SET007 Axioms: SET007+820.ax


%------------------------------------------------------------------------------
% File     : SET007+820 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Fundamental Theorem of Arithmetic
% 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    : nat_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  111 (   2 unit)
%            Number of atoms       :  655 (  81 equality)
%            Maximal formula depth :   14 (   8 average)
%            Number of connectives :  631 (  87 ~  ;   4  |; 311  &)
%                                         (  10 <=>; 219 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   41 (   0 propositional; 1-3 arity)
%            Number of functors    :   55 (   6 constant; 0-3 arity)
%            Number of variables   :  247 (   0 singleton; 244 !;   3 ?)
%            Maximal term depth    :    5 (   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_nat_3,axiom,(
    ! [A] :
      ( v1_xboole_0(A)
     => ( v1_xboole_0(k1_card_1(A))
        & v1_relat_1(k1_card_1(A))
        & v1_funct_1(k1_card_1(A))
        & v2_funct_1(k1_card_1(A))
        & v1_finset_1(k1_card_1(A))
        & v1_card_1(k1_card_1(A))
        & v1_membered(k1_card_1(A))
        & v2_membered(k1_card_1(A))
        & v3_membered(k1_card_1(A))
        & v4_membered(k1_card_1(A))
        & v5_membered(k1_card_1(A))
        & v1_polynom1(k1_card_1(A)) ) ) )).

fof(cc1_nat_3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v7_seqm_3(A) )
     => ( v1_relat_1(A)
        & v1_seq_1(A) ) ) )).

fof(rc1_nat_3,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finseq_1(A)
      & v1_seq_1(A)
      & v7_seqm_3(A)
      & v3_facirc_1(A) ) )).

fof(fc2_nat_3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A)
        & v4_ordinal2(B) )
     => ( ~ v1_xboole_0(k2_newton(A,B))
        & v4_ordinal2(k2_newton(A,B))
        & v1_xreal_0(k2_newton(A,B))
        & v2_xreal_0(k2_newton(A,B))
        & ~ v3_xreal_0(k2_newton(A,B))
        & v1_int_1(k2_newton(A,B))
        & v1_xcmplx_0(k2_newton(A,B)) ) ) )).

fof(cc2_nat_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_int_2(A)
       => ( ~ v1_xboole_0(A)
          & v1_ordinal1(A)
          & v2_ordinal1(A)
          & v3_ordinal1(A)
          & v4_ordinal2(A)
          & v1_xreal_0(A)
          & v2_xreal_0(A)
          & ~ v3_xreal_0(A)
          & v1_int_1(A)
          & v1_xcmplx_0(A)
          & v1_int_2(A) ) ) ) )).

fof(fc3_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_seq_1(A)
        & v4_ordinal2(B) )
     => ( v1_relat_1(k1_nat_3(A,B))
        & v1_funct_1(k1_nat_3(A,B))
        & v1_finseq_1(k1_nat_3(A,B))
        & v1_seq_1(k1_nat_3(A,B)) ) ) )).

fof(fc4_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v7_seqm_3(A)
        & v4_ordinal2(B) )
     => ( v1_relat_1(k1_nat_3(A,B))
        & v1_funct_1(k1_nat_3(A,B))
        & v1_finseq_1(k1_nat_3(A,B))
        & v1_seq_1(k1_nat_3(A,B))
        & v7_seqm_3(k1_nat_3(A,B))
        & v3_facirc_1(k1_nat_3(A,B)) ) ) )).

fof(rc2_nat_3,axiom,(
    ! [A] :
    ? [B] :
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v1_seq_1(B)
      & v7_seqm_3(B)
      & v1_polynom1(B)
      & v3_facirc_1(B) ) )).

fof(fc5_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A)
        & v4_ordinal2(C) )
     => ( v1_relat_1(k4_nat_3(A,B,C))
        & v1_funct_1(k4_nat_3(A,B,C))
        & v1_seq_1(k4_nat_3(A,B,C)) ) ) )).

fof(fc6_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v7_seqm_3(B)
        & m1_pboole(B,A)
        & v4_ordinal2(C) )
     => ( v1_relat_1(k4_nat_3(A,B,C))
        & v1_funct_1(k4_nat_3(A,B,C))
        & v1_seq_1(k4_nat_3(A,B,C))
        & v7_seqm_3(k4_nat_3(A,B,C))
        & v3_facirc_1(k4_nat_3(A,B,C)) ) ) )).

fof(fc7_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A) )
     => ( v1_xboole_0(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v1_relat_1(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v1_funct_1(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v2_funct_1(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v1_finset_1(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v1_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v2_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v3_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v4_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v5_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
        & v1_polynom1(k11_polynom1(k4_nat_3(A,B,np__0))) ) ) )).

fof(fc8_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A)
        & v4_ordinal2(C) )
     => ( v1_relat_1(k4_nat_3(A,B,C))
        & v1_funct_1(k4_nat_3(A,B,C))
        & v1_seq_1(k4_nat_3(A,B,C))
        & v1_polynom1(k4_nat_3(A,B,C)) ) ) )).

fof(fc9_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A)
        & v1_seq_1(C)
        & m1_pboole(C,A) )
     => ( v1_relat_1(k5_nat_3(A,B,C))
        & v1_funct_1(k5_nat_3(A,B,C))
        & v1_seq_1(k5_nat_3(A,B,C)) ) ) )).

fof(fc10_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v7_seqm_3(B)
        & m1_pboole(B,A)
        & v7_seqm_3(C)
        & m1_pboole(C,A) )
     => ( v1_relat_1(k5_nat_3(A,B,C))
        & v1_funct_1(k5_nat_3(A,B,C))
        & v1_seq_1(k5_nat_3(A,B,C))
        & v7_seqm_3(k5_nat_3(A,B,C))
        & v3_facirc_1(k5_nat_3(A,B,C)) ) ) )).

fof(fc11_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A)
        & v1_seq_1(C)
        & v1_polynom1(C)
        & m1_pboole(C,A) )
     => ( v1_relat_1(k5_nat_3(A,B,C))
        & v1_funct_1(k5_nat_3(A,B,C))
        & v1_seq_1(k5_nat_3(A,B,C))
        & v1_polynom1(k5_nat_3(A,B,C)) ) ) )).

fof(fc12_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A)
        & v1_seq_1(C)
        & m1_pboole(C,A) )
     => ( v1_relat_1(k6_nat_3(A,B,C))
        & v1_funct_1(k6_nat_3(A,B,C))
        & v1_seq_1(k6_nat_3(A,B,C)) ) ) )).

fof(fc13_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v7_seqm_3(B)
        & m1_pboole(B,A)
        & v7_seqm_3(C)
        & m1_pboole(C,A) )
     => ( v1_relat_1(k6_nat_3(A,B,C))
        & v1_funct_1(k6_nat_3(A,B,C))
        & v1_seq_1(k6_nat_3(A,B,C))
        & v7_seqm_3(k6_nat_3(A,B,C))
        & v3_facirc_1(k6_nat_3(A,B,C)) ) ) )).

fof(fc14_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A)
        & v1_seq_1(C)
        & v1_polynom1(C)
        & m1_pboole(C,A) )
     => ( v1_relat_1(k6_nat_3(A,B,C))
        & v1_funct_1(k6_nat_3(A,B,C))
        & v1_seq_1(k6_nat_3(A,B,C))
        & v1_polynom1(k6_nat_3(A,B,C)) ) ) )).

fof(fc15_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v7_seqm_3(B)
        & m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & v4_ordinal2(C) )
     => ( v1_relat_1(k9_nat_3(A,B,C))
        & v1_funct_1(k9_nat_3(A,B,C))
        & v1_seq_1(k9_nat_3(A,B,C))
        & v7_seqm_3(k9_nat_3(A,B,C))
        & v3_facirc_1(k9_nat_3(A,B,C)) ) ) )).

fof(fc16_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & v4_ordinal2(C) )
     => ( v1_relat_1(k9_nat_3(A,B,C))
        & v1_funct_1(k9_nat_3(A,B,C))
        & v1_polynom1(k9_nat_3(A,B,C)) ) ) )).

fof(fc17_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_relat_1(k11_nat_3(A))
        & v1_funct_1(k11_nat_3(A))
        & v1_seq_1(k11_nat_3(A))
        & v7_seqm_3(k11_nat_3(A))
        & v3_facirc_1(k11_nat_3(A)) ) ) )).

fof(fc18_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ( v1_relat_1(k11_nat_3(A))
        & v1_funct_1(k11_nat_3(A))
        & v1_seq_1(k11_nat_3(A))
        & v7_seqm_3(k11_nat_3(A))
        & v1_polynom1(k11_nat_3(A))
        & v3_facirc_1(k11_nat_3(A)) ) ) )).

fof(fc19_nat_3,axiom,
    ( v1_xboole_0(k11_polynom1(k11_nat_3(np__1)))
    & v1_relat_1(k11_polynom1(k11_nat_3(np__1)))
    & v1_funct_1(k11_polynom1(k11_nat_3(np__1)))
    & v2_funct_1(k11_polynom1(k11_nat_3(np__1)))
    & v1_finset_1(k11_polynom1(k11_nat_3(np__1)))
    & v1_membered(k11_polynom1(k11_nat_3(np__1)))
    & v2_membered(k11_polynom1(k11_nat_3(np__1)))
    & v3_membered(k11_polynom1(k11_nat_3(np__1)))
    & v4_membered(k11_polynom1(k11_nat_3(np__1)))
    & v5_membered(k11_polynom1(k11_nat_3(np__1)))
    & v1_polynom1(k11_polynom1(k11_nat_3(np__1))) )).

fof(fc20_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_int_2(A)
        & m1_subset_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v4_ordinal2(B) )
     => ( ~ v1_xboole_0(k11_polynom1(k11_nat_3(k3_newton(A,B))))
        & v1_finset_1(k11_polynom1(k11_nat_3(k3_newton(A,B))))
        & v1_realset1(k11_polynom1(k11_nat_3(k3_newton(A,B)))) ) ) )).

fof(fc21_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m1_subset_1(A,k5_numbers) )
     => ( ~ v1_xboole_0(k11_polynom1(k11_nat_3(A)))
        & v1_finset_1(k11_polynom1(k11_nat_3(A)))
        & v1_realset1(k11_polynom1(k11_nat_3(A))) ) ) )).

fof(fc22_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ( v1_relat_1(k12_nat_3(A))
        & v1_funct_1(k12_nat_3(A))
        & v1_seq_1(k12_nat_3(A))
        & v7_seqm_3(k12_nat_3(A))
        & v1_polynom1(k12_nat_3(A))
        & v3_facirc_1(k12_nat_3(A)) ) ) )).

fof(t1_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => ( ( r1_nat_1(A,C)
                      & r1_nat_1(B,D) )
                   => r1_nat_1(k3_xcmplx_0(A,B),k3_xcmplx_0(C,D)) ) ) ) ) ) )).

fof(t2_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,np__1)
           => r1_xreal_0(B,k2_newton(A,B)) ) ) ) )).

fof(t3_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( A != np__0
           => r1_nat_1(B,k2_newton(B,A)) ) ) ) )).

fof(t4_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => ( r1_nat_1(k2_newton(C,B),D)
                   => ( r1_xreal_0(B,A)
                      | r1_nat_1(k2_newton(C,k2_xcmplx_0(A,np__1)),D) ) ) ) ) ) ) )).

fof(t5_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( ( v1_int_2(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( r1_nat_1(C,k2_newton(A,B))
               => r1_nat_1(C,A) ) ) ) ) )).

fof(t6_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( ( v1_int_2(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( r1_nat_1(C,k3_newton(B,A))
               => C = B ) ) ) ) )).

fof(t7_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( m1_trees_4(B,k1_numbers,k5_numbers)
         => ( r2_hidden(A,k2_relat_1(B))
           => r1_nat_1(A,k10_wsierp_1(B)) ) ) ) )).

fof(t8_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_trees_4(B,k5_numbers,k12_newton)
         => ( r1_nat_1(A,k10_wsierp_1(B))
           => r2_hidden(A,k2_relat_1(B)) ) ) ) )).

fof(d1_nat_3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_seq_1(A) )
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( C = k1_nat_3(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(A)
                  & ! [D] :
                      ( r2_hidden(D,k4_finseq_1(C))
                     => k1_funct_1(C,D) = k3_newton(k1_seq_1(A,D),B) ) ) ) ) ) ) )).

fof(t9_nat_3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k2_nat_3(A,np__0) = k1_newton(k3_finseq_1(A),np__1) ) )).

fof(t10_nat_3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k2_nat_3(A,np__1) = A ) )).

fof(t11_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k2_nat_3(k6_finseq_1(k1_numbers),A) = k6_finseq_1(k1_numbers) ) )).

fof(t12_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k2_nat_3(k13_binarith(k1_numbers,B),A) = k13_binarith(k1_numbers,k3_newton(B,A)) ) ) )).

fof(t13_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => k2_nat_3(k8_finseq_1(k1_numbers,C,k13_binarith(k1_numbers,B)),A) = k8_finseq_1(k1_numbers,k2_nat_3(C,A),k2_nat_3(k13_binarith(k1_numbers,B),A)) ) ) ) )).

fof(t14_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k16_rvsum_1(k2_nat_3(B,k2_xcmplx_0(A,np__1))) = k11_binop_2(k16_rvsum_1(k2_nat_3(B,A)),k16_rvsum_1(B)) ) ) )).

fof(t15_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k16_rvsum_1(k2_nat_3(B,A)) = k3_newton(k16_rvsum_1(B),A) ) ) )).

fof(d2_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( v4_ordinal2(C)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( D = k4_nat_3(A,B,C)
              <=> ! [E] : k1_funct_1(D,E) = k3_xcmplx_0(C,k1_seq_1(B,E)) ) ) ) ) )).

fof(t16_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B,C] :
          ( ( v1_seq_1(C)
            & m1_pboole(C,B) )
         => ( A != np__0
           => k11_polynom1(C) = k11_polynom1(k4_nat_3(B,C,A)) ) ) ) )).

fof(d3_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( ( v1_seq_1(C)
            & m1_pboole(C,A) )
         => ! [D] :
              ( m1_pboole(D,A)
             => ( D = k5_nat_3(A,B,C)
              <=> ! [E] :
                    ( ( r1_xreal_0(k1_seq_1(B,E),k1_seq_1(C,E))
                     => k1_funct_1(D,E) = k1_seq_1(B,E) )
                    & ( ~ r1_xreal_0(k1_seq_1(B,E),k1_seq_1(C,E))
                     => k1_funct_1(D,E) = k1_seq_1(C,E) ) ) ) ) ) ) )).

fof(t17_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( ( v1_seq_1(C)
            & v1_polynom1(C)
            & m1_pboole(C,A) )
         => r1_tarski(k11_polynom1(k5_nat_3(A,B,C)),k2_xboole_0(k11_polynom1(B),k11_polynom1(C))) ) ) )).

fof(d4_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( ( v1_seq_1(C)
            & m1_pboole(C,A) )
         => ! [D] :
              ( m1_pboole(D,A)
             => ( D = k6_nat_3(A,B,C)
              <=> ! [E] :
                    ( ( r1_xreal_0(k1_seq_1(B,E),k1_seq_1(C,E))
                     => k1_funct_1(D,E) = k1_seq_1(C,E) )
                    & ( ~ r1_xreal_0(k1_seq_1(B,E),k1_seq_1(C,E))
                     => k1_funct_1(D,E) = k1_seq_1(B,E) ) ) ) ) ) ) )).

fof(t18_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( ( v1_seq_1(C)
            & v1_polynom1(C)
            & m1_pboole(C,A) )
         => r1_tarski(k11_polynom1(k6_nat_3(A,B,C)),k2_xboole_0(k11_polynom1(B),k11_polynom1(C))) ) ) )).

fof(d5_nat_3,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( v4_ordinal2(C)
         => ( C = k7_nat_3(A,B)
          <=> ? [D] :
                ( m1_trees_4(D,k1_numbers,k5_numbers)
                & C = k10_wsierp_1(D)
                & D = k5_relat_1(k1_uproots(k1_polynom2(A,B)),B) ) ) ) ) )).

fof(t19_nat_3,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( ( v7_seqm_3(C)
            & v1_polynom1(C)
            & m1_pboole(C,A) )
         => ( r1_xboole_0(k1_polynom2(A,B),k1_polynom2(A,C))
           => k8_nat_3(A,k9_polynom1(A,B,C)) = k24_binop_2(k8_nat_3(A,B),k8_nat_3(A,C)) ) ) ) )).

fof(d6_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( ( ~ v1_xboole_0(C)
            & v4_ordinal2(C) )
         => ! [D] :
              ( m1_pboole(D,A)
             => ( D = k9_nat_3(A,B,C)
              <=> ( k11_polynom1(D) = k11_polynom1(B)
                  & ! [E] : k1_funct_1(D,E) = k3_newton(k1_seq_1(B,E),C) ) ) ) ) ) )).

fof(t20_nat_3,axiom,(
    ! [A] : k8_nat_3(A,k16_polynom1(A)) = np__1 )).

fof(d7_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( B != np__1
              & A != np__0
              & ~ ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ( C = k10_nat_3(A,B)
                    <=> ( r1_nat_1(k2_newton(B,C),A)
                        & ~ r1_nat_1(k2_newton(B,k23_binop_2(C,np__1)),A) ) ) ) ) ) ) )).

fof(t21_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( A != np__1
       => k10_nat_3(np__1,A) = np__0 ) ) )).

fof(t22_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__1)
       => k10_nat_3(A,A) = np__1 ) ) )).

fof(t23_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( A != np__0
              & ~ r1_xreal_0(B,A)
              & B != np__1
              & k10_nat_3(A,B) != np__0 ) ) ) )).

fof(t24_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ~ ( A != np__1
              & A != B
              & k10_nat_3(B,A) != np__0 ) ) ) )).

fof(t25_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,np__1)
           => k10_nat_3(k2_newton(A,B),A) = B ) ) ) )).

fof(t26_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_nat_1(A,k2_newton(A,k10_nat_3(B,A)))
           => ( A = np__1
              | B = np__0
              | r1_nat_1(A,B) ) ) ) ) )).

fof(t27_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( A != np__1
           => ( ( B != np__0
                & k10_nat_3(B,A) = np__0 )
            <=> ~ r1_nat_1(A,B) ) ) ) ) )).

fof(t28_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v4_ordinal2(C) )
             => k10_nat_3(k3_xcmplx_0(B,C),A) = k23_binop_2(k10_nat_3(B,A),k10_nat_3(C,A)) ) ) ) )).

fof(t29_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v4_ordinal2(C) )
             => k2_wsierp_1(A,k10_nat_3(k3_xcmplx_0(B,C),A)) = k24_binop_2(k2_wsierp_1(A,k10_nat_3(B,A)),k2_wsierp_1(A,k10_nat_3(C,A))) ) ) ) )).

fof(t30_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v4_ordinal2(C) )
             => ( r1_nat_1(C,B)
               => r1_xreal_0(k10_nat_3(C,A),k10_nat_3(B,A)) ) ) ) ) )).

fof(t31_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v4_ordinal2(C) )
             => ( r1_nat_1(C,B)
               => k10_nat_3(k3_nat_1(B,C),A) = k5_binarith(k10_nat_3(B,A),k10_nat_3(C,A)) ) ) ) ) )).

fof(t32_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v4_ordinal2(C) )
             => k10_nat_3(k2_newton(C,A),B) = k3_xcmplx_0(A,k10_nat_3(C,B)) ) ) ) )).

fof(d8_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( m1_pboole(B,k12_newton)
         => ( B = k11_nat_3(A)
          <=> ! [C] :
                ( ( v1_int_2(C)
                  & m2_subset_1(C,k1_numbers,k5_numbers) )
               => k1_funct_1(B,C) = k10_nat_3(A,C) ) ) ) ) )).

fof(t33_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( r2_hidden(B,k1_relat_1(k11_nat_3(A)))
         => ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) ) ) ) )).

fof(t34_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( r2_hidden(B,k11_polynom1(k11_nat_3(A)))
         => ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) ) ) ) )).

fof(t35_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( ~ r1_xreal_0(A,B)
              & B != np__0
              & k1_funct_1(k11_nat_3(B),A) != np__0 ) ) ) )).

fof(t36_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r2_hidden(A,k11_polynom1(k11_nat_3(B)))
           => r1_nat_1(A,B) ) ) ) )).

fof(t37_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ( v1_int_2(B)
              & m2_subset_1(B,k1_numbers,k5_numbers)
              & r1_nat_1(B,A) )
           => ( v1_xboole_0(A)
              | r2_hidden(B,k11_polynom1(k11_nat_3(A))) ) ) ) ) )).

fof(t38_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ~ ( r1_nat_1(A,B)
              & k8_polynom1(k11_nat_3(B),A) = np__0 ) ) ) )).

fof(t39_nat_3,axiom,(
    r6_pboole(k12_newton,k11_nat_3(np__1),k16_polynom1(k12_newton)) )).

fof(t40_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k8_polynom1(k11_nat_3(k3_newton(B,A)),B) = A ) ) )).

fof(t41_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k8_polynom1(k11_nat_3(A),A) = np__1 ) )).

fof(t42_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ( A != np__0
           => k1_polynom2(k12_newton,k11_nat_3(k3_newton(B,A))) = k1_tarski(B) ) ) ) )).

fof(t43_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k1_polynom2(k12_newton,k11_nat_3(A)) = k1_tarski(A) ) )).

fof(t44_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ( r2_int_2(A,B)
           => r1_xboole_0(k1_polynom2(k12_newton,k11_nat_3(A)),k1_polynom2(k12_newton,k11_nat_3(B))) ) ) ) )).

fof(t45_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => r1_tarski(k1_polynom2(k12_newton,k11_nat_3(A)),k1_polynom2(k12_newton,k11_nat_3(k3_xcmplx_0(A,B)))) ) ) )).

fof(t46_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k1_polynom2(k12_newton,k11_nat_3(k24_binop_2(A,B))) = k4_subset_1(k12_newton,k1_polynom2(k12_newton,k11_nat_3(A)),k1_polynom2(k12_newton,k11_nat_3(B))) ) ) )).

fof(t47_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ( r2_int_2(A,B)
           => k4_card_1(k1_polynom2(k12_newton,k11_nat_3(k24_binop_2(A,B)))) = k23_binop_2(k4_card_1(k1_polynom2(k12_newton,k11_nat_3(A))),k4_card_1(k1_polynom2(k12_newton,k11_nat_3(B)))) ) ) ) )).

fof(t48_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => k1_polynom2(k12_newton,k11_nat_3(A)) = k1_polynom2(k12_newton,k11_nat_3(k2_newton(A,B))) ) ) )).

fof(t49_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => r6_pboole(k12_newton,k11_nat_3(k3_xcmplx_0(A,B)),k9_polynom1(k12_newton,k11_nat_3(A),k11_nat_3(B))) ) ) )).

fof(t50_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ( r1_nat_1(A,B)
           => r6_pboole(k12_newton,k11_nat_3(k3_nat_1(B,A)),k10_polynom1(k12_newton,k11_nat_3(B),k11_nat_3(A))) ) ) ) )).

fof(t51_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => r6_pboole(k12_newton,k11_nat_3(k2_newton(B,A)),k4_nat_3(k12_newton,k11_nat_3(B),A)) ) ) )).

fof(t52_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ( k1_polynom2(k12_newton,k11_nat_3(A)) = k1_xboole_0
       => A = np__1 ) ) )).

fof(t53_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => r6_pboole(k12_newton,k11_nat_3(k6_nat_1(B,A)),k5_nat_3(k12_newton,k11_nat_3(B),k11_nat_3(A))) ) ) )).

fof(t54_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => r6_pboole(k12_newton,k11_nat_3(k5_nat_1(B,A)),k6_nat_3(k12_newton,k11_nat_3(B),k11_nat_3(A))) ) ) )).

fof(d9_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ! [B] :
          ( m1_pboole(B,k12_newton)
         => ( B = k12_nat_3(A)
          <=> ( k11_polynom1(B) = k1_polynom2(k12_newton,k11_nat_3(A))
              & ! [C] :
                  ( v4_ordinal2(C)
                 => ( r2_hidden(C,k1_polynom2(k12_newton,k11_nat_3(A)))
                   => k1_funct_1(B,C) = k2_newton(C,k10_nat_3(A,C)) ) ) ) ) ) ) )).

fof(t55_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ( k10_nat_3(B,A) = np__0
           => k8_polynom1(k12_nat_3(B),A) = np__0 ) ) ) )).

fof(t56_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ( k10_nat_3(B,A) != np__0
           => k8_polynom1(k12_nat_3(B),A) = k2_wsierp_1(A,k10_nat_3(B,A)) ) ) ) )).

fof(t57_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ( k1_polynom2(k12_newton,k12_nat_3(A)) = k1_xboole_0
       => A = np__1 ) ) )).

fof(t58_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ( r2_int_2(A,B)
           => r6_pboole(k12_newton,k12_nat_3(k24_binop_2(A,B)),k9_polynom1(k12_newton,k12_nat_3(A),k12_nat_3(B))) ) ) ) )).

fof(t59_nat_3,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => k8_polynom1(k12_nat_3(k3_newton(A,B)),A) = k3_newton(A,B) ) ) )).

fof(t60_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => r6_pboole(k12_newton,k12_nat_3(k2_newton(A,B)),k9_nat_3(k12_newton,k12_nat_3(A),B)) ) ) )).

fof(t61_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => k8_nat_3(k12_newton,k12_nat_3(A)) = A ) )).

fof(dt_k1_nat_3,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_seq_1(A)
        & v4_ordinal2(B) )
     => ( v1_relat_1(k1_nat_3(A,B))
        & v1_funct_1(k1_nat_3(A,B))
        & v1_finseq_1(k1_nat_3(A,B)) ) ) )).

fof(dt_k2_nat_3,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & v4_ordinal2(B) )
     => m2_finseq_1(k2_nat_3(A,B),k1_numbers) ) )).

fof(redefinition_k2_nat_3,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & v4_ordinal2(B) )
     => k2_nat_3(A,B) = k1_nat_3(A,B) ) )).

fof(dt_k3_nat_3,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k5_numbers)
        & v4_ordinal2(B) )
     => m1_trees_4(k3_nat_3(A,B),k1_numbers,k5_numbers) ) )).

fof(redefinition_k3_nat_3,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k5_numbers)
        & v4_ordinal2(B) )
     => k3_nat_3(A,B) = k1_nat_3(A,B) ) )).

fof(dt_k4_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A)
        & v4_ordinal2(C) )
     => m1_pboole(k4_nat_3(A,B,C),A) ) )).

fof(dt_k5_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A)
        & v1_seq_1(C)
        & m1_pboole(C,A) )
     => m1_pboole(k5_nat_3(A,B,C),A) ) )).

fof(dt_k6_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A)
        & v1_seq_1(C)
        & m1_pboole(C,A) )
     => m1_pboole(k6_nat_3(A,B,C),A) ) )).

fof(dt_k7_nat_3,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => v4_ordinal2(k7_nat_3(A,B)) ) )).

fof(dt_k8_nat_3,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => m2_subset_1(k8_nat_3(A,B),k1_numbers,k5_numbers) ) )).

fof(redefinition_k8_nat_3,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => k8_nat_3(A,B) = k7_nat_3(A,B) ) )).

fof(dt_k9_nat_3,axiom,(
    ! [A,B,C] :
      ( ( v1_seq_1(B)
        & m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & v4_ordinal2(C) )
     => m1_pboole(k9_nat_3(A,B,C),A) ) )).

fof(dt_k10_nat_3,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => m2_subset_1(k10_nat_3(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k11_nat_3,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => m1_pboole(k11_nat_3(A),k12_newton) ) )).

fof(dt_k12_nat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => m1_pboole(k12_nat_3(A),k12_newton) ) )).
%------------------------------------------------------------------------------