SET007 Axioms: SET007+843.ax


%------------------------------------------------------------------------------
% File     : SET007+843 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Lucas Numbers and Generalized Fibonacci Numbers
% 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    : fib_num3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   62 (   3 unit)
%            Number of atoms       :  220 (  66 equality)
%            Maximal formula depth :   18 (   6 average)
%            Number of connectives :  174 (  16 ~  ;   0  |;  44  &)
%                                         (   2 <=>; 112 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   18 (   0 propositional; 1-3 arity)
%            Number of functors    :   31 (  11 constant; 0-5 arity)
%            Number of variables   :  115 (   0 singleton; 113 !;   2 ?)
%            Maximal term depth    :    7 (   2 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_fib_num3,axiom,
    ( ~ v1_xboole_0(k1_fib_num)
    & v1_xcmplx_0(k1_fib_num)
    & v1_xreal_0(k1_fib_num)
    & v2_xreal_0(k1_fib_num)
    & ~ v3_xreal_0(k1_fib_num) )).

fof(fc2_fib_num3,axiom,
    ( ~ v1_xboole_0(k2_fib_num)
    & v1_xcmplx_0(k2_fib_num)
    & v1_xreal_0(k2_fib_num)
    & ~ v2_xreal_0(k2_fib_num)
    & v3_xreal_0(k2_fib_num) )).

fof(fc3_fib_num3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v1_xboole_0(k1_fib_num3(A))
        & v1_ordinal1(k1_fib_num3(A))
        & v2_ordinal1(k1_fib_num3(A))
        & v3_ordinal1(k1_fib_num3(A))
        & v4_ordinal2(k1_fib_num3(A))
        & v1_xcmplx_0(k1_fib_num3(A))
        & v1_xreal_0(k1_fib_num3(A))
        & v2_xreal_0(k1_fib_num3(A))
        & ~ v3_xreal_0(k1_fib_num3(A))
        & v1_int_1(k1_fib_num3(A))
        & v1_rat_1(k1_fib_num3(A)) ) ) )).

fof(t1_fib_num3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k3_power(A,B) = np__0
           => A = np__0 ) ) ) )).

fof(t2_fib_num3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => k3_xcmplx_0(k8_square_1(A),k8_square_1(A)) = A ) )).

fof(t3_fib_num3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_xreal_0(A) )
     => k3_power(A,np__2) = k3_power(k4_xcmplx_0(A),np__2) ) )).

fof(t4_fib_num3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k1_nat_1(k5_binarith(A,np__1),np__2) = k5_binarith(k1_nat_1(A,np__2),np__1) ) )).

fof(t5_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k3_prepower(k1_nat_1(A,B),np__2) = k1_nat_1(k1_nat_1(k1_nat_1(k2_nat_1(A,A),k2_nat_1(A,B)),k2_nat_1(A,B)),k2_nat_1(B,B)) ) ) )).

fof(t6_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_xreal_0(B) )
         => k3_power(k3_power(B,A),np__2) = k3_power(B,k2_nat_1(np__2,A)) ) ) )).

fof(t7_fib_num3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k3_xcmplx_0(k2_xcmplx_0(A,B),k6_xcmplx_0(A,B)) = k6_xcmplx_0(k3_power(A,np__2),k3_power(B,np__2)) ) ) )).

fof(t8_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_xreal_0(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_xreal_0(C) )
             => k3_power(k3_xcmplx_0(B,C),A) = k3_xcmplx_0(k3_power(B,A),k3_power(C,A)) ) ) ) )).

fof(t9_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_xcmplx_0(k3_power(k1_fib_num,A),k3_power(k1_fib_num,k1_nat_1(A,np__1))) = k3_power(k1_fib_num,k1_nat_1(A,np__2)) ) )).

fof(t10_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_xcmplx_0(k3_power(k2_fib_num,A),k3_power(k2_fib_num,k1_nat_1(A,np__1))) = k3_power(k2_fib_num,k1_nat_1(A,np__2)) ) )).

fof(d1_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k1_fib_num3(A)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
                & m2_relset_1(C,k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
                & B = k1_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,A))
                & k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,np__0) = k1_domain_1(k5_numbers,k5_numbers,np__2,np__1)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,k1_nat_1(D,np__1)) = k1_domain_1(k5_numbers,k5_numbers,k2_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)),k1_nat_1(k1_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)),k2_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)))) ) ) ) ) ) )).

fof(t11_fib_num3,axiom,
    ( k1_fib_num3(np__0) = np__2
    & k1_fib_num3(np__1) = np__1
    & ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => k1_fib_num3(k1_nat_1(k1_nat_1(A,np__1),np__1)) = k1_nat_1(k1_fib_num3(A),k1_fib_num3(k1_nat_1(A,np__1))) ) )).

fof(t12_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_fib_num3(k1_nat_1(A,np__2)) = k1_nat_1(k1_fib_num3(A),k1_fib_num3(k1_nat_1(A,np__1))) ) )).

fof(t13_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_nat_1(k1_fib_num3(k1_nat_1(A,np__1)),k1_fib_num3(k1_nat_1(A,np__2))) = k1_fib_num3(k1_nat_1(A,np__3)) ) )).

fof(t14_fib_num3,axiom,(
    k1_fib_num3(np__2) = np__3 )).

fof(t15_fib_num3,axiom,(
    k1_fib_num3(np__3) = np__4 )).

fof(t16_fib_num3,axiom,(
    k1_fib_num3(np__4) = np__7 )).

fof(t17_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_xreal_0(A,k1_fib_num3(A)) ) )).

fof(t18_fib_num3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => r1_xreal_0(k1_fib_num3(A),k1_fib_num3(k1_nat_1(A,np__1))) ) )).

fof(t19_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_nat_1(np__2,k1_fib_num3(k1_nat_1(A,np__2))) = k1_nat_1(k1_fib_num3(A),k1_fib_num3(k1_nat_1(A,np__3))) ) )).

fof(t20_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_fib_num3(k1_nat_1(A,np__1)) = k1_nat_1(k3_pre_ff(A),k3_pre_ff(k1_nat_1(A,np__2))) ) )).

fof(t21_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_fib_num3(A) = k2_xcmplx_0(k3_power(k1_fib_num,A),k3_power(k2_fib_num,A)) ) )).

fof(t22_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_nat_1(k2_nat_1(np__2,k1_fib_num3(A)),k1_fib_num3(k1_nat_1(A,np__1))) = k2_nat_1(np__5,k3_pre_ff(k1_nat_1(A,np__1))) ) )).

fof(t23_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k6_xcmplx_0(k1_fib_num3(k1_nat_1(A,np__3)),k2_nat_1(np__2,k1_fib_num3(A))) = k2_nat_1(np__5,k3_pre_ff(A)) ) )).

fof(t24_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_nat_1(k1_fib_num3(A),k3_pre_ff(A)) = k2_nat_1(np__2,k3_pre_ff(k1_nat_1(A,np__1))) ) )).

fof(t25_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_nat_1(k2_nat_1(np__3,k3_pre_ff(A)),k1_fib_num3(A)) = k2_nat_1(np__2,k3_pre_ff(k1_nat_1(A,np__2))) ) )).

fof(t26_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_nat_1(np__2,k1_fib_num3(k1_nat_1(A,B))) = k1_nat_1(k2_nat_1(k1_fib_num3(A),k1_fib_num3(B)),k2_nat_1(k2_nat_1(np__5,k3_pre_ff(A)),k3_pre_ff(B))) ) ) )).

fof(t27_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_nat_1(k1_fib_num3(k1_nat_1(A,np__3)),k1_fib_num3(A)) = k6_xcmplx_0(k3_prepower(k1_fib_num3(k1_nat_1(A,np__2)),np__2),k3_prepower(k1_fib_num3(k1_nat_1(A,np__1)),np__2)) ) )).

fof(t28_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_pre_ff(k2_nat_1(np__2,A)) = k2_nat_1(k3_pre_ff(A),k1_fib_num3(A)) ) )).

fof(t29_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_nat_1(np__2,k3_pre_ff(k1_nat_1(k2_nat_1(np__2,A),np__1))) = k1_nat_1(k2_nat_1(k1_fib_num3(k1_nat_1(A,np__1)),k3_pre_ff(A)),k2_nat_1(k1_fib_num3(A),k3_pre_ff(k1_nat_1(A,np__1)))) ) )).

fof(t30_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k6_xcmplx_0(k3_xcmplx_0(np__5,k3_prepower(k3_pre_ff(A),np__2)),k3_prepower(k1_fib_num3(A),np__2)) = k3_xcmplx_0(np__4,k3_power(k4_xcmplx_0(np__1),k1_nat_1(A,np__1))) ) )).

fof(t31_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_pre_ff(k1_nat_1(k2_nat_1(np__2,A),np__1)) = k6_xcmplx_0(k2_nat_1(k3_pre_ff(k1_nat_1(A,np__1)),k1_fib_num3(k1_nat_1(A,np__1))),k2_nat_1(k3_pre_ff(A),k1_fib_num3(A))) ) )).

fof(d2_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( D = k2_fib_num3(A,B,C)
                  <=> ? [E] :
                        ( v1_funct_1(E)
                        & v1_funct_2(E,k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
                        & m2_relset_1(E,k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
                        & D = k1_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),E,C))
                        & k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),E,np__0) = k1_domain_1(k5_numbers,k5_numbers,A,B)
                        & ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),E,k1_nat_1(F,np__1)) = k1_domain_1(k5_numbers,k5_numbers,k2_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),E,F)),k1_nat_1(k1_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),E,F)),k2_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),E,F)))) ) ) ) ) ) ) ) )).

fof(t32_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k2_fib_num3(A,B,np__0) = A
            & k2_fib_num3(A,B,np__1) = B
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_fib_num3(A,B,k1_nat_1(k1_nat_1(C,np__1),np__1)) = k1_nat_1(k2_fib_num3(A,B,C),k2_fib_num3(A,B,k1_nat_1(C,np__1))) ) ) ) ) )).

fof(t33_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_prepower(k1_nat_1(k2_fib_num3(A,B,k1_nat_1(C,np__1)),k2_fib_num3(A,B,k1_nat_1(k1_nat_1(C,np__1),np__1))),np__2) = k2_xcmplx_0(k2_xcmplx_0(k3_prepower(k2_fib_num3(A,B,k1_nat_1(C,np__1)),np__2),k2_nat_1(k2_nat_1(np__2,k2_fib_num3(A,B,k1_nat_1(C,np__1))),k2_fib_num3(A,B,k1_nat_1(k1_nat_1(C,np__1),np__1)))),k3_prepower(k2_fib_num3(A,B,k1_nat_1(k1_nat_1(C,np__1),np__1)),np__2)) ) ) ) )).

fof(t34_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k1_nat_1(k2_fib_num3(A,B,C),k2_fib_num3(A,B,k1_nat_1(C,np__1))) = k2_fib_num3(A,B,k1_nat_1(C,np__2)) ) ) ) )).

fof(t35_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k1_nat_1(k2_fib_num3(A,B,k1_nat_1(C,np__1)),k2_fib_num3(A,B,k1_nat_1(C,np__2))) = k2_fib_num3(A,B,k1_nat_1(C,np__3)) ) ) ) )).

fof(t36_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k1_nat_1(k2_fib_num3(A,B,k1_nat_1(C,np__2)),k2_fib_num3(A,B,k1_nat_1(C,np__3))) = k2_fib_num3(A,B,k1_nat_1(C,np__4)) ) ) ) )).

fof(t37_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_fib_num3(np__0,np__1,A) = k3_pre_ff(A) ) )).

fof(t38_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_fib_num3(np__2,np__1,A) = k1_fib_num3(A) ) )).

fof(t39_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k1_nat_1(k2_fib_num3(A,B,C),k2_fib_num3(A,B,k1_nat_1(C,np__3))) = k2_nat_1(np__2,k2_fib_num3(A,B,k1_nat_1(C,np__2))) ) ) ) )).

fof(t40_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k1_nat_1(k2_fib_num3(A,B,C),k2_fib_num3(A,B,k1_nat_1(C,np__4))) = k2_nat_1(np__3,k2_fib_num3(A,B,k1_nat_1(C,np__2))) ) ) ) )).

fof(t41_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k6_xcmplx_0(k2_fib_num3(A,B,k1_nat_1(C,np__3)),k2_fib_num3(A,B,C)) = k2_nat_1(np__2,k2_fib_num3(A,B,k1_nat_1(C,np__1))) ) ) ) )).

fof(t42_fib_num3,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) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => k2_fib_num3(A,B,C) = k1_nat_1(k2_nat_1(k2_fib_num3(A,B,np__0),k3_pre_ff(k5_binarith(C,np__1))),k2_nat_1(k2_fib_num3(A,B,np__1),k3_pre_ff(C))) ) ) ) )).

fof(t43_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_nat_1(k2_nat_1(k3_pre_ff(B),k1_fib_num3(A)),k2_nat_1(k1_fib_num3(B),k3_pre_ff(A))) = k2_fib_num3(k3_pre_ff(np__0),k1_fib_num3(np__0),k1_nat_1(A,B)) ) ) )).

fof(t44_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_nat_1(k1_fib_num3(A),k1_fib_num3(k1_nat_1(A,np__3))) = k2_nat_1(np__2,k1_fib_num3(k1_nat_1(A,np__2))) ) )).

fof(t45_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_fib_num3(A,A,B) = k3_xcmplx_0(k7_xcmplx_0(k2_fib_num3(A,A,np__0),np__2),k1_nat_1(k3_pre_ff(B),k1_fib_num3(B))) ) ) )).

fof(t46_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_fib_num3(B,k1_nat_1(A,B),C) = k2_fib_num3(A,B,k1_nat_1(C,np__1)) ) ) ) )).

fof(t47_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k6_xcmplx_0(k2_nat_1(k2_fib_num3(A,B,k1_nat_1(C,np__2)),k2_fib_num3(A,B,C)),k3_prepower(k2_fib_num3(A,B,k1_nat_1(C,np__1)),np__2)) = k3_xcmplx_0(k3_power(k4_xcmplx_0(np__1),C),k6_xcmplx_0(k3_prepower(k2_fib_num3(A,B,np__2),np__2),k2_nat_1(k2_fib_num3(A,B,np__1),k2_fib_num3(A,B,np__3)))) ) ) ) )).

fof(t48_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_fib_num3(k2_fib_num3(A,B,C),k2_fib_num3(A,B,k1_nat_1(C,np__1)),D) = k2_fib_num3(A,B,k1_nat_1(D,C)) ) ) ) ) )).

fof(t49_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_fib_num3(A,B,k1_nat_1(C,np__1)) = k1_nat_1(k2_nat_1(A,k3_pre_ff(C)),k2_nat_1(B,k3_pre_ff(k1_nat_1(C,np__1)))) ) ) ) )).

fof(t50_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_fib_num3(np__0,A,B) = k2_nat_1(A,k3_pre_ff(B)) ) ) )).

fof(t51_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_fib_num3(A,np__0,k1_nat_1(B,np__1)) = k2_nat_1(A,k3_pre_ff(B)) ) ) )).

fof(t52_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => k1_nat_1(k2_fib_num3(A,B,E),k2_fib_num3(C,D,E)) = k2_fib_num3(k1_nat_1(A,C),k1_nat_1(B,D),E) ) ) ) ) ) )).

fof(t53_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_fib_num3(k2_nat_1(C,A),k2_nat_1(C,B),D) = k2_nat_1(C,k2_fib_num3(A,B,D)) ) ) ) ) )).

fof(t54_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_fib_num3(A,B,C) = k7_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,k4_xcmplx_0(k2_fib_num)),B),k3_power(k1_fib_num,C)),k3_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(A,k1_fib_num),B),k3_power(k2_fib_num,C))),k9_square_1(np__5)) ) ) ) )).

fof(t55_fib_num3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_fib_num3(k1_nat_1(k2_nat_1(np__2,A),np__1),k1_nat_1(k2_nat_1(np__2,A),np__1),k1_nat_1(B,np__1)) = k2_nat_1(k1_nat_1(k2_nat_1(np__2,A),np__1),k3_pre_ff(k1_nat_1(B,np__2))) ) ) )).

fof(dt_k1_fib_num3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m2_subset_1(k1_fib_num3(A),k1_numbers,k5_numbers) ) )).

fof(dt_k2_fib_num3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers) )
     => m2_subset_1(k2_fib_num3(A,B,C),k1_numbers,k5_numbers) ) )).
%------------------------------------------------------------------------------