SET007 Axioms: SET007+445.ax


%------------------------------------------------------------------------------
% File     : SET007+445 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Concept of the Triangulation
% 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    : triang_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   64 (   6 unit)
%            Number of atoms       :  394 (  38 equality)
%            Maximal formula depth :   19 (   7 average)
%            Number of connectives :  387 (  57 ~  ;   3  |; 197  &)
%                                         (  11 <=>; 119 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   40 (   1 propositional; 0-4 arity)
%            Number of functors    :   48 (   9 constant; 0-4 arity)
%            Number of variables   :  148 (   0 singleton; 134 !;  14 ?)
%            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_triang_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v1_xboole_0(k1_finseq_1(k1_nat_1(A,np__1)))
        & v1_finset_1(k1_finseq_1(k1_nat_1(A,np__1))) ) ) )).

fof(fc2_triang_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_relat_2(B)
        & v4_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => ( ~ v3_struct_0(g1_orders_2(A,B))
        & v1_orders_2(g1_orders_2(A,B))
        & v2_orders_2(g1_orders_2(A,B))
        & v3_orders_2(g1_orders_2(A,B))
        & v4_orders_2(g1_orders_2(A,B)) ) ) )).

fof(rc1_triang_1,axiom,(
    ! [A] :
    ? [B] :
      ( m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A)))
      & ~ v1_xboole_0(B) ) )).

fof(rc2_triang_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A)))
          & ~ v1_xboole_0(B)
          & v1_setfam_1(B) ) ) )).

fof(rc3_triang_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_setfam_1(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
     => ? [C] :
          ( m1_subset_1(C,B)
          & ~ v1_xboole_0(C) ) ) )).

fof(rc4_triang_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A)))
          & ~ v1_xboole_0(B)
          & ~ v2_setfam_1(B) ) ) )).

fof(rc5_triang_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_setfam_1(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
     => ? [C] :
          ( m1_subset_1(C,B)
          & ~ v1_xboole_0(C)
          & v1_finset_1(C) ) ) )).

fof(fc3_triang_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A)
        & ~ v1_xboole_0(B)
        & v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,u1_struct_0(A)) )
     => v1_finset_1(k3_orders_2(A,B,C)) ) )).

fof(fc4_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => ( ~ v1_xboole_0(k3_triang_1(A))
        & ~ v2_setfam_1(k3_triang_1(A)) ) ) )).

fof(cc1_triang_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
     => ! [C] :
          ( m1_subset_1(C,B)
         => v1_finset_1(C) ) ) )).

fof(rc6_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => ? [B] :
          ( m1_subset_1(B,k3_triang_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) )).

fof(rc7_triang_1,axiom,(
    ? [A] :
      ( m1_pboole(A,k5_numbers)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_triang_1(A) ) )).

fof(fc5_triang_1,axiom,(
    ! [A,B] :
      ( ( v1_triang_1(A)
        & m1_pboole(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ~ v1_xboole_0(k1_funct_1(k4_triang_1(A),B)) ) )).

fof(fc6_triang_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ~ v1_xboole_0(k1_funct_1(k6_triang_1,A)) ) )).

fof(cc2_triang_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_funct_1(k6_triang_1,A))
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) )).

fof(rc8_triang_1,axiom,(
    ? [A] :
      ( l1_triang_1(A)
      & v2_triang_1(A) ) )).

fof(rc9_triang_1,axiom,(
    ? [A] :
      ( l1_triang_1(A)
      & v2_triang_1(A)
      & v3_triang_1(A) ) )).

fof(fc7_triang_1,axiom,(
    ! [A] :
      ( ( v3_triang_1(A)
        & l1_triang_1(A) )
     => ( v1_relat_1(u1_triang_1(A))
        & v1_funct_1(u1_triang_1(A))
        & v1_triang_1(u1_triang_1(A)) ) ) )).

fof(fc8_triang_1,axiom,(
    ! [A,B] :
      ( ( v1_triang_1(A)
        & m1_pboole(A,k5_numbers)
        & m3_pboole(B,k5_numbers,k6_triang_1,k4_triang_1(A)) )
     => ( v2_triang_1(g1_triang_1(A,B))
        & v3_triang_1(g1_triang_1(A,B)) ) ) )).

fof(t1_triang_1,axiom,(
    ! [A] : k1_toler_1(k1_xboole_0,A) = k1_xboole_0 )).

fof(t2_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( v1_finset_1(B)
              & B != k1_xboole_0
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_hidden(C,B)
                          & r2_hidden(D,B)
                          & ~ r3_orders_2(A,C,D)
                          & ~ r3_orders_2(A,D,C) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( r2_hidden(C,B)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ( r2_hidden(D,B)
                           => r3_orders_2(A,C,D) ) ) ) ) ) ) ) )).

fof(t3_triang_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( ( r1_tarski(A,B)
              & k4_card_1(A) = k4_card_1(B) )
           => A = B ) ) ) )).

fof(d1_triang_1,axiom,(
    $true )).

fof(d2_triang_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v4_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( r3_orders_1(C,B)
           => ! [D] :
                ( m2_finseq_1(D,A)
               => ( D = k2_triang_1(A,B,C)
                <=> ( k2_relat_1(D) = B
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( ( r2_hidden(E,k4_finseq_1(D))
                                & r2_hidden(F,k4_finseq_1(D)) )
                             => ( r1_xreal_0(F,E)
                                | ( k4_finseq_4(k5_numbers,A,D,E) != k4_finseq_4(k5_numbers,A,D,F)
                                  & r2_hidden(k4_tarski(k4_finseq_4(k5_numbers,A,D,E),k4_finseq_4(k5_numbers,A,D,F)),C) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t4_triang_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v4_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ! [D] :
              ( m2_finseq_1(D,A)
             => ( ( k2_relat_1(D) = B
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(E,k4_finseq_1(D))
                              & r2_hidden(F,k4_finseq_1(D)) )
                           => ( r1_xreal_0(F,E)
                              | ( k4_finseq_4(k5_numbers,A,D,E) != k4_finseq_4(k5_numbers,A,D,F)
                                & r2_hidden(k4_tarski(k4_finseq_4(k5_numbers,A,D,E),k4_finseq_4(k5_numbers,A,D,F)),C) ) ) ) ) ) )
               => D = k2_triang_1(A,B,C) ) ) ) ) )).

fof(t5_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r2_hidden(k1_struct_0(A,B),k3_triang_1(A)) ) ) )).

fof(t6_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => r2_hidden(k1_xboole_0,k3_triang_1(A)) ) )).

fof(t7_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => ! [B,C] :
          ( ( r1_tarski(B,C)
            & r2_hidden(C,k3_triang_1(A)) )
         => r2_hidden(B,k3_triang_1(A)) ) ) )).

fof(t8_triang_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v4_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( r3_orders_1(C,B)
           => v2_funct_1(k2_triang_1(A,B,C)) ) ) ) )).

fof(t9_triang_1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v4_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( r3_orders_1(C,B)
           => k3_finseq_1(k2_triang_1(A,B,C)) = k1_card_1(B) ) ) ) )).

fof(t10_triang_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_triang_1(C,u1_struct_0(B),k3_triang_1(B)) )
             => ( k1_card_1(C) = A
               => k4_finseq_1(k2_triang_1(u1_struct_0(B),C,u1_orders_2(B))) = k2_finseq_1(A) ) ) ) ) )).

fof(d4_triang_1,axiom,(
    ! [A] :
      ( m1_pboole(A,k5_numbers)
     => ( v1_triang_1(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( ~ v1_xboole_0(k1_funct_1(A,B))
             => ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( ~ r1_xreal_0(B,C)
                      & v1_xboole_0(k1_funct_1(A,C)) ) ) ) ) ) ) )).

fof(d5_triang_1,axiom,(
    ! [A] :
      ( m1_pboole(A,k5_numbers)
     => ! [B] :
          ( m1_pboole(B,k5_numbers)
         => ( B = k4_triang_1(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k1_funct_1(B,C) = k1_funct_2(k1_funct_1(A,k1_nat_1(C,np__1)),k1_funct_1(A,C)) ) ) ) ) )).

fof(d6_triang_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_fraenkel(B,k2_finseq_1(A),k2_finseq_1(k1_nat_1(A,np__1)),k1_fraenkel(k2_finseq_1(A),k2_finseq_1(k1_nat_1(A,np__1))))
         => k5_triang_1(A,B) = B ) ) )).

fof(d8_triang_1,axiom,(
    $true )).

fof(d9_triang_1,axiom,(
    ! [A] :
      ( l1_triang_1(A)
     => ( v3_triang_1(A)
      <=> v1_triang_1(u1_triang_1(A)) ) ) )).

fof(d10_triang_1,axiom,(
    ! [A] :
      ( ( v3_triang_1(A)
        & l1_triang_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_funct_1(u1_triang_1(A),k1_nat_1(B,np__1)))
             => ! [D] :
                  ( m1_subset_1(D,k1_funct_1(k6_triang_1,B))
                 => ( k1_funct_1(u1_triang_1(A),k1_nat_1(B,np__1)) != k1_xboole_0
                   => ! [E] :
                        ( m1_subset_1(E,k1_funct_1(u1_triang_1(A),B))
                       => ( E = k7_triang_1(A,B,C,D)
                        <=> ! [F] :
                              ( ( v1_relat_1(F)
                                & v1_funct_1(F) )
                             => ! [G] :
                                  ( ( v1_relat_1(G)
                                    & v1_funct_1(G) )
                                 => ( ( F = k1_funct_1(u2_triang_1(A),B)
                                      & G = k1_funct_1(F,D) )
                                   => E = k1_funct_1(G,C) ) ) ) ) ) ) ) ) ) ) )).

fof(s1_triang_1,axiom,
    ( ( p1_s1_triang_1(f1_s1_triang_1)
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( p1_s1_triang_1(k1_nat_1(A,np__1))
           => ( r1_xreal_0(f1_s1_triang_1,A)
              | p1_s1_triang_1(A) ) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( r1_xreal_0(A,f1_s1_triang_1)
         => p1_s1_triang_1(A) ) ) )).

fof(s2_triang_1,axiom,
    ( ( v1_finset_1(f2_s2_triang_1)
      & p1_s2_triang_1(k1_subset_1(f1_s2_triang_1))
      & ! [A] :
          ( m1_subset_1(A,f1_s2_triang_1)
         => ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(f1_s2_triang_1))
             => ( ( r2_hidden(A,f2_s2_triang_1)
                  & r1_tarski(B,f2_s2_triang_1)
                  & p1_s2_triang_1(B) )
               => p1_s2_triang_1(k2_xboole_0(B,k1_tarski(A))) ) ) ) )
   => p1_s2_triang_1(f2_s2_triang_1) )).

fof(dt_m1_triang_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
     => ! [C] :
          ( m1_triang_1(C,A,B)
         => m1_subset_1(C,k1_zfmisc_1(A)) ) ) )).

fof(existence_m1_triang_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
     => ? [C] : m1_triang_1(C,A,B) ) )).

fof(redefinition_m1_triang_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_finsub_1(A))) )
     => ! [C] :
          ( m1_triang_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) )).

fof(dt_l1_triang_1,axiom,(
    $true )).

fof(existence_l1_triang_1,axiom,(
    ? [A] : l1_triang_1(A) )).

fof(abstractness_v2_triang_1,axiom,(
    ! [A] :
      ( l1_triang_1(A)
     => ( v2_triang_1(A)
       => A = g1_triang_1(u1_triang_1(A),u2_triang_1(A)) ) ) )).

fof(dt_k1_triang_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_relat_2(B)
        & v4_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => ( v1_relat_2(k1_triang_1(A,B,C))
        & v4_relat_2(k1_triang_1(A,B,C))
        & v8_relat_2(k1_triang_1(A,B,C))
        & v1_partfun1(k1_triang_1(A,B,C),C,C)
        & m2_relset_1(k1_triang_1(A,B,C),C,C) ) ) )).

fof(redefinition_k1_triang_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_relat_2(B)
        & v4_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k1_triang_1(A,B,C) = k2_wellord1(B,C) ) )).

fof(dt_k2_triang_1,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & v1_relat_2(C)
        & v4_relat_2(C)
        & v8_relat_2(C)
        & v1_partfun1(C,A,A)
        & m1_relset_1(C,A,A) )
     => m2_finseq_1(k2_triang_1(A,B,C),A) ) )).

fof(dt_k3_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => m1_subset_1(k3_triang_1(A),k1_zfmisc_1(k5_finsub_1(u1_struct_0(A)))) ) )).

fof(dt_k4_triang_1,axiom,(
    ! [A] :
      ( m1_pboole(A,k5_numbers)
     => m1_pboole(k4_triang_1(A),k5_numbers) ) )).

fof(dt_k5_triang_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_fraenkel(k2_finseq_1(A),k2_finseq_1(k1_nat_1(A,np__1)))) )
     => m2_finseq_1(k5_triang_1(A,B),k1_numbers) ) )).

fof(dt_k6_triang_1,axiom,(
    m1_pboole(k6_triang_1,k5_numbers) )).

fof(dt_k7_triang_1,axiom,(
    ! [A,B,C,D] :
      ( ( v3_triang_1(A)
        & l1_triang_1(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k1_funct_1(u1_triang_1(A),k1_nat_1(B,np__1)))
        & m1_subset_1(D,k1_funct_1(k6_triang_1,B)) )
     => m1_subset_1(k7_triang_1(A,B,C,D),k1_funct_1(u1_triang_1(A),B)) ) )).

fof(dt_k8_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => ( v2_triang_1(k8_triang_1(A))
        & v3_triang_1(k8_triang_1(A))
        & l1_triang_1(k8_triang_1(A)) ) ) )).

fof(dt_u1_triang_1,axiom,(
    ! [A] :
      ( l1_triang_1(A)
     => m1_pboole(u1_triang_1(A),k5_numbers) ) )).

fof(dt_u2_triang_1,axiom,(
    ! [A] :
      ( l1_triang_1(A)
     => m3_pboole(u2_triang_1(A),k5_numbers,k6_triang_1,k4_triang_1(u1_triang_1(A))) ) )).

fof(dt_g1_triang_1,axiom,(
    ! [A,B] :
      ( ( m1_pboole(A,k5_numbers)
        & m3_pboole(B,k5_numbers,k6_triang_1,k4_triang_1(A)) )
     => ( v2_triang_1(g1_triang_1(A,B))
        & l1_triang_1(g1_triang_1(A,B)) ) ) )).

fof(free_g1_triang_1,axiom,(
    ! [A,B] :
      ( ( m1_pboole(A,k5_numbers)
        & m3_pboole(B,k5_numbers,k6_triang_1,k4_triang_1(A)) )
     => ! [C,D] :
          ( g1_triang_1(A,B) = g1_triang_1(C,D)
         => ( A = C
            & B = D ) ) ) )).

fof(d3_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => k3_triang_1(A) = a_1_0_triang_1(A) ) )).

fof(d7_triang_1,axiom,(
    ! [A] :
      ( m1_pboole(A,k5_numbers)
     => ( A = k6_triang_1
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k1_funct_1(A,B) = a_1_1_triang_1(B) ) ) ) )).

fof(d11_triang_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( v2_triang_1(B)
            & v3_triang_1(B)
            & l1_triang_1(B) )
         => ( B = k8_triang_1(A)
          <=> ( k1_funct_1(u1_triang_1(B),np__0) = k1_tarski(k1_xboole_0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ~ r1_xreal_0(C,np__0)
                   => k1_funct_1(u1_triang_1(B),C) = a_2_0_triang_1(A,C) ) )
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_funct_1(k6_triang_1,C))
                     => ! [E] :
                          ( m1_subset_1(E,k1_funct_1(u1_triang_1(B),k1_nat_1(C,np__1)))
                         => ( r2_hidden(E,k1_funct_1(u1_triang_1(B),k1_nat_1(C,np__1)))
                           => ! [F] :
                                ( ( ~ v1_xboole_0(F)
                                  & m1_triang_1(F,u1_struct_0(A),k3_triang_1(A)) )
                               => ( k2_triang_1(u1_struct_0(A),F,u1_orders_2(A)) = E
                                 => k7_triang_1(B,C,E,D) = k5_relat_1(D,k2_triang_1(u1_struct_0(A),F,u1_orders_2(A))) ) ) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_1_0_triang_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & l1_orders_2(B) )
     => ( r2_hidden(A,a_1_0_triang_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,k5_finsub_1(u1_struct_0(B)))
            & A = C
            & r3_orders_1(u1_orders_2(B),C) ) ) ) )).

fof(fraenkel_a_1_1_triang_1,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_1_triang_1(B))
      <=> ? [C] :
            ( m2_fraenkel(C,k2_finseq_1(B),k2_finseq_1(k1_nat_1(B,np__1)),k1_fraenkel(k2_finseq_1(B),k2_finseq_1(k1_nat_1(B,np__1))))
            & A = C
            & v1_goboard1(k5_triang_1(B,C)) ) ) ) )).

fof(fraenkel_a_2_0_triang_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & l1_orders_2(B)
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_0_triang_1(B,C))
      <=> ? [D] :
            ( ~ v1_xboole_0(D)
            & m1_triang_1(D,u1_struct_0(B),k3_triang_1(B))
            & A = k2_triang_1(u1_struct_0(B),D,u1_orders_2(B))
            & k1_card_1(D) = C ) ) ) )).
%------------------------------------------------------------------------------