%------------------------------------------------------------------------------
% File : SET007+283 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Projections in Projective Planes - Part II
% 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 : projred2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 32 ( 1 unt; 0 def)
% Number of atoms : 716 ( 54 equ)
% Maximal formula atoms : 57 ( 22 avg)
% Number of connectives : 787 ( 103 ~; 66 |; 389 &)
% ( 4 <=>; 225 =>; 0 <=; 0 <~>)
% Maximal formula depth : 52 ( 23 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-5 aty)
% Number of functors : 12 ( 12 usr; 0 con; 1-4 aty)
% Number of variables : 226 ( 219 !; 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(d1_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ( r1_projred2(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,u1_incsp_1(A))
& r1_incsp_1(A,E,B)
& r1_incsp_1(A,E,C)
& r1_incsp_1(A,E,D) ) ) ) ) ) ) ).
fof(d3_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,u1_incsp_1(A),u1_incsp_1(A)) )
=> ( m1_projred2(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_incsp_1(A))
& ? [D] :
( m1_subset_1(D,u2_incsp_1(A))
& ? [E] :
( m1_subset_1(E,u2_incsp_1(A))
& ~ r1_incsp_1(A,C,D)
& ~ r1_incsp_1(A,C,E)
& B = k1_projred1(A,D,E,C) ) ) ) ) ) ) ).
fof(t1_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ( ~ ( B != C
& C != D
& D != B )
=> r1_projred2(A,B,C,D) ) ) ) ) ) ).
fof(t2_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ( r1_projred2(A,B,C,D)
=> ( r1_projred2(A,B,D,C)
& r1_projred2(A,C,B,D)
& r1_projred2(A,C,D,B)
& r1_projred2(A,D,B,C)
& r1_projred2(A,D,C,B) ) ) ) ) ) ) ).
fof(t3_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,D)
& ~ r1_incsp_1(A,B,E)
& r1_incsp_1(A,C,E)
& ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ~ ( r1_incsp_1(A,F,D)
& k1_funct_1(k1_projred1(A,D,E,B),F) = C ) ) ) ) ) ) ) ) ).
fof(t4_projred2,axiom,
$true ).
fof(t5_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,C)
& ~ r1_incsp_1(A,B,D)
& k4_relset_1(u1_incsp_1(A),u1_incsp_1(A),k1_projred1(A,C,D,B)) != k1_projred2(A,C) ) ) ) ) ) ).
fof(t6_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,C)
& ~ r1_incsp_1(A,B,D)
& k2_relat_1(k1_projred1(A,C,D,B)) != k1_projred2(A,D) ) ) ) ) ) ).
fof(t7_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(A))
=> ! [C] :
( r2_hidden(C,k1_projred2(A,B))
<=> ? [D] :
( m1_subset_1(D,u1_incsp_1(A))
& C = D
& r1_incsp_1(A,D,B) ) ) ) ) ).
fof(t8_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,C)
& ~ r1_incsp_1(A,B,D)
& ~ v2_funct_1(k1_projred1(A,C,D,B)) ) ) ) ) ) ).
fof(t9_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,C)
& ~ r1_incsp_1(A,B,D)
& k2_funct_1(k1_projred1(A,C,D,B)) != k1_projred1(A,D,C,B) ) ) ) ) ) ).
fof(t10_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_projred2(B,A)
=> m1_projred2(k2_funct_1(B),A) ) ) ).
fof(t11_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ( ~ r1_incsp_1(A,B,C)
=> k1_projred1(A,C,C,B) = k6_relat_1(k1_projred2(A,C)) ) ) ) ) ).
fof(t12_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(A))
=> m1_projred2(k6_relat_1(k1_projred2(A,B)),A) ) ) ).
fof(t13_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,C)
& ~ r1_incsp_1(A,B,D)
& ~ r1_incsp_1(A,B,E)
& k5_relat_1(k1_projred1(A,C,E,B),k1_projred1(A,E,D,B)) != k1_projred1(A,C,D,B) ) ) ) ) ) ) ).
fof(t14_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,D)
& ~ r1_incsp_1(A,B,E)
& ~ r1_incsp_1(A,C,E)
& ~ r1_incsp_1(A,C,F)
& r1_projred2(A,D,E,F)
& D != F
& ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,G,D)
& ~ r1_incsp_1(A,G,F)
& k5_relat_1(k1_projred1(A,D,E,B),k1_projred1(A,E,F,C)) = k1_projred1(A,D,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t15_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u1_incsp_1(A))
=> ! [I] :
( m1_subset_1(I,u2_incsp_1(A))
=> ! [J] :
( m1_subset_1(J,u2_incsp_1(A))
=> ! [K] :
( m1_subset_1(K,u2_incsp_1(A))
=> ! [L] :
( m1_subset_1(L,u2_incsp_1(A))
=> ! [M] :
( m1_subset_1(M,u2_incsp_1(A))
=> ! [N] :
( m1_subset_1(N,u2_incsp_1(A))
=> ! [O] :
( m1_subset_1(O,u2_incsp_1(A))
=> ! [P] :
( m1_subset_1(P,u2_incsp_1(A))
=> ( ( r1_incsp_1(A,D,I)
& r1_incsp_1(A,D,K)
& r1_incsp_1(A,D,L)
& r1_incsp_1(A,B,M)
& r1_incsp_1(A,C,M)
& r1_incsp_1(A,F,K)
& r1_incsp_1(A,F,J)
& r1_incsp_1(A,B,N)
& r1_incsp_1(A,F,N)
& r1_incsp_1(A,G,I)
& r1_incsp_1(A,G,N)
& r1_incsp_1(A,E,M)
& r1_incsp_1(A,E,O)
& r1_incsp_1(A,G,O)
& r1_incsp_1(A,H,O)
& r1_incsp_1(A,F,P)
& r1_incsp_1(A,C,P)
& r1_incsp_1(A,H,P)
& r1_incsp_1(A,H,L) )
=> ( r1_incsp_1(A,B,I)
| r1_incsp_1(A,C,J)
| r1_incsp_1(A,B,K)
| r1_incsp_1(A,C,K)
| r1_projred2(A,I,J,K)
| r1_incsp_1(A,C,L)
| I = L
| B = C
| C = E
| r1_projred2(A,J,K,M)
| L = K
| E = B
| r1_incsp_1(A,E,I)
| r1_incsp_1(A,E,L)
| k5_relat_1(k1_projred1(A,I,K,B),k1_projred1(A,K,J,C)) = k5_relat_1(k1_projred1(A,I,L,E),k1_projred1(A,L,J,C)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t16_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u1_incsp_1(A))
=> ! [I] :
( m1_subset_1(I,u1_incsp_1(A))
=> ! [J] :
( m1_subset_1(J,u1_incsp_1(A))
=> ! [K] :
( m1_subset_1(K,u2_incsp_1(A))
=> ! [L] :
( m1_subset_1(L,u2_incsp_1(A))
=> ! [M] :
( m1_subset_1(M,u2_incsp_1(A))
=> ! [N] :
( m1_subset_1(N,u2_incsp_1(A))
=> ! [O] :
( m1_subset_1(O,u2_incsp_1(A))
=> ! [P] :
( m1_subset_1(P,u2_incsp_1(A))
=> ! [Q] :
( m1_subset_1(Q,u2_incsp_1(A))
=> ! [R] :
( m1_subset_1(R,u2_incsp_1(A))
=> ( ( r3_incproj(A,E,F,K)
& r4_incproj(A,F,G,M,H)
& r4_incproj(A,E,H,L,I)
& r4_incproj(A,B,C,O,H)
& r3_incproj(A,E,J,N)
& r4_incproj(A,B,F,P,I)
& r4_incproj(A,C,I,Q,J)
& r4_incproj(A,F,J,R,D)
& r1_incsp_1(A,D,O) )
=> ( r1_incsp_1(A,B,K)
| r1_incsp_1(A,B,L)
| r1_incsp_1(A,C,M)
| r1_incsp_1(A,C,L)
| r1_incsp_1(A,C,N)
| r1_projred2(A,K,M,L)
| B = C
| C = D
| K = N
| k5_relat_1(k1_projred1(A,K,L,B),k1_projred1(A,L,M,C)) = k5_relat_1(k1_projred1(A,K,N,D),k1_projred1(A,N,M,C)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u1_incsp_1(A))
=> ! [I] :
( m1_subset_1(I,u2_incsp_1(A))
=> ! [J] :
( m1_subset_1(J,u2_incsp_1(A))
=> ! [K] :
( m1_subset_1(K,u2_incsp_1(A))
=> ! [L] :
( m1_subset_1(L,u2_incsp_1(A))
=> ! [M] :
( m1_subset_1(M,u2_incsp_1(A))
=> ! [N] :
( m1_subset_1(N,u2_incsp_1(A))
=> ! [O] :
( m1_subset_1(O,u2_incsp_1(A))
=> ! [P] :
( m1_subset_1(P,u2_incsp_1(A))
=> ( ( r3_incproj(A,D,E,I)
& r1_incsp_1(A,F,K)
& r3_incproj(A,D,F,J)
& r4_incproj(A,B,C,M,G)
& r3_incproj(A,D,H,L)
& r4_incproj(A,B,F,N,E)
& r4_incproj(A,G,E,O,H)
& r4_incproj(A,C,F,P,H) )
=> ( r1_incsp_1(A,B,I)
| r1_incsp_1(A,B,J)
| r1_incsp_1(A,C,K)
| r1_incsp_1(A,C,J)
| r1_incsp_1(A,C,L)
| r1_projred2(A,I,K,J)
| r1_projred2(A,K,J,M)
| I = L
| L = J
| B = C
| ( G != B
& G != C
& ~ r1_incsp_1(A,G,I)
& ~ r1_incsp_1(A,G,L) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u1_incsp_1(A))
=> ! [I] :
( m1_subset_1(I,u1_incsp_1(A))
=> ! [J] :
( m1_subset_1(J,u1_incsp_1(A))
=> ! [K] :
( m1_subset_1(K,u2_incsp_1(A))
=> ! [L] :
( m1_subset_1(L,u2_incsp_1(A))
=> ! [M] :
( m1_subset_1(M,u2_incsp_1(A))
=> ! [N] :
( m1_subset_1(N,u2_incsp_1(A))
=> ! [O] :
( m1_subset_1(O,u2_incsp_1(A))
=> ! [P] :
( m1_subset_1(P,u2_incsp_1(A))
=> ! [Q] :
( m1_subset_1(Q,u2_incsp_1(A))
=> ! [R] :
( m1_subset_1(R,u2_incsp_1(A))
=> ( ( r3_incproj(A,D,E,K)
& r4_incproj(A,E,F,M,G)
& r4_incproj(A,D,G,L,H)
& r4_incproj(A,B,C,O,G)
& r3_incproj(A,D,I,N)
& r4_incproj(A,B,E,P,H)
& r4_incproj(A,C,H,Q,I)
& r4_incproj(A,E,I,R,J)
& r1_incsp_1(A,J,O) )
=> ( r1_incsp_1(A,B,K)
| r1_incsp_1(A,B,L)
| r1_incsp_1(A,C,M)
| r1_incsp_1(A,C,L)
| r1_incsp_1(A,C,N)
| r1_projred2(A,K,M,L)
| B = C
| K = N
| ( ~ r1_incsp_1(A,J,K)
& ~ r1_incsp_1(A,J,N)
& C != J ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t19_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u1_incsp_1(A))
=> ! [I] :
( m1_subset_1(I,u2_incsp_1(A))
=> ! [J] :
( m1_subset_1(J,u2_incsp_1(A))
=> ! [K] :
( m1_subset_1(K,u2_incsp_1(A))
=> ! [L] :
( m1_subset_1(L,u2_incsp_1(A))
=> ! [M] :
( m1_subset_1(M,u2_incsp_1(A))
=> ! [N] :
( m1_subset_1(N,u2_incsp_1(A))
=> ! [O] :
( m1_subset_1(O,u2_incsp_1(A))
=> ! [P] :
( m1_subset_1(P,u2_incsp_1(A))
=> ( ( r3_incproj(A,E,F,I)
& r1_incsp_1(A,G,K)
& r3_incproj(A,E,G,J)
& r4_incproj(A,B,C,L,D)
& r3_incproj(A,E,H,M)
& r4_incproj(A,B,G,N,F)
& r4_incproj(A,D,F,O,H)
& r4_incproj(A,C,G,P,H) )
=> ( r1_incsp_1(A,B,I)
| r1_incsp_1(A,B,J)
| r1_incsp_1(A,C,K)
| r1_incsp_1(A,C,J)
| r1_incsp_1(A,D,I)
| r1_projred2(A,I,K,J)
| r1_projred2(A,K,J,L)
| B = C
| C = D
| D = B
| ( M != I
& M != J
& ~ r1_incsp_1(A,D,M)
& ~ r1_incsp_1(A,C,M) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u1_incsp_1(A))
=> ! [I] :
( m1_subset_1(I,u1_incsp_1(A))
=> ! [J] :
( m1_subset_1(J,u1_incsp_1(A))
=> ! [K] :
( m1_subset_1(K,u2_incsp_1(A))
=> ! [L] :
( m1_subset_1(L,u2_incsp_1(A))
=> ! [M] :
( m1_subset_1(M,u2_incsp_1(A))
=> ! [N] :
( m1_subset_1(N,u2_incsp_1(A))
=> ! [O] :
( m1_subset_1(O,u2_incsp_1(A))
=> ! [P] :
( m1_subset_1(P,u2_incsp_1(A))
=> ! [Q] :
( m1_subset_1(Q,u2_incsp_1(A))
=> ! [R] :
( m1_subset_1(R,u2_incsp_1(A))
=> ( ( r3_incproj(A,E,F,K)
& r4_incproj(A,F,G,M,H)
& r4_incproj(A,E,H,L,I)
& r4_incproj(A,B,C,N,H)
& r3_incproj(A,E,J,O)
& r4_incproj(A,B,F,P,I)
& r4_incproj(A,C,I,Q,J)
& r4_incproj(A,F,J,R,D)
& r1_incsp_1(A,D,N) )
=> ( r1_incsp_1(A,B,K)
| r1_incsp_1(A,B,L)
| r1_incsp_1(A,C,M)
| r1_incsp_1(A,C,L)
| r1_incsp_1(A,D,K)
| r1_projred2(A,K,M,L)
| B = C
| C = D
| ( ~ r1_incsp_1(A,C,O)
& ~ r1_incsp_1(A,D,O)
& K != O ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t21_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u2_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u2_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,D)
& ~ r1_incsp_1(A,C,E)
& ~ r1_incsp_1(A,B,F)
& ~ r1_incsp_1(A,C,F)
& ~ r1_projred2(A,D,E,F)
& r1_projred2(A,D,F,G)
& ~ r1_incsp_1(A,C,G)
& D != G
& B != C
& r1_incsp_1(A,B,H)
& r1_incsp_1(A,C,H)
& ! [I] :
( m1_subset_1(I,u1_incsp_1(A))
=> ~ ( r1_incsp_1(A,I,H)
& ~ r1_incsp_1(A,I,D)
& ~ r1_incsp_1(A,I,G)
& k5_relat_1(k1_projred1(A,D,F,B),k1_projred1(A,F,E,C)) = k5_relat_1(k1_projred1(A,D,G,I),k1_projred1(A,G,E,C)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u2_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u2_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,D)
& ~ r1_incsp_1(A,C,E)
& ~ r1_incsp_1(A,B,F)
& ~ r1_incsp_1(A,C,F)
& ~ r1_projred2(A,D,E,F)
& r1_projred2(A,E,F,G)
& ~ r1_incsp_1(A,B,G)
& E != G
& B != C
& r1_incsp_1(A,B,H)
& r1_incsp_1(A,C,H)
& ! [I] :
( m1_subset_1(I,u1_incsp_1(A))
=> ~ ( r1_incsp_1(A,I,H)
& ~ r1_incsp_1(A,I,E)
& ~ r1_incsp_1(A,I,G)
& k5_relat_1(k1_projred1(A,D,F,B),k1_projred1(A,F,E,C)) = k5_relat_1(k1_projred1(A,D,G,B),k1_projred1(A,G,E,I)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u2_incsp_1(A))
=> ! [I] :
( m1_subset_1(I,u2_incsp_1(A))
=> ! [J] :
( m1_subset_1(J,u2_incsp_1(A))
=> ! [K] :
( m1_subset_1(K,u2_incsp_1(A))
=> ! [L] :
( m1_subset_1(L,u2_incsp_1(A))
=> ! [M] :
( m1_subset_1(M,u2_incsp_1(A))
=> ( ( r1_incsp_1(A,D,H)
& r1_incsp_1(A,D,J)
& r1_incsp_1(A,E,I)
& r1_incsp_1(A,E,J)
& r1_incsp_1(A,B,K)
& r1_incsp_1(A,E,K)
& r1_incsp_1(A,D,L)
& r1_incsp_1(A,C,L)
& r1_incsp_1(A,F,H)
& r1_incsp_1(A,F,K)
& r1_incsp_1(A,G,I)
& r1_incsp_1(A,G,L)
& r1_incsp_1(A,F,M)
& r1_incsp_1(A,G,M) )
=> ( r1_incsp_1(A,B,H)
| r1_incsp_1(A,C,I)
| r1_incsp_1(A,B,J)
| r1_incsp_1(A,C,J)
| r1_incsp_1(A,B,I)
| r1_incsp_1(A,C,H)
| r1_projred2(A,H,I,J)
| k5_relat_1(k1_projred1(A,H,J,B),k1_projred1(A,J,I,C)) = k5_relat_1(k1_projred1(A,H,M,C),k1_projred1(A,M,I,B)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u2_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u2_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,E)
& ~ r1_incsp_1(A,C,F)
& ~ r1_incsp_1(A,B,G)
& ~ r1_incsp_1(A,C,G)
& B != C
& r1_incsp_1(A,B,H)
& r1_incsp_1(A,C,H)
& r1_incsp_1(A,D,H)
& ~ r1_incsp_1(A,D,E)
& D != C
& ~ r1_projred2(A,E,F,G)
& ! [I] :
( m1_subset_1(I,u2_incsp_1(A))
=> ~ ( r1_projred2(A,E,G,I)
& ~ r1_incsp_1(A,C,I)
& ~ r1_incsp_1(A,D,I)
& k5_relat_1(k1_projred1(A,E,G,B),k1_projred1(A,G,F,C)) = k5_relat_1(k1_projred1(A,E,I,D),k1_projred1(A,I,F,C)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t25_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(A))
=> ! [F] :
( m1_subset_1(F,u2_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u2_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u2_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,B,E)
& ~ r1_incsp_1(A,C,F)
& ~ r1_incsp_1(A,B,G)
& ~ r1_incsp_1(A,C,G)
& B != C
& r1_incsp_1(A,B,H)
& r1_incsp_1(A,C,H)
& r1_incsp_1(A,D,H)
& ~ r1_incsp_1(A,D,F)
& D != B
& ~ r1_projred2(A,E,F,G)
& ! [I] :
( m1_subset_1(I,u2_incsp_1(A))
=> ~ ( r1_projred2(A,F,G,I)
& ~ r1_incsp_1(A,B,I)
& ~ r1_incsp_1(A,D,I)
& k5_relat_1(k1_projred1(A,E,G,B),k1_projred1(A,G,F,C)) = k5_relat_1(k1_projred1(A,E,I,B),k1_projred1(A,I,F,D)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_m1_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_projred2(B,A)
=> ( v1_funct_1(B)
& m2_relset_1(B,u1_incsp_1(A),u1_incsp_1(A)) ) ) ) ).
fof(existence_m1_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& v10_incproj(A)
& l1_incsp_1(A) )
=> ? [B] : m1_projred2(B,A) ) ).
fof(dt_k1_projred2,axiom,
! [A,B] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A)
& m1_subset_1(B,u2_incsp_1(A)) )
=> m1_subset_1(k1_projred2(A,B),k1_zfmisc_1(u1_incsp_1(A))) ) ).
fof(d2_projred2,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(A))
=> k1_projred2(A,B) = a_2_0_projred2(A,B) ) ) ).
fof(fraenkel_a_2_0_projred2,axiom,
! [A,B,C] :
( ( v1_incproj(B)
& v2_incproj(B)
& v3_incproj(B)
& v4_incproj(B)
& v5_incproj(B)
& l1_incsp_1(B)
& m1_subset_1(C,u2_incsp_1(B)) )
=> ( r2_hidden(A,a_2_0_projred2(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_incsp_1(B))
& A = D
& r1_incsp_1(B,D,C) ) ) ) ).
%------------------------------------------------------------------------------