SET007 Axioms: SET007+884.ax


%------------------------------------------------------------------------------
% File     : SET007+884 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Real Valued Functions
% 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    : partfun3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  130 (   0 unit)
%            Number of atoms       : 1158 (  29 equality)
%            Maximal formula depth :   16 (   9 average)
%            Number of connectives : 1086 (  58 ~  ;   0  |; 885  &)
%                                         (   5 <=>; 138 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   0 propositional; 1-3 arity)
%            Number of functors    :   49 (   2 constant; 0-4 arity)
%            Number of variables   :  314 (   0 singleton; 309 !;   5 ?)
%            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_partfun3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k7_xcmplx_0(A,A))
        & v1_xreal_0(k7_xcmplx_0(A,A))
        & ~ v3_xreal_0(k7_xcmplx_0(A,A)) ) ) )).

fof(fc2_partfun3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k3_xcmplx_0(A,A))
        & v1_xreal_0(k3_xcmplx_0(A,A))
        & ~ v3_xreal_0(k3_xcmplx_0(A,A)) ) ) )).

fof(fc3_partfun3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k3_xcmplx_0(A,k5_xcmplx_0(A)))
        & v1_xreal_0(k3_xcmplx_0(A,k5_xcmplx_0(A)))
        & ~ v3_xreal_0(k3_xcmplx_0(A,k5_xcmplx_0(A))) ) ) )).

fof(fc4_partfun3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xcmplx_0(k8_square_1(A))
        & v1_xreal_0(k8_square_1(A))
        & ~ v3_xreal_0(k8_square_1(A)) ) ) )).

fof(fc5_partfun3,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ( ~ v1_xboole_0(k8_square_1(A))
        & v1_xcmplx_0(k8_square_1(A))
        & v1_xreal_0(k8_square_1(A))
        & v2_xreal_0(k8_square_1(A))
        & ~ v3_xreal_0(k8_square_1(A)) ) ) )).

fof(fc6_partfun3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_xboole_0(k10_relat_1(A,k1_tarski(np__0)))
        & v1_funct_1(k10_relat_1(A,k1_tarski(np__0)))
        & v1_membered(k10_relat_1(A,k1_tarski(np__0)))
        & v2_membered(k10_relat_1(A,k1_tarski(np__0)))
        & v3_membered(k10_relat_1(A,k1_tarski(np__0)))
        & v4_membered(k10_relat_1(A,k1_tarski(np__0)))
        & v5_membered(k10_relat_1(A,k1_tarski(np__0))) ) ) )).

fof(fc7_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(B)
        & v2_xreal_0(B) )
     => ( v1_relat_1(k2_funcop_1(A,B))
        & v2_relat_1(k2_funcop_1(A,B))
        & v1_funct_1(k2_funcop_1(A,B))
        & v1_partfun3(k2_funcop_1(A,B)) ) ) )).

fof(fc8_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(B)
        & v3_xreal_0(B) )
     => ( v1_relat_1(k2_funcop_1(A,B))
        & v2_relat_1(k2_funcop_1(A,B))
        & v1_funct_1(k2_funcop_1(A,B))
        & v2_partfun3(k2_funcop_1(A,B)) ) ) )).

fof(fc9_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_relat_1(k2_funcop_1(A,B))
        & v1_funct_1(k2_funcop_1(A,B))
        & v3_partfun3(k2_funcop_1(A,B)) ) ) )).

fof(fc10_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_relat_1(k2_funcop_1(A,B))
        & v1_funct_1(k2_funcop_1(A,B))
        & v4_partfun3(k2_funcop_1(A,B)) ) ) )).

fof(fc11_partfun3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_relat_1(k2_funcop_1(A,np__0))
        & ~ v2_relat_1(k2_funcop_1(A,np__0))
        & v1_funct_1(k2_funcop_1(A,np__0))
        & v3_partfun3(k2_funcop_1(A,np__0))
        & v4_partfun3(k2_funcop_1(A,np__0)) ) ) )).

fof(cc1_partfun3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_partfun3(A) )
     => ( v1_relat_1(A)
        & v2_relat_1(A)
        & v4_partfun3(A) ) ) )).

fof(cc2_partfun3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v2_partfun3(A) )
     => ( v1_relat_1(A)
        & v2_relat_1(A)
        & v3_partfun3(A) ) ) )).

fof(rc1_partfun3,axiom,(
    ! [A] :
    ? [B] :
      ( m1_relset_1(B,A,k1_numbers)
      & v1_relat_1(B)
      & v2_relat_1(B)
      & v1_funct_1(B)
      & v1_funct_2(B,A,k1_numbers)
      & v1_seq_1(B)
      & v2_partfun3(B)
      & v3_partfun3(B) ) )).

fof(rc2_partfun3,axiom,(
    ! [A] :
    ? [B] :
      ( m1_relset_1(B,A,k1_numbers)
      & v1_relat_1(B)
      & v2_relat_1(B)
      & v1_funct_1(B)
      & v1_funct_2(B,A,k1_numbers)
      & v1_seq_1(B)
      & v1_partfun3(B)
      & v4_partfun3(B) ) )).

fof(rc3_partfun3,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v2_relat_1(A)
      & v1_funct_1(A)
      & v1_seq_1(A) ) )).

fof(fc12_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k3_seq_1(B,C))
        & v1_funct_1(k3_seq_1(B,C))
        & v1_seq_1(k3_seq_1(B,C))
        & v3_partfun3(k3_seq_1(B,C)) ) ) )).

fof(fc13_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k3_seq_1(B,C))
        & v1_funct_1(k3_seq_1(B,C))
        & v1_seq_1(k3_seq_1(B,C))
        & v4_partfun3(k3_seq_1(B,C)) ) ) )).

fof(fc14_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k3_seq_1(B,C))
        & v2_relat_1(k3_seq_1(B,C))
        & v1_funct_1(k3_seq_1(B,C))
        & v1_seq_1(k3_seq_1(B,C))
        & v1_partfun3(k3_seq_1(B,C))
        & v4_partfun3(k3_seq_1(B,C)) ) ) )).

fof(fc15_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k3_seq_1(B,C))
        & v2_relat_1(k3_seq_1(B,C))
        & v1_funct_1(k3_seq_1(B,C))
        & v1_seq_1(k3_seq_1(B,C))
        & v1_partfun3(k3_seq_1(B,C))
        & v4_partfun3(k3_seq_1(B,C)) ) ) )).

fof(fc16_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v2_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k3_seq_1(B,C))
        & v2_relat_1(k3_seq_1(B,C))
        & v1_funct_1(k3_seq_1(B,C))
        & v1_seq_1(k3_seq_1(B,C))
        & v2_partfun3(k3_seq_1(B,C))
        & v3_partfun3(k3_seq_1(B,C)) ) ) )).

fof(fc17_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k3_seq_1(B,C))
        & v2_relat_1(k3_seq_1(B,C))
        & v1_funct_1(k3_seq_1(B,C))
        & v1_seq_1(k3_seq_1(B,C))
        & v2_partfun3(k3_seq_1(B,C))
        & v3_partfun3(k3_seq_1(B,C)) ) ) )).

fof(fc18_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k4_seq_1(B,C))
        & v1_funct_1(k4_seq_1(B,C))
        & v1_seq_1(k4_seq_1(B,C))
        & v4_partfun3(k4_seq_1(B,C)) ) ) )).

fof(fc19_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k4_seq_1(B,C))
        & v1_funct_1(k4_seq_1(B,C))
        & v1_seq_1(k4_seq_1(B,C))
        & v3_partfun3(k4_seq_1(B,C)) ) ) )).

fof(fc20_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k4_seq_1(B,C))
        & v2_relat_1(k4_seq_1(B,C))
        & v1_funct_1(k4_seq_1(B,C))
        & v1_seq_1(k4_seq_1(B,C))
        & v1_partfun3(k4_seq_1(B,C))
        & v4_partfun3(k4_seq_1(B,C)) ) ) )).

fof(fc21_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k4_seq_1(B,C))
        & v2_relat_1(k4_seq_1(B,C))
        & v1_funct_1(k4_seq_1(B,C))
        & v1_seq_1(k4_seq_1(B,C))
        & v2_partfun3(k4_seq_1(B,C))
        & v3_partfun3(k4_seq_1(B,C)) ) ) )).

fof(fc22_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k4_seq_1(B,C))
        & v2_relat_1(k4_seq_1(B,C))
        & v1_funct_1(k4_seq_1(B,C))
        & v1_seq_1(k4_seq_1(B,C))
        & v2_partfun3(k4_seq_1(B,C))
        & v3_partfun3(k4_seq_1(B,C)) ) ) )).

fof(fc23_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v2_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k4_seq_1(B,C))
        & v2_relat_1(k4_seq_1(B,C))
        & v1_funct_1(k4_seq_1(B,C))
        & v1_seq_1(k4_seq_1(B,C))
        & v1_partfun3(k4_seq_1(B,C))
        & v4_partfun3(k4_seq_1(B,C)) ) ) )).

fof(fc24_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C))
        & v4_partfun3(k5_seq_1(B,C)) ) ) )).

fof(fc25_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C))
        & v4_partfun3(k5_seq_1(B,C)) ) ) )).

fof(fc26_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C))
        & v3_partfun3(k5_seq_1(B,C)) ) ) )).

fof(fc27_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C))
        & v3_partfun3(k5_seq_1(B,C)) ) ) )).

fof(fc28_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v2_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v2_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C))
        & v2_partfun3(k5_seq_1(B,C))
        & v3_partfun3(k5_seq_1(B,C)) ) ) )).

fof(fc29_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v2_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C))
        & v2_partfun3(k5_seq_1(B,C))
        & v3_partfun3(k5_seq_1(B,C)) ) ) )).

fof(fc30_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v2_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C))
        & v1_partfun3(k5_seq_1(B,C))
        & v4_partfun3(k5_seq_1(B,C)) ) ) )).

fof(fc31_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v2_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v2_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C))
        & v1_partfun3(k5_seq_1(B,C))
        & v4_partfun3(k5_seq_1(B,C)) ) ) )).

fof(fc32_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v2_relat_1(B)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k1_numbers)
        & v2_relat_1(C)
        & v1_funct_1(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,C))
        & v2_relat_1(k5_seq_1(B,C))
        & v1_funct_1(k5_seq_1(B,C))
        & v1_seq_1(k5_seq_1(B,C)) ) ) )).

fof(fc33_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k5_seq_1(B,B))
        & v1_funct_1(k5_seq_1(B,B))
        & v1_seq_1(k5_seq_1(B,B))
        & v4_partfun3(k5_seq_1(B,B)) ) ) )).

fof(fc34_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & ~ v2_xreal_0(B)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B))
        & v4_partfun3(k12_seq_1(C,B)) ) ) )).

fof(fc35_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & ~ v3_xreal_0(B)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B))
        & v4_partfun3(k12_seq_1(C,B)) ) ) )).

fof(fc36_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & ~ v2_xreal_0(B)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B))
        & v3_partfun3(k12_seq_1(C,B)) ) ) )).

fof(fc37_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & ~ v3_xreal_0(B)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B))
        & v3_partfun3(k12_seq_1(C,B)) ) ) )).

fof(fc38_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v2_xreal_0(B)
        & v1_funct_1(C)
        & v2_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v2_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B))
        & v2_partfun3(k12_seq_1(C,B))
        & v3_partfun3(k12_seq_1(C,B)) ) ) )).

fof(fc39_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v3_xreal_0(B)
        & v1_funct_1(C)
        & v1_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v2_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B))
        & v2_partfun3(k12_seq_1(C,B))
        & v3_partfun3(k12_seq_1(C,B)) ) ) )).

fof(fc40_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v2_xreal_0(B)
        & v1_funct_1(C)
        & v1_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v2_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B))
        & v1_partfun3(k12_seq_1(C,B))
        & v4_partfun3(k12_seq_1(C,B)) ) ) )).

fof(fc41_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v3_xreal_0(B)
        & v1_funct_1(C)
        & v2_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v2_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B))
        & v1_partfun3(k12_seq_1(C,B))
        & v4_partfun3(k12_seq_1(C,B)) ) ) )).

fof(fc42_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_xreal_0(B)
        & v2_relat_1(C)
        & v1_funct_1(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k12_seq_1(C,B))
        & v2_relat_1(k12_seq_1(C,B))
        & v1_funct_1(k12_seq_1(C,B))
        & v1_seq_1(k12_seq_1(C,B)) ) ) )).

fof(fc43_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C))
        & v4_partfun3(k1_rfunct_1(B,C)) ) ) )).

fof(fc44_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C))
        & v4_partfun3(k1_rfunct_1(B,C)) ) ) )).

fof(fc45_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v4_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C))
        & v3_partfun3(k1_rfunct_1(B,C)) ) ) )).

fof(fc46_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v3_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C))
        & v3_partfun3(k1_rfunct_1(B,C)) ) ) )).

fof(fc47_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v2_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v2_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C))
        & v2_partfun3(k1_rfunct_1(B,C))
        & v3_partfun3(k1_rfunct_1(B,C)) ) ) )).

fof(fc48_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v2_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C))
        & v2_partfun3(k1_rfunct_1(B,C))
        & v3_partfun3(k1_rfunct_1(B,C)) ) ) )).

fof(fc49_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v2_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C))
        & v1_partfun3(k1_rfunct_1(B,C))
        & v4_partfun3(k1_rfunct_1(B,C)) ) ) )).

fof(fc50_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v2_partfun3(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v2_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C))
        & v1_partfun3(k1_rfunct_1(B,C))
        & v4_partfun3(k1_rfunct_1(B,C)) ) ) )).

fof(fc51_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,B))
        & v1_funct_1(k1_rfunct_1(B,B))
        & v4_partfun3(k1_rfunct_1(B,B)) ) ) )).

fof(fc52_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v2_relat_1(B)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k1_numbers)
        & v2_relat_1(C)
        & v1_funct_1(C)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_relat_1(k1_rfunct_1(B,C))
        & v2_relat_1(k1_rfunct_1(B,C))
        & v1_funct_1(k1_rfunct_1(B,C)) ) ) )).

fof(fc53_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k10_pscomp_1(A,B))
        & v1_funct_1(k10_pscomp_1(A,B))
        & v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
        & v1_seq_1(k10_pscomp_1(A,B))
        & v3_partfun3(k10_pscomp_1(A,B)) ) ) )).

fof(fc54_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k10_pscomp_1(A,B))
        & v1_funct_1(k10_pscomp_1(A,B))
        & v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
        & v1_seq_1(k10_pscomp_1(A,B))
        & v4_partfun3(k10_pscomp_1(A,B)) ) ) )).

fof(fc55_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k10_pscomp_1(A,B))
        & v2_relat_1(k10_pscomp_1(A,B))
        & v1_funct_1(k10_pscomp_1(A,B))
        & v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
        & v1_seq_1(k10_pscomp_1(A,B))
        & v1_partfun3(k10_pscomp_1(A,B))
        & v4_partfun3(k10_pscomp_1(A,B)) ) ) )).

fof(fc56_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k10_pscomp_1(A,B))
        & v2_relat_1(k10_pscomp_1(A,B))
        & v1_funct_1(k10_pscomp_1(A,B))
        & v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
        & v1_seq_1(k10_pscomp_1(A,B))
        & v2_partfun3(k10_pscomp_1(A,B))
        & v3_partfun3(k10_pscomp_1(A,B)) ) ) )).

fof(fc57_partfun3,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k10_pscomp_1(A,B))
        & v2_relat_1(k10_pscomp_1(A,B))
        & v1_funct_1(k10_pscomp_1(A,B))
        & v1_funct_2(k10_pscomp_1(A,B),A,k1_numbers)
        & v1_seq_1(k10_pscomp_1(A,B)) ) ) )).

fof(fc58_partfun3,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k15_seq_1(B))
        & v2_relat_1(k15_seq_1(B))
        & v1_funct_1(k15_seq_1(B))
        & v1_seq_1(k15_seq_1(B)) ) ) )).

fof(fc59_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k15_seq_1(B))
        & v1_funct_1(k15_seq_1(B))
        & v1_seq_1(k15_seq_1(B))
        & v4_partfun3(k15_seq_1(B)) ) ) )).

fof(fc60_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k15_seq_1(B))
        & v1_funct_1(k15_seq_1(B))
        & v1_seq_1(k15_seq_1(B))
        & v3_partfun3(k15_seq_1(B)) ) ) )).

fof(fc61_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k15_seq_1(B))
        & v2_relat_1(k15_seq_1(B))
        & v1_funct_1(k15_seq_1(B))
        & v1_seq_1(k15_seq_1(B))
        & v2_partfun3(k15_seq_1(B))
        & v3_partfun3(k15_seq_1(B)) ) ) )).

fof(fc62_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k15_seq_1(B))
        & v2_relat_1(k15_seq_1(B))
        & v1_funct_1(k15_seq_1(B))
        & v1_seq_1(k15_seq_1(B))
        & v1_partfun3(k15_seq_1(B))
        & v4_partfun3(k15_seq_1(B)) ) ) )).

fof(fc63_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k20_seq_1(B))
        & v1_funct_1(k20_seq_1(B))
        & v1_seq_1(k20_seq_1(B))
        & v4_partfun3(k20_seq_1(B)) ) ) )).

fof(fc64_partfun3,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k20_seq_1(B))
        & v2_relat_1(k20_seq_1(B))
        & v1_funct_1(k20_seq_1(B))
        & v1_seq_1(k20_seq_1(B))
        & v1_partfun3(k20_seq_1(B))
        & v4_partfun3(k20_seq_1(B)) ) ) )).

fof(fc65_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v3_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k3_rfunct_1(B))
        & v1_funct_1(k3_rfunct_1(B))
        & v3_partfun3(k3_rfunct_1(B)) ) ) )).

fof(fc66_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k3_rfunct_1(B))
        & v1_funct_1(k3_rfunct_1(B))
        & v4_partfun3(k3_rfunct_1(B)) ) ) )).

fof(fc67_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k3_rfunct_1(B))
        & v2_relat_1(k3_rfunct_1(B))
        & v1_funct_1(k3_rfunct_1(B))
        & v1_partfun3(k3_rfunct_1(B))
        & v4_partfun3(k3_rfunct_1(B)) ) ) )).

fof(fc68_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v2_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k3_rfunct_1(B))
        & v2_relat_1(k3_rfunct_1(B))
        & v1_funct_1(k3_rfunct_1(B))
        & v2_partfun3(k3_rfunct_1(B))
        & v3_partfun3(k3_rfunct_1(B)) ) ) )).

fof(fc69_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v2_relat_1(B)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k3_rfunct_1(B))
        & v2_relat_1(k3_rfunct_1(B))
        & v1_funct_1(k3_rfunct_1(B)) ) ) )).

fof(fc70_partfun3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k1_partfun3(A))
        & v1_funct_1(k1_partfun3(A))
        & v1_seq_1(k1_partfun3(A)) ) ) )).

fof(fc71_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v4_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k1_partfun3(B))
        & v1_funct_1(k1_partfun3(B))
        & v1_seq_1(k1_partfun3(B))
        & v4_partfun3(k1_partfun3(B)) ) ) )).

fof(fc72_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & v1_partfun3(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k1_partfun3(B))
        & v2_relat_1(k1_partfun3(B))
        & v1_funct_1(k1_partfun3(B))
        & v1_seq_1(k1_partfun3(B))
        & v1_partfun3(k1_partfun3(B))
        & v4_partfun3(k1_partfun3(B)) ) ) )).

fof(rc4_partfun3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_relset_1(B,u1_struct_0(A),k1_numbers)
          & v1_relat_1(B)
          & v2_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,u1_struct_0(A),k1_numbers)
          & v1_seq_1(B)
          & v9_pscomp_1(B,A)
          & v1_partfun3(B)
          & v4_partfun3(B) ) ) )).

fof(rc5_partfun3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_relset_1(B,u1_struct_0(A),k1_numbers)
          & v1_relat_1(B)
          & v2_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,u1_struct_0(A),k1_numbers)
          & v1_seq_1(B)
          & v9_pscomp_1(B,A)
          & v2_partfun3(B)
          & v3_partfun3(B) ) ) )).

fof(t1_partfun3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v2_funct_1(A)
            & r1_tarski(B,k1_relat_1(k2_funct_1(A))) )
         => k9_relat_1(A,k9_relat_1(k2_funct_1(A),B)) = B ) ) )).

fof(d1_partfun3,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v1_partfun3(A)
      <=> ! [B] :
            ( v1_xreal_0(B)
           => ~ ( r2_hidden(B,k2_relat_1(A))
                & r1_xreal_0(B,np__0) ) ) ) ) )).

fof(d2_partfun3,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v2_partfun3(A)
      <=> ! [B] :
            ( v1_xreal_0(B)
           => ~ ( r2_hidden(B,k2_relat_1(A))
                & r1_xreal_0(np__0,B) ) ) ) ) )).

fof(d3_partfun3,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v3_partfun3(A)
      <=> ! [B] :
            ( v1_xreal_0(B)
           => ( r2_hidden(B,k2_relat_1(A))
             => r1_xreal_0(B,np__0) ) ) ) ) )).

fof(d4_partfun3,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v4_partfun3(A)
      <=> ! [B] :
            ( v1_xreal_0(B)
           => ( r2_hidden(B,k2_relat_1(A))
             => r1_xreal_0(np__0,B) ) ) ) ) )).

fof(t2_partfun3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => k1_relat_1(k3_rfunct_1(A)) = k1_relat_1(A) ) )).

fof(t3_partfun3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k1_numbers) )
         => ! [C] :
              ( ( v2_relat_1(C)
                & v1_funct_1(C)
                & m2_relset_1(C,A,k1_numbers) )
             => k4_relset_1(A,k1_numbers,k2_rfunct_1(A,k1_numbers,B,C)) = k5_subset_1(A,k4_relset_1(A,k1_numbers,B),k4_relset_1(A,k1_numbers,C)) ) ) ) )).

fof(d5_partfun3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k1_partfun3(A)
          <=> ( k1_relat_1(B) = k1_relat_1(A)
              & ! [C] :
                  ( r2_hidden(C,k1_relat_1(B))
                 => k1_funct_1(B,C) = k9_square_1(k1_seq_1(A,C)) ) ) ) ) ) )).

fof(dt_k1_partfun3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_relat_1(k1_partfun3(A))
        & v1_funct_1(k1_partfun3(A)) ) ) )).

fof(dt_k2_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => ( v1_funct_1(k2_partfun3(A,B,C))
        & m2_relset_1(k2_partfun3(A,B,C),A,k1_numbers) ) ) )).

fof(redefinition_k2_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => k2_partfun3(A,B,C) = k1_partfun3(C) ) )).

fof(dt_k3_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_funct_1(k3_partfun3(A,B,C))
        & v1_funct_2(k3_partfun3(A,B,C),A,k1_numbers)
        & m2_relset_1(k3_partfun3(A,B,C),A,k1_numbers) ) ) )).

fof(commutativity_k3_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => k3_partfun3(A,B,C) = k3_partfun3(A,C,B) ) )).

fof(redefinition_k3_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => k3_partfun3(A,B,C) = k3_seq_1(B,C) ) )).

fof(dt_k4_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_funct_1(k4_partfun3(A,B,C))
        & v1_funct_2(k4_partfun3(A,B,C),A,k1_numbers)
        & m2_relset_1(k4_partfun3(A,B,C),A,k1_numbers) ) ) )).

fof(redefinition_k4_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => k4_partfun3(A,B,C) = k4_seq_1(B,C) ) )).

fof(dt_k5_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_funct_1(k5_partfun3(A,B,C))
        & v1_funct_2(k5_partfun3(A,B,C),A,k1_numbers)
        & m2_relset_1(k5_partfun3(A,B,C),A,k1_numbers) ) ) )).

fof(commutativity_k5_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => k5_partfun3(A,B,C) = k5_partfun3(A,C,B) ) )).

fof(redefinition_k5_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => k5_partfun3(A,B,C) = k5_seq_1(B,C) ) )).

fof(dt_k6_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_funct_1(k6_partfun3(A,B))
        & v1_funct_2(k6_partfun3(A,B),A,k1_numbers)
        & m2_relset_1(k6_partfun3(A,B),A,k1_numbers) ) ) )).

fof(redefinition_k6_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => k6_partfun3(A,B) = k15_seq_1(B) ) )).

fof(dt_k7_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_funct_1(k7_partfun3(A,B))
        & v1_funct_2(k7_partfun3(A,B),A,k1_numbers)
        & m2_relset_1(k7_partfun3(A,B),A,k1_numbers) ) ) )).

fof(redefinition_k7_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => k7_partfun3(A,B) = k20_seq_1(B) ) )).

fof(dt_k8_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_funct_1(k8_partfun3(A,B))
        & v1_funct_2(k8_partfun3(A,B),A,k1_numbers)
        & m2_relset_1(k8_partfun3(A,B),A,k1_numbers) ) ) )).

fof(redefinition_k8_partfun3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => k8_partfun3(A,B) = k1_partfun3(B) ) )).

fof(dt_k9_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_xreal_0(C) )
     => ( v1_funct_1(k9_partfun3(A,B,C))
        & v1_funct_2(k9_partfun3(A,B,C),A,k1_numbers)
        & m2_relset_1(k9_partfun3(A,B,C),A,k1_numbers) ) ) )).

fof(redefinition_k9_partfun3,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v1_xreal_0(C) )
     => k9_partfun3(A,B,C) = k12_seq_1(B,C) ) )).

fof(dt_k10_partfun3,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_funct_1(k10_partfun3(A,B))
        & v1_funct_2(k10_partfun3(A,B),A,k1_numbers)
        & m2_relset_1(k10_partfun3(A,B),A,k1_numbers) ) ) )).

fof(redefinition_k10_partfun3,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers) )
     => k10_partfun3(A,B) = k3_rfunct_1(B) ) )).

fof(dt_k11_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v2_relat_1(C)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => ( v1_funct_1(k11_partfun3(A,B,C))
        & v1_funct_2(k11_partfun3(A,B,C),A,k1_numbers)
        & m2_relset_1(k11_partfun3(A,B,C),A,k1_numbers) ) ) )).

fof(redefinition_k11_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,A,k1_numbers)
        & m1_relset_1(B,A,k1_numbers)
        & v2_relat_1(C)
        & v1_funct_1(C)
        & v1_funct_2(C,A,k1_numbers)
        & m1_relset_1(C,A,k1_numbers) )
     => k11_partfun3(A,B,C) = k1_rfunct_1(B,C) ) )).

fof(dt_k12_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k12_partfun3(A,B,C))
        & v1_funct_2(k12_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k12_partfun3(A,B,C),A)
        & m2_relset_1(k12_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) )).

fof(commutativity_k12_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => k12_partfun3(A,B,C) = k12_partfun3(A,C,B) ) )).

fof(redefinition_k12_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => k12_partfun3(A,B,C) = k3_seq_1(B,C) ) )).

fof(dt_k13_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k13_partfun3(A,B,C))
        & v1_funct_2(k13_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k13_partfun3(A,B,C),A)
        & m2_relset_1(k13_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) )).

fof(redefinition_k13_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => k13_partfun3(A,B,C) = k4_seq_1(B,C) ) )).

fof(dt_k14_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k14_partfun3(A,B,C))
        & v1_funct_2(k14_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k14_partfun3(A,B,C),A)
        & m2_relset_1(k14_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) )).

fof(commutativity_k14_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => k14_partfun3(A,B,C) = k14_partfun3(A,C,B) ) )).

fof(redefinition_k14_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => k14_partfun3(A,B,C) = k5_seq_1(B,C) ) )).

fof(dt_k15_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k15_partfun3(A,B))
        & v1_funct_2(k15_partfun3(A,B),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k15_partfun3(A,B),A)
        & m2_relset_1(k15_partfun3(A,B),u1_struct_0(A),k1_numbers) ) ) )).

fof(redefinition_k15_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers) )
     => k15_partfun3(A,B) = k15_seq_1(B) ) )).

fof(dt_k16_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k16_partfun3(A,B))
        & v1_funct_2(k16_partfun3(A,B),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k16_partfun3(A,B),A)
        & m2_relset_1(k16_partfun3(A,B),u1_struct_0(A),k1_numbers) ) ) )).

fof(redefinition_k16_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers) )
     => k16_partfun3(A,B) = k20_seq_1(B) ) )).

fof(dt_k17_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & v4_partfun3(B)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k17_partfun3(A,B))
        & v1_funct_2(k17_partfun3(A,B),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k17_partfun3(A,B),A)
        & m2_relset_1(k17_partfun3(A,B),u1_struct_0(A),k1_numbers) ) ) )).

fof(redefinition_k17_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & v4_partfun3(B)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers) )
     => k17_partfun3(A,B) = k1_partfun3(B) ) )).

fof(dt_k18_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_xreal_0(C) )
     => ( v1_funct_1(k18_partfun3(A,B,C))
        & v1_funct_2(k18_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k18_partfun3(A,B,C),A)
        & m2_relset_1(k18_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) )).

fof(redefinition_k18_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v1_xreal_0(C) )
     => k18_partfun3(A,B,C) = k12_seq_1(B,C) ) )).

fof(dt_k19_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v2_relat_1(B)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k19_partfun3(A,B))
        & v1_funct_2(k19_partfun3(A,B),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k19_partfun3(A,B),A)
        & m2_relset_1(k19_partfun3(A,B),u1_struct_0(A),k1_numbers) ) ) )).

fof(redefinition_k19_partfun3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v2_relat_1(B)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers) )
     => k19_partfun3(A,B) = k3_rfunct_1(B) ) )).

fof(dt_k20_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v2_relat_1(C)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => ( v1_funct_1(k20_partfun3(A,B,C))
        & v1_funct_2(k20_partfun3(A,B,C),u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(k20_partfun3(A,B,C),A)
        & m2_relset_1(k20_partfun3(A,B,C),u1_struct_0(A),k1_numbers) ) ) )).

fof(redefinition_k20_partfun3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(B,A)
        & m1_relset_1(B,u1_struct_0(A),k1_numbers)
        & v2_relat_1(C)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k1_numbers)
        & v9_pscomp_1(C,A)
        & m1_relset_1(C,u1_struct_0(A),k1_numbers) )
     => k20_partfun3(A,B,C) = k1_rfunct_1(B,C) ) )).
%------------------------------------------------------------------------------