SET007 Axioms: SET007+239.ax
%------------------------------------------------------------------------------
% File : SET007+239 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Construction of Analytical Projective Space
% 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 : anproj_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 17 unt; 0 def)
% Number of atoms : 676 ( 49 equ)
% Maximal formula atoms : 28 ( 10 avg)
% Number of connectives : 717 ( 105 ~; 5 |; 423 &)
% ( 19 <=>; 165 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 25 ( 23 usr; 1 prp; 0-4 aty)
% Number of functors : 21 ( 21 usr; 2 con; 0-3 aty)
% Number of variables : 184 ( 166 !; 18 ?)
% 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_anproj_1,axiom,
? [A] :
( l2_rlvect_1(A)
& ~ v3_struct_0(A)
& v2_rlvect_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& ~ v3_realset2(A) ) ).
fof(fc1_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> ~ v1_xboole_0(k1_anproj_1(A)) ) ).
fof(fc2_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> ~ v1_xboole_0(k4_anproj_1(A)) ) ).
fof(fc3_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> ( ~ v3_struct_0(k6_anproj_1(A))
& v1_collsp(k6_anproj_1(A)) ) ) ).
fof(d1_anproj_1,axiom,
$true ).
fof(d2_anproj_1,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))
=> ( r1_anproj_1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& k3_rlvect_1(A,B,D) = k3_rlvect_1(A,C,E)
& D != np__0
& E != np__0 ) ) ) ) ) ) ).
fof(t1_anproj_1,axiom,
$true ).
fof(t2_anproj_1,axiom,
$true ).
fof(t3_anproj_1,axiom,
$true ).
fof(t4_anproj_1,axiom,
$true ).
fof(t5_anproj_1,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))
=> ( r1_anproj_1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& D != np__0
& B = k3_rlvect_1(A,C,D) ) ) ) ) ) ).
fof(t6_anproj_1,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))
=> ( ( r1_anproj_1(A,B,C)
& r1_anproj_1(A,C,D) )
=> r1_anproj_1(A,B,D) ) ) ) ) ) ).
fof(t7_anproj_1,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))
=> ( r1_anproj_1(A,B,k1_rlvect_1(A))
<=> B = k1_rlvect_1(A) ) ) ) ).
fof(d3_anproj_1,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))
=> ( r2_anproj_1(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m1_subset_1(F,k1_numbers)
& ? [G] :
( m1_subset_1(G,k1_numbers)
& k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,E),k3_rlvect_1(A,C,F)),k3_rlvect_1(A,D,G)) = k1_rlvect_1(A)
& ~ ( E = np__0
& F = np__0
& G = np__0 ) ) ) ) ) ) ) ) ) ).
fof(t8_anproj_1,axiom,
$true ).
fof(t9_anproj_1,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,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( r1_anproj_1(A,B,C)
& r1_anproj_1(A,D,E)
& r1_anproj_1(A,F,G)
& r2_anproj_1(A,B,D,F) )
=> r2_anproj_1(A,C,E,G) ) ) ) ) ) ) ) ) ).
fof(t10_anproj_1,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))
=> ( r2_anproj_1(A,B,C,D)
=> ( r2_anproj_1(A,B,D,C)
& r2_anproj_1(A,C,B,D)
& r2_anproj_1(A,D,C,B)
& r2_anproj_1(A,D,B,C)
& r2_anproj_1(A,C,D,B) ) ) ) ) ) ) ).
fof(t11_anproj_1,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))
=> ( ( v8_rlvect_1(B,A)
& v8_rlvect_1(C,A) )
=> ( r1_anproj_1(A,B,C)
| ( r2_anproj_1(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m1_subset_1(F,k1_numbers)
& D = k4_rlvect_1(A,k3_rlvect_1(A,B,E),k3_rlvect_1(A,C,F)) ) ) ) ) ) ) ) ) ) ).
fof(t12_anproj_1,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,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( k4_rlvect_1(A,k3_rlvect_1(A,B,D),k3_rlvect_1(A,C,E)) = k4_rlvect_1(A,k3_rlvect_1(A,B,F),k3_rlvect_1(A,C,G))
& v8_rlvect_1(B,A)
& v8_rlvect_1(C,A) )
=> ( r1_anproj_1(A,B,C)
| ( D = F
& E = G ) ) ) ) ) ) ) ) ) ) ).
fof(t13_anproj_1,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,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ! [I] :
( m1_subset_1(I,k1_numbers)
=> ! [J] :
( m1_subset_1(J,k1_numbers)
=> ( k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,E),k3_rlvect_1(A,C,F)),k3_rlvect_1(A,D,G)) = k4_rlvect_1(A,k4_rlvect_1(A,k3_rlvect_1(A,B,H),k3_rlvect_1(A,C,I)),k3_rlvect_1(A,D,J))
=> ( r2_anproj_1(A,B,C,D)
| ( E = H
& F = I
& G = J ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t14_anproj_1,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)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ! [I] :
( m1_subset_1(I,k1_numbers)
=> ~ ( ~ r1_anproj_1(A,B,C)
& D = k4_rlvect_1(A,k3_rlvect_1(A,B,F),k3_rlvect_1(A,C,G))
& E = k4_rlvect_1(A,k3_rlvect_1(A,B,H),k3_rlvect_1(A,C,I))
& k5_real_1(k4_real_1(F,I),k4_real_1(H,G)) = np__0
& v8_rlvect_1(B,A)
& v8_rlvect_1(C,A)
& ~ r1_anproj_1(A,D,E)
& D != k1_rlvect_1(A)
& E != k1_rlvect_1(A) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_anproj_1,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))
=> ( ~ ( B != k1_rlvect_1(A)
& C != k1_rlvect_1(A)
& D != k1_rlvect_1(A) )
=> r2_anproj_1(A,B,C,D) ) ) ) ) ) ).
fof(t16_anproj_1,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))
=> ( ~ ( ~ r1_anproj_1(A,B,C)
& ~ r1_anproj_1(A,D,B)
& ~ r1_anproj_1(A,C,D) )
=> r2_anproj_1(A,D,B,C) ) ) ) ) ) ).
fof(t17_anproj_1,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))
=> ( ~ r2_anproj_1(A,B,C,D)
=> ( v8_rlvect_1(B,A)
& v8_rlvect_1(C,A)
& v8_rlvect_1(D,A)
& ~ r1_anproj_1(A,B,C)
& ~ r1_anproj_1(A,C,D)
& ~ r1_anproj_1(A,D,B) ) ) ) ) ) ) ).
fof(t18_anproj_1,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))
=> ( k4_rlvect_1(A,B,C) = k1_rlvect_1(A)
=> r1_anproj_1(A,B,C) ) ) ) ) ).
fof(t19_anproj_1,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,u1_struct_0(A))
=> ( ( r2_anproj_1(A,B,C,D)
& r2_anproj_1(A,B,C,E)
& r2_anproj_1(A,B,C,F)
& v8_rlvect_1(B,A)
& v8_rlvect_1(C,A) )
=> ( r1_anproj_1(A,B,C)
| r2_anproj_1(A,D,E,F) ) ) ) ) ) ) ) ) ).
fof(t20_anproj_1,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,u1_struct_0(A))
=> ~ ( ~ r2_anproj_1(A,B,C,D)
& r2_anproj_1(A,B,C,E)
& r2_anproj_1(A,C,D,F)
& ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ~ ( r2_anproj_1(A,B,D,G)
& r2_anproj_1(A,E,F,G)
& v8_rlvect_1(G,A) ) ) ) ) ) ) ) ) ) ).
fof(t21_anproj_1,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))
=> ( ( v8_rlvect_1(B,A)
& v8_rlvect_1(C,A) )
=> ( r1_anproj_1(A,B,C)
| ! [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))
& v8_rlvect_1(F,A)
& r2_anproj_1(A,D,E,F)
& ~ r1_anproj_1(A,D,F)
& ~ r1_anproj_1(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t22_anproj_1,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))
=> ( ~ r2_anproj_1(A,B,C,D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( v8_rlvect_1(E,A)
& v8_rlvect_1(F,A)
& ~ r1_anproj_1(A,E,F)
& ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ~ ( v8_rlvect_1(G,A)
& ~ r2_anproj_1(A,E,F,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_anproj_1,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,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ~ ( r2_anproj_1(A,B,C,D)
& r2_anproj_1(A,E,F,D)
& r2_anproj_1(A,B,E,G)
& r2_anproj_1(A,C,F,G)
& r2_anproj_1(A,B,F,H)
& r2_anproj_1(A,C,E,H)
& r2_anproj_1(A,G,D,H)
& v8_rlvect_1(G,A)
& v8_rlvect_1(D,A)
& v8_rlvect_1(H,A)
& ~ r2_anproj_1(A,B,C,F)
& ~ r2_anproj_1(A,B,C,E)
& ~ r2_anproj_1(A,B,E,F)
& ~ r2_anproj_1(A,C,E,F) ) ) ) ) ) ) ) ) ) ).
fof(d4_anproj_1,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] :
( B = k1_anproj_1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( C != k1_rlvect_1(A)
& m1_subset_1(C,u1_struct_0(A)) ) ) ) ) ).
fof(t24_anproj_1,axiom,
$true ).
fof(t25_anproj_1,axiom,
$true ).
fof(t26_anproj_1,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))
=> ( r2_hidden(B,k1_anproj_1(A))
<=> v8_rlvect_1(B,A) ) ) ) ).
fof(d5_anproj_1,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] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,k1_anproj_1(A),k1_anproj_1(A))
& m2_relset_1(B,k1_anproj_1(A),k1_anproj_1(A)) )
=> ( B = k2_anproj_1(A)
<=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
<=> ( r2_hidden(C,k1_anproj_1(A))
& r2_hidden(D,k1_anproj_1(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& C = E
& D = F
& r1_anproj_1(A,E,F) ) ) ) ) ) ) ) ).
fof(t27_anproj_1,axiom,
$true ).
fof(t28_anproj_1,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,C] :
( r2_hidden(k4_tarski(B,C),k2_anproj_1(A))
=> ( m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) ) ) ) ).
fof(t29_anproj_1,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))
=> ( r2_hidden(k4_tarski(B,C),k2_anproj_1(A))
<=> ( v8_rlvect_1(B,A)
& v8_rlvect_1(C,A)
& r1_anproj_1(A,B,C) ) ) ) ) ) ).
fof(d6_anproj_1,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))
=> k3_anproj_1(A,B) = k6_eqrel_1(k1_anproj_1(A),k2_anproj_1(A),B) ) ) ).
fof(d7_anproj_1,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] :
( B = k4_anproj_1(A)
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k1_anproj_1(A))))
& C = k8_eqrel_1(k1_anproj_1(A),k2_anproj_1(A))
& B = C ) ) ) ).
fof(d8_anproj_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ( v3_realset2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> B = k1_rlvect_1(A) ) ) ) ).
fof(t30_anproj_1,axiom,
$true ).
fof(t31_anproj_1,axiom,
$true ).
fof(t32_anproj_1,axiom,
$true ).
fof(t33_anproj_1,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) )
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& r2_hidden(B,k1_anproj_1(A)) ) ) ) ).
fof(t34_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v8_rlvect_1(B,A)
=> m1_subset_1(k3_anproj_1(A,B),k4_anproj_1(A)) ) ) ) ).
fof(t35_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( v8_rlvect_1(B,A)
& v8_rlvect_1(C,A) )
=> ( k3_anproj_1(A,B) = k3_anproj_1(A,C)
<=> r1_anproj_1(A,B,C) ) ) ) ) ) ).
fof(d9_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> ! [B] :
( m1_collsp(B,k4_anproj_1(A))
=> ( B = k5_anproj_1(A)
<=> ! [C,D,E] :
( r2_hidden(k3_mcart_1(C,D,E),B)
<=> ? [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))
& C = k3_anproj_1(A,F)
& D = k3_anproj_1(A,G)
& E = k3_anproj_1(A,H)
& v8_rlvect_1(F,A)
& v8_rlvect_1(G,A)
& v8_rlvect_1(H,A)
& r2_anproj_1(A,F,G,H) ) ) ) ) ) ) ) ).
fof(d10_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> k6_anproj_1(A) = g1_collsp(k4_anproj_1(A),k5_anproj_1(A)) ) ).
fof(t36_anproj_1,axiom,
$true ).
fof(t37_anproj_1,axiom,
$true ).
fof(t38_anproj_1,axiom,
$true ).
fof(t39_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> ( u1_struct_0(k6_anproj_1(A)) = k4_anproj_1(A)
& u1_collsp(k6_anproj_1(A)) = k5_anproj_1(A) ) ) ).
fof(t40_anproj_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_rlvect_1(D)
& ~ v3_realset2(D)
& l2_rlvect_1(D) )
=> ~ ( r2_hidden(k3_mcart_1(A,B,C),u1_collsp(k6_anproj_1(D)))
& ! [E] :
( m1_subset_1(E,u1_struct_0(D))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(D))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(D))
=> ~ ( A = k3_anproj_1(D,E)
& B = k3_anproj_1(D,F)
& C = k3_anproj_1(D,G)
& v8_rlvect_1(E,D)
& v8_rlvect_1(F,D)
& v8_rlvect_1(G,D)
& r2_anproj_1(D,E,F,G) ) ) ) ) ) ) ).
fof(t41_anproj_1,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)
& ~ v3_realset2(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))
=> ( ( v8_rlvect_1(B,A)
& v8_rlvect_1(C,A)
& v8_rlvect_1(D,A) )
=> ( r2_hidden(k3_mcart_1(k3_anproj_1(A,B),k3_anproj_1(A,C),k3_anproj_1(A,D)),u1_collsp(k6_anproj_1(A)))
<=> r2_anproj_1(A,B,C,D) ) ) ) ) ) ) ).
fof(t42_anproj_1,axiom,
! [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)
& ~ v3_realset2(B)
& l2_rlvect_1(B) )
=> ( m1_subset_1(A,u1_struct_0(k6_anproj_1(B)))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& v8_rlvect_1(C,B)
& A = k3_anproj_1(B,C) ) ) ) ).
fof(symmetry_r1_anproj_1,axiom,
! [A,B,C] :
( ( ~ 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)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r1_anproj_1(A,B,C)
=> r1_anproj_1(A,C,B) ) ) ).
fof(reflexivity_r1_anproj_1,axiom,
! [A,B,C] :
( ( ~ 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)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> r1_anproj_1(A,B,B) ) ).
fof(dt_k1_anproj_1,axiom,
$true ).
fof(dt_k2_anproj_1,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) )
=> ( v3_relat_2(k2_anproj_1(A))
& v8_relat_2(k2_anproj_1(A))
& v1_partfun1(k2_anproj_1(A),k1_anproj_1(A),k1_anproj_1(A))
& m2_relset_1(k2_anproj_1(A),k1_anproj_1(A),k1_anproj_1(A)) ) ) ).
fof(dt_k3_anproj_1,axiom,
! [A,B] :
( ( ~ 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)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k3_anproj_1(A,B),k1_zfmisc_1(k1_anproj_1(A))) ) ).
fof(dt_k4_anproj_1,axiom,
$true ).
fof(dt_k5_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> m1_collsp(k5_anproj_1(A),k4_anproj_1(A)) ) ).
fof(dt_k6_anproj_1,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)
& ~ v3_realset2(A)
& l2_rlvect_1(A) )
=> ( v1_collsp(k6_anproj_1(A))
& l1_collsp(k6_anproj_1(A)) ) ) ).
%------------------------------------------------------------------------------