SET007 Axioms: SET007+290.ax


%------------------------------------------------------------------------------
% File     : SET007+290 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  879 (  94 equality)
%            Maximal formula depth :   39 (  11 average)
%            Number of connectives : 1018 ( 226 ~  ;  19  |; 502  &)
%                                         (  28 <=>; 243 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   36 (   0 propositional; 1-5 arity)
%            Number of functors    :   32 (   2 constant; 0-3 arity)
%            Number of variables   :  256 (   0 singleton; 233 !;  23 ?)
%            Maximal term depth    :    4 (   1 average)
% 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) ) ) ) )).
%------------------------------------------------------------------------------