SET007 Axioms: SET007+275.ax
%------------------------------------------------------------------------------
% File : SET007+275 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Incidence Projective Space (a reduction theorem in a plane)
% 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 : projred1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 30 ( 3 unt; 0 def)
% Number of atoms : 502 ( 28 equ)
% Maximal formula atoms : 42 ( 16 avg)
% Number of connectives : 532 ( 60 ~; 10 |; 321 &)
% ( 3 <=>; 138 =>; 0 <=; 0 <~>)
% Maximal formula depth : 59 ( 18 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-5 aty)
% Number of functors : 8 ( 8 usr; 0 con; 1-4 aty)
% Number of variables : 177 ( 131 !; 46 ?)
% 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_projred1,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,u1_incsp_1(A))
=> r1_incsp_1(A,C,B) ) ) ) ).
fof(t2_projred1,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,u1_incsp_1(A))
=> ~ ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> r1_incsp_1(A,B,C) ) ) ) ).
fof(t3_projred1,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))
=> ~ ( B != C
& ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ~ ( r1_incsp_1(A,D,B)
& ~ r1_incsp_1(A,D,C)
& r1_incsp_1(A,E,C)
& ~ r1_incsp_1(A,E,B) ) ) ) ) ) ) ) ).
fof(t4_projred1,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,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ~ ( B != C
& ! [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)
& ~ r1_incsp_1(A,C,D) ) ) ) ) ) ) ) ).
fof(t5_projred1,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,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)
& C != D
& D != E
& E != C ) ) ) ) ) ).
fof(t6_projred1,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,u1_incsp_1(A))
& ~ r1_incsp_1(A,D,B)
& ~ r1_incsp_1(A,D,C) ) ) ) ) ).
fof(t7_projred1,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,u1_incsp_1(A))
& r1_incsp_1(A,C,B) ) ) ) ).
fof(t8_projred1,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,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,u1_incsp_1(A))
& r1_incsp_1(A,E,D)
& E != B
& E != C ) ) ) ) ) ).
fof(t9_projred1,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,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(A))
=> ? [D] :
( m1_subset_1(D,u2_incsp_1(A))
& ~ r1_incsp_1(A,B,D)
& ~ r1_incsp_1(A,C,D) ) ) ) ) ).
fof(t10_projred1,axiom,
$true ).
fof(t11_projred1,axiom,
$true ).
fof(t12_projred1,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,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,u2_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u2_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u2_incsp_1(A))
=> ! [I] :
( m1_subset_1(I,u2_incsp_1(A))
=> ~ ( r1_incsp_1(A,B,F)
& r1_incsp_1(A,B,G)
& F != G
& r1_incsp_1(A,C,F)
& B != C
& r1_incsp_1(A,D,G)
& r1_incsp_1(A,E,G)
& D != E
& r1_incsp_1(A,C,H)
& r1_incsp_1(A,D,H)
& r1_incsp_1(A,C,I)
& r1_incsp_1(A,E,I)
& H = I ) ) ) ) ) ) ) ) ) ) ).
fof(t13_projred1,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,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))
=> ( r4_incproj(A,B,C,E,D)
=> ( r4_incproj(A,B,D,E,C)
& r4_incproj(A,C,B,E,D)
& r4_incproj(A,C,D,E,B)
& r4_incproj(A,D,B,E,C)
& r4_incproj(A,D,C,E,B) ) ) ) ) ) ) ) ).
fof(t14_projred1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_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,u1_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))
=> ! [S] :
( m1_subset_1(S,u2_incsp_1(A))
=> ! [T] :
( m1_subset_1(T,u2_incsp_1(A))
=> ~ ( r4_incproj(A,B,C,L,D)
& r4_incproj(A,B,F,M,E)
& r4_incproj(A,B,H,N,G)
& r4_incproj(A,H,F,O,K)
& r4_incproj(A,H,I,P,D)
& r4_incproj(A,F,J,Q,D)
& r4_incproj(A,K,E,R,G)
& r4_incproj(A,C,I,S,G)
& r4_incproj(A,C,J,T,E)
& r1_incproj(L,M,N)
& B != H
& B != C
& B != E
& F != E
& ! [U] :
( m1_subset_1(U,u2_incsp_1(A))
=> ~ r4_incproj(A,I,J,U,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_projred1,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,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))
& r1_incsp_1(A,C,B)
& r1_incsp_1(A,D,B)
& r1_incsp_1(A,E,B)
& r1_incsp_1(A,F,B)
& r2_incproj(C,D,E,F) ) ) ) ) )
=> ! [B] :
( m1_subset_1(B,u2_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))
& r1_incsp_1(A,C,B)
& r1_incsp_1(A,D,B)
& r1_incsp_1(A,E,B)
& r1_incsp_1(A,F,B)
& r2_incproj(C,D,E,F) ) ) ) ) ) ) ) ).
fof(t16_projred1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v9_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,C,M)
& ~ r1_incsp_1(A,D,M)
& ~ r1_incsp_1(A,B,L)
& ~ r1_incsp_1(A,E,L)
& ~ r1_incsp_1(A,B,N)
& ~ r1_incsp_1(A,D,N)
& ~ r1_incsp_1(A,C,O)
& ~ r1_incsp_1(A,E,O)
& r4_incproj(A,F,B,M,E)
& r4_incproj(A,F,C,L,D)
& r4_incproj(A,G,C,N,E)
& r4_incproj(A,G,B,O,D)
& r4_incproj(A,H,B,I,C)
& r4_incproj(A,H,D,J,E)
& r3_incproj(A,F,G,K)
& ~ r1_incsp_1(A,H,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_projred1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v9_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))
& ? [F] :
( m1_subset_1(F,u2_incsp_1(A))
& r1_incsp_1(A,B,C)
& r1_incsp_1(A,B,D)
& r1_incsp_1(A,B,E)
& r1_incsp_1(A,B,F)
& r2_incproj(C,D,E,F) ) ) ) ) ) ) ).
fof(t18_projred1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v9_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,u2_incsp_1(A))
& r1_incsp_1(A,B,F)
& r1_incsp_1(A,C,F)
& r1_incsp_1(A,D,F)
& r1_incsp_1(A,E,F)
& r2_incproj(B,C,D,E) ) ) ) ) ) ) ).
fof(t19_projred1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v9_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_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))
& r1_incsp_1(A,C,B)
& r1_incsp_1(A,D,B)
& r1_incsp_1(A,E,B)
& r1_incsp_1(A,F,B)
& r2_incproj(C,D,E,F) ) ) ) ) ) ) ).
fof(d1_projred1,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,u1_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,D,B)
& ~ r1_incsp_1(A,D,C)
& ~ ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,u1_incsp_1(A),u1_incsp_1(A)) )
=> ( E = k1_projred1(A,B,C,D)
<=> ( r1_tarski(k4_relset_1(u1_incsp_1(A),u1_incsp_1(A),E),u1_incsp_1(A))
& ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ( r2_hidden(F,k4_relset_1(u1_incsp_1(A),u1_incsp_1(A),E))
<=> r1_incsp_1(A,F,B) ) )
& ! [F] :
( m1_subset_1(F,u1_incsp_1(A))
=> ! [G] :
( m1_subset_1(G,u1_incsp_1(A))
=> ( ( r1_incsp_1(A,F,B)
& r1_incsp_1(A,G,C) )
=> ( k1_funct_1(E,F) = G
<=> ? [H] :
( m1_subset_1(H,u2_incsp_1(A))
& r1_incsp_1(A,D,H)
& r1_incsp_1(A,F,H)
& r1_incsp_1(A,G,H) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_projred1,axiom,
$true ).
fof(t21_projred1,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)
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ( r1_incsp_1(A,D,C)
=> k1_funct_1(k1_projred1(A,C,C,B),D) = D ) ) ) ) ) ) ).
fof(t22_projred1,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,C,D)
=> ( r1_incsp_1(A,B,D)
| r1_incsp_1(A,B,E)
| m1_subset_1(k1_funct_1(k1_projred1(A,D,E,B),C),u1_incsp_1(A)) ) ) ) ) ) ) ) ).
fof(t23_projred1,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))
=> ( ( r1_incsp_1(A,C,E)
& D = k1_funct_1(k1_projred1(A,E,F,B),C) )
=> ( r1_incsp_1(A,B,E)
| r1_incsp_1(A,B,F)
| r1_incsp_1(A,D,F) ) ) ) ) ) ) ) ) ).
fof(t24_projred1,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))
=> ( r2_hidden(C,k2_relat_1(k1_projred1(A,D,E,B)))
=> ( r1_incsp_1(A,B,D)
| r1_incsp_1(A,B,E)
| r1_incsp_1(A,C,E) ) ) ) ) ) ) ) ).
fof(t25_projred1,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)
& ~ ( k1_relat_1(k5_relat_1(k1_projred1(A,D,E,B),k1_projred1(A,E,F,C))) = k4_relset_1(u1_incsp_1(A),u1_incsp_1(A),k1_projred1(A,D,E,B))
& k2_relat_1(k5_relat_1(k1_projred1(A,D,E,B),k1_projred1(A,E,F,C))) = k2_relat_1(k1_projred1(A,E,F,C)) ) ) ) ) ) ) ) ) ).
fof(t26_projred1,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,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))
=> ( ( r1_incsp_1(A,E,C)
& r1_incsp_1(A,F,C)
& k1_funct_1(k1_projred1(A,C,D,B),E) = G
& k1_funct_1(k1_projred1(A,C,D,B),F) = H
& G = H )
=> ( r1_incsp_1(A,B,C)
| r1_incsp_1(A,B,D)
| E = F ) ) ) ) ) ) ) ) ) ) ).
fof(t27_projred1,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,C,D)
& r1_incsp_1(A,C,E) )
=> ( r1_incsp_1(A,B,D)
| r1_incsp_1(A,B,E)
| k1_funct_1(k1_projred1(A,D,E,B),C) = C ) ) ) ) ) ) ) ).
fof(t28_projred1,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))
=> ~ ( ~ r1_incsp_1(A,B,E)
& ~ r1_incsp_1(A,B,F)
& ~ r1_incsp_1(A,C,F)
& ~ r1_incsp_1(A,C,G)
& r1_incsp_1(A,D,E)
& r1_incsp_1(A,D,F)
& r1_incsp_1(A,D,G)
& E != G
& ! [H] :
( m1_subset_1(H,u1_incsp_1(A))
=> ~ ( ~ r1_incsp_1(A,H,E)
& ~ r1_incsp_1(A,H,G)
& k5_relat_1(k1_projred1(A,E,F,B),k1_projred1(A,F,G,C)) = k1_projred1(A,E,G,H) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_projred1,axiom,
! [A,B,C,D] :
( ( 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)
& m1_subset_1(B,u2_incsp_1(A))
& m1_subset_1(C,u2_incsp_1(A))
& m1_subset_1(D,u1_incsp_1(A)) )
=> ( v1_funct_1(k1_projred1(A,B,C,D))
& m2_relset_1(k1_projred1(A,B,C,D),u1_incsp_1(A),u1_incsp_1(A)) ) ) ).
%------------------------------------------------------------------------------