SET007 Axioms: SET007+408.ax
%------------------------------------------------------------------------------
% File : SET007+408 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Projective Planes
% 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 : projpl_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 50 ( 0 unt; 0 def)
% Number of atoms : 691 ( 55 equ)
% Maximal formula atoms : 39 ( 13 avg)
% Number of connectives : 728 ( 87 ~; 15 |; 336 &)
% ( 15 <=>; 275 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 16 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 21 ( 20 usr; 0 prp; 1-5 aty)
% Number of functors : 9 ( 9 usr; 0 con; 1-5 aty)
% Number of variables : 278 ( 255 !; 23 ?)
% 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_projpl_1,axiom,
! [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_projpl_1(A,B,C,D)
<=> ( ~ r1_incsp_1(A,B,D)
& ~ r1_incsp_1(A,C,D) ) ) ) ) ) ) ).
fof(d2_projpl_1,axiom,
! [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))
=> ( r2_projpl_1(A,B,C,D)
<=> ( r1_incsp_1(A,B,C)
& r1_incsp_1(A,B,D) ) ) ) ) ) ) ).
fof(d3_projpl_1,axiom,
! [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))
=> ( r3_projpl_1(A,B,C,D,E)
<=> ( r1_incsp_1(A,B,C)
& r1_incsp_1(A,B,D)
& r1_incsp_1(A,B,E) ) ) ) ) ) ) ) ).
fof(t1_projpl_1,axiom,
! [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))
=> ( ( r3_incproj(A,B,C,E)
=> r3_incproj(A,C,B,E) )
& ( 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) ) )
& ( r2_projpl_1(A,B,E,F)
=> r2_projpl_1(A,B,F,E) )
& ( r3_projpl_1(A,B,E,F,G)
=> ( r3_projpl_1(A,B,E,G,F)
& r3_projpl_1(A,B,F,E,G)
& r3_projpl_1(A,B,F,G,E)
& r3_projpl_1(A,B,G,E,F)
& r3_projpl_1(A,B,G,F,E) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_projpl_1,axiom,
! [A] :
( l1_incsp_1(A)
=> ( v1_projpl_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,C,D)
& r1_incsp_1(A,B,E)
& r1_incsp_1(A,C,E)
& B != C
& D != E ) ) ) ) ) ) ) ).
fof(t2_projpl_1,axiom,
! [A] :
( l1_incsp_1(A)
=> ( v1_projpl_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))
=> ~ ( r3_incproj(A,B,C,D)
& r3_incproj(A,B,C,E)
& B != C
& D != E ) ) ) ) ) ) ) ).
fof(t3_projpl_1,axiom,
! [A] :
( l1_incsp_1(A)
=> ( v1_projpl_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_projpl_1(A,B,D,E)
& r2_projpl_1(A,C,D,E)
& B != C
& D != E ) ) ) ) ) ) ) ).
fof(t4_projpl_1,axiom,
! [A] :
( l1_incsp_1(A)
=> ( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A) )
<=> ( v1_projpl_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))
& r3_incproj(A,B,C,D) ) ) )
& ~ ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> r1_incsp_1(A,B,C) ) )
& ! [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))
& r1_incproj(C,D,E)
& r4_incproj(A,C,D,B,E) ) ) ) )
& ! [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,u2_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))
=> ~ ( r4_incproj(A,B,C,G,F)
& r4_incproj(A,D,E,H,F)
& r3_incproj(A,B,D,I)
& r3_incproj(A,C,E,J)
& ~ r1_incsp_1(A,F,I)
& ~ r1_incsp_1(A,F,J)
& G != H
& ! [K] :
( m1_subset_1(K,u1_incsp_1(A))
=> ~ r2_projpl_1(A,K,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_projpl_1,axiom,
! [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))
=> ( r4_projpl_1(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,u2_incsp_1(A))
& r4_incproj(A,B,C,E,D) ) ) ) ) ) ) ).
fof(t5_projpl_1,axiom,
! [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))
=> ( r4_projpl_1(A,B,C,D)
<=> ? [E] :
( m1_subset_1(E,u2_incsp_1(A))
& r1_incsp_1(A,B,E)
& r1_incsp_1(A,C,E)
& r1_incsp_1(A,D,E) ) ) ) ) ) ) ).
fof(t6_projpl_1,axiom,
! [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))
=> ( ~ r4_projpl_1(A,B,C,D)
<=> ! [E] :
( m1_subset_1(E,u2_incsp_1(A))
=> ~ ( r1_incsp_1(A,B,E)
& r1_incsp_1(A,C,E)
& r1_incsp_1(A,D,E) ) ) ) ) ) ) ) ).
fof(d6_projpl_1,axiom,
! [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))
=> ( r5_projpl_1(A,B,C,D,E)
<=> ( ~ r4_projpl_1(A,B,C,D)
& ~ r4_projpl_1(A,C,D,E)
& ~ r4_projpl_1(A,D,E,B)
& ~ r4_projpl_1(A,E,B,C) ) ) ) ) ) ) ) ).
fof(t7_projpl_1,axiom,
! [A] :
( l1_incsp_1(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 ) ) ) ) ).
fof(t8_projpl_1,axiom,
! [A] :
( l1_incsp_1(A)
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(A))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(A))
=> ~ ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A)
& r1_incsp_1(A,B,C)
& ! [D] :
( m1_subset_1(D,u1_incsp_1(A))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ~ ( r3_incproj(A,D,E,C)
& r1_incproj(B,D,E) ) ) ) ) ) ) ) ).
fof(t9_projpl_1,axiom,
! [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))
=> ~ ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A)
& r1_incsp_1(A,B,C)
& C != D
& ! [E] :
( m1_subset_1(E,u1_incsp_1(A))
=> ~ ( r1_incsp_1(A,E,C)
& ~ r1_incsp_1(A,E,D)
& B != E ) ) ) ) ) ) ) ).
fof(t10_projpl_1,axiom,
! [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))
=> ~ ( v1_projpl_1(A)
& r3_incproj(A,B,C,E)
& B != C
& ~ r1_incsp_1(A,D,E)
& r4_projpl_1(A,B,C,D) ) ) ) ) ) ) ).
fof(t11_projpl_1,axiom,
! [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))
=> ( r4_projpl_1(A,B,C,D)
=> ( r4_projpl_1(A,B,D,C)
& r4_projpl_1(A,C,B,D)
& r4_projpl_1(A,C,D,B)
& r4_projpl_1(A,D,B,C)
& r4_projpl_1(A,D,C,B) ) ) ) ) ) ) ).
fof(t12_projpl_1,axiom,
! [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))
=> ( ~ r4_projpl_1(A,B,C,D)
=> ( ~ r4_projpl_1(A,B,D,C)
& ~ r4_projpl_1(A,C,B,D)
& ~ r4_projpl_1(A,C,D,B)
& ~ r4_projpl_1(A,D,B,C)
& ~ r4_projpl_1(A,D,C,B) ) ) ) ) ) ) ).
fof(t13_projpl_1,axiom,
! [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))
=> ( r5_projpl_1(A,B,C,D,E)
=> ( r5_projpl_1(A,B,D,C,E)
& r5_projpl_1(A,C,B,D,E)
& r5_projpl_1(A,C,D,B,E)
& r5_projpl_1(A,D,B,C,E)
& r5_projpl_1(A,D,C,B,E)
& r5_projpl_1(A,B,C,E,D)
& r5_projpl_1(A,B,D,E,C)
& r5_projpl_1(A,C,B,E,D)
& r5_projpl_1(A,C,D,E,B)
& r5_projpl_1(A,D,B,E,C)
& r5_projpl_1(A,D,C,E,B)
& r5_projpl_1(A,B,E,C,D)
& r5_projpl_1(A,B,E,D,C)
& r5_projpl_1(A,C,E,B,D)
& r5_projpl_1(A,C,E,D,B)
& r5_projpl_1(A,D,E,B,C)
& r5_projpl_1(A,D,E,C,B)
& r5_projpl_1(A,E,B,C,D)
& r5_projpl_1(A,E,B,D,C)
& r5_projpl_1(A,E,C,B,D)
& r5_projpl_1(A,E,C,D,B)
& r5_projpl_1(A,E,D,B,C)
& r5_projpl_1(A,E,D,C,B) ) ) ) ) ) ) ) ).
fof(t14_projpl_1,axiom,
! [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))
=> ( ( v1_projpl_1(A)
& r3_incproj(A,B,C,F)
& r3_incproj(A,D,E,G)
& r1_projpl_1(A,B,C,G)
& r1_projpl_1(A,D,E,F) )
=> ( B = C
| D = E
| r5_projpl_1(A,B,C,D,E) ) ) ) ) ) ) ) ) ) ).
fof(t15_projpl_1,axiom,
! [A] :
( l1_incsp_1(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))
=> ~ r5_projpl_1(A,B,C,D,E) ) ) ) ) ) ) ).
fof(d7_projpl_1,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,k4_zfmisc_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A)))
=> ( m1_projpl_1(B,A)
<=> r5_projpl_1(A,k8_mcart_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),B),k9_mcart_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),B),k10_mcart_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),B),k11_mcart_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),B)) ) ) ) ).
fof(d8_projpl_1,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))
=> ( D = k1_projpl_1(A,B,C)
<=> r3_incproj(A,B,C,D) ) ) ) ) ) ) ).
fof(t16_projpl_1,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))
=> ( B != C
=> ( r1_incsp_1(A,B,k1_projpl_1(A,B,C))
& r1_incsp_1(A,C,k1_projpl_1(A,B,C))
& k1_projpl_1(A,B,C) = k1_projpl_1(A,C,B)
& ( ( r1_incsp_1(A,B,D)
& r1_incsp_1(A,C,D) )
=> D = k1_projpl_1(A,B,C) ) ) ) ) ) ) ) ).
fof(t17_projpl_1,axiom,
! [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))
=> r4_projpl_1(A,B,C,D) ) ) )
& ! [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))
& r3_incproj(A,B,C,D) ) ) )
& ! [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(t18_projpl_1,axiom,
! [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))
& r5_projpl_1(A,B,C,D,E) ) ) ) )
& ! [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))
=> r4_projpl_1(A,B,C,D) ) ) ) ) ) ).
fof(t19_projpl_1,axiom,
! [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))
=> ~ ( ~ r4_projpl_1(A,B,C,D)
& r3_incproj(A,B,C,E)
& r3_incproj(A,B,D,F)
& E = F ) ) ) ) ) ) ) ).
fof(t20_projpl_1,axiom,
! [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))
=> ( ( r5_projpl_1(A,B,C,D,E)
& r3_incproj(A,B,C,F)
& r3_incproj(A,B,D,G)
& r3_incproj(A,B,E,H) )
=> r1_incproj(F,G,H) ) ) ) ) ) ) ) ) ) ).
fof(t21_projpl_1,axiom,
! [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))
=> ( ( v1_projpl_1(A)
& r3_projpl_1(A,B,F,G,H)
& r1_incproj(F,G,H)
& r2_projpl_1(A,C,I,F)
& r2_projpl_1(A,D,I,G)
& r2_projpl_1(A,E,I,H) )
=> ( r1_incsp_1(A,B,I)
| r1_incproj(C,D,E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_projpl_1,axiom,
! [A] :
( l1_incsp_1(A)
=> ( ( v1_projpl_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))
& r3_incproj(A,B,C,D) ) ) )
& ! [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))
& r2_projpl_1(A,D,B,C) ) ) ) )
=> ( ! [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))
=> ~ r5_projpl_1(A,B,C,D,E) ) ) ) )
| ! [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))
& r1_incproj(C,D,E)
& r4_incproj(A,C,D,B,E) ) ) ) ) ) ) ) ).
fof(t23_projpl_1,axiom,
! [A] :
( l1_incsp_1(A)
=> ( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A)
& l1_incsp_1(A) )
<=> ( v1_projpl_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))
& r3_incproj(A,B,C,D) ) ) )
& ! [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))
& r2_projpl_1(A,D,B,C) ) ) )
& ? [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))
& r5_projpl_1(A,B,C,D,E) ) ) ) ) ) ) ) ).
fof(d9_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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))
=> ( D = k2_projpl_1(A,B,C)
<=> r2_projpl_1(A,D,B,C) ) ) ) ) ) ) ).
fof(t24_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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))
=> ( C != D
=> ( r1_incsp_1(A,k2_projpl_1(A,C,D),C)
& r1_incsp_1(A,k2_projpl_1(A,C,D),D)
& k2_projpl_1(A,C,D) = k2_projpl_1(A,D,C)
& ( ( r1_incsp_1(A,B,C)
& r1_incsp_1(A,B,D) )
=> B = k2_projpl_1(A,C,D) ) ) ) ) ) ) ) ).
fof(t25_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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)
=> ( D = E
| r1_incsp_1(A,C,D)
| B = k2_projpl_1(A,D,E)
| ( r1_incsp_1(A,k2_projpl_1(A,k1_projpl_1(A,C,B),E),E)
& ~ r1_incsp_1(A,k2_projpl_1(A,k1_projpl_1(A,C,B),E),D) ) ) ) ) ) ) ) ) ).
fof(t26_projpl_1,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))
=> ( ~ r4_projpl_1(A,B,C,D)
=> r1_incproj(B,C,D) ) ) ) ) ) ).
fof(t27_projpl_1,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))
=> ( r5_projpl_1(A,B,C,D,E)
=> r2_incproj(B,C,D,E) ) ) ) ) ) ) ).
fof(t28_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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))
=> ~ ( k1_projpl_1(A,B,D) = k1_projpl_1(A,C,E)
& B != D
& C != E
& D != E
& k1_projpl_1(A,B,D) != k1_projpl_1(A,D,E) ) ) ) ) ) ) ).
fof(t29_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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))
=> ~ ( k1_projpl_1(A,B,D) = k1_projpl_1(A,C,E)
& B != D
& C != E
& D != E
& ~ r1_incsp_1(A,B,k1_projpl_1(A,D,E)) ) ) ) ) ) ) ).
fof(t30_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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))
=> ( ( r1_incsp_1(A,C,E)
& r1_incsp_1(A,D,E) )
=> ( C = D
| r1_incsp_1(A,B,E)
| ( k1_projpl_1(A,C,B) != k1_projpl_1(A,D,B)
& k1_projpl_1(A,B,C) != k1_projpl_1(A,B,D) ) ) ) ) ) ) ) ) ).
fof(t31_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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,D)
& r1_incsp_1(A,B,E) )
=> ( D = E
| r1_incsp_1(A,B,C)
| ( k2_projpl_1(A,C,D) != k2_projpl_1(A,C,E)
& k2_projpl_1(A,D,C) != k2_projpl_1(A,E,C) ) ) ) ) ) ) ) ) ).
fof(t32_projpl_1,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))
=> ( ( r1_incsp_1(A,D,k1_projpl_1(A,B,C))
& r1_incsp_1(A,D,k1_projpl_1(A,B,E)) )
=> ( D = B
| E = B
| B = C
| r1_incsp_1(A,E,k1_projpl_1(A,B,C)) ) ) ) ) ) ) ) ).
fof(t33_projpl_1,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))
=> ( r1_incsp_1(A,D,k1_projpl_1(A,B,C))
=> ( B = D
| r1_incsp_1(A,C,k1_projpl_1(A,B,D)) ) ) ) ) ) ) ).
fof(t34_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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))
=> ~ ( D != E
& r1_incsp_1(A,D,F)
& r1_incsp_1(A,E,F)
& ~ r1_incsp_1(A,B,F)
& ~ r1_incsp_1(A,C,F)
& k1_projpl_1(A,B,D) = k1_projpl_1(A,C,E) ) ) ) ) ) ) ) ).
fof(t35_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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))
=> ~ ( r1_incsp_1(A,B,k1_projpl_1(A,C,D))
& B != D
& C != D
& ~ r1_incsp_1(A,C,k1_projpl_1(A,D,B)) ) ) ) ) ) ).
fof(t36_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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))
=> ~ ( r1_incsp_1(A,B,k1_projpl_1(A,C,D))
& C != B
& C != D
& ~ r1_incsp_1(A,D,k1_projpl_1(A,C,B)) ) ) ) ) ) ).
fof(t37_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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,u2_incsp_1(A))
=> ! [H] :
( m1_subset_1(H,u2_incsp_1(A))
=> ~ ( r1_incsp_1(A,C,H)
& r1_incsp_1(A,D,H)
& r1_incsp_1(A,B,G)
& ~ r1_incsp_1(A,B,H)
& C != D
& E != B
& F != B
& r1_incsp_1(A,E,k1_projpl_1(A,B,C))
& r1_incsp_1(A,F,k1_projpl_1(A,B,D))
& ! [I] :
( m1_subset_1(I,u1_incsp_1(A))
=> ~ ( r1_incsp_1(A,I,k1_projpl_1(A,E,F))
& r1_incsp_1(A,I,G)
& I != B ) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_m1_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A) )
=> ! [B] :
( m1_projpl_1(B,A)
=> m1_subset_1(B,k4_zfmisc_1(u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A),u1_incsp_1(A))) ) ) ).
fof(existence_m1_projpl_1,axiom,
! [A] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A) )
=> ? [B] : m1_projpl_1(B,A) ) ).
fof(dt_k1_projpl_1,axiom,
! [A,B,C] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& l1_incsp_1(A)
& m1_subset_1(B,u1_incsp_1(A))
& m1_subset_1(C,u1_incsp_1(A)) )
=> m1_subset_1(k1_projpl_1(A,B,C),u2_incsp_1(A)) ) ).
fof(dt_k2_projpl_1,axiom,
! [A,B,C] :
( ( v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_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(k2_projpl_1(A,B,C),u1_incsp_1(A)) ) ).
%------------------------------------------------------------------------------