2_analmetr(A,D,E,B,C)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ( r2_analmetr(A,k4_rlvect_1(A,k3_rlvect_1(A,B,G),k3_rlvect_1(A,C,H)),k4_rlvect_1(A,k3_rlvect_1(A,B,k4_real_1(F,H)),k3_rlvect_1(A,C,k1_real_1(k4_real_1(F,G)))),B,C)
& r2_analmetr(A,k6_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,G),k3_rlvect_1(A,C,H)),D),k6_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,k4_real_1(F,H)),k3_rlvect_1(A,C,k1_real_1(k4_real_1(F,G)))),E),B,C) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_euclmetr,axiom,
$true ).
fof(t21_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& l2_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( ( r1_analmetr(B,C,D)
& A = k2_analmetr(B,C,D) )
=> v5_conaffm(A) ) ) ) ) ) ).
fof(t22_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ( ( r5_analmetr(A,B,C,B,D)
& r5_analmetr(A,B,E,B,F)
& r5_analmetr(A,B,G,B,H)
& r5_analmetr(A,C,E,D,F)
& r5_analmetr(A,C,G,D,H)
& B = D )
=> ( r4_analmetr(A,B,G,B,C)
| r4_analmetr(A,B,C,B,E)
| r5_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& l2_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( ( r1_analmetr(B,C,D)
& A = k2_analmetr(B,C,D) )
=> v7_euclmetr(A) ) ) ) ) ) ).
fof(t24_euclmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& l2_rlvect_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( ( r1_analmetr(B,C,D)
& A = k2_analmetr(B,C,D) )
=> ( ~ v3_struct_0(A)
& v3_analmetr(A)
& v1_euclmetr(A)
& v2_euclmetr(A)
& v3_euclmetr(A)
& v4_euclmetr(A)
& v5_euclmetr(A)
& v6_euclmetr(A)
& v7_euclmetr(A)
& l1_analmetr(A) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------