SET007 Axioms: SET007+290.ax
%------------------------------------------------------------------------------
% File : SET007+290 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Projective Closure and Projective Horizon of an Affine 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 : afproj [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 87 ( 0 unt; 0 def)
% Number of atoms : 879 ( 94 equ)
% Maximal formula atoms : 36 ( 10 avg)
% Number of connectives : 1018 ( 226 ~; 19 |; 502 &)
% ( 28 <=>; 243 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 36 ( 35 usr; 0 prp; 1-5 aty)
% Number of functors : 32 ( 32 usr; 2 con; 0-3 aty)
% Number of variables : 256 ( 233 !; 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(fc1_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v1_incsp_1(k13_afproj(A))
& v1_incproj(k13_afproj(A))
& v2_incproj(k13_afproj(A))
& v3_incproj(k13_afproj(A))
& v4_incproj(k13_afproj(A))
& v5_incproj(k13_afproj(A)) ) ) ).
fof(rc1_afproj,axiom,
? [A] :
( l1_incsp_1(A)
& v1_incsp_1(A)
& v1_incproj(A)
& v2_incproj(A)
& v3_incproj(A)
& v4_incproj(A)
& v5_incproj(A)
& v6_incproj(A) ) ).
fof(fc2_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v1_incsp_1(k13_afproj(A))
& v1_incproj(k13_afproj(A))
& v2_incproj(k13_afproj(A))
& v3_incproj(k13_afproj(A))
& v4_incproj(k13_afproj(A))
& v5_incproj(k13_afproj(A))
& v6_incproj(k13_afproj(A)) ) ) ).
fof(t1_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A)
& B = u1_struct_0(A) )
=> v1_aff_4(B,A) ) ) ) ).
fof(t2_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A)
& v1_aff_4(B,A) )
=> B = u1_struct_0(A) ) ) ) ).
fof(t3_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A)
& v1_aff_4(B,A)
& v1_aff_4(C,A) )
=> B = C ) ) ) ) ).
fof(t4_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( B = u1_struct_0(A)
& v1_aff_4(B,A) )
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) ) ) ) ) ).
fof(t5_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(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)))
=> ( ( r1_aff_4(A,B,D)
& r1_aff_4(A,B,E)
& r1_aff_4(A,C,D)
& r1_aff_4(A,C,E)
& v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_4(D,A)
& v1_aff_4(E,A) )
=> ( r4_aff_1(A,B,C)
| r1_aff_4(A,D,E) ) ) ) ) ) ) ) ).
fof(t6_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(B,A)
& r1_aff_4(A,C,B)
& r1_aff_4(A,B,D) )
=> r1_aff_4(A,C,D) ) ) ) ) ) ).
fof(t7_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( r2_hidden(B,k1_afproj(A))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = C
& v1_aff_1(C,A) ) ) ) ).
fof(t8_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( r2_hidden(B,k2_afproj(A))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = C
& v1_aff_4(C,A) ) ) ) ).
fof(d5_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_1(B,A)
=> k5_afproj(A,B) = k6_eqrel_1(k1_afproj(A),k3_afproj(A),B) ) ) ) ).
fof(d6_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_4(B,A)
=> k6_afproj(A,B) = k6_eqrel_1(k2_afproj(A),k4_afproj(A),B) ) ) ) ).
fof(t9_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_1(B,A)
=> ! [C] :
( r2_hidden(C,k5_afproj(A,B))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& C = D
& v1_aff_1(D,A)
& r1_aff_4(A,B,D) ) ) ) ) ) ).
fof(t10_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_aff_4(B,A)
=> ! [C] :
( r2_hidden(C,k6_afproj(A,B))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& C = D
& v1_aff_4(D,A)
& r1_aff_4(A,B,D) ) ) ) ) ) ).
fof(t11_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A) )
=> ( k5_afproj(A,B) = k5_afproj(A,C)
<=> r4_aff_1(A,B,C) ) ) ) ) ) ).
fof(t12_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A) )
=> ( k5_afproj(A,B) = k5_afproj(A,C)
<=> r1_aff_4(A,B,C) ) ) ) ) ) ).
fof(t13_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_4(B,A)
& v1_aff_4(C,A) )
=> ( k6_afproj(A,B) = k6_afproj(A,C)
<=> r1_aff_4(A,B,C) ) ) ) ) ) ).
fof(d7_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k7_afproj(A) = k8_eqrel_1(k1_afproj(A),k3_afproj(A)) ) ).
fof(d8_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k8_afproj(A) = k8_eqrel_1(k2_afproj(A),k4_afproj(A)) ) ).
fof(t14_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( r2_hidden(B,k7_afproj(A))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = k5_afproj(A,C)
& v1_aff_1(C,A) ) ) ) ).
fof(t15_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( r2_hidden(B,k8_afproj(A))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = k6_afproj(A,C)
& v1_aff_4(C,A) ) ) ) ).
fof(t16_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> r1_subset_1(u1_struct_0(A),k7_afproj(A)) ) ).
fof(t17_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> r1_xboole_0(k1_afproj(A),k8_afproj(A)) ) ) ).
fof(t18_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( r2_hidden(B,k2_zfmisc_1(k1_afproj(A),k1_tarski(np__1)))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = k4_tarski(C,np__1)
& v1_aff_1(C,A) ) ) ) ).
fof(t19_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( r2_hidden(B,k2_zfmisc_1(k8_afproj(A),k1_tarski(np__2)))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = k4_tarski(k6_afproj(A,C),np__2)
& v1_aff_4(C,A) ) ) ) ).
fof(d9_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k9_afproj(A) = k2_xboole_0(u1_struct_0(A),k7_afproj(A)) ) ).
fof(d10_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k10_afproj(A) = k2_xboole_0(k2_zfmisc_1(k1_afproj(A),k1_tarski(np__1)),k2_zfmisc_1(k8_afproj(A),k1_tarski(np__2))) ) ).
fof(d11_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m2_relset_1(B,k9_afproj(A),k10_afproj(A))
=> ( B = k11_afproj(A)
<=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
<=> ~ ( ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_1(E,A)
& D = k4_tarski(E,np__1)
& ( ( r2_hidden(C,u1_struct_0(A))
& r2_hidden(C,E) )
| C = k5_afproj(A,E) ) ) )
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_aff_1(E,A)
& v1_aff_4(F,A)
& C = k5_afproj(A,E)
& D = k4_tarski(k6_afproj(A,F),np__2)
& r1_aff_4(A,E,F) ) ) ) ) ) ) ) ) ).
fof(d12_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m2_relset_1(B,k7_afproj(A),k8_afproj(A))
=> ( B = k12_afproj(A)
<=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
& C = k5_afproj(A,E)
& D = k6_afproj(A,F)
& v1_aff_1(E,A)
& v1_aff_4(F,A)
& r1_aff_4(A,E,F) ) ) ) ) ) ) ).
fof(d13_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k13_afproj(A) = g1_incsp_1(k9_afproj(A),k10_afproj(A),k11_afproj(A)) ) ).
fof(d14_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k14_afproj(A) = g1_incsp_1(k7_afproj(A),k8_afproj(A),k12_afproj(A)) ) ).
fof(t20_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k13_afproj(A)))
<=> ~ ( ~ m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( B = k5_afproj(A,C)
& v1_aff_1(C,A) ) ) ) ) ) ).
fof(t21_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k14_afproj(A)))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = k5_afproj(A,C)
& v1_aff_1(C,A) ) ) ) ).
fof(t22_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k14_afproj(A)))
=> m1_subset_1(B,u1_incsp_1(k13_afproj(A))) ) ) ).
fof(t23_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(k13_afproj(A)))
<=> ~ ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ ( B = k4_tarski(C,np__1)
& v1_aff_1(C,A) )
& ~ ( B = k4_tarski(k6_afproj(A,C),np__2)
& v1_aff_4(C,A) ) ) ) ) ) ).
fof(t24_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(k14_afproj(A)))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = k6_afproj(A,C)
& v1_aff_4(C,A) ) ) ) ).
fof(t25_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(k14_afproj(A)))
=> m1_subset_1(k4_tarski(B,np__2),u2_incsp_1(k13_afproj(A))) ) ) ).
fof(t26_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(k13_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k13_afproj(A)))
=> ( ( B = D
& k4_tarski(C,np__1) = E )
=> ( r1_incsp_1(k13_afproj(A),D,E)
<=> ( v1_aff_1(C,A)
& r2_hidden(B,C) ) ) ) ) ) ) ) ) ).
fof(t27_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(k13_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k13_afproj(A)))
=> ~ ( B = D
& k4_tarski(k6_afproj(A,C),np__2) = E
& v1_aff_4(C,A)
& r1_incsp_1(k13_afproj(A),D,E) ) ) ) ) ) ) ).
fof(t28_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(k13_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k13_afproj(A)))
=> ( ( D = k5_afproj(A,B)
& k4_tarski(C,np__1) = E
& v1_aff_1(B,A)
& v1_aff_1(C,A) )
=> ( r1_incsp_1(k13_afproj(A),D,E)
<=> r1_aff_4(A,B,C) ) ) ) ) ) ) ) ).
fof(t29_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(k13_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k13_afproj(A)))
=> ( ( D = k5_afproj(A,B)
& E = k4_tarski(k6_afproj(A,C),np__2)
& v1_aff_1(B,A)
& v1_aff_4(C,A) )
=> ( r1_incsp_1(k13_afproj(A),D,E)
<=> r1_aff_4(A,B,C) ) ) ) ) ) ) ) ).
fof(t30_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k13_afproj(A)))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(k13_afproj(A)))
=> ( ( v1_aff_1(B,A)
& C = k5_afproj(A,B)
& D = k4_tarski(B,np__1) )
=> r1_incsp_1(k13_afproj(A),C,D) ) ) ) ) ) ).
fof(t31_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(k13_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k13_afproj(A)))
=> ( ( v1_aff_1(B,A)
& v1_aff_4(C,A)
& r1_tarski(B,C)
& D = k5_afproj(A,B)
& E = k4_tarski(k6_afproj(A,C),np__2) )
=> r1_incsp_1(k13_afproj(A),D,E) ) ) ) ) ) ) ).
fof(t32_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(k13_afproj(A)))
=> ! [F] :
( m1_subset_1(F,u2_incsp_1(k13_afproj(A)))
=> ( ( v1_aff_4(B,A)
& r1_tarski(C,B)
& r4_aff_1(A,D,C)
& E = k5_afproj(A,D)
& F = k4_tarski(k6_afproj(A,B),np__2) )
=> r1_incsp_1(k13_afproj(A),E,F) ) ) ) ) ) ) ) ).
fof(t33_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k13_afproj(A)))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(k13_afproj(A)))
=> ~ ( D = k4_tarski(k6_afproj(A,B),np__2)
& v1_aff_4(B,A)
& r1_incsp_1(k13_afproj(A),C,D)
& m1_subset_1(C,u1_struct_0(A)) ) ) ) ) ) ).
fof(t34_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k13_afproj(A)))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(k13_afproj(A)))
=> ( ( D = k4_tarski(B,np__1)
& v1_aff_1(B,A)
& r1_incsp_1(k13_afproj(A),C,D) )
=> ( m1_subset_1(C,u1_struct_0(A))
| C = k5_afproj(A,B) ) ) ) ) ) ) ).
fof(t35_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k13_afproj(A)))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(k13_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k13_afproj(A)))
=> ( ( E = k4_tarski(B,np__1)
& v1_aff_1(B,A)
& r1_incsp_1(k13_afproj(A),C,E)
& r1_incsp_1(k13_afproj(A),D,E) )
=> ( D = C
| m1_subset_1(C,u1_struct_0(A))
| m1_subset_1(D,u1_struct_0(A)) ) ) ) ) ) ) ) ).
fof(t36_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(k14_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k14_afproj(A)))
=> ( ( D = k5_afproj(A,B)
& E = k6_afproj(A,C)
& v1_aff_1(B,A)
& v1_aff_4(C,A) )
=> ( r1_incsp_1(k14_afproj(A),D,E)
<=> r1_aff_4(A,B,C) ) ) ) ) ) ) ) ).
fof(t37_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k14_afproj(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k13_afproj(A)))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(k14_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k13_afproj(A)))
=> ( ( C = B
& E = k4_tarski(D,np__2) )
=> ( r1_incsp_1(k14_afproj(A),B,D)
<=> r1_incsp_1(k13_afproj(A),C,E) ) ) ) ) ) ) ) ).
fof(t38_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k14_afproj(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k14_afproj(A)))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(k14_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k14_afproj(A)))
=> ~ ( r1_incsp_1(k14_afproj(A),B,D)
& r1_incsp_1(k14_afproj(A),B,E)
& r1_incsp_1(k14_afproj(A),C,D)
& r1_incsp_1(k14_afproj(A),C,E)
& B != C
& D != E ) ) ) ) ) ) ).
fof(t39_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u2_incsp_1(k14_afproj(A)))
=> ? [C] :
( m1_subset_1(C,u1_incsp_1(k14_afproj(A)))
& ? [D] :
( m1_subset_1(D,u1_incsp_1(k14_afproj(A)))
& ? [E] :
( m1_subset_1(E,u1_incsp_1(k14_afproj(A)))
& r1_incsp_1(k14_afproj(A),C,B)
& r1_incsp_1(k14_afproj(A),D,B)
& r1_incsp_1(k14_afproj(A),E,B)
& C != D
& D != E
& E != C ) ) ) ) ) ).
fof(t40_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k14_afproj(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k14_afproj(A)))
=> ? [D] :
( m1_subset_1(D,u2_incsp_1(k14_afproj(A)))
& r1_incsp_1(k14_afproj(A),B,D)
& r1_incsp_1(k14_afproj(A),C,D) ) ) ) ) ).
fof(t41_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k14_afproj(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k14_afproj(A)))
=> ! [D] :
( m1_subset_1(D,u2_incsp_1(k13_afproj(A)))
=> ~ ( B != C
& r2_hidden(k4_tarski(B,D),u3_incsp_1(k13_afproj(A)))
& r2_hidden(k4_tarski(C,D),u3_incsp_1(k13_afproj(A)))
& ! [E] :
( m1_subset_1(E,u2_incsp_1(k14_afproj(A)))
=> D != k4_tarski(E,np__2) ) ) ) ) ) ) ).
fof(t42_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k13_afproj(A)))
=> ! [C] :
( m1_subset_1(C,u2_incsp_1(k14_afproj(A)))
=> ( r2_hidden(k4_tarski(B,k4_tarski(C,np__2)),u3_incsp_1(k13_afproj(A)))
=> m1_subset_1(B,u1_incsp_1(k14_afproj(A))) ) ) ) ) ).
fof(t43_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,u2_incsp_1(k13_afproj(A)))
=> ! [F] :
( m1_subset_1(F,u2_incsp_1(k13_afproj(A)))
=> ~ ( v1_aff_4(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r1_tarski(C,B)
& r1_tarski(D,B)
& E = k4_tarski(C,np__1)
& F = k4_tarski(D,np__1)
& ! [G] :
( m1_subset_1(G,u1_incsp_1(k13_afproj(A)))
=> ~ ( r1_incsp_1(k13_afproj(A),G,E)
& r1_incsp_1(k13_afproj(A),G,F) ) ) ) ) ) ) ) ) ) ).
fof(t44_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_incsp_1(k14_afproj(A)))
=> ! [C] :
( m1_subset_1(C,u1_incsp_1(k14_afproj(A)))
=> ! [D] :
( m1_subset_1(D,u1_incsp_1(k14_afproj(A)))
=> ! [E] :
( m1_subset_1(E,u1_incsp_1(k14_afproj(A)))
=> ! [F] :
( m1_subset_1(F,u1_incsp_1(k14_afproj(A)))
=> ! [G] :
( m1_subset_1(G,u2_incsp_1(k14_afproj(A)))
=> ! [H] :
( m1_subset_1(H,u2_incsp_1(k14_afproj(A)))
=> ! [I] :
( m1_subset_1(I,u2_incsp_1(k14_afproj(A)))
=> ! [J] :
( m1_subset_1(J,u2_incsp_1(k14_afproj(A)))
=> ~ ( r1_incsp_1(k14_afproj(A),B,G)
& r1_incsp_1(k14_afproj(A),C,G)
& r1_incsp_1(k14_afproj(A),D,H)
& r1_incsp_1(k14_afproj(A),E,H)
& r1_incsp_1(k14_afproj(A),F,G)
& r1_incsp_1(k14_afproj(A),F,H)
& r1_incsp_1(k14_afproj(A),B,I)
& r1_incsp_1(k14_afproj(A),D,I)
& r1_incsp_1(k14_afproj(A),C,J)
& r1_incsp_1(k14_afproj(A),E,J)
& ~ r1_incsp_1(k14_afproj(A),F,I)
& ~ r1_incsp_1(k14_afproj(A),F,J)
& G != H
& ! [K] :
( m1_subset_1(K,u1_incsp_1(k14_afproj(A)))
=> ~ ( r1_incsp_1(k14_afproj(A),K,I)
& r1_incsp_1(k14_afproj(A),K,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t45_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v6_incproj(k13_afproj(A))
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) ) ) ) ).
fof(t46_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( ~ ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v1_incproj(k14_afproj(A))
& v2_incproj(k14_afproj(A))
& v3_incproj(k14_afproj(A))
& v4_incproj(k14_afproj(A))
& v5_incproj(k14_afproj(A))
& l1_incsp_1(k14_afproj(A)) ) ) ) ).
fof(t47_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ~ ( v1_incproj(k14_afproj(A))
& v2_incproj(k14_afproj(A))
& v3_incproj(k14_afproj(A))
& v4_incproj(k14_afproj(A))
& v5_incproj(k14_afproj(A))
& l1_incsp_1(k14_afproj(A))
& ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) ) ) ).
fof(t48_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(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))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(A))
=> ( ( v1_aff_1(B,A)
& v1_aff_1(C,A)
& r2_hidden(D,B)
& r2_hidden(D,C)
& r2_hidden(E,B)
& r2_hidden(F,B)
& r2_hidden(G,B)
& r2_hidden(H,C)
& r2_hidden(I,C)
& r2_hidden(J,C)
& r2_analoaf(A,E,I,F,H)
& r2_analoaf(A,F,J,G,I) )
=> ( B = C
| D = E
| D = H
| D = F
| D = I
| D = G
| D = J
| ( E != F
& F != G
& E != G )
| r2_analoaf(A,E,J,G,H) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t49_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v11_incproj(k13_afproj(A))
=> v2_aff_2(A) ) ) ).
fof(t50_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(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))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(A))
=> ! [K] :
( m1_subset_1(K,u1_struct_0(A))
=> ( ( r2_hidden(E,B)
& r2_hidden(E,C)
& r2_hidden(E,D)
& r2_hidden(F,B)
& r2_hidden(I,B)
& r2_hidden(G,C)
& r2_hidden(J,C)
& r2_hidden(H,D)
& r2_hidden(K,D)
& v1_aff_1(B,A)
& v1_aff_1(C,A)
& v1_aff_1(D,A)
& r2_analoaf(A,F,G,I,J)
& r2_analoaf(A,F,H,I,K) )
=> ( E = F
| E = G
| E = H
| B = C
| B = D
| ( E != I
& F != I )
| r2_analoaf(A,G,H,J,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t51_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v10_incproj(k13_afproj(A))
=> v4_aff_2(A) ) ) ).
fof(t52_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v9_incproj(k13_afproj(A))
=> v1_translac(A) ) ) ).
fof(dt_k1_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> m1_subset_1(k1_afproj(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k2_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> m1_subset_1(k2_afproj(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k3_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v3_relat_2(k3_afproj(A))
& v8_relat_2(k3_afproj(A))
& v1_partfun1(k3_afproj(A),k1_afproj(A),k1_afproj(A))
& m2_relset_1(k3_afproj(A),k1_afproj(A),k1_afproj(A)) ) ) ).
fof(dt_k4_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v3_relat_2(k4_afproj(A))
& v8_relat_2(k4_afproj(A))
& v1_partfun1(k4_afproj(A),k2_afproj(A),k2_afproj(A))
& m2_relset_1(k4_afproj(A),k2_afproj(A),k2_afproj(A)) ) ) ).
fof(dt_k5_afproj,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k5_afproj(A,B),k1_zfmisc_1(k1_afproj(A))) ) ).
fof(dt_k6_afproj,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k6_afproj(A,B),k1_zfmisc_1(k2_afproj(A))) ) ).
fof(dt_k7_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ~ v1_xboole_0(k7_afproj(A)) ) ).
fof(dt_k8_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ~ v1_xboole_0(k8_afproj(A)) ) ).
fof(dt_k9_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ~ v1_xboole_0(k9_afproj(A)) ) ).
fof(dt_k10_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ~ v1_xboole_0(k10_afproj(A)) ) ).
fof(dt_k11_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> m2_relset_1(k11_afproj(A),k9_afproj(A),k10_afproj(A)) ) ).
fof(dt_k12_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> m2_relset_1(k12_afproj(A),k7_afproj(A),k8_afproj(A)) ) ).
fof(dt_k13_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v1_incsp_1(k13_afproj(A))
& l1_incsp_1(k13_afproj(A)) ) ) ).
fof(dt_k14_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> ( v1_incsp_1(k14_afproj(A))
& l1_incsp_1(k14_afproj(A)) ) ) ).
fof(d1_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k1_afproj(A) = a_1_0_afproj(A) ) ).
fof(d2_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k2_afproj(A) = a_1_1_afproj(A) ) ).
fof(d3_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k3_afproj(A) = a_1_2_afproj(A) ) ).
fof(d4_afproj,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& l1_analoaf(A) )
=> k4_afproj(A) = a_1_3_afproj(A) ) ).
fof(fraenkel_a_1_0_afproj,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v1_diraf(B)
& l1_analoaf(B) )
=> ( r2_hidden(A,a_1_0_afproj(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& v1_aff_1(C,B) ) ) ) ).
fof(fraenkel_a_1_1_afproj,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v1_diraf(B)
& l1_analoaf(B) )
=> ( r2_hidden(A,a_1_1_afproj(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& v1_aff_4(C,B) ) ) ) ).
fof(fraenkel_a_1_2_afproj,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v1_diraf(B)
& l1_analoaf(B) )
=> ( r2_hidden(A,a_1_2_afproj(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = k4_tarski(C,D)
& v1_aff_1(C,B)
& v1_aff_1(D,B)
& r1_aff_4(B,C,D) ) ) ) ).
fof(fraenkel_a_1_3_afproj,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v3_realset2(B)
& v1_diraf(B)
& l1_analoaf(B) )
=> ( r2_hidden(A,a_1_3_afproj(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = k4_tarski(C,D)
& v1_aff_4(C,B)
& v1_aff_4(D,B)
& r1_aff_4(B,C,D) ) ) ) ).
%------------------------------------------------------------------------------