%------------------------------------------------------------------------------
% File     : SWW473+3 : TPTP v8.2.0. Released v5.3.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 383, 1000 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : hoare_1000_fofmg_l383 [Bla11]

% Status   : Theorem
% Rating   : 0.36 v8.2.0, 0.44 v8.1.0, 0.42 v7.5.0, 0.44 v7.4.0, 0.37 v7.3.0, 0.38 v7.2.0, 0.34 v7.1.0, 0.43 v7.0.0, 0.47 v6.4.0, 0.50 v6.3.0, 0.33 v6.2.0, 0.40 v6.1.0, 0.63 v6.0.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.68 v5.3.0
% Syntax   : Number of formulae    : 1415 ( 383 unt;   0 def)
%            Number of atoms       : 3616 ( 905 equ)
%            Maximal formula atoms :   13 (   2 avg)
%            Number of connectives : 2518 ( 317   ~;  77   |; 306   &)
%                                         ( 253 <=>;1565  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   5 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of predicates  :    9 (   8 usr;   0 prp; 1-2 aty)
%            Number of functors    :  372 ( 372 usr; 105 con; 0-2 aty)
%            Number of variables   : 3314 (3205   !; 109   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 17:58:47
%          : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (106)
fof(gsy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000t__a,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_fun_a_bool(cOMBB_bool_bool_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_fun_pname_bool(cOMBB_647938656_pname(B_1_1,B_2_1)) ) ).

fof(gsy_c_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000t__a,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun949378684l_bool(B_1_1)
     => is_fun_a_bool(cOMBB_1273333599bool_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_COMBK_000tc__HOL__Obool_000t__a,axiom,
    ! [B_1_1] :
      ( is_bool(B_1_1)
     => is_fun_a_bool(cOMBK_bool_a(B_1_1)) ) ).

fof(gsy_c_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_fun_a_bool(cOMBS_a_bool_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_fun_pname_bool(cOMBS_568398431l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__HOL__Obo,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun949378684l_bool(B_2_1)
     => is_fun949378684l_bool(cOMBS_1035972772l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool_000t,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun1661590463l_bool(B_2_1)
     => is_fun1661590463l_bool(cOMBS_350070575l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_Finite__Set_Ofinite_000t__a,axiom,
    is_fun949378684l_bool(finite_finite_a) ).

fof(gsy_c_Finite__Set_Ofinite_000tc__Com__Opname,hypothesis,
    is_fun1661590463l_bool(finite_finite_pname) ).

fof(gsy_c_Finite__Set_Ofolding__one_000t__a,axiom,
    ! [B_1_1,B_2_1] : is_bool(finite_folding_one_a(B_1_1,B_2_1)) ).

fof(gsy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] : is_bool(finite1282449217_pname(B_1_1,B_2_1)) ).

fof(gsy_c_Finite__Set_Ofolding__one_000tc__Int__Oint,axiom,
    ! [B_1_1,B_2_1] : is_bool(finite1626084323ne_int(B_1_1,B_2_1)) ).

fof(gsy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,axiom,
    ! [B_1_1,B_2_1] : is_bool(finite988810631ne_nat(B_1_1,B_2_1)) ).

fof(gsy_c_Finite__Set_Ofolding__one__idem_000t__a,axiom,
    ! [B_1_1,B_2_1] : is_bool(finite1819937229idem_a(B_1_1,B_2_1)) ).

fof(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] : is_bool(finite89670078_pname(B_1_1,B_2_1)) ).

fof(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint,axiom,
    ! [B_1_1,B_2_1] : is_bool(finite1432773856em_int(B_1_1,B_2_1)) ).

fof(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,axiom,
    ! [B_1_1,B_2_1] : is_bool(finite795500164em_nat(B_1_1,B_2_1)) ).

fof(gsy_c_HOL_Oundefined_000t__a,axiom,
    is_a(undefined_a(x_a)) ).

fof(gsy_c_HOL_Oundefined_000tc__Com__Opname,axiom,
    is_pname(undefined_pname(pname)) ).

fof(gsy_c_HOL_Oundefined_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    is_fun_a_bool(undefined_fun_a_bool(fun(x_a,bool))) ).

fof(gsy_c_HOL_Oundefined_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    is_fun_pname_bool(undefi17486888e_bool(fun(pname,bool))) ).

fof(gsy_c_HOL_Oundefined_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool,axiom,
    is_fun949378684l_bool(undefi1699038445l_bool(fun(fun(x_a,bool),bool))) ).

fof(gsy_c_HOL_Oundefined_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc_,axiom,
    is_fun1661590463l_bool(undefi64961550l_bool(fun(fun(pname,bool),bool))) ).

fof(gsy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,axiom,
    is_bool(bot_bot_bool) ).

fof(gsy_c_Orderings_Obot__class_Obot_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    is_fun_a_bool(bot_bot_fun_a_bool) ).

fof(gsy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    is_fun_pname_bool(bot_bo844097828e_bool) ).

fof(gsy_c_Set_OCollect_000t__a,axiom,
    ! [B_1_1] :
      ( is_fun_a_bool(B_1_1)
     => is_fun_a_bool(collect_a(B_1_1)) ) ).

fof(gsy_c_Set_OCollect_000tc__Com__Opname,axiom,
    ! [B_1_1] :
      ( is_fun_pname_bool(B_1_1)
     => is_fun_pname_bool(collect_pname(B_1_1)) ) ).

fof(gsy_c_Set_OCollect_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1] :
      ( is_fun949378684l_bool(B_1_1)
     => is_fun949378684l_bool(collect_fun_a_bool(B_1_1)) ) ).

fof(gsy_c_Set_OCollect_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1] :
      ( is_fun1661590463l_bool(B_1_1)
     => is_fun1661590463l_bool(collec1974731493e_bool(B_1_1)) ) ).

fof(gsy_c_Set_Oimage_000t__a_000t__a,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_fun_a_bool(image_a_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000t__a_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_fun_pname_bool(image_a_pname(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_fun949378684l_bool(image_a_fun_a_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_fun1661590463l_bool(image_819518260e_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__Com__Opname_000t__a,hypothesis,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_fun_a_bool(image_pname_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_fun_pname_bool(image_pname_pname(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_fun949378684l_bool(image_112932426a_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_fun1661590463l_bool(image_47868345e_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__Int__Oint_000t__a,axiom,
    ! [B_1_1,B_2_1] : is_fun_a_bool(image_int_a(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__Int__Oint_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] : is_fun_pname_bool(image_int_pname(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__Int__Oint_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] : is_fun949378684l_bool(image_int_fun_a_bool(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__Int__Oint_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] : is_fun1661590463l_bool(image_637364955e_bool(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__Nat__Onat_000t__a,axiom,
    ! [B_1_1,B_2_1] : is_fun_a_bool(image_nat_a(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__Nat__Onat_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] : is_fun_pname_bool(image_nat_pname(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] : is_fun949378684l_bool(image_nat_fun_a_bool(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] : is_fun1661590463l_bool(image_1655916159e_bool(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__fun_It__a_Mtc__HOL__Obool_J_000t__a,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun949378684l_bool(B_2_1)
     => is_fun_a_bool(image_fun_a_bool_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun949378684l_bool(B_2_1)
     => is_fun_pname_bool(image_1854862208_pname(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000t__a,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun1661590463l_bool(B_2_1)
     => is_fun_a_bool(image_876012084bool_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Com__Opnam,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun1661590463l_bool(B_2_1)
     => is_fun_pname_bool(image_1283814551_pname(B_1_1,B_2_1)) ) ).

fof(gsy_c_Set_Oimage_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000t__a,axiom,
    ! [B_1_1,B_2_1] : is_fun_a_bool(image_fun_int_bool_a(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] : is_fun_pname_bool(image_62782965_pname(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000t__a,axiom,
    ! [B_1_1,B_2_1] : is_fun_a_bool(image_fun_nat_bool_a(B_1_1,B_2_1)) ).

fof(gsy_c_Set_Oimage_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] : is_fun_pname_bool(image_1921560913_pname(B_1_1,B_2_1)) ).

fof(gsy_c_fFalse,axiom,
    is_bool(fFalse) ).

fof(gsy_c_fTrue,axiom,
    is_bool(fTrue) ).

fof(gsy_c_hAPP_000t__a_000t__a,axiom,
    ! [B_1_1,B_2_1] :
      ( is_a(B_2_1)
     => is_a(hAPP_a_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000t__a_000tc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] :
      ( ( is_fun_a_bool(B_1_1)
        & is_a(B_2_1) )
     => is_bool(hAPP_a_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] :
      ( is_a(B_2_1)
     => is_fun_a_bool(hAPP_a_fun_a_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] :
      ( is_a(B_2_1)
     => is_fun_pname_bool(hAPP_a93125764e_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000t__a_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,hypothesis,
    ! [B_1_1,B_2_1] :
      ( is_a(B_2_1)
     => is_fun949378684l_bool(hAPP_a85458249l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__Com__Opname_000t__a,hypothesis,
    ! [B_1_1,B_2_1] :
      ( is_pname(B_2_1)
     => is_a(hAPP_pname_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__Com__Opname_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] :
      ( is_pname(B_2_1)
     => is_pname(hAPP_pname_pname(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__Com__Opname_000tc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] :
      ( ( is_fun_pname_bool(B_1_1)
        & is_pname(B_2_1) )
     => is_bool(hAPP_pname_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] :
      ( is_pname(B_2_1)
     => is_fun_a_bool(hAPP_p1534023578a_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] :
      ( is_pname(B_2_1)
     => is_fun_pname_bool(hAPP_p61793385e_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obo,hypothesis,
    ! [B_1_1,B_2_1] :
      ( is_pname(B_2_1)
     => is_fun1661590463l_bool(hAPP_p338031245l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] :
      ( is_bool(B_2_1)
     => is_bool(hAPP_bool_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__Int__Oint_000t__a,axiom,
    ! [B_1_1,B_2_1] : is_a(hAPP_int_a(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_int_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Int__Oint_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] : is_fun_a_bool(hAPP_int_fun_a_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] : is_fun_pname_bool(hAPP_i7355787e_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL,axiom,
    ! [B_1_1,B_2_1] : is_fun949378684l_bool(hAPP_i888906320l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] : is_fun1661590463l_bool(hAPP_i1587699435l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,hypothesis,
    ! [B_1_1,B_2_1] : is_bool(hAPP_nat_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] : is_fun_a_bool(hAPP_nat_fun_a_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,axiom,
    ! [B_1_1,B_2_1] : is_fun_pname_bool(hAPP_n1025906991e_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL,axiom,
    ! [B_1_1,B_2_1] : is_fun949378684l_bool(hAPP_n1414589940l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] : is_fun1661590463l_bool(hAPP_n850744903l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000t__a,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_a(hAPP_fun_a_bool_a(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool,hypothesis,
    ! [B_1_1,B_2_1] :
      ( ( is_fun949378684l_bool(B_1_1)
        & is_fun_a_bool(B_2_1) )
     => is_bool(hAPP_fun_a_bool_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HOL__Obool_J,hypothesis,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_fun_a_bool(hAPP_f2050579477a_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL,hypothesis,
    ! [B_1_1,B_2_1] :
      ( is_fun_a_bool(B_2_1)
     => is_fun949378684l_bool(hAPP_f1631501043l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Com__Opname,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_pname(hAPP_f1297739591_pname(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool,hypothesis,
    ! [B_1_1,B_2_1] :
      ( ( is_fun1661590463l_bool(B_1_1)
        & is_fun_pname_bool(B_2_1) )
     => is_bool(hAPP_f1664156314l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__Com__Op,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_fun_pname_bool(hAPP_f759274231e_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun_pname_bool(B_2_1)
     => is_fun1661590463l_bool(hAPP_f434788991l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_f448129468l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_f54304608l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun949378684l_bool(B_2_1)
     => is_bool(hAPP_f621171935l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc___001,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun949378684l_bool(B_2_1)
     => is_fun949378684l_bool(hAPP_f2117159681l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Oboo,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun1661590463l_bool(B_2_1)
     => is_bool(hAPP_f1935102916l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Oboo_002,axiom,
    ! [B_1_1,B_2_1] :
      ( is_fun1661590463l_bool(B_2_1)
     => is_fun1661590463l_bool(hAPP_f559147733l_bool(B_1_1,B_2_1)) ) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_f215623910l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_f1637334154l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_f292226953l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_f389811538l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HO,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_f1593524852l_bool(B_1_1,B_2_1)) ).

fof(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HO,axiom,
    ! [B_1_1,B_2_1] : is_bool(hAPP_f937997336l_bool(B_1_1,B_2_1)) ).

fof(gsy_v_G,hypothesis,
    is_fun_a_bool(g) ).

fof(gsy_v_P,axiom,
    ! [B_1_1] :
      ( is_fun_a_bool(B_1_1)
     => is_fun949378684l_bool(p(B_1_1)) ) ).

fof(gsy_v_U,hypothesis,
    is_fun_pname_bool(u) ).

fof(gsy_v_mgt,axiom,
    ! [B_1_1] : is_a(mgt(B_1_1)) ).

fof(gsy_v_pn,hypothesis,
    is_pname(pn) ).

fof(gsy_v_wt,axiom,
    ! [B_1_1] : is_bool(wt(B_1_1)) ).

%----Relevant facts (1199)
fof(fact_0_assms_I1_J,axiom,
    ! [Ts,G] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Ts),G))
     => hBOOL(hAPP_fun_a_bool_bool(p(G),Ts)) ) ).

fof(fact_1_finite__Collect__subsets,axiom,
    ! [A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_f1593524852l_bool(finite2095298929l_bool,collec1409689523l_bool(hAPP_f317860125l_bool(cOMBC_348868698l_bool(ord_le1848167016l_bool),A)))) ) ).

fof(fact_2_finite__Collect__subsets,axiom,
    ! [A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,collec1015864663l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(ord_le1454342156l_bool),A)))) ) ).

fof(fact_3_finite__Collect__subsets,axiom,
    ! [A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,collec1613912337l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(ord_le675606854l_bool),A)))) ) ).

fof(fact_4_finite__Collect__subsets,axiom,
    ! [A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,collec707592106l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(ord_le1375614389l_bool),A)))) ) ).

fof(fact_5_finite__Collect__subsets,axiom,
    ! [A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(ord_le1311769555a_bool),A)))) ) ).

fof(fact_6_finite__Collect__subsets,axiom,
    ! [A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(ord_le313189616e_bool),A)))) ) ).

fof(fact_7_finite__Collect__subsets,axiom,
    ! [A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(ord_le1568362934t_bool),A)))) ) ).

fof(fact_8_finite__Collect__subsets,axiom,
    ! [A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(hAPP_f284875647l_bool(cOMBC_605892544l_bool(ord_le951220754t_bool),A)))) ) ).

fof(fact_9_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_1512837979t_bool(H,F_1))) ) ).

fof(fact_10_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_2129980159t_bool(H,F_1))) ) ).

fof(fact_11_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_47868345e_bool(H,F_1))) ) ).

fof(fact_12_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_112932426a_bool(H,F_1))) ) ).

fof(fact_13_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_nat_a(H,F_1))) ) ).

fof(fact_14_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_1556378401t_bool(H,F_1))) ) ).

fof(fact_15_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_26036933t_bool(H,F_1))) ) ).

fof(fact_16_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_1655916159e_bool(H,F_1))) ) ).

fof(fact_17_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_nat_fun_a_bool(H,F_1))) ) ).

fof(fact_18_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_int_a(H,F_1))) ) ).

fof(fact_19_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_275383677t_bool(H,F_1))) ) ).

fof(fact_20_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_892525857t_bool(H,F_1))) ) ).

fof(fact_21_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_637364955e_bool(H,F_1))) ) ).

fof(fact_22_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_int_fun_a_bool(H,F_1))) ) ).

fof(fact_23_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_a_pname(H,F_1))) ) ).

fof(fact_24_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_62782965_pname(H,F_1))) ) ).

fof(fact_25_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1921560913_pname(H,F_1))) ) ).

fof(fact_26_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1283814551_pname(H,F_1))) ) ).

fof(fact_27_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1854862208_pname(H,F_1))) ) ).

fof(fact_28_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_a_nat(H,F_1))) ) ).

fof(fact_29_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1431733819ol_nat(H,F_1))) ) ).

fof(fact_30_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_496248727ol_nat(H,F_1))) ) ).

fof(fact_31_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1551609309ol_nat(H,F_1))) ) ).

fof(fact_32_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_fun_a_bool_nat(H,F_1))) ) ).

fof(fact_33_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_a_int(H,F_1))) ) ).

fof(fact_34_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_2069007511ol_int(H,F_1))) ) ).

fof(fact_35_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_1133522419ol_int(H,F_1))) ) ).

fof(fact_36_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_41399353ol_int(H,F_1))) ) ).

fof(fact_37_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_fun_a_bool_int(H,F_1))) ) ).

fof(fact_38_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(H,F_1))) ) ).

fof(fact_39_finite__imageI,axiom,
    ! [H,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(H,F_1))) ) ).

fof(fact_40_finite_OinsertI,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,A_2),A))) ) ).

fof(fact_41_finite_OinsertI,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_2),A))) ) ).

fof(fact_42_finite_OinsertI,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,A_2),A))) ) ).

fof(fact_43_finite_OinsertI,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,A_2),A))) ) ).

fof(fact_44_finite_OinsertI,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A))) ) ).

fof(fact_45_finite_OinsertI,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A))) ) ).

fof(fact_46_finite_OinsertI,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A))) ) ).

fof(fact_47_finite_OinsertI,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A))) ) ).

fof(fact_48_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_pname_pname(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

fof(fact_49_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_a_a(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

fof(fact_50_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_int_bool_a(F,A))),hAPP_f521865025ol_nat(finite1876863882t_bool,A))) ) ).

fof(fact_51_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_nat_bool_a(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).

fof(fact_52_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_876012084bool_a(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).

fof(fact_53_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_a_bool_a(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).

fof(fact_54_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_pname_nat(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

fof(fact_55_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_a_nat(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

fof(fact_56_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_1431733819ol_nat(F,A))),hAPP_f521865025ol_nat(finite1876863882t_bool,A))) ) ).

fof(fact_57_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_496248727ol_nat(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).

fof(fact_58_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_1551609309ol_nat(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).

fof(fact_59_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_fun_a_bool_nat(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).

fof(fact_60_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_pname_int(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

fof(fact_61_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_a_int(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

fof(fact_62_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_2069007511ol_int(F,A))),hAPP_f521865025ol_nat(finite1876863882t_bool,A))) ) ).

fof(fact_63_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_1133522419ol_int(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).

fof(fact_64_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_41399353ol_int(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).

fof(fact_65_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_fun_a_bool_int(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).

fof(fact_66_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_a_pname(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

fof(fact_67_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_nat_pname(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).

fof(fact_68_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_int_pname(F,A))),hAPP_f957591787ol_nat(finite_card_int,A))) ) ).

fof(fact_69_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

fof(fact_70_card__image__le,axiom,
    ! [F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,image_nat_int(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).

fof(fact_71_card__mono,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f521865025ol_nat(finite1876863882t_bool,A)),hAPP_f521865025ol_nat(finite1876863882t_bool,B))) ) ) ).

fof(fact_72_card__mono,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A)),hAPP_f696928925ol_nat(finite346522414t_bool,B))) ) ) ).

fof(fact_73_card__mono,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A)),hAPP_f55526627ol_nat(finite1340463720e_bool,B))) ) ) ).

fof(fact_74_card__mono,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)),hAPP_f2009550088ol_nat(finite1306199131a_bool,B))) ) ) ).

fof(fact_75_card__mono,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,B))) ) ) ).

fof(fact_76_card__mono,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,B))) ) ) ).

fof(fact_77_card__mono,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,B))) ) ) ).

fof(fact_78_card__mono,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,A)),hAPP_f957591787ol_nat(finite_card_int,B))) ) ) ).

fof(fact_79_card__seteq,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f521865025ol_nat(finite1876863882t_bool,B)),hAPP_f521865025ol_nat(finite1876863882t_bool,A)))
         => A = B ) ) ) ).

fof(fact_80_card__seteq,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,B)),hAPP_f696928925ol_nat(finite346522414t_bool,A)))
         => A = B ) ) ) ).

fof(fact_81_card__seteq,axiom,
    ! [A,B] :
      ( ( is_fun1661590463l_bool(A)
        & is_fun1661590463l_bool(B) )
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
       => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,B)),hAPP_f55526627ol_nat(finite1340463720e_bool,A)))
           => A = B ) ) ) ) ).

fof(fact_82_card__seteq,axiom,
    ! [A,B] :
      ( ( is_fun949378684l_bool(A)
        & is_fun949378684l_bool(B) )
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
       => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,B)),hAPP_f2009550088ol_nat(finite1306199131a_bool,A)))
           => A = B ) ) ) ) ).

fof(fact_83_card__seteq,axiom,
    ! [A,B] :
      ( ( is_fun_pname_bool(A)
        & is_fun_pname_bool(B) )
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,B)),hAPP_f921600141ol_nat(finite_card_pname,A)))
           => A = B ) ) ) ) ).

fof(fact_84_card__seteq,axiom,
    ! [A,B] :
      ( ( is_fun_a_bool(A)
        & is_fun_a_bool(B) )
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
         => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,B)),hAPP_fun_a_bool_nat(finite_card_a,A)))
           => A = B ) ) ) ) ).

fof(fact_85_card__seteq,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,B)),hAPP_f22106695ol_nat(finite_card_nat,A)))
         => A = B ) ) ) ).

fof(fact_86_card__seteq,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,B)),hAPP_f957591787ol_nat(finite_card_int,A)))
         => A = B ) ) ) ).

fof(fact_87_card__insert__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f521865025ol_nat(finite1876863882t_bool,A)),hAPP_f521865025ol_nat(finite1876863882t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,X_1),A)))) ) ).

fof(fact_88_card__insert__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A)),hAPP_f696928925ol_nat(finite346522414t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_1),A)))) ) ).

fof(fact_89_card__insert__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A)),hAPP_f55526627ol_nat(finite1340463720e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_1),A)))) ) ).

fof(fact_90_card__insert__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)),hAPP_f2009550088ol_nat(finite1306199131a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,X_1),A)))) ) ).

fof(fact_91_card__insert__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)))) ) ).

fof(fact_92_card__insert__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)))) ) ).

fof(fact_93_card__insert__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,A)),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)))) ) ).

fof(fact_94_card__insert__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)))) ) ).

fof(fact_95_card__insert__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A))
         => hAPP_f521865025ol_nat(finite1876863882t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,X_1),A)) = hAPP_f521865025ol_nat(finite1876863882t_bool,A) )
        & ( ~ hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A))
         => hAPP_f521865025ol_nat(finite1876863882t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f521865025ol_nat(finite1876863882t_bool,A)) ) ) ) ).

fof(fact_96_card__insert__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),A))
         => hAPP_f696928925ol_nat(finite346522414t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_1),A)) = hAPP_f696928925ol_nat(finite346522414t_bool,A) )
        & ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),A))
         => hAPP_f696928925ol_nat(finite346522414t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A)) ) ) ) ).

fof(fact_97_card__insert__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),A))
         => hAPP_f55526627ol_nat(finite1340463720e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_1),A)) = hAPP_f55526627ol_nat(finite1340463720e_bool,A) )
        & ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),A))
         => hAPP_f55526627ol_nat(finite1340463720e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A)) ) ) ) ).

fof(fact_98_card__insert__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( ( hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),A))
         => hAPP_f2009550088ol_nat(finite1306199131a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,X_1),A)) = hAPP_f2009550088ol_nat(finite1306199131a_bool,A) )
        & ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),A))
         => hAPP_f2009550088ol_nat(finite1306199131a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)) ) ) ) ).

fof(fact_99_card__insert__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_f921600141ol_nat(finite_card_pname,A) )
        & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A)) ) ) ) ).

fof(fact_100_card__insert__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_f22106695ol_nat(finite_card_nat,A) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A)) ) ) ) ).

fof(fact_101_card__insert__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_f957591787ol_nat(finite_card_int,A) )
        & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,A)) ) ) ) ).

fof(fact_102_card__insert__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_fun_a_bool_nat(finite_card_a,A) )
        & ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A)) ) ) ) ).

fof(fact_103_card__insert__disjoint,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( ~ hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A))
       => hAPP_f521865025ol_nat(finite1876863882t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f521865025ol_nat(finite1876863882t_bool,A)) ) ) ).

fof(fact_104_card__insert__disjoint,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),A))
       => hAPP_f696928925ol_nat(finite346522414t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A)) ) ) ).

fof(fact_105_card__insert__disjoint,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),A))
       => hAPP_f55526627ol_nat(finite1340463720e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A)) ) ) ).

fof(fact_106_card__insert__disjoint,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),A))
       => hAPP_f2009550088ol_nat(finite1306199131a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,X_1),A)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)) ) ) ).

fof(fact_107_card__insert__disjoint,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A)) ) ) ).

fof(fact_108_card__insert__disjoint,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A)) ) ) ).

fof(fact_109_card__insert__disjoint,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,A)) ) ) ).

fof(fact_110_card__insert__disjoint,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A)) ) ) ).

fof(fact_111_finite__Collect__conjI,axiom,
    ! [Q_3,Pa] :
      ( ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa)))
        | hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_3))) )
     => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,Pa),Q_3)))) ) ).

fof(fact_112_finite__Collect__conjI,axiom,
    ! [Q_3,Pa] :
      ( ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(Pa)))
        | hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(Q_3))) )
     => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(cOMBS_1912792529l_bool(cOMBB_1974511970t_bool(fconj,Pa),Q_3)))) ) ).

fof(fact_113_finite__Collect__conjI,axiom,
    ! [Q_3,Pa] :
      ( ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa)))
        | hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_3))) )
     => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,Pa),Q_3)))) ) ).

fof(fact_114_finite__Collect__conjI,axiom,
    ! [Q_3,Pa] :
      ( ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa)))
        | hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_3))) )
     => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,Pa),Q_3)))) ) ).

fof(fact_115_finite__Collect__conjI,axiom,
    ! [Q_3,Pa] :
      ( ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa)))
        | hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_3))) )
     => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,Pa),Q_3)))) ) ).

fof(fact_116_finite__Collect__conjI,axiom,
    ! [Q_3,Pa] :
      ( ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa)))
        | hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_3))) )
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,Pa),Q_3)))) ) ).

fof(fact_117_finite__Collect__conjI,axiom,
    ! [Q_3,Pa] :
      ( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
        | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_3))) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,Pa),Q_3)))) ) ).

fof(fact_118_finite__Collect__conjI,axiom,
    ! [Q_3,Pa] :
      ( ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(Pa)))
        | hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(Q_3))) )
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,Pa),Q_3)))) ) ).

fof(fact_119_Suc__diff__le,axiom,
    ! [N,M_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) ) ).

fof(fact_120_finite__Collect__le__nat,axiom,
    ! [K] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),K)))) ).

fof(fact_121_card__Collect__le__nat,axiom,
    ! [Na] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),Na))) = hAPP_nat_nat(suc,Na) ).

fof(fact_122_Suc__inject,axiom,
    ! [X,Y] :
      ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y)
     => X = Y ) ).

fof(fact_123_nat_Oinject,axiom,
    ! [Nat_3,Nat_4] :
      ( hAPP_nat_nat(suc,Nat_3) = hAPP_nat_nat(suc,Nat_4)
    <=> Nat_3 = Nat_4 ) ).

fof(fact_124_Suc__n__not__n,axiom,
    ! [N] : hAPP_nat_nat(suc,N) != N ).

fof(fact_125_n__not__Suc__n,axiom,
    ! [N] : N != hAPP_nat_nat(suc,N) ).

fof(fact_126_le__antisym,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
       => M_1 = N ) ) ).

fof(fact_127_le__trans,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1)) ) ) ).

fof(fact_128_eq__imp__le,axiom,
    ! [M_1,N] :
      ( M_1 = N
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

fof(fact_129_nat__le__linear,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1)) ) ).

fof(fact_130_le__refl,axiom,
    ! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).

fof(fact_131_diff__commute,axiom,
    ! [I_1,J,K_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),K_1)),J) ).

fof(fact_132_finite__Collect__disjI,axiom,
    ! [Pa,Q_3] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa)))
        & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_3))) ) ) ).

fof(fact_133_finite__Collect__disjI,axiom,
    ! [Pa,Q_3] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(cOMBS_1912792529l_bool(cOMBB_1974511970t_bool(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(Pa)))
        & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,collect_fun_int_bool(Q_3))) ) ) ).

fof(fact_134_finite__Collect__disjI,axiom,
    ! [Pa,Q_3] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa)))
        & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_3))) ) ) ).

fof(fact_135_finite__Collect__disjI,axiom,
    ! [Pa,Q_3] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa)))
        & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_3))) ) ) ).

fof(fact_136_finite__Collect__disjI,axiom,
    ! [Pa,Q_3] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa)))
        & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_3))) ) ) ).

fof(fact_137_finite__Collect__disjI,axiom,
    ! [Pa,Q_3] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa)))
        & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_3))) ) ) ).

fof(fact_138_finite__Collect__disjI,axiom,
    ! [Pa,Q_3] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
        & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_3))) ) ) ).

fof(fact_139_finite__Collect__disjI,axiom,
    ! [Pa,Q_3] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fdisj,Pa),Q_3))))
    <=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(Pa)))
        & hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(Q_3))) ) ) ).

fof(fact_140_finite__insert,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,hAPP_f1948010709l_bool(hAPP_f587666462l_bool(insert_fun_int_bool,A_2),A)))
    <=> hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A)) ) ).

fof(fact_141_finite__insert,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,hAPP_f1246832597l_bool(hAPP_f633452666l_bool(insert_fun_nat_bool,A_2),A)))
    <=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ).

fof(fact_142_finite__insert,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,hAPP_f559147733l_bool(hAPP_f1320879424l_bool(insert1325755072e_bool,A_2),A)))
    <=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ).

fof(fact_143_finite__insert,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,hAPP_f2117159681l_bool(hAPP_f1531278575l_bool(insert_fun_a_bool,A_2),A)))
    <=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ).

fof(fact_144_finite__insert,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A)))
    <=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ).

fof(fact_145_finite__insert,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A)))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ).

fof(fact_146_finite__insert,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A)))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ).

fof(fact_147_finite__insert,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A)))
    <=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ).

fof(fact_148_finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,A),B))
     => ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A)) ) ) ).

fof(fact_149_finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
     => ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ) ).

fof(fact_150_finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ) ).

fof(fact_151_finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ) ).

fof(fact_152_finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ) ).

fof(fact_153_finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ) ).

fof(fact_154_finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

fof(fact_155_finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

fof(fact_156_rev__finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,A),B))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A)) ) ) ).

fof(fact_157_rev__finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ) ).

fof(fact_158_rev__finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ) ).

fof(fact_159_rev__finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ) ).

fof(fact_160_rev__finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ) ).

fof(fact_161_rev__finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ) ).

fof(fact_162_rev__finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

fof(fact_163_rev__finite__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

fof(fact_164_Suc__leD,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

fof(fact_165_le__SucE,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
       => M_1 = hAPP_nat_nat(suc,N) ) ) ).

fof(fact_166_le__SucI,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N))) ) ).

fof(fact_167_Suc__le__mono,axiom,
    ! [Na,M] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),M)) ) ).

fof(fact_168_le__Suc__eq,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
        | M = hAPP_nat_nat(suc,Na) ) ) ).

fof(fact_169_not__less__eq__eq,axiom,
    ! [M,Na] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),M)) ) ).

fof(fact_170_Suc__n__not__le__n,axiom,
    ! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),N)) ).

fof(fact_171_Suc__diff__diff,axiom,
    ! [M_1,N,K_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),N)),hAPP_nat_nat(suc,K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),K_1) ).

fof(fact_172_diff__Suc__Suc,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) ).

fof(fact_173_le__diff__iff,axiom,
    ! [Na,K,M] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ) ) ).

fof(fact_174_Nat_Odiff__diff__eq,axiom,
    ! [N,K_1,M_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
       => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) ) ) ).

fof(fact_175_eq__diff__iff,axiom,
    ! [Na,K,M] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
       => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K)
        <=> M = Na ) ) ) ).

fof(fact_176_diff__diff__cancel,axiom,
    ! [I_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),N))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I_1)) = I_1 ) ).

fof(fact_177_diff__le__mono,axiom,
    ! [L_1,M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),L_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),L_1))) ) ).

fof(fact_178_diff__le__mono2,axiom,
    ! [L_1,M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M_1))) ) ).

fof(fact_179_diff__le__self,axiom,
    ! [M_1,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),M_1)) ).

fof(fact_180_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_a_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

fof(fact_181_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_int_bool_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

fof(fact_182_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_nat_bool_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

fof(fact_183_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_876012084bool_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

fof(fact_184_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_a_bool_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

fof(fact_185_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_1512837979t_bool(F,A)))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B)) ) ) ).

fof(fact_186_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_2129980159t_bool(F,A)))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) ) ) ).

fof(fact_187_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_47868345e_bool(F,A)))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) ) ) ).

fof(fact_188_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_112932426a_bool(F,A)))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) ) ) ).

fof(fact_189_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_pname_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

fof(fact_190_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_nat_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

fof(fact_191_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_1556378401t_bool(F,A)))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B)) ) ) ).

fof(fact_192_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_26036933t_bool(F,A)))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) ) ) ).

fof(fact_193_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_1655916159e_bool(F,A)))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) ) ) ).

fof(fact_194_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_nat_fun_a_bool(F,A)))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) ) ) ).

fof(fact_195_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_nat_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

fof(fact_196_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_int_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

fof(fact_197_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_275383677t_bool(F,A)))
       => hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B)) ) ) ).

fof(fact_198_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_892525857t_bool(F,A)))
       => hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) ) ) ).

fof(fact_199_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_637364955e_bool(F,A)))
       => hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) ) ) ).

fof(fact_200_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_int_fun_a_bool(F,A)))
       => hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) ) ) ).

fof(fact_201_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_int_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

fof(fact_202_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_a_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

fof(fact_203_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_62782965_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

fof(fact_204_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1921560913_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

fof(fact_205_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1283814551_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

fof(fact_206_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1854862208_pname(F,A)))
       => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) ) ) ).

fof(fact_207_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_a_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

fof(fact_208_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_1431733819ol_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

fof(fact_209_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_496248727ol_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

fof(fact_210_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_1551609309ol_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

fof(fact_211_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_fun_a_bool_nat(F,A)))
       => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) ) ) ).

fof(fact_212_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_a_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

fof(fact_213_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_2069007511ol_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

fof(fact_214_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_1133522419ol_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

fof(fact_215_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_41399353ol_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

fof(fact_216_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_fun_a_bool_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

fof(fact_217_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
       => hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).

fof(fact_218_finite__surj,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
       => hBOOL(hAPP_f448129468l_bool(finite_finite_int,B)) ) ) ).

fof(fact_219_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_int_bool_a(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,C_16),A))
              & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,C_16))
              & B = image_fun_int_bool_a(F,C_16) ) ) ) ) ).

fof(fact_220_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_nat_bool_a(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_16),A))
              & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_16))
              & B = image_fun_nat_bool_a(F,C_16) ) ) ) ) ).

fof(fact_221_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_876012084bool_a(F,A)))
         => ? [C_16] :
              ( is_fun1661590463l_bool(C_16)
              & hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_16),A))
              & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_16))
              & B = image_876012084bool_a(F,C_16) ) ) ) ) ).

fof(fact_222_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_fun_a_bool_a(F,A)))
         => ? [C_16] :
              ( is_fun949378684l_bool(C_16)
              & hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_16),A))
              & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_16))
              & B = image_fun_a_bool_a(F,C_16) ) ) ) ) ).

fof(fact_223_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_a_a(F,A)))
         => ? [C_16] :
              ( is_fun_a_bool(C_16)
              & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
              & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
              & B = image_a_a(F,C_16) ) ) ) ) ).

fof(fact_224_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_a_fun_int_bool(F,A)))
       => ? [C_16] :
            ( is_fun_a_bool(C_16)
            & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & B = image_a_fun_int_bool(F,C_16) ) ) ) ).

fof(fact_225_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_a_fun_nat_bool(F,A)))
       => ? [C_16] :
            ( is_fun_a_bool(C_16)
            & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & B = image_a_fun_nat_bool(F,C_16) ) ) ) ).

fof(fact_226_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun1661590463l_bool(B)
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
       => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_819518260e_bool(F,A)))
         => ? [C_16] :
              ( is_fun_a_bool(C_16)
              & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
              & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
              & B = image_819518260e_bool(F,C_16) ) ) ) ) ).

fof(fact_227_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun949378684l_bool(B)
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
       => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_a_fun_a_bool(F,A)))
         => ? [C_16] :
              ( is_fun_a_bool(C_16)
              & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
              & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
              & B = image_a_fun_a_bool(F,C_16) ) ) ) ) ).

fof(fact_228_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_a_pname(F,A)))
         => ? [C_16] :
              ( is_fun_a_bool(C_16)
              & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
              & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
              & B = image_a_pname(F,C_16) ) ) ) ) ).

fof(fact_229_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_62782965_pname(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,C_16),A))
              & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,C_16))
              & B = image_62782965_pname(F,C_16) ) ) ) ) ).

fof(fact_230_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1921560913_pname(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_16),A))
              & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_16))
              & B = image_1921560913_pname(F,C_16) ) ) ) ) ).

fof(fact_231_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1283814551_pname(F,A)))
         => ? [C_16] :
              ( is_fun1661590463l_bool(C_16)
              & hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_16),A))
              & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_16))
              & B = image_1283814551_pname(F,C_16) ) ) ) ) ).

fof(fact_232_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_1854862208_pname(F,A)))
         => ? [C_16] :
              ( is_fun949378684l_bool(C_16)
              & hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_16),A))
              & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_16))
              & B = image_1854862208_pname(F,C_16) ) ) ) ) ).

fof(fact_233_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_a_nat(F,A)))
       => ? [C_16] :
            ( is_fun_a_bool(C_16)
            & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & B = image_a_nat(F,C_16) ) ) ) ).

fof(fact_234_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_1431733819ol_nat(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,C_16),A))
            & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,C_16))
            & B = image_1431733819ol_nat(F,C_16) ) ) ) ).

fof(fact_235_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_496248727ol_nat(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_16),A))
            & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_16))
            & B = image_496248727ol_nat(F,C_16) ) ) ) ).

fof(fact_236_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_1551609309ol_nat(F,A)))
       => ? [C_16] :
            ( is_fun1661590463l_bool(C_16)
            & hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_16),A))
            & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_16))
            & B = image_1551609309ol_nat(F,C_16) ) ) ) ).

fof(fact_237_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_fun_a_bool_nat(F,A)))
       => ? [C_16] :
            ( is_fun949378684l_bool(C_16)
            & hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_16),A))
            & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_16))
            & B = image_fun_a_bool_nat(F,C_16) ) ) ) ).

fof(fact_238_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),image_pname_nat(F,A)))
       => ? [C_16] :
            ( is_fun_pname_bool(C_16)
            & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & B = image_pname_nat(F,C_16) ) ) ) ).

fof(fact_239_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_a_int(F,A)))
       => ? [C_16] :
            ( is_fun_a_bool(C_16)
            & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_16),A))
            & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_16))
            & B = image_a_int(F,C_16) ) ) ) ).

fof(fact_240_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_2069007511ol_int(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,C_16),A))
            & hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,C_16))
            & B = image_2069007511ol_int(F,C_16) ) ) ) ).

fof(fact_241_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_1133522419ol_int(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_16),A))
            & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_16))
            & B = image_1133522419ol_int(F,C_16) ) ) ) ).

fof(fact_242_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_41399353ol_int(F,A)))
       => ? [C_16] :
            ( is_fun1661590463l_bool(C_16)
            & hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_16),A))
            & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_16))
            & B = image_41399353ol_int(F,C_16) ) ) ) ).

fof(fact_243_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_fun_a_bool_int(F,A)))
       => ? [C_16] :
            ( is_fun949378684l_bool(C_16)
            & hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_16),A))
            & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_16))
            & B = image_fun_a_bool_int(F,C_16) ) ) ) ).

fof(fact_244_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_pname_int(F,A)))
       => ? [C_16] :
            ( is_fun_pname_bool(C_16)
            & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & B = image_pname_int(F,C_16) ) ) ) ).

fof(fact_245_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_1512837979t_bool(F,A)))
       => ? [C_16] :
            ( is_fun_pname_bool(C_16)
            & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & B = image_1512837979t_bool(F,C_16) ) ) ) ).

fof(fact_246_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_2129980159t_bool(F,A)))
       => ? [C_16] :
            ( is_fun_pname_bool(C_16)
            & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
            & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
            & B = image_2129980159t_bool(F,C_16) ) ) ) ).

fof(fact_247_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun1661590463l_bool(B)
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
       => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_47868345e_bool(F,A)))
         => ? [C_16] :
              ( is_fun_pname_bool(C_16)
              & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
              & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
              & B = image_47868345e_bool(F,C_16) ) ) ) ) ).

fof(fact_248_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun949378684l_bool(B)
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
       => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_112932426a_bool(F,A)))
         => ? [C_16] :
              ( is_fun_pname_bool(C_16)
              & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
              & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
              & B = image_112932426a_bool(F,C_16) ) ) ) ) ).

fof(fact_249_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_pname_pname(F,A)))
         => ? [C_16] :
              ( is_fun_pname_bool(C_16)
              & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
              & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
              & B = image_pname_pname(F,C_16) ) ) ) ) ).

fof(fact_250_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_nat_a(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
              & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
              & B = image_nat_a(F,C_16) ) ) ) ) ).

fof(fact_251_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_1556378401t_bool(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & B = image_1556378401t_bool(F,C_16) ) ) ) ).

fof(fact_252_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_26036933t_bool(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & B = image_26036933t_bool(F,C_16) ) ) ) ).

fof(fact_253_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun1661590463l_bool(B)
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
       => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_1655916159e_bool(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
              & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
              & B = image_1655916159e_bool(F,C_16) ) ) ) ) ).

fof(fact_254_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun949378684l_bool(B)
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
       => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_nat_fun_a_bool(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
              & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
              & B = image_nat_fun_a_bool(F,C_16) ) ) ) ) ).

fof(fact_255_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_nat_pname(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
              & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
              & B = image_nat_pname(F,C_16) ) ) ) ) ).

fof(fact_256_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_int_a(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
              & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
              & B = image_int_a(F,C_16) ) ) ) ) ).

fof(fact_257_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,B))
     => ( hBOOL(hAPP_f215623910l_bool(hAPP_f317860125l_bool(ord_le1848167016l_bool,B),image_275383677t_bool(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
            & B = image_275383677t_bool(F,C_16) ) ) ) ).

fof(fact_258_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
     => ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B),image_892525857t_bool(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
            & B = image_892525857t_bool(F,C_16) ) ) ) ).

fof(fact_259_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun1661590463l_bool(B)
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
       => ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B),image_637364955e_bool(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
              & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
              & B = image_637364955e_bool(F,C_16) ) ) ) ) ).

fof(fact_260_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun949378684l_bool(B)
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
       => ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B),image_int_fun_a_bool(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
              & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
              & B = image_int_fun_a_bool(F,C_16) ) ) ) ) ).

fof(fact_261_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
       => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),image_int_pname(F,A)))
         => ? [C_16] :
              ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_16),A))
              & hBOOL(hAPP_f448129468l_bool(finite_finite_int,C_16))
              & B = image_int_pname(F,C_16) ) ) ) ) ).

fof(fact_262_finite__subset__image,axiom,
    ! [F,A,B] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
         => ? [C_16] :
              ( is_fun_pname_bool(C_16)
              & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_16),A))
              & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_16))
              & B = image_pname_a(F,C_16) ) ) ) ) ).

fof(fact_263_finite__subset__image,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
       => ? [C_16] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_16),A))
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_16))
            & B = image_nat_int(F,C_16) ) ) ) ).

fof(fact_264_lift__Suc__mono__le,axiom,
    ! [Na,N_3,F] :
      ( ! [N_1] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,N_1)),hAPP_n1025906991e_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,Na)),hAPP_n1025906991e_bool(F,N_3))) ) ) ).

fof(fact_265_lift__Suc__mono__le,axiom,
    ! [Na,N_3,F] :
      ( ! [N_1] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,N_1)),hAPP_nat_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,Na)),hAPP_nat_bool(F,N_3))) ) ) ).

fof(fact_266_lift__Suc__mono__le,axiom,
    ! [Na,N_3,F] :
      ( ! [N_1] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,N_1)),hAPP_nat_fun_a_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,Na)),hAPP_nat_fun_a_bool(F,N_3))) ) ) ).

fof(fact_267_lift__Suc__mono__le,axiom,
    ! [Na,N_3,F] :
      ( ! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_3))) ) ) ).

fof(fact_268_lift__Suc__mono__le,axiom,
    ! [Na,N_3,F] :
      ( ! [N_1] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,N_1)),hAPP_nat_int(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,Na)),hAPP_nat_int(F,N_3))) ) ) ).

fof(fact_269_lift__Suc__mono__le,axiom,
    ! [Na,N_3,F] :
      ( ! [N_1] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,N_1)),hAPP_n1699378549t_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,Na)),hAPP_n1699378549t_bool(F,N_3))) ) ) ).

fof(fact_270_lift__Suc__mono__le,axiom,
    ! [Na,N_3,F] :
      ( ! [N_1] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_n1082236369t_bool(F,N_1)),hAPP_n1082236369t_bool(F,hAPP_nat_nat(suc,N_1))))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_n1082236369t_bool(F,Na)),hAPP_n1082236369t_bool(F,N_3))) ) ) ).

fof(fact_271_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_1556378401t_bool(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_f274826591t_bool(cOMBC_620423604l_bool(cOMBB_1123824708ol_nat(fequal_fun_int_bool,F)),hAPP_n1082236369t_bool(F,X_2)))))) ) ) ) ).

fof(fact_272_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_26036933t_bool(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_2026977092ol_nat(fequal_fun_nat_bool,F)),hAPP_n1699378549t_bool(F,X_2)))))) ) ) ) ).

fof(fact_273_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_1655916159e_bool(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_f1066163005t_bool(cOMBC_386238098l_bool(cOMBB_2117322052ol_nat(fequal533582459e_bool,F)),hAPP_n1025906991e_bool(F,X_2)))))) ) ) ) ).

fof(fact_274_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_nat_fun_a_bool(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_f282463732t_bool(cOMBC_1928494297l_bool(cOMBB_1823939024ol_nat(fequal_fun_a_bool,F)),hAPP_nat_fun_a_bool(F,X_2)))))) ) ) ) ).

fof(fact_275_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_int_a(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_a_fun_int_bool(cOMBC_int_a_bool(cOMBB_375279840ol_int(fequal_a,F)),hAPP_int_a(F,X_2)))))) ) ) ) ).

fof(fact_276_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,image_275383677t_bool(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(cOMBB_1761098400ol_int(fequal_fun_int_bool,F)),hAPP_i1948725293t_bool(F,X_2)))))) ) ) ) ).

fof(fact_277_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_892525857t_bool(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_f183368031t_bool(cOMBC_1848398772l_bool(cOMBB_516767136ol_int(fequal_fun_nat_bool,F)),hAPP_i418383825t_bool(F,X_2)))))) ) ) ) ).

fof(fact_278_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_637364955e_bool(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_f449020825t_bool(cOMBC_316919022l_bool(cOMBB_607112096ol_int(fequal533582459e_bool,F)),hAPP_i7355787e_bool(F,X_2)))))) ) ) ) ).

fof(fact_279_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_int_fun_a_bool(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
            & ~ hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),hAPP_f1812805200t_bool(cOMBC_1753430397l_bool(cOMBB_313729068ol_int(fequal_fun_a_bool,F)),hAPP_int_fun_a_bool(F,X_2)))))) ) ) ) ).

fof(fact_280_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(F,A)))
       => ? [X_2] :
            ( is_pname(X_2)
            & hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
            & ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A)),hAPP_a93125764e_bool(cOMBC_pname_a_bool(cOMBB_1897541054_pname(fequal_a,F)),hAPP_pname_a(F,X_2)))))) ) ) ) ).

fof(fact_281_pigeonhole__infinite,axiom,
    ! [F,A] :
      ( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(F,A)))
       => ? [X_2] :
            ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
            & ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),hAPP_i418383825t_bool(cOMBC_nat_int_bool(cOMBB_1628441366ol_nat(fequal_int,F)),hAPP_nat_int(F,X_2)))))) ) ) ) ).

fof(fact_282_image__eqI,axiom,
    ! [A,B_3,F,X_1] :
      ( B_3 = hAPP_nat_int(F,X_1)
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),image_nat_int(F,A))) ) ) ).

fof(fact_283_image__eqI,axiom,
    ! [A,B_3,F,X_1] :
      ( B_3 = hAPP_pname_a(F,X_1)
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),image_pname_a(F,A))) ) ) ).

fof(fact_284_equalityI,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
       => A = B ) ) ).

fof(fact_285_equalityI,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
       => A = B ) ) ).

fof(fact_286_equalityI,axiom,
    ! [A,B] :
      ( ( is_fun_a_bool(A)
        & is_fun_a_bool(B) )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A))
         => A = B ) ) ) ).

fof(fact_287_subsetD,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B)) ) ) ).

fof(fact_288_subsetD,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B)) ) ) ).

fof(fact_289_subsetD,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B)) ) ) ).

fof(fact_290_subsetD,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B)) ) ) ).

fof(fact_291_insertCI,axiom,
    ! [B_3,A_2,B] :
      ( ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),B))
       => A_2 = B_3 )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),B))) ) ).

fof(fact_292_insertCI,axiom,
    ! [B_3,A_2,B] :
      ( ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),B))
       => A_2 = B_3 )
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),B))) ) ).

fof(fact_293_insertCI,axiom,
    ! [B_3,A_2,B] :
      ( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B))
       => A_2 = B_3 )
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),B))) ) ).

fof(fact_294_insertCI,axiom,
    ! [B_3,A_2,B] :
      ( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B))
       => A_2 = B_3 )
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_3),B))) ) ).

fof(fact_295_insertE,axiom,
    ! [A_2,B_3,A] :
      ( ( is_a(A_2)
        & is_a(B_3) )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),A)))
       => ( A_2 != B_3
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A)) ) ) ) ).

fof(fact_296_insertE,axiom,
    ! [A_2,B_3,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),A)))
     => ( A_2 != B_3
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A)) ) ) ).

fof(fact_297_insertE,axiom,
    ! [A_2,B_3,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),A)))
     => ( A_2 != B_3
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A)) ) ) ).

fof(fact_298_insertE,axiom,
    ! [A_2,B_3,A] :
      ( ( is_pname(A_2)
        & is_pname(B_3) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_3),A)))
       => ( A_2 != B_3
         => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A)) ) ) ) ).

fof(fact_299_zero__induct__lemma,axiom,
    ! [I_2,Pa,K] :
      ( hBOOL(hAPP_nat_bool(Pa,K))
     => ( ! [N_1] :
            ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_1)))
           => hBOOL(hAPP_nat_bool(Pa,N_1)) )
       => hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K),I_2))) ) ) ).

fof(fact_300_Suc__le__D,axiom,
    ! [N,M_4] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M_4))
     => ? [M_2] : M_4 = hAPP_nat_nat(suc,M_2) ) ).

fof(fact_301_order__refl,axiom,
    ! [X_1] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),X_1)) ).

fof(fact_302_order__refl,axiom,
    ! [X_1] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),X_1)) ).

fof(fact_303_order__refl,axiom,
    ! [X_11] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_11),X_11)) ).

fof(fact_304_order__refl,axiom,
    ! [X_11] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_11),X_11)) ).

fof(fact_305_order__refl,axiom,
    ! [X_1] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),X_1)) ).

fof(fact_306_linorder__linear,axiom,
    ! [X_10,Y_9] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_10),Y_9))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_9),X_10)) ) ).

fof(fact_307_linorder__linear,axiom,
    ! [X_10,Y_9] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_10),Y_9))
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_9),X_10)) ) ).

fof(fact_308_order__eq__iff,axiom,
    ! [X_1,Y_1] :
      ( X_1 = Y_1
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1)) ) ) ).

fof(fact_309_order__eq__iff,axiom,
    ! [X_1,Y_1] :
      ( X_1 = Y_1
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1)) ) ) ).

fof(fact_310_order__eq__iff,axiom,
    ! [X_1,Y_1] :
      ( X_1 = Y_1
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
        & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1)) ) ) ).

fof(fact_311_order__eq__iff,axiom,
    ! [X_1,Y_1] :
      ( X_1 = Y_1
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1)) ) ) ).

fof(fact_312_order__eq__iff,axiom,
    ! [X_1,Y_1] :
      ( ( is_fun_a_bool(X_1)
        & is_fun_a_bool(Y_1) )
     => ( X_1 = Y_1
      <=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
          & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1)) ) ) ) ).

fof(fact_313_order__eq__refl,axiom,
    ! [X_1,Y_1] :
      ( X_1 = Y_1
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1)) ) ).

fof(fact_314_order__eq__refl,axiom,
    ! [X_1,Y_1] :
      ( X_1 = Y_1
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1)) ) ).

fof(fact_315_order__eq__refl,axiom,
    ! [X_9,Y_8] :
      ( X_9 = Y_8
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_9),Y_8)) ) ).

fof(fact_316_order__eq__refl,axiom,
    ! [X_9,Y_8] :
      ( X_9 = Y_8
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_9),Y_8)) ) ).

fof(fact_317_order__eq__refl,axiom,
    ! [X_1,Y_1] :
      ( X_1 = Y_1
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1)) ) ).

fof(fact_318_order__antisym__conv,axiom,
    ! [Y_1,X_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
      <=> X_1 = Y_1 ) ) ).

fof(fact_319_order__antisym__conv,axiom,
    ! [Y_1,X_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
      <=> X_1 = Y_1 ) ) ).

fof(fact_320_order__antisym__conv,axiom,
    ! [Y_1,X_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),X_1))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_1))
      <=> X_1 = Y_1 ) ) ).

fof(fact_321_order__antisym__conv,axiom,
    ! [Y_1,X_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1))
      <=> X_1 = Y_1 ) ) ).

fof(fact_322_order__antisym__conv,axiom,
    ! [Y_1,X_1] :
      ( ( is_fun_a_bool(Y_1)
        & is_fun_a_bool(X_1) )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
        <=> X_1 = Y_1 ) ) ) ).

fof(fact_323_ord__eq__le__trans,axiom,
    ! [C_6,A_2,B_3] :
      ( A_2 = B_3
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_3),C_6))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),C_6)) ) ) ).

fof(fact_324_ord__eq__le__trans,axiom,
    ! [C_6,A_2,B_3] :
      ( A_2 = B_3
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_3),C_6))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),C_6)) ) ) ).

fof(fact_325_ord__eq__le__trans,axiom,
    ! [C_15,A_21,B_19] :
      ( A_21 = B_19
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),C_15))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_21),C_15)) ) ) ).

fof(fact_326_ord__eq__le__trans,axiom,
    ! [C_15,A_21,B_19] :
      ( A_21 = B_19
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),C_15))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_21),C_15)) ) ) ).

fof(fact_327_ord__eq__le__trans,axiom,
    ! [C_6,A_2,B_3] :
      ( A_2 = B_3
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_3),C_6))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),C_6)) ) ) ).

fof(fact_328_xt1_I3_J,axiom,
    ! [C_6,A_2,B_3] :
      ( A_2 = B_3
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_6),B_3))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_6),A_2)) ) ) ).

fof(fact_329_xt1_I3_J,axiom,
    ! [C_6,A_2,B_3] :
      ( A_2 = B_3
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_6),B_3))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_6),A_2)) ) ) ).

fof(fact_330_xt1_I3_J,axiom,
    ! [C_14,A_20,B_18] :
      ( A_20 = B_18
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_14),B_18))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_14),A_20)) ) ) ).

fof(fact_331_xt1_I3_J,axiom,
    ! [C_14,A_20,B_18] :
      ( A_20 = B_18
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_14),B_18))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_14),A_20)) ) ) ).

fof(fact_332_xt1_I3_J,axiom,
    ! [C_6,A_2,B_3] :
      ( A_2 = B_3
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_6),B_3))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_6),A_2)) ) ) ).

fof(fact_333_ord__le__eq__trans,axiom,
    ! [C_6,A_2,B_3] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),B_3))
     => ( B_3 = C_6
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),C_6)) ) ) ).

fof(fact_334_ord__le__eq__trans,axiom,
    ! [C_6,A_2,B_3] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),B_3))
     => ( B_3 = C_6
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),C_6)) ) ) ).

fof(fact_335_ord__le__eq__trans,axiom,
    ! [C_13,A_19,B_17] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),B_17))
     => ( B_17 = C_13
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_19),C_13)) ) ) ).

fof(fact_336_ord__le__eq__trans,axiom,
    ! [C_13,A_19,B_17] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_19),B_17))
     => ( B_17 = C_13
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_19),C_13)) ) ) ).

fof(fact_337_ord__le__eq__trans,axiom,
    ! [C_6,A_2,B_3] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),B_3))
     => ( B_3 = C_6
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),C_6)) ) ) ).

fof(fact_338_xt1_I4_J,axiom,
    ! [C_6,B_3,A_2] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_3),A_2))
     => ( B_3 = C_6
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_6),A_2)) ) ) ).

fof(fact_339_xt1_I4_J,axiom,
    ! [C_6,B_3,A_2] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_3),A_2))
     => ( B_3 = C_6
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_6),A_2)) ) ) ).

fof(fact_340_xt1_I4_J,axiom,
    ! [C_12,B_16,A_18] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_16),A_18))
     => ( B_16 = C_12
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),A_18)) ) ) ).

fof(fact_341_xt1_I4_J,axiom,
    ! [C_12,B_16,A_18] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_16),A_18))
     => ( B_16 = C_12
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_12),A_18)) ) ) ).

fof(fact_342_xt1_I4_J,axiom,
    ! [C_6,B_3,A_2] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_3),A_2))
     => ( B_3 = C_6
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_6),A_2)) ) ) ).

fof(fact_343_order__antisym,axiom,
    ! [X_1,Y_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
       => X_1 = Y_1 ) ) ).

fof(fact_344_order__antisym,axiom,
    ! [X_1,Y_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
       => X_1 = Y_1 ) ) ).

fof(fact_345_order__antisym,axiom,
    ! [X_8,Y_7] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_8),Y_7))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),X_8))
       => X_8 = Y_7 ) ) ).

fof(fact_346_order__antisym,axiom,
    ! [X_8,Y_7] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_8),Y_7))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_8))
       => X_8 = Y_7 ) ) ).

fof(fact_347_order__antisym,axiom,
    ! [X_1,Y_1] :
      ( ( is_fun_a_bool(X_1)
        & is_fun_a_bool(Y_1) )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1))
         => X_1 = Y_1 ) ) ) ).

fof(fact_348_order__trans,axiom,
    ! [Z_2,X_1,Y_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),Z_2))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Z_2)) ) ) ).

fof(fact_349_order__trans,axiom,
    ! [Z_2,X_1,Y_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),Z_2))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Z_2)) ) ) ).

fof(fact_350_order__trans,axiom,
    ! [Z_6,X_7,Y_6] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_7),Y_6))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),Z_6))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_7),Z_6)) ) ) ).

fof(fact_351_order__trans,axiom,
    ! [Z_6,X_7,Y_6] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),Y_6))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_6),Z_6))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),Z_6)) ) ) ).

fof(fact_352_order__trans,axiom,
    ! [Z_2,X_1,Y_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),Z_2))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Z_2)) ) ) ).

fof(fact_353_xt1_I5_J,axiom,
    ! [Y_1,X_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_1))
       => X_1 = Y_1 ) ) ).

fof(fact_354_xt1_I5_J,axiom,
    ! [Y_1,X_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_1))
       => X_1 = Y_1 ) ) ).

fof(fact_355_xt1_I5_J,axiom,
    ! [Y_5,X_6] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_5),X_6))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_6),Y_5))
       => X_6 = Y_5 ) ) ).

fof(fact_356_xt1_I5_J,axiom,
    ! [Y_5,X_6] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),X_6))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_6),Y_5))
       => X_6 = Y_5 ) ) ).

fof(fact_357_xt1_I5_J,axiom,
    ! [Y_1,X_1] :
      ( ( is_fun_a_bool(Y_1)
        & is_fun_a_bool(X_1) )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_1),Y_1))
         => X_1 = Y_1 ) ) ) ).

fof(fact_358_xt1_I6_J,axiom,
    ! [Z_2,Y_1,X_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Y_1))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),X_1)) ) ) ).

fof(fact_359_xt1_I6_J,axiom,
    ! [Z_2,Y_1,X_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_1),X_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),Y_1))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),X_1)) ) ) ).

fof(fact_360_xt1_I6_J,axiom,
    ! [Z_5,Y_4,X_5] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_4),X_5))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_5),Y_4))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_5),X_5)) ) ) ).

fof(fact_361_xt1_I6_J,axiom,
    ! [Z_5,Y_4,X_5] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_4),X_5))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),Y_4))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),X_5)) ) ) ).

fof(fact_362_xt1_I6_J,axiom,
    ! [Z_2,Y_1,X_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_1),X_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Z_2),Y_1))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Z_2),X_1)) ) ) ).

fof(fact_363_linorder__le__cases,axiom,
    ! [X_4,Y_3] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_4),Y_3))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),X_4)) ) ).

fof(fact_364_linorder__le__cases,axiom,
    ! [X_4,Y_3] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_4),Y_3))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_3),X_4)) ) ).

fof(fact_365_insertI1,axiom,
    ! [A_2,B] : hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B))) ).

fof(fact_366_insertI1,axiom,
    ! [A_2,B] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B))) ).

fof(fact_367_insertI1,axiom,
    ! [A_2,B] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B))) ).

fof(fact_368_insertI1,axiom,
    ! [A_2,B] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B))) ).

fof(fact_369_insert__compr,axiom,
    ! [A_2,B] : hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B))) ).

fof(fact_370_insert__compr,axiom,
    ! [A_2,B] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fdisj,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2)),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),B))) ).

fof(fact_371_insert__compr,axiom,
    ! [A_2,B] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B))) ).

fof(fact_372_insert__compr,axiom,
    ! [A_2,B] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_2)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),B))) ).

fof(fact_373_insert__Collect,axiom,
    ! [A_2,Pa] : hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),collect_a(Pa)) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fimplies,cOMBB_bool_bool_a(fNot,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2))),Pa)) ).

fof(fact_374_insert__Collect,axiom,
    ! [A_2,Pa] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),collect_int(Pa)) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fimplies,cOMBB_bool_bool_int(fNot,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2))),Pa)) ).

fof(fact_375_insert__Collect,axiom,
    ! [A_2,Pa] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),collect_nat(Pa)) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fimplies,cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2))),Pa)) ).

fof(fact_376_mem__def,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
    <=> hBOOL(hAPP_int_bool(A,X_1)) ) ).

fof(fact_377_mem__def,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_nat_bool(A,X_1)) ) ).

fof(fact_378_mem__def,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
    <=> hBOOL(hAPP_a_bool(A,X_1)) ) ).

fof(fact_379_mem__def,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
    <=> hBOOL(hAPP_pname_bool(A,X_1)) ) ).

fof(fact_380_Collect__def,axiom,
    ! [Pa] : collect_int(Pa) = Pa ).

fof(fact_381_Collect__def,axiom,
    ! [Pa] : collect_nat(Pa) = Pa ).

fof(fact_382_insert__absorb2,axiom,
    ! [X_1,A] : hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A) ).

fof(fact_383_insert__commute,axiom,
    ! [X_1,Y_1,A] : hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,Y_1),A)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,Y_1),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) ).

fof(fact_384_insert__iff,axiom,
    ! [A_2,B_3,A] :
      ( ( is_a(A_2)
        & is_a(B_3) )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),A)))
      <=> ( A_2 = B_3
          | hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A)) ) ) ) ).

fof(fact_385_insert__iff,axiom,
    ! [A_2,B_3,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),A)))
    <=> ( A_2 = B_3
        | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A)) ) ) ).

fof(fact_386_insert__iff,axiom,
    ! [A_2,B_3,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),A)))
    <=> ( A_2 = B_3
        | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A)) ) ) ).

fof(fact_387_insert__iff,axiom,
    ! [A_2,B_3,A] :
      ( ( is_pname(A_2)
        & is_pname(B_3) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_3),A)))
      <=> ( A_2 = B_3
          | hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A)) ) ) ) ).

fof(fact_388_insert__code,axiom,
    ! [Y_1,A,X_1] :
      ( ( is_a(Y_1)
        & is_a(X_1) )
     => ( hBOOL(hAPP_a_bool(hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,Y_1),A),X_1))
      <=> ( Y_1 = X_1
          | hBOOL(hAPP_a_bool(A,X_1)) ) ) ) ).

fof(fact_389_insert__ident,axiom,
    ! [B,X_1,A] :
      ( ( is_fun_a_bool(B)
        & is_fun_a_bool(A) )
     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
         => ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),B)
          <=> A = B ) ) ) ) ).

fof(fact_390_insert__ident,axiom,
    ! [B,X_1,A] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)
        <=> A = B ) ) ) ).

fof(fact_391_insert__ident,axiom,
    ! [B,X_1,A] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)
        <=> A = B ) ) ) ).

fof(fact_392_insert__ident,axiom,
    ! [B,X_1,A] :
      ( ( is_fun_pname_bool(B)
        & is_fun_pname_bool(A) )
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
         => ( hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),B)
          <=> A = B ) ) ) ) ).

fof(fact_393_insertI2,axiom,
    ! [B_3,A_2,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),B))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),B))) ) ).

fof(fact_394_insertI2,axiom,
    ! [B_3,A_2,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),B))) ) ).

fof(fact_395_insertI2,axiom,
    ! [B_3,A_2,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),B))) ) ).

fof(fact_396_insertI2,axiom,
    ! [B_3,A_2,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,B_3),B))) ) ).

fof(fact_397_insert__absorb,axiom,
    ! [A_2,A] :
      ( is_fun_a_bool(A)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A))
       => hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A) = A ) ) ).

fof(fact_398_insert__absorb,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A))
     => hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A) = A ) ).

fof(fact_399_insert__absorb,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A))
     => hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A) = A ) ).

fof(fact_400_insert__absorb,axiom,
    ! [A_2,A] :
      ( is_fun_pname_bool(A)
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A))
       => hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),A) = A ) ) ).

fof(fact_401_subset__refl,axiom,
    ! [A] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),A)) ).

fof(fact_402_subset__refl,axiom,
    ! [A] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),A)) ).

fof(fact_403_subset__refl,axiom,
    ! [A] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),A)) ).

fof(fact_404_set__eq__subset,axiom,
    ! [A,B] :
      ( A = B
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ) ).

fof(fact_405_set__eq__subset,axiom,
    ! [A,B] :
      ( A = B
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).

fof(fact_406_set__eq__subset,axiom,
    ! [A,B] :
      ( ( is_fun_a_bool(A)
        & is_fun_a_bool(B) )
     => ( A = B
      <=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
          & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ) ) ).

fof(fact_407_equalityD1,axiom,
    ! [A,B] :
      ( A = B
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ).

fof(fact_408_equalityD1,axiom,
    ! [A,B] :
      ( A = B
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).

fof(fact_409_equalityD1,axiom,
    ! [A,B] :
      ( A = B
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ).

fof(fact_410_equalityD2,axiom,
    ! [A,B] :
      ( A = B
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ).

fof(fact_411_equalityD2,axiom,
    ! [A,B] :
      ( A = B
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ).

fof(fact_412_equalityD2,axiom,
    ! [A,B] :
      ( A = B
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ).

fof(fact_413_in__mono,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

fof(fact_414_in__mono,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

fof(fact_415_in__mono,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B)) ) ) ).

fof(fact_416_in__mono,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B)) ) ) ).

fof(fact_417_set__rev__mp,axiom,
    ! [B,X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

fof(fact_418_set__rev__mp,axiom,
    ! [B,X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

fof(fact_419_set__rev__mp,axiom,
    ! [B,X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B)) ) ) ).

fof(fact_420_set__rev__mp,axiom,
    ! [B,X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B)) ) ) ).

fof(fact_421_set__mp,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B)) ) ) ).

fof(fact_422_set__mp,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).

fof(fact_423_set__mp,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B)) ) ) ).

fof(fact_424_set__mp,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B)) ) ) ).

fof(fact_425_subset__trans,axiom,
    ! [C_11,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),C_11))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),C_11)) ) ) ).

fof(fact_426_subset__trans,axiom,
    ! [C_11,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C_11))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C_11)) ) ) ).

fof(fact_427_subset__trans,axiom,
    ! [C_11,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),C_11))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),C_11)) ) ) ).

fof(fact_428_equalityE,axiom,
    ! [A,B] :
      ( A = B
     => ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
         => ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A)) ) ) ).

fof(fact_429_equalityE,axiom,
    ! [A,B] :
      ( A = B
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
         => ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).

fof(fact_430_equalityE,axiom,
    ! [A,B] :
      ( A = B
     => ~ ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
         => ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ) ).

fof(fact_431_image__iff,axiom,
    ! [Z_2,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Z_2),image_nat_int(F,A)))
    <=> ? [X_2] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
          & Z_2 = hAPP_nat_int(F,X_2) ) ) ).

fof(fact_432_image__iff,axiom,
    ! [Z_2,F,A] :
      ( is_a(Z_2)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,Z_2),image_pname_a(F,A)))
      <=> ? [X_2] :
            ( is_pname(X_2)
            & hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
            & Z_2 = hAPP_pname_a(F,X_2) ) ) ) ).

fof(fact_433_imageI,axiom,
    ! [F,X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(F,X_1)),image_nat_int(F,A))) ) ).

fof(fact_434_imageI,axiom,
    ! [F,X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_1)),image_pname_a(F,A))) ) ).

fof(fact_435_rev__image__eqI,axiom,
    ! [B_3,F,X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( B_3 = hAPP_nat_int(F,X_1)
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),image_nat_int(F,A))) ) ) ).

fof(fact_436_rev__image__eqI,axiom,
    ! [B_3,F,X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( B_3 = hAPP_pname_a(F,X_1)
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),image_pname_a(F,A))) ) ) ).

fof(fact_437_insert__compr__raw,axiom,
    ! [X_2,Xa] : hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),Xa) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),X_2)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),Xa))) ).

fof(fact_438_insert__compr__raw,axiom,
    ! [X_2,Xa] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),Xa) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fdisj,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),X_2)),hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),Xa))) ).

fof(fact_439_insert__compr__raw,axiom,
    ! [X_2,Xa] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),Xa) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_2)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),Xa))) ).

fof(fact_440_insert__compr__raw,axiom,
    ! [X_2,Xa] : hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),Xa) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),X_2)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),Xa))) ).

fof(fact_441_le__fun__def,axiom,
    ! [F,G_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G_1))
    <=> ! [X_2] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G_1,X_2))) ) ).

fof(fact_442_le__fun__def,axiom,
    ! [F,G_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
    <=> ! [X_2] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G_1,X_2))) ) ).

fof(fact_443_le__fun__def,axiom,
    ! [F,G_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
    <=> ! [X_2] :
          ( is_a(X_2)
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_2)),hAPP_a_bool(G_1,X_2))) ) ) ).

fof(fact_444_le__funD,axiom,
    ! [X_1,F,G_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G_1,X_1))) ) ).

fof(fact_445_le__funD,axiom,
    ! [X_1,F,G_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G_1,X_1))) ) ).

fof(fact_446_le__funD,axiom,
    ! [X_1,F,G_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_1)),hAPP_a_bool(G_1,X_1))) ) ).

fof(fact_447_le__funE,axiom,
    ! [X_1,F,G_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G_1,X_1))) ) ).

fof(fact_448_le__funE,axiom,
    ! [X_1,F,G_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G_1,X_1))) ) ).

fof(fact_449_le__funE,axiom,
    ! [X_1,F,G_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
     => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_1)),hAPP_a_bool(G_1,X_1))) ) ).

fof(fact_450_subset__insertI,axiom,
    ! [B,A_2] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B))) ).

fof(fact_451_subset__insertI,axiom,
    ! [B,A_2] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B))) ).

fof(fact_452_subset__insertI,axiom,
    ! [B,A_2] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B))) ).

fof(fact_453_insert__subset,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),B))
    <=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
        & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ) ).

fof(fact_454_insert__subset,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
        & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ).

fof(fact_455_insert__subset,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
        & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).

fof(fact_456_insert__subset,axiom,
    ! [X_1,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),B))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
        & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ) ).

fof(fact_457_subset__insert,axiom,
    ! [B,X_1,A] :
      ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),B)))
      <=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ) ).

fof(fact_458_subset__insert,axiom,
    ! [B,X_1,A] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)))
      <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ).

fof(fact_459_subset__insert,axiom,
    ! [B,X_1,A] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
      <=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).

fof(fact_460_subset__insert,axiom,
    ! [B,X_1,A] :
      ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),B)))
      <=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ) ).

fof(fact_461_subset__insertI2,axiom,
    ! [B_3,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),B))) ) ).

fof(fact_462_subset__insertI2,axiom,
    ! [B_3,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),B))) ) ).

fof(fact_463_subset__insertI2,axiom,
    ! [B_3,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),B))) ) ).

fof(fact_464_insert__mono,axiom,
    ! [A_2,C_11,D_4] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_11),D_4))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),C_11)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),D_4))) ) ).

fof(fact_465_insert__mono,axiom,
    ! [A_2,C_11,D_4] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C_11),D_4))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),C_11)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),D_4))) ) ).

fof(fact_466_insert__mono,axiom,
    ! [A_2,C_11,D_4] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_11),D_4))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),C_11)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),D_4))) ) ).

fof(fact_467_image__insert,axiom,
    ! [F,A_2,B] : image_nat_int(F,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_nat_int(F,A_2)),image_nat_int(F,B)) ).

fof(fact_468_image__insert,axiom,
    ! [F,A_2,B] : image_pname_a(F,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(F,A_2)),image_pname_a(F,B)) ).

fof(fact_469_insert__image,axiom,
    ! [F,X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,hAPP_nat_int(F,X_1)),image_nat_int(F,A)) = image_nat_int(F,A) ) ).

fof(fact_470_insert__image,axiom,
    ! [F,X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(F,X_1)),image_pname_a(F,A)) = image_pname_a(F,A) ) ).

fof(fact_471_subset__image__iff,axiom,
    ! [B,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),image_nat_int(F,A)))
    <=> ? [AA] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,AA),A))
          & B = image_nat_int(F,AA) ) ) ).

fof(fact_472_subset__image__iff,axiom,
    ! [B,F,A] :
      ( is_fun_a_bool(B)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
      <=> ? [AA] :
            ( is_fun_pname_bool(AA)
            & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,AA),A))
            & B = image_pname_a(F,AA) ) ) ) ).

fof(fact_473_image__mono,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(F,A)),image_nat_int(F,B))) ) ).

fof(fact_474_image__mono,axiom,
    ! [F,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A)),image_pname_a(F,B))) ) ).

fof(fact_475_imageE,axiom,
    ! [B_3,F,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),image_nat_int(F,A)))
     => ~ ! [X_2] :
            ( B_3 = hAPP_nat_int(F,X_2)
           => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) ) ) ).

fof(fact_476_imageE,axiom,
    ! [B_3,F,A] :
      ( is_a(B_3)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),image_pname_a(F,A)))
       => ~ ! [X_2] :
              ( is_pname(X_2)
             => ( B_3 = hAPP_pname_a(F,X_2)
               => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) ) ) ) ) ).

fof(fact_477_subsetI,axiom,
    ! [B,A] :
      ( ! [X_2] :
          ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ).

fof(fact_478_subsetI,axiom,
    ! [B,A] :
      ( ! [X_2] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) )
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).

fof(fact_479_subsetI,axiom,
    ! [B,A] :
      ( ! [X_2] :
          ( is_a(X_2)
         => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
           => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B)) ) )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ).

fof(fact_480_subsetI,axiom,
    ! [B,A] :
      ( ! [X_2] :
          ( is_pname(X_2)
         => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
           => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B)) ) )
     => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ).

fof(fact_481_image__subsetI,axiom,
    ! [F,B,A] :
      ( ! [X_2] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(F,X_2)),B)) )
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(F,A)),B)) ) ).

fof(fact_482_image__subsetI,axiom,
    ! [F,B,A] :
      ( ! [X_2] :
          ( is_pname(X_2)
         => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
           => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_2)),B)) ) )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A)),B)) ) ).

fof(fact_483_le__funI,axiom,
    ! [F,G_1] :
      ( ! [X_2] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_2)),hAPP_int_bool(G_1,X_2)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G_1)) ) ).

fof(fact_484_le__funI,axiom,
    ! [F,G_1] :
      ( ! [X_2] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_2)),hAPP_nat_bool(G_1,X_2)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1)) ) ).

fof(fact_485_le__funI,axiom,
    ! [F,G_1] :
      ( ! [X_2] :
          ( is_a(X_2)
         => hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_2)),hAPP_a_bool(G_1,X_2))) )
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1)) ) ).

fof(fact_486_finite__nat__set__iff__bounded__le,axiom,
    ! [N_2] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_2))
    <=> ? [M_2] :
        ! [X_2] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M_2)) ) ) ).

fof(fact_487_assms_I3_J,axiom,
    ! [G,C_6] :
      ( hBOOL(wt(C_6))
     => ( ! [X_2] :
            ( is_pname(X_2)
           => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),u))
             => hBOOL(hAPP_fun_a_bool_bool(p(G),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(mgt_call,X_2)),bot_bot_fun_a_bool))) ) )
       => hBOOL(hAPP_fun_a_bool_bool(p(G),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,mgt(C_6)),bot_bot_fun_a_bool))) ) ) ).

fof(fact_488_diff__eq__diff__less__eq,axiom,
    ! [A_2,B_3,C_6,D_1] :
      ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_3) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_6),D_1)
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),B_3))
      <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_6),D_1)) ) ) ).

fof(fact_489_less__eq__nat_Osimps_I2_J,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),Na))
    <=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,M)),Na)) ) ).

fof(fact_490_emptyE,axiom,
    ! [A_2] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),bot_bot_fun_int_bool)) ).

fof(fact_491_emptyE,axiom,
    ! [A_2] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),bot_bot_fun_nat_bool)) ).

fof(fact_492_emptyE,axiom,
    ! [A_2] : ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),bot_bot_fun_a_bool)) ).

fof(fact_493_emptyE,axiom,
    ! [A_2] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),bot_bo844097828e_bool)) ).

fof(fact_494_finite__Diff,axiom,
    ! [B,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ).

fof(fact_495_finite__Diff,axiom,
    ! [B,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).

fof(fact_496_finite__Diff,axiom,
    ! [B,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B))) ) ).

fof(fact_497_finite_OemptyI,axiom,
    hBOOL(hAPP_f448129468l_bool(finite_finite_int,bot_bot_fun_int_bool)) ).

fof(fact_498_finite_OemptyI,axiom,
    hBOOL(hAPP_f54304608l_bool(finite_finite_nat,bot_bot_fun_nat_bool)) ).

fof(fact_499_finite_OemptyI,axiom,
    hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,bot_bo844097828e_bool)) ).

fof(fact_500_finite_OemptyI,axiom,
    hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,bot_bot_fun_a_bool)) ).

fof(fact_501_empty__subsetI,axiom,
    ! [A] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A)) ).

fof(fact_502_empty__subsetI,axiom,
    ! [A] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A)) ).

fof(fact_503_empty__subsetI,axiom,
    ! [A] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,bot_bot_fun_a_bool),A)) ).

fof(fact_504_equals0D,axiom,
    ! [A_2,A] :
      ( A = bot_bot_fun_int_bool
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A)) ) ).

fof(fact_505_equals0D,axiom,
    ! [A_2,A] :
      ( A = bot_bot_fun_nat_bool
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A)) ) ).

fof(fact_506_equals0D,axiom,
    ! [A_2,A] :
      ( A = bot_bot_fun_a_bool
     => ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A)) ) ).

fof(fact_507_equals0D,axiom,
    ! [A_2,A] :
      ( A = bot_bo844097828e_bool
     => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A)) ) ).

fof(fact_508_Collect__empty__eq,axiom,
    ! [Pa] :
      ( collect_int(Pa) = bot_bot_fun_int_bool
    <=> ! [X_2] : ~ hBOOL(hAPP_int_bool(Pa,X_2)) ) ).

fof(fact_509_Collect__empty__eq,axiom,
    ! [Pa] :
      ( collect_nat(Pa) = bot_bot_fun_nat_bool
    <=> ! [X_2] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).

fof(fact_510_Collect__empty__eq,axiom,
    ! [Pa] :
      ( collect_a(Pa) = bot_bot_fun_a_bool
    <=> ! [X_2] :
          ( is_a(X_2)
         => ~ hBOOL(hAPP_a_bool(Pa,X_2)) ) ) ).

fof(fact_511_Diff__cancel,axiom,
    ! [A] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),A) = bot_bot_fun_nat_bool ).

fof(fact_512_Diff__cancel,axiom,
    ! [A] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),A) = bot_bot_fun_int_bool ).

fof(fact_513_Diff__cancel,axiom,
    ! [A] : hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),A) = bot_bot_fun_a_bool ).

fof(fact_514_Diff__empty,axiom,
    ! [A] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),bot_bot_fun_nat_bool) = A ).

fof(fact_515_Diff__empty,axiom,
    ! [A] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),bot_bot_fun_int_bool) = A ).

fof(fact_516_Diff__empty,axiom,
    ! [A] :
      ( is_fun_a_bool(A)
     => hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),bot_bot_fun_a_bool) = A ) ).

fof(fact_517_empty__iff,axiom,
    ! [C_6] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),bot_bot_fun_int_bool)) ).

fof(fact_518_empty__iff,axiom,
    ! [C_6] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),bot_bot_fun_nat_bool)) ).

fof(fact_519_empty__iff,axiom,
    ! [C_6] : ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),bot_bot_fun_a_bool)) ).

fof(fact_520_empty__iff,axiom,
    ! [C_6] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),bot_bo844097828e_bool)) ).

fof(fact_521_empty__Collect__eq,axiom,
    ! [Pa] :
      ( bot_bot_fun_int_bool = collect_int(Pa)
    <=> ! [X_2] : ~ hBOOL(hAPP_int_bool(Pa,X_2)) ) ).

fof(fact_522_empty__Collect__eq,axiom,
    ! [Pa] :
      ( bot_bot_fun_nat_bool = collect_nat(Pa)
    <=> ! [X_2] : ~ hBOOL(hAPP_nat_bool(Pa,X_2)) ) ).

fof(fact_523_empty__Collect__eq,axiom,
    ! [Pa] :
      ( bot_bot_fun_a_bool = collect_a(Pa)
    <=> ! [X_2] :
          ( is_a(X_2)
         => ~ hBOOL(hAPP_a_bool(Pa,X_2)) ) ) ).

fof(fact_524_empty__Diff,axiom,
    ! [A] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,bot_bot_fun_nat_bool),A) = bot_bot_fun_nat_bool ).

fof(fact_525_empty__Diff,axiom,
    ! [A] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,bot_bot_fun_int_bool),A) = bot_bot_fun_int_bool ).

fof(fact_526_empty__Diff,axiom,
    ! [A] : hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,bot_bot_fun_a_bool),A) = bot_bot_fun_a_bool ).

fof(fact_527_ex__in__conv,axiom,
    ! [A] :
      ( ? [X_2] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
    <=> A != bot_bot_fun_int_bool ) ).

fof(fact_528_ex__in__conv,axiom,
    ! [A] :
      ( ? [X_2] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
    <=> A != bot_bot_fun_nat_bool ) ).

fof(fact_529_ex__in__conv,axiom,
    ! [A] :
      ( is_fun_a_bool(A)
     => ( ? [X_2] :
            ( is_a(X_2)
            & hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) )
      <=> A != bot_bot_fun_a_bool ) ) ).

fof(fact_530_ex__in__conv,axiom,
    ! [A] :
      ( is_fun_pname_bool(A)
     => ( ? [X_2] :
            ( is_pname(X_2)
            & hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) )
      <=> A != bot_bo844097828e_bool ) ) ).

fof(fact_531_all__not__in__conv,axiom,
    ! [A] :
      ( ! [X_2] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A))
    <=> A = bot_bot_fun_int_bool ) ).

fof(fact_532_all__not__in__conv,axiom,
    ! [A] :
      ( ! [X_2] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
    <=> A = bot_bot_fun_nat_bool ) ).

fof(fact_533_all__not__in__conv,axiom,
    ! [A] :
      ( is_fun_a_bool(A)
     => ( ! [X_2] :
            ( is_a(X_2)
           => ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) )
      <=> A = bot_bot_fun_a_bool ) ) ).

fof(fact_534_all__not__in__conv,axiom,
    ! [A] :
      ( is_fun_pname_bool(A)
     => ( ! [X_2] :
            ( is_pname(X_2)
           => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) )
      <=> A = bot_bo844097828e_bool ) ) ).

fof(fact_535_bot__apply,axiom,
    ! [X_1] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

fof(fact_536_bot__apply,axiom,
    ! [X_1] :
      ( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

fof(fact_537_bot__apply,axiom,
    ! [X_1] :
      ( hBOOL(hAPP_a_bool(bot_bot_fun_a_bool,X_1))
    <=> hBOOL(bot_bot_bool) ) ).

fof(fact_538_empty__def,axiom,
    bot_bot_fun_int_bool = collect_int(cOMBK_bool_int(fFalse)) ).

fof(fact_539_empty__def,axiom,
    bot_bot_fun_nat_bool = collect_nat(cOMBK_bool_nat(fFalse)) ).

fof(fact_540_empty__def,axiom,
    bot_bot_fun_a_bool = collect_a(cOMBK_bool_a(fFalse)) ).

fof(fact_541_bot__fun__def,axiom,
    ! [X_2] :
      ( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

fof(fact_542_bot__fun__def,axiom,
    ! [X_2] :
      ( hBOOL(hAPP_int_bool(bot_bot_fun_int_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

fof(fact_543_bot__fun__def,axiom,
    ! [X_2] :
      ( hBOOL(hAPP_a_bool(bot_bot_fun_a_bool,X_2))
    <=> hBOOL(bot_bot_bool) ) ).

fof(fact_544_insert__Diff,axiom,
    ! [A_2,A] :
      ( is_fun_a_bool(A)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A))
       => hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool))) = A ) ) ).

fof(fact_545_insert__Diff,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A))
     => hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool))) = A ) ).

fof(fact_546_insert__Diff,axiom,
    ! [A_2,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A))
     => hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool))) = A ) ).

fof(fact_547_insert__Diff,axiom,
    ! [A_2,A] :
      ( is_fun_pname_bool(A)
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A))
       => hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool))) = A ) ) ).

fof(fact_548_Diff__insert__absorb,axiom,
    ! [X_1,A] :
      ( is_fun_a_bool(A)
     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = A ) ) ).

fof(fact_549_Diff__insert__absorb,axiom,
    ! [X_1,A] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
     => hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = A ) ).

fof(fact_550_Diff__insert__absorb,axiom,
    ! [X_1,A] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = A ) ).

fof(fact_551_Diff__insert__absorb,axiom,
    ! [X_1,A] :
      ( is_fun_pname_bool(A)
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) = A ) ) ).

fof(fact_552_insert__Diff__single,axiom,
    ! [A_2,A] : hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool))) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A) ).

fof(fact_553_insert__Diff__single,axiom,
    ! [A_2,A] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool))) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A) ).

fof(fact_554_insert__Diff__single,axiom,
    ! [A_2,A] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool))) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A) ).

fof(fact_555_Diff__insert2,axiom,
    ! [A,A_2,B] : hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B)) = hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool))),B) ).

fof(fact_556_Diff__insert2,axiom,
    ! [A,A_2,B] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool))),B) ).

fof(fact_557_Diff__insert2,axiom,
    ! [A,A_2,B] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool))),B) ).

fof(fact_558_Diff__insert,axiom,
    ! [A,A_2,B] : hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B)) = hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool)) ).

fof(fact_559_Diff__insert,axiom,
    ! [A,A_2,B] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B)) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)) ).

fof(fact_560_Diff__insert,axiom,
    ! [A,A_2,B] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B)) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool)) ).

fof(fact_561_diff__single__insert,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))),B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),B))) ) ) ).

fof(fact_562_diff__single__insert,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B))) ) ) ).

fof(fact_563_diff__single__insert,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B))) ) ) ).

fof(fact_564_diff__single__insert,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))),B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),B))) ) ) ).

fof(fact_565_subset__insert__iff,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),B)))
    <=> ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))),B)) )
        & ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ) ) ).

fof(fact_566_subset__insert__iff,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),B)))
    <=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))),B)) )
        & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) ) ) ) ).

fof(fact_567_subset__insert__iff,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),B)))
    <=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))),B)) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ) ).

fof(fact_568_subset__insert__iff,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),B)))
    <=> ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))),B)) )
        & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ) ) ).

fof(fact_569_finite__Diff2,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
      <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ) ).

fof(fact_570_finite__Diff2,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
      <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).

fof(fact_571_finite__Diff2,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
      <=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ) ).

fof(fact_572_insert__Diff1,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
     => hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),B) = hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B) ) ).

fof(fact_573_insert__Diff1,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
     => hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) ) ).

fof(fact_574_insert__Diff1,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
     => hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) ) ).

fof(fact_575_insert__Diff1,axiom,
    ! [A,X_1,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
     => hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),B) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B) ) ).

fof(fact_576_insert__Diff__if,axiom,
    ! [A,X_1,B] :
      ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
       => hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),B) = hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B) )
      & ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))
       => hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)),B) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)) ) ) ).

fof(fact_577_insert__Diff__if,axiom,
    ! [A,X_1,B] :
      ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) )
      & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),B))
       => hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)),B) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)) ) ) ).

fof(fact_578_insert__Diff__if,axiom,
    ! [A,X_1,B] :
      ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) )
      & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B))
       => hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)),B) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) ) ) ).

fof(fact_579_insert__Diff__if,axiom,
    ! [A,X_1,B] :
      ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
       => hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),B) = hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B) )
      & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))
       => hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)),B) = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)) ) ) ).

fof(fact_580_double__diff,axiom,
    ! [C_11,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),C_11))
       => hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,B),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_11),A)) = A ) ) ).

fof(fact_581_double__diff,axiom,
    ! [C_11,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C_11))
       => hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_11),A)) = A ) ) ).

fof(fact_582_double__diff,axiom,
    ! [C_11,A,B] :
      ( is_fun_a_bool(A)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
       => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),C_11))
         => hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,B),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,C_11),A)) = A ) ) ) ).

fof(fact_583_Diff__mono,axiom,
    ! [D_4,B,A,C_11] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),C_11))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,D_4),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,C_11),D_4))) ) ) ).

fof(fact_584_Diff__mono,axiom,
    ! [D_4,B,A,C_11] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C_11))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,D_4),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,C_11),D_4))) ) ) ).

fof(fact_585_Diff__mono,axiom,
    ! [D_4,B,A,C_11] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),C_11))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,D_4),B))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,C_11),D_4))) ) ) ).

fof(fact_586_Diff__subset,axiom,
    ! [A,B] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)),A)) ).

fof(fact_587_Diff__subset,axiom,
    ! [A,B] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)),A)) ).

fof(fact_588_Diff__subset,axiom,
    ! [A,B] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)),A)) ).

fof(fact_589_singleton__inject,axiom,
    ! [A_2,B_3] :
      ( ( is_a(A_2)
        & is_a(B_3) )
     => ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),bot_bot_fun_a_bool)
       => A_2 = B_3 ) ) ).

fof(fact_590_singleton__inject,axiom,
    ! [A_2,B_3] :
      ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),bot_bot_fun_nat_bool)
     => A_2 = B_3 ) ).

fof(fact_591_singleton__inject,axiom,
    ! [A_2,B_3] :
      ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),bot_bot_fun_int_bool)
     => A_2 = B_3 ) ).

fof(fact_592_singletonE,axiom,
    ! [B_3,A_2] :
      ( ( is_a(B_3)
        & is_a(A_2) )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool)))
       => B_3 = A_2 ) ) ).

fof(fact_593_singletonE,axiom,
    ! [B_3,A_2] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool)))
     => B_3 = A_2 ) ).

fof(fact_594_singletonE,axiom,
    ! [B_3,A_2] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_3),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)))
     => B_3 = A_2 ) ).

fof(fact_595_singletonE,axiom,
    ! [B_3,A_2] :
      ( ( is_pname(B_3)
        & is_pname(A_2) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_3),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool)))
       => B_3 = A_2 ) ) ).

fof(fact_596_doubleton__eq__iff,axiom,
    ! [A_2,B_3,C_6,D_1] :
      ( ( is_a(A_2)
        & is_a(B_3)
        & is_a(C_6)
        & is_a(D_1) )
     => ( hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,B_3),bot_bot_fun_a_bool)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,C_6),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,D_1),bot_bot_fun_a_bool))
      <=> ( ( A_2 = C_6
            & B_3 = D_1 )
          | ( A_2 = D_1
            & B_3 = C_6 ) ) ) ) ).

fof(fact_597_doubleton__eq__iff,axiom,
    ! [A_2,B_3,C_6,D_1] :
      ( hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,B_3),bot_bot_fun_nat_bool)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,C_6),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,D_1),bot_bot_fun_nat_bool))
    <=> ( ( A_2 = C_6
          & B_3 = D_1 )
        | ( A_2 = D_1
          & B_3 = C_6 ) ) ) ).

fof(fact_598_doubleton__eq__iff,axiom,
    ! [A_2,B_3,C_6,D_1] :
      ( hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,B_3),bot_bot_fun_int_bool)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,C_6),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,D_1),bot_bot_fun_int_bool))
    <=> ( ( A_2 = C_6
          & B_3 = D_1 )
        | ( A_2 = D_1
          & B_3 = C_6 ) ) ) ).

fof(fact_599_singleton__iff,axiom,
    ! [B_3,A_2] :
      ( ( is_a(B_3)
        & is_a(A_2) )
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_3),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool)))
      <=> B_3 = A_2 ) ) ).

fof(fact_600_singleton__iff,axiom,
    ! [B_3,A_2] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_3),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool)))
    <=> B_3 = A_2 ) ).

fof(fact_601_singleton__iff,axiom,
    ! [B_3,A_2] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_3),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool)))
    <=> B_3 = A_2 ) ).

fof(fact_602_singleton__iff,axiom,
    ! [B_3,A_2] :
      ( ( is_pname(B_3)
        & is_pname(A_2) )
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_3),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),bot_bo844097828e_bool)))
      <=> B_3 = A_2 ) ) ).

fof(fact_603_insert__not__empty,axiom,
    ! [A_2,A] : hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A) != bot_bot_fun_a_bool ).

fof(fact_604_insert__not__empty,axiom,
    ! [A_2,A] : hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A) != bot_bot_fun_nat_bool ).

fof(fact_605_insert__not__empty,axiom,
    ! [A_2,A] : hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A) != bot_bot_fun_int_bool ).

fof(fact_606_empty__not__insert,axiom,
    ! [A_2,A] : bot_bot_fun_a_bool != hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),A) ).

fof(fact_607_empty__not__insert,axiom,
    ! [A_2,A] : bot_bot_fun_nat_bool != hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),A) ).

fof(fact_608_empty__not__insert,axiom,
    ! [A_2,A] : bot_bot_fun_int_bool != hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),A) ).

fof(fact_609_subset__empty,axiom,
    ! [A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),bot_bot_fun_nat_bool))
    <=> A = bot_bot_fun_nat_bool ) ).

fof(fact_610_subset__empty,axiom,
    ! [A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),bot_bot_fun_int_bool))
    <=> A = bot_bot_fun_int_bool ) ).

fof(fact_611_subset__empty,axiom,
    ! [A] :
      ( is_fun_a_bool(A)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),bot_bot_fun_a_bool))
      <=> A = bot_bot_fun_a_bool ) ) ).

fof(fact_612_image__is__empty,axiom,
    ! [F,A] :
      ( image_nat_int(F,A) = bot_bot_fun_int_bool
    <=> A = bot_bot_fun_nat_bool ) ).

fof(fact_613_image__is__empty,axiom,
    ! [F,A] :
      ( is_fun_pname_bool(A)
     => ( image_pname_a(F,A) = bot_bot_fun_a_bool
      <=> A = bot_bo844097828e_bool ) ) ).

fof(fact_614_image__empty,axiom,
    ! [F] : image_nat_int(F,bot_bot_fun_nat_bool) = bot_bot_fun_int_bool ).

fof(fact_615_image__empty,axiom,
    ! [F] : image_pname_a(F,bot_bo844097828e_bool) = bot_bot_fun_a_bool ).

fof(fact_616_empty__is__image,axiom,
    ! [F,A] :
      ( bot_bot_fun_int_bool = image_nat_int(F,A)
    <=> A = bot_bot_fun_nat_bool ) ).

fof(fact_617_empty__is__image,axiom,
    ! [F,A] :
      ( is_fun_pname_bool(A)
     => ( bot_bot_fun_a_bool = image_pname_a(F,A)
      <=> A = bot_bo844097828e_bool ) ) ).

fof(fact_618_le__bot,axiom,
    ! [A_2] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),bot_bot_fun_nat_bool))
     => A_2 = bot_bot_fun_nat_bool ) ).

fof(fact_619_le__bot,axiom,
    ! [A_2] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),bot_bot_fun_int_bool))
     => A_2 = bot_bot_fun_int_bool ) ).

fof(fact_620_le__bot,axiom,
    ! [A_17] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_17),bot_bot_nat))
     => A_17 = bot_bot_nat ) ).

fof(fact_621_le__bot,axiom,
    ! [A_2] :
      ( is_fun_a_bool(A_2)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),bot_bot_fun_a_bool))
       => A_2 = bot_bot_fun_a_bool ) ) ).

fof(fact_622_bot__unique,axiom,
    ! [A_2] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_2),bot_bot_fun_nat_bool))
    <=> A_2 = bot_bot_fun_nat_bool ) ).

fof(fact_623_bot__unique,axiom,
    ! [A_2] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_2),bot_bot_fun_int_bool))
    <=> A_2 = bot_bot_fun_int_bool ) ).

fof(fact_624_bot__unique,axiom,
    ! [A_2] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),bot_bot_nat))
    <=> A_2 = bot_bot_nat ) ).

fof(fact_625_bot__unique,axiom,
    ! [A_2] :
      ( is_fun_a_bool(A_2)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_2),bot_bot_fun_a_bool))
      <=> A_2 = bot_bot_fun_a_bool ) ) ).

fof(fact_626_bot__least,axiom,
    ! [A_2] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_2)) ).

fof(fact_627_bot__least,axiom,
    ! [A_2] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,bot_bot_fun_int_bool),A_2)) ).

fof(fact_628_bot__least,axiom,
    ! [A_16] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,bot_bot_nat),A_16)) ).

fof(fact_629_bot__least,axiom,
    ! [A_2] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,bot_bot_fun_a_bool),A_2)) ).

fof(fact_630_Collect__conv__if,axiom,
    ! [Pa,A_2] :
      ( ( hBOOL(hAPP_a_bool(Pa,A_2))
       => collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2)),Pa)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) )
      & ( ~ hBOOL(hAPP_a_bool(Pa,A_2))
       => collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2)),Pa)) = bot_bot_fun_a_bool ) ) ).

fof(fact_631_Collect__conv__if,axiom,
    ! [Pa,A_2] :
      ( ( hBOOL(hAPP_int_bool(Pa,A_2))
       => collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2)),Pa)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) )
      & ( ~ hBOOL(hAPP_int_bool(Pa,A_2))
       => collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2)),Pa)) = bot_bot_fun_int_bool ) ) ).

fof(fact_632_Collect__conv__if,axiom,
    ! [Pa,A_2] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_2))
       => collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2)),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_2))
       => collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2)),Pa)) = bot_bot_fun_nat_bool ) ) ).

fof(fact_633_Collect__conv__if2,axiom,
    ! [Pa,A_2] :
      ( ( hBOOL(hAPP_a_bool(Pa,A_2))
       => collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(fequal_a,A_2)),Pa)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) )
      & ( ~ hBOOL(hAPP_a_bool(Pa,A_2))
       => collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(fequal_a,A_2)),Pa)) = bot_bot_fun_a_bool ) ) ).

fof(fact_634_Collect__conv__if2,axiom,
    ! [Pa,A_2] :
      ( ( hBOOL(hAPP_int_bool(Pa,A_2))
       => collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(fequal_int,A_2)),Pa)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) )
      & ( ~ hBOOL(hAPP_int_bool(Pa,A_2))
       => collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(fequal_int,A_2)),Pa)) = bot_bot_fun_int_bool ) ) ).

fof(fact_635_Collect__conv__if2,axiom,
    ! [Pa,A_2] :
      ( ( hBOOL(hAPP_nat_bool(Pa,A_2))
       => collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(fequal_nat,A_2)),Pa)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) )
      & ( ~ hBOOL(hAPP_nat_bool(Pa,A_2))
       => collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(fequal_nat,A_2)),Pa)) = bot_bot_fun_nat_bool ) ) ).

fof(fact_636_singleton__conv,axiom,
    ! [A_2] : collect_a(hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_2)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) ).

fof(fact_637_singleton__conv,axiom,
    ! [A_2] : collect_int(hAPP_i1948725293t_bool(cOMBC_int_int_bool(fequal_int),A_2)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) ).

fof(fact_638_singleton__conv,axiom,
    ! [A_2] : collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ).

fof(fact_639_singleton__conv2,axiom,
    ! [A_2] : collect_a(hAPP_a_fun_a_bool(fequal_a,A_2)) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),bot_bot_fun_a_bool) ).

fof(fact_640_singleton__conv2,axiom,
    ! [A_2] : collect_int(hAPP_i1948725293t_bool(fequal_int,A_2)) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),bot_bot_fun_int_bool) ).

fof(fact_641_singleton__conv2,axiom,
    ! [A_2] : collect_nat(hAPP_n1699378549t_bool(fequal_nat,A_2)) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),bot_bot_fun_nat_bool) ).

fof(fact_642_card__Suc__Diff1,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) = hAPP_fun_a_bool_nat(finite_card_a,A) ) ) ).

fof(fact_643_card__Suc__Diff1,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) = hAPP_f957591787ol_nat(finite_card_int,A) ) ) ).

fof(fact_644_card__Suc__Diff1,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) = hAPP_f22106695ol_nat(finite_card_nat,A) ) ) ).

fof(fact_645_card__Suc__Diff1,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) = hAPP_f921600141ol_nat(finite_card_pname,A) ) ) ).

fof(fact_646_card__insert,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ).

fof(fact_647_card__insert,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(suc,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ).

fof(fact_648_card__insert,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ).

fof(fact_649_card__insert,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ).

fof(fact_650_card__Diff1__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).

fof(fact_651_card__Diff1__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))),hAPP_f957591787ol_nat(finite_card_int,A))) ) ).

fof(fact_652_card__Diff1__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).

fof(fact_653_card__Diff1__le,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).

fof(fact_654_finite__Diff__insert,axiom,
    ! [A,A_2,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B))))
    <=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B))) ) ).

fof(fact_655_finite__Diff__insert,axiom,
    ! [A,A_2,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B))))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ).

fof(fact_656_finite__Diff__insert,axiom,
    ! [A,A_2,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B))))
    <=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ).

fof(fact_657_finite__Diff__insert,axiom,
    ! [A,A_2,B] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B))))
    <=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B))) ) ).

fof(fact_658_image__diff__subset,axiom,
    ! [F,A,B] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,image_nat_int(F,A)),image_nat_int(F,B))),image_nat_int(F,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))) ).

fof(fact_659_image__diff__subset,axiom,
    ! [F,A,B] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,image_pname_a(F,A)),image_pname_a(F,B))),image_pname_a(F,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))) ).

fof(fact_660_subset__singletonD,axiom,
    ! [A,X_1] :
      ( is_fun_a_bool(A)
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))
       => ( A = bot_bot_fun_a_bool
          | A = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool) ) ) ) ).

fof(fact_661_subset__singletonD,axiom,
    ! [A,X_1] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))
     => ( A = bot_bot_fun_nat_bool
        | A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool) ) ) ).

fof(fact_662_subset__singletonD,axiom,
    ! [A,X_1] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))
     => ( A = bot_bot_fun_int_bool
        | A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool) ) ) ).

fof(fact_663_nat__case__Suc,axiom,
    ! [F1,F2,Nat_3] : hAPP_nat_nat(nat_case_nat(F1,F2),hAPP_nat_nat(suc,Nat_3)) = hAPP_nat_nat(F2,Nat_3) ).

fof(fact_664_nat__case__Suc,axiom,
    ! [F1,F2,Nat_3] :
      ( hBOOL(hAPP_nat_bool(nat_case_bool(F1,F2),hAPP_nat_nat(suc,Nat_3)))
    <=> hBOOL(hAPP_nat_bool(F2,Nat_3)) ) ).

fof(fact_665_image__constant,axiom,
    ! [C_6,X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
     => image_nat_int(cOMBK_int_nat(C_6),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,C_6),bot_bot_fun_int_bool) ) ).

fof(fact_666_image__constant,axiom,
    ! [C_6,X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
     => image_pname_a(cOMBK_a_pname(C_6),A) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,C_6),bot_bot_fun_a_bool) ) ).

fof(fact_667_image__constant__conv,axiom,
    ! [C_6,A] :
      ( ( A = bot_bot_fun_nat_bool
       => image_nat_int(cOMBK_int_nat(C_6),A) = bot_bot_fun_int_bool )
      & ( A != bot_bot_fun_nat_bool
       => image_nat_int(cOMBK_int_nat(C_6),A) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,C_6),bot_bot_fun_int_bool) ) ) ).

fof(fact_668_image__constant__conv,axiom,
    ! [C_6,A] :
      ( is_fun_pname_bool(A)
     => ( ( A = bot_bo844097828e_bool
         => image_pname_a(cOMBK_a_pname(C_6),A) = bot_bot_fun_a_bool )
        & ( A != bot_bo844097828e_bool
         => image_pname_a(cOMBK_a_pname(C_6),A) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,C_6),bot_bot_fun_a_bool) ) ) ) ).

fof(fact_669_diff__eq__diff__eq,axiom,
    ! [A_2,B_3,C_6,D_1] :
      ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),B_3) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_6),D_1)
     => ( A_2 = B_3
      <=> C_6 = D_1 ) ) ).

fof(fact_670_card__Diff__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
       => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A)),hAPP_f957591787ol_nat(finite_card_int,B)) ) ) ).

fof(fact_671_card__Diff__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
       => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,B)) ) ) ).

fof(fact_672_card__Diff__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A))
       => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,B)) ) ) ).

fof(fact_673_card__Diff__subset,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A))
       => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,B)) ) ) ).

fof(fact_674_diff__card__le__card__Diff,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,B))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A)),hAPP_f957591787ol_nat(finite_card_int,B))),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))) ) ).

fof(fact_675_diff__card__le__card__Diff,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,B))),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))) ) ).

fof(fact_676_diff__card__le__card__Diff,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,B))),hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))) ) ).

fof(fact_677_diff__card__le__card__Diff,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,B))),hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))) ) ).

fof(fact_678_finite__subset__induct,axiom,
    ! [Pa,A,F_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F_1),A))
       => ( hBOOL(hAPP_fun_a_bool_bool(Pa,bot_bot_fun_a_bool))
         => ( ! [A_14,F_2] :
                ( ( is_a(A_14)
                  & is_fun_a_bool(F_2) )
               => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_2))
                 => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_14),A))
                   => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_14),F_2))
                     => ( hBOOL(hAPP_fun_a_bool_bool(Pa,F_2))
                       => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_14),F_2))) ) ) ) ) )
           => hBOOL(hAPP_fun_a_bool_bool(Pa,F_1)) ) ) ) ) ).

fof(fact_679_finite__subset__induct,axiom,
    ! [Pa,A,F_1] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F_1),A))
       => ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
         => ( ! [A_14,F_2] :
                ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
               => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_14),A))
                 => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_14),F_2))
                   => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                     => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_14),F_2))) ) ) ) )
           => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).

fof(fact_680_finite__subset__induct,axiom,
    ! [Pa,A,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F_1),A))
       => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
         => ( ! [A_14,F_2] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_14),A))
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_14),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_14),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).

fof(fact_681_finite__subset__induct,axiom,
    ! [Pa,A,F_1] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,F_1),A))
       => ( hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool))
         => ( ! [A_14,F_2] :
                ( ( is_pname(A_14)
                  & is_fun_pname_bool(F_2) )
               => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
                 => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_14),A))
                   => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_14),F_2))
                     => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                       => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_14),F_2))) ) ) ) ) )
           => hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ) ).

fof(fact_682_assms_I2_J,axiom,
    ! [Pn,G] :
      ( hBOOL(hAPP_fun_a_bool_bool(p(hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(mgt_call,Pn)),G)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,mgt(the_com(body(Pn)))),bot_bot_fun_a_bool)))
     => hBOOL(hAPP_fun_a_bool_bool(p(G),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(mgt_call,Pn)),bot_bot_fun_a_bool))) ) ).

fof(fact_683_finite__empty__induct,axiom,
    ! [Pa,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(Pa,A))
       => ( ! [A_14,A_15] :
              ( ( is_a(A_14)
                & is_fun_a_bool(A_15) )
             => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_15))
               => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_14),A_15))
                 => ( hBOOL(hAPP_fun_a_bool_bool(Pa,A_15))
                   => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A_15),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_14),bot_bot_fun_a_bool)))) ) ) ) )
         => hBOOL(hAPP_fun_a_bool_bool(Pa,bot_bot_fun_a_bool)) ) ) ) ).

fof(fact_684_finite__empty__induct,axiom,
    ! [Pa,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(Pa,A))
       => ( ! [A_14,A_15] :
              ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_15))
             => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_14),A_15))
               => ( hBOOL(hAPP_f448129468l_bool(Pa,A_15))
                 => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A_15),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_14),bot_bot_fun_int_bool)))) ) ) )
         => hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool)) ) ) ) ).

fof(fact_685_finite__empty__induct,axiom,
    ! [Pa,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,A))
       => ( ! [A_14,A_15] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_15))
             => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_14),A_15))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,A_15))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A_15),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_14),bot_bot_fun_nat_bool)))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool)) ) ) ) ).

fof(fact_686_finite__empty__induct,axiom,
    ! [Pa,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(Pa,A))
       => ( ! [A_14,A_15] :
              ( ( is_pname(A_14)
                & is_fun_pname_bool(A_15) )
             => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_15))
               => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_14),A_15))
                 => ( hBOOL(hAPP_f1664156314l_bool(Pa,A_15))
                   => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A_15),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_14),bot_bo844097828e_bool)))) ) ) ) )
         => hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool)) ) ) ) ).

fof(fact_687_finite__induct,axiom,
    ! [Pa,F_1] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
     => ( hBOOL(hAPP_fun_a_bool_bool(Pa,bot_bot_fun_a_bool))
       => ( ! [X_2,F_2] :
              ( ( is_a(X_2)
                & is_fun_a_bool(F_2) )
             => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_2))
               => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),F_2))
                 => ( hBOOL(hAPP_fun_a_bool_bool(Pa,F_2))
                   => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),F_2))) ) ) ) )
         => hBOOL(hAPP_fun_a_bool_bool(Pa,F_1)) ) ) ) ).

fof(fact_688_finite__induct,axiom,
    ! [Pa,F_1] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(Pa,bot_bot_fun_int_bool))
       => ( ! [X_2,F_2] :
              ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
             => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),F_2))
               => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                 => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ).

fof(fact_689_finite__induct,axiom,
    ! [Pa,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
       => ( ! [X_2,F_2] :
              ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
             => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
               => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                 => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) )
         => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ).

fof(fact_690_finite__induct,axiom,
    ! [Pa,F_1] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
     => ( hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool))
       => ( ! [X_2,F_2] :
              ( ( is_pname(X_2)
                & is_fun_pname_bool(F_2) )
             => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
               => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),F_2))
                 => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                   => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),F_2))) ) ) ) )
         => hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ).

fof(fact_691_finite_Osimps,axiom,
    ! [A_2] :
      ( is_fun_a_bool(A_2)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_2))
      <=> ( A_2 = bot_bot_fun_a_bool
          | ? [A_15,A_14] :
              ( is_fun_a_bool(A_15)
              & is_a(A_14)
              & A_2 = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_14),A_15)
              & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_15)) ) ) ) ) ).

fof(fact_692_finite_Osimps,axiom,
    ! [A_2] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_2))
    <=> ( A_2 = bot_bot_fun_int_bool
        | ? [A_15,A_14] :
            ( A_2 = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_14),A_15)
            & hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_15)) ) ) ) ).

fof(fact_693_finite_Osimps,axiom,
    ! [A_2] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_2))
    <=> ( A_2 = bot_bot_fun_nat_bool
        | ? [A_15,A_14] :
            ( A_2 = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_14),A_15)
            & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_15)) ) ) ) ).

fof(fact_694_finite_Osimps,axiom,
    ! [A_2] :
      ( is_fun_pname_bool(A_2)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_2))
      <=> ( A_2 = bot_bo844097828e_bool
          | ? [A_15,A_14] :
              ( is_fun_pname_bool(A_15)
              & is_pname(A_14)
              & A_2 = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_14),A_15)
              & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_15)) ) ) ) ) ).

fof(fact_695_finite__less__ub,axiom,
    ! [U,F] :
      ( ! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(F,N_1)))
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(cOMBB_800536526ol_nat(ord_less_eq_nat,F)),U)))) ) ).

fof(fact_696_the__elem__eq,axiom,
    ! [X_1] :
      ( is_a(X_1)
     => hAPP_fun_a_bool_a(the_elem_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = X_1 ) ).

fof(fact_697_the__elem__eq,axiom,
    ! [X_1] : hAPP_f22106695ol_nat(the_elem_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ).

fof(fact_698_the__elem__eq,axiom,
    ! [X_1] : hAPP_f1594865479ol_int(the_elem_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = X_1 ).

fof(fact_699_nonempty__iff,axiom,
    ! [A] :
      ( is_fun_a_bool(A)
     => ( A != bot_bot_fun_a_bool
      <=> ? [X_2,B_15] :
            ( is_a(X_2)
            & is_fun_a_bool(B_15)
            & A = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),B_15)
            & ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B_15)) ) ) ) ).

fof(fact_700_nonempty__iff,axiom,
    ! [A] :
      ( A != bot_bot_fun_int_bool
    <=> ? [X_2,B_15] :
          ( A = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),B_15)
          & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),B_15)) ) ) ).

fof(fact_701_nonempty__iff,axiom,
    ! [A] :
      ( A != bot_bot_fun_nat_bool
    <=> ? [X_2,B_15] :
          ( A = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),B_15)
          & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B_15)) ) ) ).

fof(fact_702_nonempty__iff,axiom,
    ! [A] :
      ( is_fun_pname_bool(A)
     => ( A != bot_bo844097828e_bool
      <=> ? [X_2,B_15] :
            ( is_pname(X_2)
            & is_fun_pname_bool(B_15)
            & A = hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),B_15)
            & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B_15)) ) ) ) ).

fof(fact_703_assms_I4_J,axiom,
    ! [Pn] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Pn),u))
     => hBOOL(wt(the_com(body(Pn)))) ) ).

fof(fact_704_DiffE,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => ~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A))
         => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B)) ) ) ).

fof(fact_705_DiffE,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A))
         => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B)) ) ) ).

fof(fact_706_DiffE,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))
     => ~ ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A))
         => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B)) ) ) ).

fof(fact_707_DiffE,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
     => ~ ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A))
         => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B)) ) ) ).

fof(fact_708_DiffI,axiom,
    ! [B,C_6,A] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A))
     => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B))
       => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))) ) ) ).

fof(fact_709_DiffI,axiom,
    ! [B,C_6,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A))
     => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B))
       => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ) ).

fof(fact_710_DiffI,axiom,
    ! [B,C_6,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A))
     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B))
       => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B))) ) ) ).

fof(fact_711_DiffI,axiom,
    ! [B,C_6,A] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A))
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B))
       => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B))) ) ) ).

fof(fact_712_DiffD2,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B)) ) ).

fof(fact_713_DiffD2,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B)) ) ).

fof(fact_714_DiffD2,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))
     => ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B)) ) ).

fof(fact_715_DiffD2,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
     => ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B)) ) ).

fof(fact_716_DiffD1,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
     => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A)) ) ).

fof(fact_717_DiffD1,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
     => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A)) ) ).

fof(fact_718_DiffD1,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))
     => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A)) ) ).

fof(fact_719_DiffD1,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
     => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A)) ) ).

fof(fact_720_Diff__iff,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B)))
    <=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),A))
        & ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_6),B)) ) ) ).

fof(fact_721_Diff__iff,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
    <=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),A))
        & ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_6),B)) ) ) ).

fof(fact_722_Diff__iff,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B)))
    <=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),A))
        & ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_6),B)) ) ) ).

fof(fact_723_Diff__iff,axiom,
    ! [C_6,A,B] :
      ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B)))
    <=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),A))
        & ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_6),B)) ) ) ).

fof(fact_724_set__diff__eq,axiom,
    ! [A,B] : hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),A)),cOMBB_bool_bool_int(fNot,hAPP_f1805168059t_bool(cOMBC_94739984l_bool(member_int),B)))) ).

fof(fact_725_set__diff__eq,axiom,
    ! [A,B] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),cOMBB_bool_bool_nat(fNot,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B)))) ).

fof(fact_726_set__diff__eq,axiom,
    ! [A,B] : hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),A)),cOMBB_bool_bool_a(fNot,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B)))) ).

fof(fact_727_set__diff__eq,axiom,
    ! [A,B] : hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A)),cOMBB_647938656_pname(fNot,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),B)))) ).

fof(fact_728_folding__one_Oinsert__remove,axiom,
    ! [X_1,A,F,F_1] :
      ( is_a(X_1)
     => ( hBOOL(finite_folding_one_a(F,F_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
         => ( ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = bot_bot_fun_a_bool
             => hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = X_1 )
            & ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) != bot_bot_fun_a_bool
             => hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_a_a(hAPP_a_fun_a_a(F,X_1),hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ) ) ) ) ).

fof(fact_729_folding__one_Oinsert__remove,axiom,
    ! [X_1,A,F,F_1] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = bot_bot_fun_int_bool
           => hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = X_1 )
          & ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) != bot_bot_fun_int_bool
           => hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ).

fof(fact_730_folding__one_Oinsert__remove,axiom,
    ! [X_1,A,F,F_1] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool
           => hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = X_1 )
          & ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool
           => hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ).

fof(fact_731_folding__one_Oinsert__remove,axiom,
    ! [X_1,A,F,F_1] :
      ( is_pname(X_1)
     => ( hBOOL(finite1282449217_pname(F,F_1))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
         => ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) = bot_bo844097828e_bool
             => hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = X_1 )
            & ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) != bot_bo844097828e_bool
             => hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_1),hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ) ) ) ) ).

fof(fact_732_folding__one_Oremove,axiom,
    ! [X_1,A,F,F_1] :
      ( is_a(X_1)
     => ( hBOOL(finite_folding_one_a(F,F_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
         => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
           => ( ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = bot_bot_fun_a_bool
               => hAPP_fun_a_bool_a(F_1,A) = X_1 )
              & ( hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) != bot_bot_fun_a_bool
               => hAPP_fun_a_bool_a(F_1,A) = hAPP_a_a(hAPP_a_fun_a_a(F,X_1),hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ) ) ) ) ) ).

fof(fact_733_folding__one_Oremove,axiom,
    ! [X_1,A,F,F_1] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = bot_bot_fun_int_bool
             => hAPP_f1594865479ol_int(F_1,A) = X_1 )
            & ( hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) != bot_bot_fun_int_bool
             => hAPP_f1594865479ol_int(F_1,A) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ) ) ) ).

fof(fact_734_folding__one_Oremove,axiom,
    ! [X_1,A,F,F_1] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = bot_bot_fun_nat_bool
             => hAPP_f22106695ol_nat(F_1,A) = X_1 )
            & ( hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) != bot_bot_fun_nat_bool
             => hAPP_f22106695ol_nat(F_1,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ) ) ) ).

fof(fact_735_folding__one_Oremove,axiom,
    ! [X_1,A,F,F_1] :
      ( is_pname(X_1)
     => ( hBOOL(finite1282449217_pname(F,F_1))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
         => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
           => ( ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) = bot_bo844097828e_bool
               => hAPP_f1297739591_pname(F_1,A) = X_1 )
              & ( hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)) != bot_bo844097828e_bool
               => hAPP_f1297739591_pname(F_1,A) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_1),hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ) ) ) ) ) ).

fof(fact_736_card__Diff__singleton__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),one_one_nat) )
        & ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
         => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))) = hAPP_fun_a_bool_nat(finite_card_a,A) ) ) ) ).

fof(fact_737_card__Diff__singleton__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A)),one_one_nat) )
        & ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_f957591787ol_nat(finite_card_int,A) ) ) ) ).

fof(fact_738_card__Diff__singleton__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),one_one_nat) )
        & ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_f22106695ol_nat(finite_card_nat,A) ) ) ) ).

fof(fact_739_card__Diff__singleton__if,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),one_one_nat) )
        & ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
         => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))) = hAPP_f921600141ol_nat(finite_card_pname,A) ) ) ) ).

fof(fact_740_card__Diff__singleton,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),one_one_nat) ) ) ).

fof(fact_741_card__Diff__singleton,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,A)),one_one_nat) ) ) ).

fof(fact_742_card__Diff__singleton,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),one_one_nat) ) ) ).

fof(fact_743_card__Diff__singleton,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),one_one_nat) ) ) ).

fof(fact_744_one__reorient,axiom,
    ! [X_1] :
      ( one_one_int = X_1
    <=> X_1 = one_one_int ) ).

fof(fact_745_one__reorient,axiom,
    ! [X_1] :
      ( one_one_nat = X_1
    <=> X_1 = one_one_nat ) ).

fof(fact_746_diff__Suc__1,axiom,
    ! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N)),one_one_nat) = N ).

fof(fact_747_diff__Suc__eq__diff__pred,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N) ).

fof(fact_748_folding__one_Osingleton,axiom,
    ! [X_1,F,F_1] :
      ( is_a(X_1)
     => ( hBOOL(finite_folding_one_a(F,F_1))
       => hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)) = X_1 ) ) ).

fof(fact_749_folding__one_Osingleton,axiom,
    ! [X_1,F,F_1] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)) = X_1 ) ).

fof(fact_750_folding__one_Osingleton,axiom,
    ! [X_1,F,F_1] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)) = X_1 ) ).

fof(fact_751_card__Diff__insert,axiom,
    ! [B,A_2,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A))
       => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),B))
         => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,A_2),B))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),B))),one_one_nat) ) ) ) ).

fof(fact_752_card__Diff__insert,axiom,
    ! [B,A_2,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_2),B))
         => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,A_2),B))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),B))),one_one_nat) ) ) ) ).

fof(fact_753_card__Diff__insert,axiom,
    ! [B,A_2,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),B))
         => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,A_2),B))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))),one_one_nat) ) ) ) ).

fof(fact_754_card__Diff__insert,axiom,
    ! [B,A_2,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A))
       => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),B))
         => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,A_2),B))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),B))),one_one_nat) ) ) ) ).

fof(fact_755_folding__one_Oinsert,axiom,
    ! [X_1,A,F,F_1] :
      ( is_fun_a_bool(A)
     => ( hBOOL(finite_folding_one_a(F,F_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
         => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
           => ( A != bot_bot_fun_a_bool
             => hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_a_a(hAPP_a_fun_a_a(F,X_1),hAPP_fun_a_bool_a(F_1,A)) ) ) ) ) ) ).

fof(fact_756_folding__one_Oinsert,axiom,
    ! [X_1,A,F,F_1] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
         => ( A != bot_bot_fun_int_bool
           => hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) ) ) ) ) ).

fof(fact_757_folding__one_Oinsert,axiom,
    ! [X_1,A,F,F_1] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
         => ( A != bot_bot_fun_nat_bool
           => hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) ) ) ) ) ).

fof(fact_758_folding__one_Oinsert,axiom,
    ! [X_1,A,F,F_1] :
      ( is_fun_pname_bool(A)
     => ( hBOOL(finite1282449217_pname(F,F_1))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
         => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
           => ( A != bot_bo844097828e_bool
             => hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_1),hAPP_f1297739591_pname(F_1,A)) ) ) ) ) ) ).

fof(fact_759_folding__one_Oclosed,axiom,
    ! [A,F,F_1] :
      ( is_fun_a_bool(A)
     => ( hBOOL(finite_folding_one_a(F,F_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
         => ( A != bot_bot_fun_a_bool
           => ( ! [X_2,Y_2] :
                  ( ( is_a(X_2)
                    & is_a(Y_2) )
                 => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_a_a(hAPP_a_fun_a_a(F,X_2),Y_2)),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,Y_2),bot_bot_fun_a_bool)))) )
             => hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_fun_a_bool_a(F_1,A)),A)) ) ) ) ) ) ).

fof(fact_760_folding__one_Oclosed,axiom,
    ! [A,F,F_1] :
      ( hBOOL(finite1626084323ne_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( A != bot_bot_fun_int_bool
         => ( ! [X_2,Y_2] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(F,X_2),Y_2)),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,Y_2),bot_bot_fun_int_bool))))
           => hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_f1594865479ol_int(F_1,A)),A)) ) ) ) ) ).

fof(fact_761_folding__one_Oclosed,axiom,
    ! [A,F,F_1] :
      ( hBOOL(finite988810631ne_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( A != bot_bot_fun_nat_bool
         => ( ! [X_2,Y_2] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_2),Y_2)),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,Y_2),bot_bot_fun_nat_bool))))
           => hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_f22106695ol_nat(F_1,A)),A)) ) ) ) ) ).

fof(fact_762_folding__one_Oclosed,axiom,
    ! [A,F,F_1] :
      ( is_fun_pname_bool(A)
     => ( hBOOL(finite1282449217_pname(F,F_1))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
         => ( A != bot_bo844097828e_bool
           => ( ! [X_2,Y_2] :
                  ( ( is_pname(X_2)
                    & is_pname(Y_2) )
                 => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_pname_pname(hAPP_p1170154830_pname(F,X_2),Y_2)),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,Y_2),bot_bo844097828e_bool)))) )
             => hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_f1297739591_pname(F_1,A)),A)) ) ) ) ) ) ).

fof(fact_763_card_Oremove,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
       => hAPP_fun_a_bool_nat(finite_card_a,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ) ).

fof(fact_764_card_Oremove,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
       => hAPP_f957591787ol_nat(finite_card_int,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ) ).

fof(fact_765_card_Oremove,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
       => hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ) ).

fof(fact_766_card_Oremove,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
       => hAPP_f921600141ol_nat(finite_card_pname,A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ) ).

fof(fact_767_card_Oinsert__remove,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
     => hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_fun_a_bool_nat(finite_card_a,hAPP_f2050579477a_bool(hAPP_f1791771145a_bool(minus_1762468890a_bool,A),hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),bot_bot_fun_a_bool)))) ) ).

fof(fact_768_card_Oinsert__remove,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
     => hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f957591787ol_nat(finite_card_int,hAPP_f1805168059t_bool(hAPP_f1223193598t_bool(minus_1449998731t_bool,A),hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),bot_bot_fun_int_bool)))) ) ).

fof(fact_769_card_Oinsert__remove,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
     => hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f22106695ol_nat(finite_card_nat,hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),bot_bot_fun_nat_bool)))) ) ).

fof(fact_770_card_Oinsert__remove,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
     => hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_f921600141ol_nat(finite_card_pname,hAPP_f759274231e_bool(hAPP_f1388330588e_bool(minus_1015773161e_bool,A),hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),bot_bo844097828e_bool)))) ) ).

fof(fact_771_folding__one__idem_Osubset__idem,axiom,
    ! [B,A,F,F_1] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( B != bot_bot_fun_int_bool
         => ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B),A))
           => hAPP_int_int(hAPP_int_fun_int_int(F,hAPP_f1594865479ol_int(F_1,B)),hAPP_f1594865479ol_int(F_1,A)) = hAPP_f1594865479ol_int(F_1,A) ) ) ) ) ).

fof(fact_772_folding__one__idem_Osubset__idem,axiom,
    ! [B,A,F,F_1] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( B != bot_bot_fun_nat_bool
         => ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
           => hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_f22106695ol_nat(F_1,B)),hAPP_f22106695ol_nat(F_1,A)) = hAPP_f22106695ol_nat(F_1,A) ) ) ) ) ).

fof(fact_773_folding__one__idem_Osubset__idem,axiom,
    ! [B,A,F,F_1] :
      ( is_fun_pname_bool(B)
     => ( hBOOL(finite89670078_pname(F,F_1))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
         => ( B != bot_bo844097828e_bool
           => ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A))
             => hAPP_pname_pname(hAPP_p1170154830_pname(F,hAPP_f1297739591_pname(F_1,B)),hAPP_f1297739591_pname(F_1,A)) = hAPP_f1297739591_pname(F_1,A) ) ) ) ) ) ).

fof(fact_774_folding__one__idem_Osubset__idem,axiom,
    ! [B,A,F,F_1] :
      ( is_fun_a_bool(B)
     => ( hBOOL(finite1819937229idem_a(F,F_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
         => ( B != bot_bot_fun_a_bool
           => ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A))
             => hAPP_a_a(hAPP_a_fun_a_a(F,hAPP_fun_a_bool_a(F_1,B)),hAPP_fun_a_bool_a(F_1,A)) = hAPP_fun_a_bool_a(F_1,A) ) ) ) ) ) ).

fof(fact_775_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1,A,F,F_1] :
      ( is_fun_a_bool(A)
     => ( hBOOL(finite1819937229idem_a(F,F_1))
       => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
         => ( A != bot_bot_fun_a_bool
           => hAPP_fun_a_bool_a(F_1,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_1),A)) = hAPP_a_a(hAPP_a_fun_a_a(F,X_1),hAPP_fun_a_bool_a(F_1,A)) ) ) ) ) ).

fof(fact_776_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1,A,F,F_1] :
      ( hBOOL(finite1432773856em_int(F,F_1))
     => ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
       => ( A != bot_bot_fun_int_bool
         => hAPP_f1594865479ol_int(F_1,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_1),A)) = hAPP_int_int(hAPP_int_fun_int_int(F,X_1),hAPP_f1594865479ol_int(F_1,A)) ) ) ) ).

fof(fact_777_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1,A,F,F_1] :
      ( hBOOL(finite795500164em_nat(F,F_1))
     => ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
       => ( A != bot_bot_fun_nat_bool
         => hAPP_f22106695ol_nat(F_1,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_1),A)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_1),hAPP_f22106695ol_nat(F_1,A)) ) ) ) ).

fof(fact_778_folding__one__idem_Oinsert__idem,axiom,
    ! [X_1,A,F,F_1] :
      ( is_fun_pname_bool(A)
     => ( hBOOL(finite89670078_pname(F,F_1))
       => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
         => ( A != bot_bo844097828e_bool
           => hAPP_f1297739591_pname(F_1,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_1),A)) = hAPP_pname_pname(hAPP_p1170154830_pname(F,X_1),hAPP_f1297739591_pname(F_1,A)) ) ) ) ) ).

fof(fact_779_finite__ne__induct,axiom,
    ! [Pa,F_1] :
      ( is_fun_a_bool(F_1)
     => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
       => ( F_1 != bot_bot_fun_a_bool
         => ( ! [X_2] :
                ( is_a(X_2)
               => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),bot_bot_fun_a_bool))) )
           => ( ! [X_2,F_2] :
                  ( ( is_a(X_2)
                    & is_fun_a_bool(F_2) )
                 => ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_2))
                   => ( F_2 != bot_bot_fun_a_bool
                     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),F_2))
                       => ( hBOOL(hAPP_fun_a_bool_bool(Pa,F_2))
                         => hBOOL(hAPP_fun_a_bool_bool(Pa,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,X_2),F_2))) ) ) ) ) )
             => hBOOL(hAPP_fun_a_bool_bool(Pa,F_1)) ) ) ) ) ) ).

fof(fact_780_finite__ne__induct,axiom,
    ! [Pa,F_1] :
      ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_1))
     => ( F_1 != bot_bot_fun_int_bool
       => ( ! [X_2] : hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),bot_bot_fun_int_bool)))
         => ( ! [X_2,F_2] :
                ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,F_2))
               => ( F_2 != bot_bot_fun_int_bool
                 => ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),F_2))
                   => ( hBOOL(hAPP_f448129468l_bool(Pa,F_2))
                     => hBOOL(hAPP_f448129468l_bool(Pa,hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(insert_int,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f448129468l_bool(Pa,F_1)) ) ) ) ) ).

fof(fact_781_finite__ne__induct,axiom,
    ! [Pa,F_1] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
     => ( F_1 != bot_bot_fun_nat_bool
       => ( ! [X_2] : hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),bot_bot_fun_nat_bool)))
         => ( ! [X_2,F_2] :
                ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
               => ( F_2 != bot_bot_fun_nat_bool
                 => ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),F_2))
                   => ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
                     => hBOOL(hAPP_f54304608l_bool(Pa,hAPP_f800510211t_bool(hAPP_n1512601776t_bool(insert_nat,X_2),F_2))) ) ) ) )
           => hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).

fof(fact_782_finite__ne__induct,axiom,
    ! [Pa,F_1] :
      ( is_fun_pname_bool(F_1)
     => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
       => ( F_1 != bot_bo844097828e_bool
         => ( ! [X_2] :
                ( is_pname(X_2)
               => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),bot_bo844097828e_bool))) )
           => ( ! [X_2,F_2] :
                  ( ( is_pname(X_2)
                    & is_fun_pname_bool(F_2) )
                 => ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
                   => ( F_2 != bot_bo844097828e_bool
                     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),F_2))
                       => ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
                         => hBOOL(hAPP_f1664156314l_bool(Pa,hAPP_f759274231e_bool(hAPP_p905327722e_bool(insert_pname,X_2),F_2))) ) ) ) ) )
             => hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ) ) ).

fof(fact_783_the__elem__def,axiom,
    ! [X_3] : hAPP_fun_a_bool_a(the_elem_a,X_3) = hAPP_fun_a_bool_a(the_a,cOMBB_1273333599bool_a(hAPP_f1631501043l_bool(fequal_fun_a_bool,X_3),hAPP_f1038053735a_bool(cOMBC_394484100a_bool(insert_a),bot_bot_fun_a_bool))) ).

fof(fact_784_the__elem__def,axiom,
    ! [X_3] : hAPP_f22106695ol_nat(the_elem_nat,X_3) = hAPP_f22106695ol_nat(the_nat,cOMBB_955900739ol_nat(hAPP_f103356543l_bool(fequal_fun_nat_bool,X_3),hAPP_f1570313510t_bool(cOMBC_178881787t_bool(insert_nat),bot_bot_fun_nat_bool))) ).

fof(fact_785_the__elem__def,axiom,
    ! [X_3] : hAPP_f1594865479ol_int(the_elem_int,X_3) = hAPP_f1594865479ol_int(the_int,cOMBB_1418110531ol_int(hAPP_f284875647l_bool(fequal_fun_int_bool,X_3),hAPP_f1048215610t_bool(cOMBC_1683390479t_bool(insert_int),bot_bot_fun_int_bool))) ).

fof(fact_786_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_13,B_14,C_10] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),B_14)),C_10) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_13),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_14),C_10)) ).

fof(fact_787_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A_13,B_14,C_10] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),B_14)),C_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_13),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_14),C_10)) ).

fof(fact_788_add__left__cancel,axiom,
    ! [A_2,B_3,C_6] :
      ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),C_6)
    <=> B_3 = C_6 ) ).

fof(fact_789_add__left__cancel,axiom,
    ! [A_2,B_3,C_6] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),B_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),C_6)
    <=> B_3 = C_6 ) ).

fof(fact_790_add__right__cancel,axiom,
    ! [B_3,A_2,C_6] :
      ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_3),A_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),A_2)
    <=> B_3 = C_6 ) ).

fof(fact_791_add__right__cancel,axiom,
    ! [B_3,A_2,C_6] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),A_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),A_2)
    <=> B_3 = C_6 ) ).

fof(fact_792_add__left__imp__eq,axiom,
    ! [A_12,B_13,C_9] :
      ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_12),B_13) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_12),C_9)
     => B_13 = C_9 ) ).

fof(fact_793_add__left__imp__eq,axiom,
    ! [A_12,B_13,C_9] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_12),B_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_12),C_9)
     => B_13 = C_9 ) ).

fof(fact_794_add__imp__eq,axiom,
    ! [A_11,B_12,C_8] :
      ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_11),B_12) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_11),C_8)
     => B_12 = C_8 ) ).

fof(fact_795_add__imp__eq,axiom,
    ! [A_11,B_12,C_8] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_11),B_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_11),C_8)
     => B_12 = C_8 ) ).

fof(fact_796_add__right__imp__eq,axiom,
    ! [B_11,A_10,C_7] :
      ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_11),A_10) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_7),A_10)
     => B_11 = C_7 ) ).

fof(fact_797_add__right__imp__eq,axiom,
    ! [B_11,A_10,C_7] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_11),A_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_7),A_10)
     => B_11 = C_7 ) ).

fof(fact_798_nat__add__commute,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1) ).

fof(fact_799_nat__add__left__commute,axiom,
    ! [X,Y,Z] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

fof(fact_800_nat__add__assoc,axiom,
    ! [M_1,N,K_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)) ).

fof(fact_801_nat__add__left__cancel,axiom,
    ! [K,M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na)
    <=> M = Na ) ).

fof(fact_802_nat__add__right__cancel,axiom,
    ! [M,K,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Na),K)
    <=> M = Na ) ).

fof(fact_803_add__le__cancel__right,axiom,
    ! [A_2,C_6,B_3] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),C_6)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_3),C_6)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),B_3)) ) ).

fof(fact_804_add__le__cancel__right,axiom,
    ! [A_2,C_6,B_3] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),C_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),C_6)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),B_3)) ) ).

fof(fact_805_add__le__cancel__left,axiom,
    ! [C_6,A_2,B_3] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),A_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_6),B_3)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),B_3)) ) ).

fof(fact_806_add__le__cancel__left,axiom,
    ! [C_6,A_2,B_3] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),A_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_6),B_3)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_2),B_3)) ) ).

fof(fact_807_add__right__mono,axiom,
    ! [C_5,A_9,B_10] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_10))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_9),C_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_10),C_5))) ) ).

fof(fact_808_add__right__mono,axiom,
    ! [C_5,A_9,B_10] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_10))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_9),C_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_10),C_5))) ) ).

fof(fact_809_add__left__mono,axiom,
    ! [C_4,A_8,B_9] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_8),B_9))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),A_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),B_9))) ) ).

fof(fact_810_add__left__mono,axiom,
    ! [C_4,A_8,B_9] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_8),B_9))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_4),A_8)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_4),B_9))) ) ).

fof(fact_811_add__mono,axiom,
    ! [C_3,D_3,A_7,B_8] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_7),B_8))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_3),D_3))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_7),C_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_8),D_3))) ) ) ).

fof(fact_812_add__mono,axiom,
    ! [C_3,D_3,A_7,B_8] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_7),B_8))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_3),D_3))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_7),C_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_8),D_3))) ) ) ).

fof(fact_813_add__le__imp__le__right,axiom,
    ! [A_6,C_2,B_7] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_6),C_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_7),C_2)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_6),B_7)) ) ).

fof(fact_814_add__le__imp__le__right,axiom,
    ! [A_6,C_2,B_7] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_6),C_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_7),C_2)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_6),B_7)) ) ).

fof(fact_815_add__le__imp__le__left,axiom,
    ! [C_1,A_5,B_6] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),A_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),B_6)))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_5),B_6)) ) ).

fof(fact_816_add__le__imp__le__left,axiom,
    ! [C_1,A_5,B_6] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_1),A_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_1),B_6)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),B_6)) ) ).

fof(fact_817_diff__add__cancel,axiom,
    ! [A_4,B_5] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_4),B_5)),B_5) = A_4 ).

fof(fact_818_add__diff__cancel,axiom,
    ! [A_3,B_4] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_4)),B_4) = A_3 ).

fof(fact_819_add__Suc__right,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)) ).

fof(fact_820_add__Suc,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)) ).

fof(fact_821_add__Suc__shift,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(suc,N)) ).

fof(fact_822_le__add2,axiom,
    ! [N,M_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N))) ).

fof(fact_823_le__add1,axiom,
    ! [N,M_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1))) ).

fof(fact_824_le__iff__add,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
    <=> ? [K_2] : Na = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2) ) ).

fof(fact_825_nat__add__left__cancel__le,axiom,
    ! [K,M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

fof(fact_826_trans__le__add1,axiom,
    ! [M_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M_1))) ) ).

fof(fact_827_trans__le__add2,axiom,
    ! [M_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),J))) ) ).

fof(fact_828_add__le__mono1,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1))) ) ).

fof(fact_829_add__le__mono,axiom,
    ! [K_1,L_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).

fof(fact_830_add__leD2,axiom,
    ! [M_1,K_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ).

fof(fact_831_add__leD1,axiom,
    ! [M_1,K_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

fof(fact_832_add__leE,axiom,
    ! [M_1,K_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),N))
     => ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
         => ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) ) ) ).

fof(fact_833_diff__add__inverse2,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)),N) = M_1 ).

fof(fact_834_diff__add__inverse,axiom,
    ! [N,M_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1)),N) = M_1 ).

fof(fact_835_diff__diff__left,axiom,
    ! [I_1,J,K_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) ).

fof(fact_836_diff__cancel,axiom,
    ! [K_1,M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) ).

fof(fact_837_diff__cancel2,axiom,
    ! [M_1,K_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) ).

fof(fact_838_diff__diff__right,axiom,
    ! [I_1,K_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J) ) ).

fof(fact_839_le__diff__conv,axiom,
    ! [J_1,K,I_2] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K)),I_2))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K))) ) ).

fof(fact_840_le__add__diff,axiom,
    ! [M_1,K_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1)),K_1))) ) ).

fof(fact_841_le__add__diff__inverse,axiom,
    ! [N,M_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) = M_1 ) ).

fof(fact_842_add__diff__assoc,axiom,
    ! [I_1,K_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1) ) ).

fof(fact_843_le__diff__conv2,axiom,
    ! [I_2,K,J_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)),J_1)) ) ) ).

fof(fact_844_le__add__diff__inverse2,axiom,
    ! [N,M_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),N) = M_1 ) ).

fof(fact_845_le__imp__diff__is__add,axiom,
    ! [K,I_2,J_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_2) = K
      <=> J_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),I_2) ) ) ).

fof(fact_846_diff__add__assoc,axiom,
    ! [I_1,K_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)) ) ).

fof(fact_847_add__diff__assoc2,axiom,
    ! [I_1,K_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_1)),K_1) ) ).

fof(fact_848_diff__add__assoc2,axiom,
    ! [I_1,K_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_1)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1) ) ).

fof(fact_849_Suc__eq__plus1__left,axiom,
    ! [N] : hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

fof(fact_850_Suc__eq__plus1,axiom,
    ! [N] : hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

fof(fact_851_diff__Suc__diff__eq1,axiom,
    ! [M_1,K_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_1)),hAPP_nat_nat(suc,J)) ) ).

fof(fact_852_diff__Suc__diff__eq2,axiom,
    ! [M_1,K_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))),M_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),M_1)) ) ).

fof(fact_853_termination__basic__simps_I4_J,axiom,
    ! [Y,X,Z] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

fof(fact_854_termination__basic__simps_I3_J,axiom,
    ! [Z,X,Y] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

fof(fact_855_lessI,axiom,
    ! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,N))) ).

fof(fact_856_Suc__mono,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N))) ) ).

fof(fact_857_finite__Collect__less__nat,axiom,
    ! [K] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),K)))) ).

fof(fact_858_less__not__refl,axiom,
    ! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).

fof(fact_859_nat__neq__iff,axiom,
    ! [M,Na] :
      ( M != Na
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M)) ) ) ).

fof(fact_860_linorder__neqE__nat,axiom,
    ! [X,Y] :
      ( X != Y
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).

fof(fact_861_less__irrefl__nat,axiom,
    ! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).

fof(fact_862_less__not__refl2,axiom,
    ! [N,M_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M_1))
     => M_1 != N ) ).

fof(fact_863_less__not__refl3,axiom,
    ! [S,T] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
     => S != T ) ).

fof(fact_864_nat__less__cases,axiom,
    ! [Pa,M,Na] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M)) )
     => ( ( M = Na
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M)) )
       => ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M))
           => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M)) )
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M)) ) ) ) ).

fof(fact_865_not__less__eq,axiom,
    ! [M,Na] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M))) ) ).

fof(fact_866_less__Suc__eq,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
        | M = Na ) ) ).

fof(fact_867_Suc__less__eq,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

fof(fact_868_not__less__less__Suc__eq,axiom,
    ! [Na,M] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M)))
      <=> Na = M ) ) ).

fof(fact_869_less__antisym,axiom,
    ! [N,M_1] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M_1)))
       => M_1 = N ) ) ).

fof(fact_870_less__SucI,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N))) ) ).

fof(fact_871_Suc__lessI,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => ( hAPP_nat_nat(suc,M_1) != N
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),N)) ) ) ).

fof(fact_872_less__trans__Suc,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I_1)),K_1)) ) ) ).

fof(fact_873_less__SucE,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N)))
     => ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
       => M_1 = N ) ) ).

fof(fact_874_Suc__lessD,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ).

fof(fact_875_Suc__less__SucD,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N)))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ).

fof(fact_876_not__add__less1,axiom,
    ! [I_1,J] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),I_1)) ).

fof(fact_877_not__add__less2,axiom,
    ! [J,I_1] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_1)),I_1)) ).

fof(fact_878_nat__add__left__cancel__less,axiom,
    ! [K,M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

fof(fact_879_trans__less__add1,axiom,
    ! [M_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M_1))) ) ).

fof(fact_880_trans__less__add2,axiom,
    ! [M_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),J))) ) ).

fof(fact_881_add__less__mono1,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1))) ) ).

fof(fact_882_add__less__mono,axiom,
    ! [K_1,L_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L_1))) ) ) ).

fof(fact_883_less__add__eq__less,axiom,
    ! [M_1,N,K_1,L_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),L_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N)
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ) ).

fof(fact_884_add__lessD1,axiom,
    ! [I_1,J,K_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),K_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1)) ) ).

fof(fact_885_termination__basic__simps_I1_J,axiom,
    ! [Z,X,Y] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

fof(fact_886_termination__basic__simps_I2_J,axiom,
    ! [Y,X,Z] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z))) ) ).

fof(fact_887_nat__less__le,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
        & M != Na ) ) ).

fof(fact_888_le__eq__less__or__eq,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
        | M = Na ) ) ).

fof(fact_889_less__imp__le__nat,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

fof(fact_890_le__neq__implies__less,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => ( M_1 != N
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ) ).

fof(fact_891_less__or__eq__imp__le,axiom,
    ! [M_1,N] :
      ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
        | M_1 = N )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N)) ) ).

fof(fact_892_termination__basic__simps_I5_J,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).

fof(fact_893_less__imp__diff__less,axiom,
    ! [N,J,K_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K_1))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),N)),K_1)) ) ).

fof(fact_894_diff__less__mono2,axiom,
    ! [L_1,M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_1),M_1))) ) ) ).

fof(fact_895_finite__nat__set__iff__bounded,axiom,
    ! [N_2] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_2))
    <=> ? [M_2] :
        ! [X_2] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),M_2)) ) ) ).

fof(fact_896_card__Collect__less__nat,axiom,
    ! [Na] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),Na))) = Na ).

fof(fact_897_finite__M__bounded__by__nat,axiom,
    ! [Pa,I_2] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,Pa),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),I_2))))) ).

fof(fact_898_less__add__Suc1,axiom,
    ! [I_1,M_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),M_1)))) ).

fof(fact_899_less__add__Suc2,axiom,
    ! [I_1,M_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),I_1)))) ).

fof(fact_900_less__iff__Suc__add,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
    <=> ? [K_2] : Na = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)) ) ).

fof(fact_901_less__eq__Suc__le,axiom,
    ! [Na,M] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),M)) ) ).

fof(fact_902_less__Suc__eq__le,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

fof(fact_903_Suc__le__eq,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),Na))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

fof(fact_904_le__imp__less__Suc,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N))) ) ).

fof(fact_905_Suc__leI,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N)) ) ).

fof(fact_906_le__less__Suc__eq,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M)))
      <=> Na = M ) ) ).

fof(fact_907_Suc__le__lessD,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) ) ).

fof(fact_908_diff__less__Suc,axiom,
    ! [M_1,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),hAPP_nat_nat(suc,M_1))) ).

fof(fact_909_less__diff__conv,axiom,
    ! [I_2,J_1,K] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K)),J_1)) ) ).

fof(fact_910_add__diff__inverse,axiom,
    ! [M_1,N] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) = M_1 ) ).

fof(fact_911_diff__less__mono,axiom,
    ! [C,A_1,B_2] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_2),C))) ) ) ).

fof(fact_912_less__diff__iff,axiom,
    ! [Na,K,M] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
       => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K)))
        <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ) ) ).

fof(fact_913_less__eq__Suc__le__raw,axiom,
    ! [X_2] : hAPP_n1699378549t_bool(ord_less_nat,X_2) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_2)) ).

fof(fact_914_mono__nat__linear__lb,axiom,
    ! [M,K,F] :
      ( ! [M_2,N_1] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_1))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_2)),hAPP_nat_nat(F,N_1))) )
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,M)),K)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).

fof(fact_915_inc__induct,axiom,
    ! [Pa,I_2,J_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( hBOOL(hAPP_nat_bool(Pa,J_1))
       => ( ! [I] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1))
             => ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,I)))
               => hBOOL(hAPP_nat_bool(Pa,I)) ) )
         => hBOOL(hAPP_nat_bool(Pa,I_2)) ) ) ) ).

fof(fact_916_less__imp__Suc__add,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => ? [K_2] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)) ) ).

fof(fact_917_bounded__nat__set__is__finite,axiom,
    ! [Na,N_2] :
      ( ! [X_2] :
          ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Na)) )
     => hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_2)) ) ).

fof(fact_918_less__mono__imp__le__mono,axiom,
    ! [I_2,J_1,F] :
      ( ! [I,J_2] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
         => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I)),hAPP_nat_nat(F,J_2))) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_1))) ) ) ).

fof(fact_919_lessE,axiom,
    ! [I_1,K_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1))
     => ( K_1 != hAPP_nat_nat(suc,I_1)
       => ~ ! [J_2] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_2))
             => K_1 != hAPP_nat_nat(suc,J_2) ) ) ) ).

fof(fact_920_Suc__lessE,axiom,
    ! [I_1,K_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I_1)),K_1))
     => ~ ! [J_2] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_2))
           => K_1 != hAPP_nat_nat(suc,J_2) ) ) ).

fof(fact_921_less__zeroE,axiom,
    ! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

fof(fact_922_le0,axiom,
    ! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

fof(fact_923_zero__less__Suc,axiom,
    ! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N))) ).

fof(fact_924_le__0__eq,axiom,
    ! [Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat))
    <=> Na = zero_zero_nat ) ).

fof(fact_925_less__eq__nat_Osimps_I1_J,axiom,
    ! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).

fof(fact_926_diffs0__imp__equal,axiom,
    ! [M_1,N] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) = zero_zero_nat
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M_1) = zero_zero_nat
       => M_1 = N ) ) ).

fof(fact_927_diff__self__eq__0,axiom,
    ! [M_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),M_1) = zero_zero_nat ).

fof(fact_928_minus__nat_Odiff__0,axiom,
    ! [M_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),zero_zero_nat) = M_1 ).

fof(fact_929_diff__0__eq__0,axiom,
    ! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

fof(fact_930_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

fof(fact_931_add__eq__self__zero,axiom,
    ! [M_1,N] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N) = M_1
     => N = zero_zero_nat ) ).

fof(fact_932_add__is__0,axiom,
    ! [M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),Na) = zero_zero_nat
    <=> ( M = zero_zero_nat
        & Na = zero_zero_nat ) ) ).

fof(fact_933_Nat_Oadd__0__right,axiom,
    ! [M_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),zero_zero_nat) = M_1 ).

fof(fact_934_plus__nat_Oadd__0,axiom,
    ! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

fof(fact_935_gr0I,axiom,
    ! [N] :
      ( N != zero_zero_nat
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).

fof(fact_936_gr__implies__not0,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
     => N != zero_zero_nat ) ).

fof(fact_937_less__nat__zero__code,axiom,
    ! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

fof(fact_938_neq0__conv,axiom,
    ! [Na] :
      ( Na != zero_zero_nat
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).

fof(fact_939_not__less0,axiom,
    ! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).

fof(fact_940_Suc__neq__Zero,axiom,
    ! [M_1] : hAPP_nat_nat(suc,M_1) != zero_zero_nat ).

fof(fact_941_Zero__neq__Suc,axiom,
    ! [M_1] : zero_zero_nat != hAPP_nat_nat(suc,M_1) ).

fof(fact_942_nat_Osimps_I3_J,axiom,
    ! [Nat_2] : hAPP_nat_nat(suc,Nat_2) != zero_zero_nat ).

fof(fact_943_Suc__not__Zero,axiom,
    ! [M_1] : hAPP_nat_nat(suc,M_1) != zero_zero_nat ).

fof(fact_944_nat_Osimps_I2_J,axiom,
    ! [Nat_1] : zero_zero_nat != hAPP_nat_nat(suc,Nat_1) ).

fof(fact_945_Zero__not__Suc,axiom,
    ! [M_1] : zero_zero_nat != hAPP_nat_nat(suc,M_1) ).

fof(fact_946_gr0__conv__Suc,axiom,
    ! [Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
    <=> ? [M_2] : Na = hAPP_nat_nat(suc,M_2) ) ).

fof(fact_947_less__Suc0,axiom,
    ! [Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))
    <=> Na = zero_zero_nat ) ).

fof(fact_948_less__Suc__eq__0__disj,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,Na)))
    <=> ( M = zero_zero_nat
        | ? [J_2] :
            ( M = hAPP_nat_nat(suc,J_2)
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),Na)) ) ) ) ).

fof(fact_949_one__is__add,axiom,
    ! [M,Na] :
      ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),Na)
    <=> ( ( M = hAPP_nat_nat(suc,zero_zero_nat)
          & Na = zero_zero_nat )
        | ( M = zero_zero_nat
          & Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

fof(fact_950_add__is__1,axiom,
    ! [M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),Na) = hAPP_nat_nat(suc,zero_zero_nat)
    <=> ( ( M = hAPP_nat_nat(suc,zero_zero_nat)
          & Na = zero_zero_nat )
        | ( M = zero_zero_nat
          & Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).

fof(fact_951_add__gr__0,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
        | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ) ).

fof(fact_952_zero__less__diff,axiom,
    ! [Na,M] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),M)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

fof(fact_953_diff__less,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),M_1)) ) ) ).

fof(fact_954_diff__add__0,axiom,
    ! [N,M_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M_1)) = zero_zero_nat ).

fof(fact_955_diff__is__0__eq,axiom,
    ! [M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

fof(fact_956_diff__is__0__eq_H,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N) = zero_zero_nat ) ).

fof(fact_957_One__nat__def,axiom,
    one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).

fof(fact_958_diff__Suc,axiom,
    ! [M,Na] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,Na)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),Na)) ).

fof(fact_959_Suc__pred,axiom,
    ! [N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))) = N ) ).

fof(fact_960_diff__Suc__less,axiom,
    ! [I_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,I_1))),N)) ) ).

fof(fact_961_nat__diff__split,axiom,
    ! [Pa,A_2,B_3] :
      ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_2),B_3)))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_3))
         => hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) )
        & ! [D_2] :
            ( A_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),D_2)
           => hBOOL(hAPP_nat_bool(Pa,D_2)) ) ) ) ).

fof(fact_962_nat__diff__split__asm,axiom,
    ! [Pa,A_2,B_3] :
      ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_2),B_3)))
    <=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_3))
            & ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) )
          | ? [D_2] :
              ( A_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),D_2)
              & ~ hBOOL(hAPP_nat_bool(Pa,D_2)) ) ) ) ).

fof(fact_963_card__less__Suc,axiom,
    ! [I_2,M_3] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,suc)),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),I_2))))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),hAPP_nat_nat(suc,I_2))))) ) ).

fof(fact_964_card__less,axiom,
    ! [I_2,M_3] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),hAPP_nat_nat(suc,I_2))))) != zero_zero_nat ) ).

fof(fact_965_card__less__Suc2,axiom,
    ! [I_2,M_3] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_3))
     => hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,suc)),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),I_2)))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),M_3)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),hAPP_nat_nat(suc,I_2))))) ) ).

fof(fact_966_Suc__diff__1,axiom,
    ! [N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ).

fof(fact_967_Suc__pred_H,axiom,
    ! [N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ).

fof(fact_968_add__eq__if,axiom,
    ! [N,M_1] :
      ( ( M_1 = zero_zero_nat
       => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N) = N )
      & ( M_1 != zero_zero_nat
       => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N)) ) ) ).

fof(fact_969_ex__least__nat__less,axiom,
    ! [Na,Pa] :
      ( ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(Pa,Na))
       => ? [K_2] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),Na))
            & ! [I] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2))
               => ~ hBOOL(hAPP_nat_bool(Pa,I)) )
            & hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),one_one_nat))) ) ) ) ).

fof(fact_970_ex__least__nat__le,axiom,
    ! [Na,Pa] :
      ( ~ hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
     => ( hBOOL(hAPP_nat_bool(Pa,Na))
       => ? [K_2] :
            ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),Na))
            & ! [I] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2))
               => ~ hBOOL(hAPP_nat_bool(Pa,I)) )
            & hBOOL(hAPP_nat_bool(Pa,K_2)) ) ) ) ).

fof(fact_971_less__imp__add__positive,axiom,
    ! [I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ? [K_2] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
          & hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_2) = J ) ) ).

fof(fact_972_gr0__implies__Suc,axiom,
    ! [N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M_2] : N = hAPP_nat_nat(suc,M_2) ) ).

fof(fact_973_nat_Oexhaust,axiom,
    ! [Y] :
      ( Y != zero_zero_nat
     => ~ ! [Nat] : Y != hAPP_nat_nat(suc,Nat) ) ).

fof(fact_974_zero__induct,axiom,
    ! [Pa,K] :
      ( hBOOL(hAPP_nat_bool(Pa,K))
     => ( ! [N_1] :
            ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_1)))
           => hBOOL(hAPP_nat_bool(Pa,N_1)) )
       => hBOOL(hAPP_nat_bool(Pa,zero_zero_nat)) ) ) ).

fof(fact_975_not0__implies__Suc,axiom,
    ! [N] :
      ( N != zero_zero_nat
     => ? [M_2] : N = hAPP_nat_nat(suc,M_2) ) ).

fof(fact_976_nat__induct,axiom,
    ! [Na,Pa] :
      ( hBOOL(hAPP_nat_bool(Pa,zero_zero_nat))
     => ( ! [N_1] :
            ( hBOOL(hAPP_nat_bool(Pa,N_1))
           => hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_1))) )
       => hBOOL(hAPP_nat_bool(Pa,Na)) ) ) ).

fof(fact_977_expand__Suc,axiom,
    ! [V] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V)))
     => number_number_of_nat(V) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,number_number_of_nat(V)),one_one_nat)) ) ).

fof(fact_978_mult__0,axiom,
    ! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

fof(fact_979_mult__0__right,axiom,
    ! [M_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),zero_zero_nat) = zero_zero_nat ).

fof(fact_980_mult__is__0,axiom,
    ! [M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na) = zero_zero_nat
    <=> ( M = zero_zero_nat
        | Na = zero_zero_nat ) ) ).

fof(fact_981_mult__cancel1,axiom,
    ! [K,M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)
    <=> ( M = Na
        | K = zero_zero_nat ) ) ).

fof(fact_982_mult__cancel2,axiom,
    ! [M,K,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)
    <=> ( M = Na
        | K = zero_zero_nat ) ) ).

fof(fact_983_Suc__mult__cancel1,axiom,
    ! [K,M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na)
    <=> M = Na ) ).

fof(fact_984_add__mult__distrib,axiom,
    ! [M_1,N,K_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)) ).

fof(fact_985_add__mult__distrib2,axiom,
    ! [K_1,M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)) ).

fof(fact_986_mult__le__mono,axiom,
    ! [K_1,L_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),L_1))) ) ) ).

fof(fact_987_mult__le__mono2,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),J))) ) ).

fof(fact_988_mult__le__mono1,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),K_1))) ) ).

fof(fact_989_le__cube,axiom,
    ! [M_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),M_1)))) ).

fof(fact_990_le__square,axiom,
    ! [M_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),M_1))) ).

fof(fact_991_diff__mult__distrib,axiom,
    ! [M_1,N,K_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)) ).

fof(fact_992_diff__mult__distrib2,axiom,
    ! [K_1,M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N)) ).

fof(fact_993_nat__mult__eq__1__iff,axiom,
    ! [M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na) = one_one_nat
    <=> ( M = one_one_nat
        & Na = one_one_nat ) ) ).

fof(fact_994_nat__mult__1__right,axiom,
    ! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

fof(fact_995_nat__1__eq__mult__iff,axiom,
    ! [M,Na] :
      ( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na)
    <=> ( M = one_one_nat
        & Na = one_one_nat ) ) ).

fof(fact_996_nat__mult__1,axiom,
    ! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

fof(fact_997_mult__eq__1__iff,axiom,
    ! [M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na) = hAPP_nat_nat(suc,zero_zero_nat)
    <=> ( M = hAPP_nat_nat(suc,zero_zero_nat)
        & Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).

fof(fact_998_mult__less__mono2,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),J))) ) ) ).

fof(fact_999_mult__less__mono1,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),K_1))) ) ) ).

fof(fact_1000_mult__less__cancel2,axiom,
    ! [M,K,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ) ).

fof(fact_1001_mult__less__cancel1,axiom,
    ! [K,M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ) ).

fof(fact_1002_nat__0__less__mult__iff,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ) ).

fof(fact_1003_Suc__mult__less__cancel1,axiom,
    ! [K,M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

fof(fact_1004_mult__Suc__right,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)) ).

fof(fact_1005_mult__Suc,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M_1)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)) ).

fof(fact_1006_Suc__mult__le__cancel1,axiom,
    ! [K,M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

fof(fact_1007_mult__eq__self__implies__10,axiom,
    ! [M_1,N] :
      ( M_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)
     => ( N = one_one_nat
        | M_1 = zero_zero_nat ) ) ).

fof(fact_1008_n__less__m__mult__n,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N))) ) ) ).

fof(fact_1009_n__less__n__mult__m,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M_1))) ) ) ).

fof(fact_1010_one__less__mult,axiom,
    ! [M_1,N] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N))) ) ) ).

fof(fact_1011_one__le__mult__iff,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))
        & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),Na)) ) ) ).

fof(fact_1012_mult__le__cancel1,axiom,
    ! [K,M,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ) ).

fof(fact_1013_mult__le__cancel2,axiom,
    ! [M,K,Na] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)))
    <=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
       => hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ) ).

fof(fact_1014_mult__eq__if,axiom,
    ! [N,M_1] :
      ( ( M_1 = zero_zero_nat
       => hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N) = zero_zero_nat )
      & ( M_1 != zero_zero_nat
       => hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),one_one_nat)),N)) ) ) ).

fof(fact_1015_nat__less__add__iff2,axiom,
    ! [U,M,Na,I_2,J_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_2)),U)),Na))) ) ) ).

fof(fact_1016_nat__mult__commute,axiom,
    ! [M_1,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M_1) ).

fof(fact_1017_nat__mult__assoc,axiom,
    ! [M_1,N,K_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_1)) ).

fof(fact_1018_less__number__of__int__code,axiom,
    ! [K,L] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K)),number_number_of_int(L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).

fof(fact_1019_zmult__1,axiom,
    ! [Z] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z) = Z ).

fof(fact_1020_zmult__1__right,axiom,
    ! [Z] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),one_one_int) = Z ).

fof(fact_1021_zmult__commute,axiom,
    ! [Z,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z),W) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z) ).

fof(fact_1022_times__numeral__code_I5_J,axiom,
    ! [V,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,number_number_of_int(V)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V),W)) ).

fof(fact_1023_zmult__assoc,axiom,
    ! [Z1,Z2,Z3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3)) ).

fof(fact_1024_less__eq__number__of__int__code,axiom,
    ! [K,L] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K)),number_number_of_int(L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).

fof(fact_1025_zdiff__zmult__distrib,axiom,
    ! [Z1,Z2,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ).

fof(fact_1026_zdiff__zmult__distrib2,axiom,
    ! [W,Z1,Z2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ).

fof(fact_1027_zadd__zmult__distrib2,axiom,
    ! [W,Z1,Z2] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) ).

fof(fact_1028_plus__numeral__code_I9_J,axiom,
    ! [V,W] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,number_number_of_int(V)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),W)) ).

fof(fact_1029_zadd__zmult__distrib,axiom,
    ! [Z1,Z2,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) ).

fof(fact_1030_zmult__zless__mono2,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),J))) ) ) ).

fof(fact_1031_pos__zmult__eq__1__iff,axiom,
    ! [Na,M] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
     => ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),Na) = one_one_int
      <=> ( M = one_one_int
          & Na = one_one_int ) ) ) ).

fof(fact_1032_odd__nonzero,axiom,
    ! [Z] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ).

fof(fact_1033_zadd__0__right,axiom,
    ! [Z] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),zero_zero_int) = Z ).

fof(fact_1034_zadd__0,axiom,
    ! [Z] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z) = Z ).

fof(fact_1035_int__0__less__1,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).

fof(fact_1036_int__one__le__iff__zero__less,axiom,
    ! [Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) ) ).

fof(fact_1037_less__bin__lemma,axiom,
    ! [K,L] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),L)),zero_zero_int)) ) ).

fof(fact_1038_le__imp__0__less,axiom,
    ! [Z] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).

fof(fact_1039_odd__less__0,axiom,
    ! [Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_2)),Z_2)),zero_zero_int))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) ) ).

fof(fact_1040_zadd__left__mono,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),I_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),J))) ) ).

fof(fact_1041_zadd__assoc,axiom,
    ! [Z1,Z2,Z3] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) ).

fof(fact_1042_zadd__left__commute,axiom,
    ! [X,Y,Z] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z)) ).

fof(fact_1043_zadd__commute,axiom,
    ! [Z,W] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z) ).

fof(fact_1044_zle__refl,axiom,
    ! [W] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).

fof(fact_1045_zle__linear,axiom,
    ! [Z,W] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) ) ).

fof(fact_1046_zle__trans,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),K_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1)) ) ) ).

fof(fact_1047_zle__antisym,axiom,
    ! [Z,W] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z))
       => Z = W ) ) ).

fof(fact_1048_zle__diff1__eq,axiom,
    ! [W_1,Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).

fof(fact_1049_zless__linear,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
      | X = Y
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ).

fof(fact_1050_zless__le,axiom,
    ! [Z_2,W_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),W_1))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W_1))
        & Z_2 != W_1 ) ) ).

fof(fact_1051_zadd__strict__right__mono,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_1),K_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J),K_1))) ) ).

fof(fact_1052_zadd__zless__mono,axiom,
    ! [Z_4,Z,W_2,W] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_2),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z))) ) ) ).

fof(fact_1053_zle__add1__eq__le,axiom,
    ! [W_1,Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)) ) ).

fof(fact_1054_add1__zle__eq,axiom,
    ! [W_1,Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z_2))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) ) ).

fof(fact_1055_zless__imp__add1__zle,axiom,
    ! [W,Z] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)) ) ).

fof(fact_1056_zless__add1__eq,axiom,
    ! [W_1,Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))
    <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
        | W_1 = Z_2 ) ) ).

fof(fact_1057_nat__mult__eq__cancel__disj,axiom,
    ! [K,M,Na] :
      ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)
    <=> ( K = zero_zero_nat
        | M = Na ) ) ).

fof(fact_1058_left__add__mult__distrib,axiom,
    ! [I_1,U_1,J,K_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),J)),U_1)),K_1) ).

fof(fact_1059_nat__mult__eq__cancel1,axiom,
    ! [M,Na,K] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)
      <=> M = Na ) ) ).

fof(fact_1060_nat__mult__less__cancel1,axiom,
    ! [M,Na,K] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ) ).

fof(fact_1061_nat__mult__le__cancel1,axiom,
    ! [M,Na,K] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ) ).

fof(fact_1062_nat__le__add__iff1,axiom,
    ! [U,M,Na,J_1,I_2] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_2),J_1)),U)),M)),Na)) ) ) ).

fof(fact_1063_nat__diff__add__eq1,axiom,
    ! [U_1,M_1,N,J,I_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U_1)),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U_1)),M_1)),N) ) ).

fof(fact_1064_nat__eq__add__iff1,axiom,
    ! [U,M,Na,J_1,I_2] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)
      <=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_2),J_1)),U)),M) = Na ) ) ).

fof(fact_1065_nat__le__add__iff2,axiom,
    ! [U,M,Na,I_2,J_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_2)),U)),Na))) ) ) ).

fof(fact_1066_nat__diff__add__eq2,axiom,
    ! [U_1,M_1,N,I_1,J] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
     => hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U_1)),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U_1)),N)) ) ).

fof(fact_1067_nat__eq__add__iff2,axiom,
    ! [U,M,Na,I_2,J_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1))
     => ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)
      <=> M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_2)),U)),Na) ) ) ).

fof(fact_1068_nat__less__add__iff1,axiom,
    ! [U,M,Na,J_1,I_2] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_2),U)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_1),U)),Na)))
      <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_2),J_1)),U)),M)),Na)) ) ) ).

fof(fact_1069_zdiv__mono2__neg__lemma,axiom,
    ! [B_2,Q_2,R_2,B_1,Q_1,R_1] :
      ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)),zero_zero_int))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),B_2))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_2)) ) ) ) ) ) ) ).

fof(fact_1070_unique__quotient__lemma__neg,axiom,
    ! [B_2,Q_1,R_1,Q_2,R_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_1)),R_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_2)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),zero_zero_int))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_1))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_1)) ) ) ) ) ).

fof(fact_1071_number__of__is__id,axiom,
    ! [K_1] : number_number_of_int(K_1) = K_1 ).

fof(fact_1072_int__0__neq__1,axiom,
    zero_zero_int != one_one_int ).

fof(fact_1073_self__quotient__aux1,axiom,
    ! [R_2,Q_2,A_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1))
     => ( A_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),Q_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),A_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_2)) ) ) ) ).

fof(fact_1074_self__quotient__aux2,axiom,
    ! [R_2,Q_2,A_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1))
     => ( A_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),Q_2))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),one_one_int)) ) ) ) ).

fof(fact_1075_q__pos__lemma,axiom,
    ! [B_1,Q_1,R_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)) ) ) ) ).

fof(fact_1076_q__neg__lemma,axiom,
    ! [B_1,Q_1,R_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)),zero_zero_int))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)) ) ) ) ).

fof(fact_1077_unique__quotient__lemma,axiom,
    ! [B_2,Q_1,R_1,Q_2,R_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_1)),R_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_2)))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_2))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
           => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_2)) ) ) ) ) ).

fof(fact_1078_zdiv__mono2__lemma,axiom,
    ! [B_2,Q_2,R_2,B_1,Q_1,R_1] :
      ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),Q_1)),R_1)))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
           => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
             => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),B_2))
               => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_1)) ) ) ) ) ) ) ).

fof(fact_1079_int__less__induct,axiom,
    ! [Pa,I_2,K] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K))
     => ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),one_one_int)))
       => ( ! [I] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),K))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

fof(fact_1080_int__le__induct,axiom,
    ! [Pa,I_2,K] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K))
     => ( hBOOL(hAPP_int_bool(Pa,K))
       => ( ! [I] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

fof(fact_1081_int__gr__induct,axiom,
    ! [Pa,K,I_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_2))
     => ( hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),one_one_int)))
       => ( ! [I] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

fof(fact_1082_int__ge__induct,axiom,
    ! [Pa,K,I_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_2))
     => ( hBOOL(hAPP_int_bool(Pa,K))
       => ( ! [I] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

fof(fact_1083_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).

fof(fact_1084_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

fof(fact_1085_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).

fof(fact_1086_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

fof(fact_1087_transfer__nat__int__set__cong,axiom,
    ! [P_1,Pa] :
      ( ! [X_2] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => ( hBOOL(hAPP_int_bool(Pa,X_2))
          <=> hBOOL(hAPP_int_bool(P_1,X_2)) ) )
     => collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),Pa)) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),P_1)) ) ).

fof(fact_1088_decr__mult__lemma,axiom,
    ! [K,Pa,D_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ! [X_2] :
            ( hBOOL(hAPP_int_bool(Pa,X_2))
           => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X_2] :
              ( hBOOL(hAPP_int_bool(Pa,X_2))
             => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) ) ) ) ).

fof(fact_1089_conj__le__cong,axiom,
    ! [P_1,Pa,X_1] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(Pa)
        <=> hBOOL(P_1) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
          & hBOOL(Pa) )
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
          & hBOOL(P_1) ) ) ) ).

fof(fact_1090_imp__le__cong,axiom,
    ! [P_1,Pa,X_1] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
       => ( hBOOL(Pa)
        <=> hBOOL(P_1) ) )
     => ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => hBOOL(Pa) )
      <=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
         => hBOOL(P_1) ) ) ) ).

fof(fact_1091_incr__mult__lemma,axiom,
    ! [K,Pa,D_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ! [X_2] :
            ( hBOOL(hAPP_int_bool(Pa,X_2))
           => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))) )
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X_2] :
              ( hBOOL(hAPP_int_bool(Pa,X_2))
             => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) ) ) ) ).

fof(fact_1092_int__induct,axiom,
    ! [I_2,Pa,K] :
      ( hBOOL(hAPP_int_bool(Pa,K))
     => ( ! [I] :
            ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I))
           => ( hBOOL(hAPP_int_bool(Pa,I))
             => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),one_one_int))) ) )
       => ( ! [I] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K))
             => ( hBOOL(hAPP_int_bool(Pa,I))
               => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I),one_one_int))) ) )
         => hBOOL(hAPP_int_bool(Pa,I_2)) ) ) ) ).

fof(fact_1093_minusinfinity,axiom,
    ! [Pa,P1,D_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ! [X_2,K_2] :
            ( hBOOL(hAPP_int_bool(P1,X_2))
          <=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),D_1)))) )
       => ( ? [Z_3] :
            ! [X_2] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z_3))
             => ( hBOOL(hAPP_int_bool(Pa,X_2))
              <=> hBOOL(hAPP_int_bool(P1,X_2)) ) )
         => ( ? [X1] : hBOOL(hAPP_int_bool(P1,X1))
           => ? [X1] : hBOOL(hAPP_int_bool(Pa,X1)) ) ) ) ) ).

fof(fact_1094_plusinfinity,axiom,
    ! [Pa,P_1,D_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
     => ( ! [X_2,K_2] :
            ( hBOOL(hAPP_int_bool(P_1,X_2))
          <=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),D_1)))) )
       => ( ? [Z_3] :
            ! [X_2] :
              ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),X_2))
             => ( hBOOL(hAPP_int_bool(Pa,X_2))
              <=> hBOOL(hAPP_int_bool(P_1,X_2)) ) )
         => ( ? [X1] : hBOOL(hAPP_int_bool(P_1,X1))
           => ? [X1] : hBOOL(hAPP_int_bool(Pa,X1)) ) ) ) ) ).

fof(fact_1095_tsub__def,axiom,
    ! [Y,X] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
       => hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
       => hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = zero_zero_int ) ) ).

fof(fact_1096_zero__zle__int,axiom,
    ! [N] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))) ).

fof(fact_1097_int__less__0__conv,axiom,
    ! [K_1] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_1)),zero_zero_int)) ).

fof(fact_1098_int__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).

fof(fact_1099_int__0,axiom,
    hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).

fof(fact_1100_int__eq__0__conv,axiom,
    ! [Na] :
      ( hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int
    <=> Na = zero_zero_nat ) ).

fof(fact_1101_zless__int,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na)) ) ).

fof(fact_1102_zle__int,axiom,
    ! [M,Na] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),Na)) ) ).

fof(fact_1103_zadd__int,axiom,
    ! [M_1,N] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N)) ).

fof(fact_1104_zadd__int__left,axiom,
    ! [M_1,N,Z] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N)),Z)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N))),Z) ).

fof(fact_1105_zle__iff__zadd,axiom,
    ! [W_1,Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2))
    <=> ? [N_1] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,N_1)) ) ).

fof(fact_1106_zmult__int,axiom,
    ! [M_1,N] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)) ).

fof(fact_1107_int__mult,axiom,
    ! [M_1,N] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N)) ).

fof(fact_1108_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
    ! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)) ).

fof(fact_1109_transfer__int__nat__relations_I2_J,axiom,
    ! [X_1,Y_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) ) ).

fof(fact_1110_transfer__int__nat__relations_I3_J,axiom,
    ! [X_1,Y_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_1)) ) ).

fof(fact_1111_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
    ! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) ).

fof(fact_1112_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
    ! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)) ).

fof(fact_1113_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [A] : hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A)) ).

fof(fact_1114_transfer__nat__int__set__relations_I1_J,axiom,
    ! [A] :
      ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
    <=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A))) ) ).

fof(fact_1115_transfer__int__nat__numerals_I2_J,axiom,
    one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).

fof(fact_1116_transfer__int__nat__numerals_I1_J,axiom,
    zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).

fof(fact_1117_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
    ! [Pa] : collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),Pa)) = image_nat_int(semiri1621563631at_int,collect_nat(cOMBB_int_bool_nat(Pa,semiri1621563631at_int))) ).

fof(fact_1118_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
    ! [Z] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z))) ).

fof(fact_1119_transfer__int__nat__quantifiers_I2_J,axiom,
    ! [Pa] :
      ( ? [X_2] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
          & hBOOL(hAPP_int_bool(Pa,X_2)) )
    <=> ? [X_2] : hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

fof(fact_1120_transfer__int__nat__quantifiers_I1_J,axiom,
    ! [Pa] :
      ( ! [X_2] :
          ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))
         => hBOOL(hAPP_int_bool(Pa,X_2)) )
    <=> ! [X_2] : hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,X_2))) ) ).

fof(fact_1121_int__le__0__conv,axiom,
    ! [Na] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int))
    <=> Na = zero_zero_nat ) ).

fof(fact_1122_int__Suc0__eq__1,axiom,
    hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).

fof(fact_1123_zless__iff__Suc__zadd,axiom,
    ! [W_1,Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))
    <=> ? [N_1] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))) ) ).

fof(fact_1124_int__Suc,axiom,
    ! [M_1] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M_1)) ).

fof(fact_1125_zdiff__int,axiom,
    ! [N,M_1] :
      ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M_1))
     => hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N)) ) ).

fof(fact_1126_zero__less__int__conv,axiom,
    ! [Na] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))
    <=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).

fof(fact_1127_zmult__zless__mono2__lemma,axiom,
    ! [K_1,I_1,J] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J))
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_1)),I_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_1)),J))) ) ) ).

fof(fact_1128_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
       => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y))) ) ) ).

fof(fact_1129_zdiff__int__split,axiom,
    ! [Pa,X_1,Y_1] :
      ( hBOOL(hAPP_int_bool(Pa,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_1))))
    <=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_1))
         => hBOOL(hAPP_int_bool(Pa,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_1)))) )
        & ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1))
         => hBOOL(hAPP_int_bool(Pa,zero_zero_int)) ) ) ) ).

fof(fact_1130_tsub__eq,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
     => hAPP_int_int(hAPP_int_fun_int_int(nat_tsub,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) ).

fof(fact_1131_zero__less__imp__eq__int,axiom,
    ! [K_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1))
     => ? [N_1] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
          & K_1 = hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ).

fof(fact_1132_int__int__eq,axiom,
    ! [M,Na] :
      ( hAPP_nat_int(semiri1621563631at_int,M) = hAPP_nat_int(semiri1621563631at_int,Na)
    <=> M = Na ) ).

fof(fact_1133_transfer__nat__int__set__relations_I4_J,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).

fof(fact_1134_transfer__nat__int__set__relations_I5_J,axiom,
    ! [A,B] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).

fof(fact_1135_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
    bot_bot_fun_int_bool = image_nat_int(semiri1621563631at_int,bot_bot_fun_nat_bool) ).

fof(fact_1136_transfer__nat__int__set__relations_I3_J,axiom,
    ! [A,B] :
      ( A = B
    <=> image_nat_int(semiri1621563631at_int,A) = image_nat_int(semiri1621563631at_int,B) ) ).

fof(fact_1137_transfer__nat__int__set__relations_I2_J,axiom,
    ! [X_1,A] :
      ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
    <=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_1)),image_nat_int(semiri1621563631at_int,A))) ) ).

fof(fact_1138_int__if__cong,axiom,
    ! [X_1,Y_1,Pa] :
      ( ( hBOOL(Pa)
       => hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(Pa),X_1),Y_1)) )
      & ( ~ hBOOL(Pa)
       => hAPP_nat_int(semiri1621563631at_int,Y_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(Pa),X_1),Y_1)) ) ) ).

fof(fact_1139_transfer__int__nat__relations_I1_J,axiom,
    ! [X_1,Y_1] :
      ( hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_1)
    <=> X_1 = Y_1 ) ).

fof(fact_1140_nonneg__int__cases,axiom,
    ! [K_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
     => ~ ! [N_1] : K_1 != hAPP_nat_int(semiri1621563631at_int,N_1) ) ).

fof(fact_1141_nonneg__eq__int,axiom,
    ! [Z] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z))
     => ~ ! [M_2] : Z != hAPP_nat_int(semiri1621563631at_int,M_2) ) ).

fof(fact_1142_zero__le__imp__eq__int,axiom,
    ! [K_1] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1))
     => ? [N_1] : K_1 = hAPP_nat_int(semiri1621563631at_int,N_1) ) ).

fof(fact_1143_int__diff__cases,axiom,
    ! [Z] :
      ~ ! [M_2,N_1] : Z != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_2)),hAPP_nat_int(semiri1621563631at_int,N_1)) ).

fof(fact_1144_decr__lemma,axiom,
    ! [X,Z,D] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z)) ) ).

fof(fact_1145_zabs__less__one__iff,axiom,
    ! [Z_2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z_2)),one_one_int))
    <=> Z_2 = zero_zero_int ) ).

fof(fact_1146_neg__def,axiom,
    ! [Z_1] :
      ( hBOOL(hAPP_int_bool(nat_neg,Z_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)) ) ).

fof(fact_1147_not__neg__eq__ge__0,axiom,
    ! [X_1] :
      ( ~ hBOOL(hAPP_int_bool(nat_neg,X_1))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ).

fof(fact_1148_not__neg__1,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,one_one_int)) ).

fof(fact_1149_not__neg__0,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,zero_zero_int)) ).

fof(fact_1150_abs__zmult__eq__1,axiom,
    ! [M_1,N] :
      ( abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_1),N)) = one_one_int
     => abs_abs_int(M_1) = one_one_int ) ).

fof(fact_1151_not__neg__int,axiom,
    ! [N] : ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N))) ).

fof(fact_1152_abs__int__eq,axiom,
    ! [M_1] : abs_abs_int(hAPP_nat_int(semiri1621563631at_int,M_1)) = hAPP_nat_int(semiri1621563631at_int,M_1) ).

fof(fact_1153_neg__imp__number__of__eq__0,axiom,
    ! [V] :
      ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
     => number_number_of_nat(V) = zero_zero_nat ) ).

fof(fact_1154_eq__nat__number__of,axiom,
    ! [V_3,V_2] :
      ( number_number_of_nat(V_3) = number_number_of_nat(V_2)
    <=> ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_3)))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_2)),zero_zero_int)) )
        & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_3)))
         => ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_2)))
             => number_number_of_int(V_3) = zero_zero_int )
            & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_2)))
             => V_3 = V_2 ) ) ) ) ) ).

fof(fact_1155_nat__number__of__add__left,axiom,
    ! [V_1,K_1,V] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V_1)),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V_1)),K_1) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_1)))
           => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V_1)),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),K_1) )
          & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V_1)))
           => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V_1)),K_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),V_1))),K_1) ) ) ) ) ).

fof(fact_1156_int__nat__number__of,axiom,
    ! [V] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V)) = zero_zero_int )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(V)) = number_number_of_int(V) ) ) ).

fof(fact_1157_incr__lemma,axiom,
    ! [Z,X,D] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
     => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D)))) ) ).

fof(fact_1158_int__val__lemma,axiom,
    ! [K,F,Na] :
      ( ! [I] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat))),hAPP_nat_int(F,I)))),one_one_int)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
         => ? [I] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
              & hAPP_nat_int(F,I) = K ) ) ) ) ).

fof(fact_1159_nat0__intermed__int__val,axiom,
    ! [K,F,Na] :
      ( ! [I] :
          ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na))
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat))),hAPP_nat_int(F,I)))),one_one_int)) )
     => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
         => ? [I] :
              ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
              & hAPP_nat_int(F,I) = K ) ) ) ) ).

fof(fact_1160_nat__intermed__int__val,axiom,
    ! [K,F,Na,M] :
      ( ! [I] :
          ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),I))
            & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na)) )
         => hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat))),hAPP_nat_int(F,I)))),one_one_int)) )
     => ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),Na))
       => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,M)),K))
         => ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
           => ? [I] :
                ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),I))
                & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))
                & hAPP_nat_int(F,I) = K ) ) ) ) ) ).

fof(fact_1161_Suc__nat__number__of__add,axiom,
    ! [N,V] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(V)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(succ(V))),N) ) ) ).

fof(fact_1162_succ__def,axiom,
    ! [K_1] : succ(K_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int) ).

fof(fact_1163_Suc__nat__number__of,axiom,
    ! [V] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => hAPP_nat_nat(suc,number_number_of_nat(V)) = one_one_nat )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(V)))
       => hAPP_nat_nat(suc,number_number_of_nat(V)) = number_number_of_nat(succ(V)) ) ) ).

fof(fact_1164_nat__number__of__Bit1,axiom,
    ! [W] :
      ( ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(W)))
       => number_number_of_nat(bit1(W)) = zero_zero_nat )
      & ( ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(W)))
       => number_number_of_nat(bit1(W)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(W)),number_number_of_nat(W))) ) ) ).

fof(fact_1165_nat__1__add__number__of,axiom,
    ! [V] :
      ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
       => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),number_number_of_nat(V)) = one_one_nat )
      & ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
       => hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),number_number_of_nat(V)) = number_number_of_nat(succ(V)) ) ) ).

fof(fact_1166_succ__Pls,axiom,
    succ(pls) = bit1(pls) ).

fof(fact_1167_neg__number__of__Bit1,axiom,
    ! [W_1] :
      ( hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(bit1(W_1))))
    <=> hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(W_1))) ) ).

fof(fact_1168_not__neg__number__of__Pls,axiom,
    ~ hBOOL(hAPP_int_bool(nat_neg,number_number_of_int(pls))) ).

fof(fact_1169_transfer__int__nat__numerals_I4_J,axiom,
    number_number_of_int(bit1(bit1(pls))) = hAPP_nat_int(semiri1621563631at_int,number_number_of_nat(bit1(bit1(pls)))) ).

fof(fact_1170_numeral__1__eq__Suc__0,axiom,
    number_number_of_nat(bit1(pls)) = hAPP_nat_nat(suc,zero_zero_nat) ).

fof(fact_1171_numeral__3__eq__3,axiom,
    number_number_of_nat(bit1(bit1(pls))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ).

fof(fact_1172_Suc3__eq__add__3,axiom,
    ! [N] : hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,number_number_of_nat(bit1(bit1(pls)))),N) ).

fof(fact_1173_Numeral1__eq1__nat,axiom,
    one_one_nat = number_number_of_nat(bit1(pls)) ).

fof(fact_1174_nat__numeral__1__eq__1,axiom,
    number_number_of_nat(bit1(pls)) = one_one_nat ).

fof(fact_1175_one__is__num__one,axiom,
    one_one_int = number_number_of_int(bit1(pls)) ).

fof(fact_1176_zero__is__num__zero,axiom,
    zero_zero_int = number_number_of_int(pls) ).

fof(fact_1177_Bit1__def,axiom,
    ! [K_1] : bit1(K_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),K_1)),K_1) ).

fof(fact_1178_nat__number__of__Pls,axiom,
    number_number_of_nat(pls) = zero_zero_nat ).

fof(fact_1179_semiring__norm_I113_J,axiom,
    zero_zero_nat = number_number_of_nat(pls) ).

fof(fact_1180_Pls__def,axiom,
    pls = zero_zero_int ).

fof(fact_1181_rel__simps_I51_J,axiom,
    ! [K,L] :
      ( bit1(K) = bit1(L)
    <=> K = L ) ).

fof(fact_1182_rel__simps_I46_J,axiom,
    ! [K_1] : bit1(K_1) != pls ).

fof(fact_1183_rel__simps_I39_J,axiom,
    ! [L_1] : pls != bit1(L_1) ).

fof(fact_1184_diff__bin__simps_I1_J,axiom,
    ! [K_1] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),pls) = K_1 ).

fof(fact_1185_add__Pls,axiom,
    ! [K_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,pls),K_1) = K_1 ).

fof(fact_1186_add__Pls__right,axiom,
    ! [K_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),pls) = K_1 ).

fof(fact_1187_mult__Pls,axiom,
    ! [W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W) = pls ).

fof(fact_1188_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(bit1(pls))))) ).

fof(fact_1189_rel__simps_I19_J,axiom,
    hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).

fof(fact_1190_rel__simps_I22_J,axiom,
    ! [K] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit1(K)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).

fof(fact_1191_rel__simps_I34_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit1(L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).

fof(fact_1192_less__eq__int__code_I16_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit1(K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) ) ).

fof(fact_1193_rel__simps_I2_J,axiom,
    ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).

fof(fact_1194_rel__simps_I12_J,axiom,
    ! [K] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).

fof(fact_1195_rel__simps_I17_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),bit1(L)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).

fof(fact_1196_less__int__code_I16_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit1(K2)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) ) ).

fof(fact_1197_rel__simps_I29_J,axiom,
    ! [K] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),pls))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).

fof(fact_1198_rel__simps_I5_J,axiom,
    ! [K] :
      ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit1(K)))
    <=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).

%----Helper facts (103)
fof(help_fNot_1_1_U,axiom,
    ! [P] :
      ( ~ hBOOL(hAPP_bool_bool(fNot,P))
      | ~ hBOOL(P) ) ).

fof(help_fNot_2_1_U,axiom,
    ! [P] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(fNot,P)) ) ).

fof(help_fconj_1_1_U,axiom,
    ! [Q,P] :
      ( ~ hBOOL(P)
      | ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).

fof(help_fconj_2_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(P) ) ).

fof(help_fconj_3_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
      | hBOOL(Q) ) ).

fof(help_fdisj_1_1_U,axiom,
    ! [Q,P] :
      ( ~ hBOOL(P)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

fof(help_fdisj_2_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

fof(help_fdisj_3_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q))
      | hBOOL(P)
      | hBOOL(Q) ) ).

fof(help_fFalse_1_1_U,axiom,
    ~ hBOOL(fFalse) ).

fof(help_fFalse_1_1_T,axiom,
    ! [P] :
      ( is_bool(P)
     => ( P = fTrue
        | P = fFalse ) ) ).

fof(help_fimplies_1_1_U,axiom,
    ! [Q,P] :
      ( hBOOL(P)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

fof(help_fimplies_2_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

fof(help_fimplies_3_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q))
      | ~ hBOOL(P)
      | hBOOL(Q) ) ).

fof(help_fequal_1_1_fequal_000t__a_T,axiom,
    ! [X,Y] :
      ( ( is_a(X)
        & is_a(Y) )
     => ( ~ hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y))
        | X = Y ) ) ).

fof(help_fequal_2_1_fequal_000t__a_T,axiom,
    ! [X,Y] :
      ( X != Y
      | hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y)) ) ).

fof(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
    ! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

fof(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
    ! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

fof(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
    ! [P] :
      ( is_bool(P)
     => ( P = fTrue
        | P = fFalse ) ) ).

fof(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
    ! [P] : hAPP_nat_nat(cOMBI_nat,P) = P ).

fof(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X,Y] :
      ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y))
      | X = Y ) ).

fof(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
    ! [X,Y] :
      ( X != Y
      | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) ) ).

fof(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X,Y] :
      ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
      | X = Y ) ).

fof(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
    ! [X,Y] :
      ( X != Y
      | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).

fof(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
    ! [X,Y] :
      ( ( is_pname(X)
        & is_pname(Y) )
     => ( ~ hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y))
        | X = Y ) ) ).

fof(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
    ! [X,Y] :
      ( X != Y
      | hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y)) ) ).

fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000t__a_U,axiom,
    ! [P,Q] :
      ( is_bool(P)
     => hAPP_a_bool(cOMBK_bool_a(P),Q) = P ) ).

fof(help_COMBK_1_1_COMBK_000t__a_000tc__Com__Opname_U,axiom,
    ! [P,Q] :
      ( is_a(P)
     => hAPP_pname_a(cOMBK_a_pname(P),Q) = P ) ).

fof(help_COMBC_1_1_COMBC_000t__a_000t__a_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_a_bool(hAPP_a_fun_a_bool(cOMBC_a_a_bool(P),Q),R) = hAPP_a_bool(hAPP_a_fun_a_bool(P,R),Q) ).

fof(help_COMBK_1_1_COMBK_000tc__Int__Oint_000tc__Nat__Onat_U,axiom,
    ! [P,Q] : hAPP_nat_int(cOMBK_int_nat(P),Q) = P ).

fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P,Q] :
      ( is_bool(P)
     => hAPP_int_bool(cOMBK_bool_int(P),Q) = P ) ).

fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P,Q] :
      ( is_bool(P)
     => hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P ) ).

fof(help_fequal_1_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T,axiom,
    ! [X,Y] :
      ( ( is_fun_a_bool(X)
        & is_fun_a_bool(Y) )
     => ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y))
        | X = Y ) ) ).

fof(help_fequal_2_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T,axiom,
    ! [X,Y] :
      ( X != Y
      | hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y)) ) ).

fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000t__a_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_int_bool(hAPP_a_fun_int_bool(cOMBC_int_a_bool(P),Q),R) = hAPP_a_bool(hAPP_int_fun_a_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000t__a_U,axiom,
    ! [P,Q,R] : hAPP_a_bool(cOMBB_bool_bool_a(P,Q),R) = hAPP_bool_bool(P,hAPP_a_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_a_bool(cOMBS_a_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_a_fun_bool_bool(P,R),hAPP_a_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Com__Opname_000t__a_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_pname_bool(hAPP_a93125764e_bool(cOMBC_pname_a_bool(P),Q),R) = hAPP_a_bool(hAPP_p1534023578a_bool(P,R),Q) ).

fof(help_fequal_1_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T,axiom,
    ! [X,Y] :
      ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(fequal_fun_int_bool,X),Y))
      | X = Y ) ).

fof(help_fequal_2_1_fequal_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_T,axiom,
    ! [X,Y] :
      ( X != Y
      | hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(fequal_fun_int_bool,X),Y)) ) ).

fof(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X,Y] :
      ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y))
      | X = Y ) ).

fof(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
    ! [X,Y] :
      ( X != Y
      | hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)) ) ).

fof(help_fequal_1_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
    ! [X,Y] :
      ( ( is_fun_pname_bool(X)
        & is_fun_pname_bool(Y) )
     => ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y))
        | X = Y ) ) ).

fof(help_fequal_2_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
    ! [X,Y] :
      ( X != Y
      | hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y)) ) ).

fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P,Q,R] : hAPP_nat_bool(cOMBB_int_bool_nat(P,Q),R) = hAPP_int_bool(P,hAPP_nat_int(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_int_bool(hAPP_i1948725293t_bool(cOMBC_int_int_bool(P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_nat_bool(hAPP_i418383825t_bool(cOMBC_nat_int_bool(P),Q),R) = hAPP_int_bool(hAPP_n1082236369t_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_nat_bool(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U,axiom,
    ! [P,Q,R] : hAPP_int_bool(cOMBB_bool_bool_int(P,Q),R) = hAPP_bool_bool(P,hAPP_int_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
    ! [P,Q,R] : hAPP_nat_bool(cOMBB_bool_bool_nat(P,Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_int_bool(cOMBS_int_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_nat_bool(cOMBS_nat_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname_U,axiom,
    ! [P,Q,R] : hAPP_pname_bool(cOMBB_647938656_pname(P,Q),R) = hAPP_bool_bool(P,hAPP_pname_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_pname_bool(cOMBS_568398431l_bool(P,Q),R) = hAPP_bool_bool(hAPP_p393069232l_bool(P,R),hAPP_pname_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
    ! [P,Q,R] : hAPP_pname_bool(hAPP_p61793385e_bool(cOMBC_1149511130e_bool(P),Q),R) = hAPP_pname_bool(hAPP_p61793385e_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Int__Oint,axiom,
    ! [P,Q,R] : hAPP_int_fun_a_bool(cOMBB_375279840ol_int(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_int_a(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000t__,axiom,
    ! [P,Q,R] : hAPP_a_bool(cOMBB_1273333599bool_a(P,Q),R) = hAPP_fun_a_bool_bool(P,hAPP_a_fun_a_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Oboo,axiom,
    ! [P,Q,R] : hAPP_a_bool(hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opna,axiom,
    ! [P,Q,R] : hAPP_p1534023578a_bool(cOMBB_1897541054_pname(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_pname_a(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__,axiom,
    ! [P,Q,R] : hAPP_int_bool(hAPP_f1812805200t_bool(cOMBC_1753430397l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_i888906320l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__,axiom,
    ! [P,Q,R] : hAPP_nat_bool(hAPP_f282463732t_bool(cOMBC_1928494297l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_n1414589940l_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
    ! [P,Q,R] : hAPP_a_fun_bool_bool(cOMBB_1972296269bool_a(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_a_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc_,axiom,
    ! [P,Q,R] : hAPP_fun_a_bool_bool(cOMBS_1035972772l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f198738859l_bool(P,R),hAPP_fun_a_bool_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P,Q,R] : hAPP_n1082236369t_bool(cOMBB_1628441366ol_nat(P,Q),R) = hAPP_i1948725293t_bool(P,hAPP_nat_int(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P,Q,R] : hAPP_n1699378549t_bool(cOMBB_800536526ol_nat(P,Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a,axiom,
    ! [P,Q,R] : hAPP_a_fun_a_bool(hAPP_f1038053735a_bool(cOMBC_394484100a_bool(P),Q),R) = hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P,Q,R] : hAPP_int_bool(cOMBB_1418110531ol_int(P,Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P,Q,R] : hAPP_nat_bool(cOMBB_955900739ol_nat(P,Q),R) = hAPP_f54304608l_bool(P,hAPP_n1699378549t_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P,Q,R] : hAPP_int_bool(hAPP_f1805168059t_bool(cOMBC_94739984l_bool(P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P,Q,R] : hAPP_int_bool(hAPP_f183368031t_bool(cOMBC_1848398772l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_i1718399025l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_,axiom,
    ! [P,Q,R] : hAPP_nat_bool(hAPP_f274826591t_bool(cOMBC_620423604l_bool(P),Q),R) = hAPP_f448129468l_bool(hAPP_n609083369l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
    ! [P,Q,R] : hAPP_nat_bool(hAPP_f800510211t_bool(cOMBC_226598744l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_003,axiom,
    ! [P,Q,R] : hAPP_i68813070l_bool(cOMBB_1652995168ol_int(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_004,axiom,
    ! [P,Q,R] : hAPP_n1006566506l_bool(cOMBB_1015721476ol_nat(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P,Q,R] : hAPP_f448129468l_bool(cOMBS_1912792529l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1721100032l_bool(P,R),hAPP_f448129468l_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
    ! [P,Q,R] : hAPP_f54304608l_bool(cOMBS_1187019125l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1748468828l_bool(P,R),hAPP_f54304608l_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,axiom,
    ! [P,Q,R] : hAPP_int_bool(hAPP_f449020825t_bool(cOMBC_316919022l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_i1587699435l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,axiom,
    ! [P,Q,R] : hAPP_nat_bool(hAPP_f1066163005t_bool(cOMBC_386238098l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_n850744903l_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_005,axiom,
    ! [P,Q,R] : hAPP_p393069232l_bool(cOMBB_675860798_pname(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_pname_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__O,axiom,
    ! [P,Q,R] : hAPP_f1664156314l_bool(cOMBS_350070575l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1476298914l_bool(P,R),hAPP_f1664156314l_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob,axiom,
    ! [P,Q,R] : hAPP_pname_bool(hAPP_f759274231e_bool(cOMBC_1058051404l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_p338031245l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HO,axiom,
    ! [P,Q,R] : hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_006,axiom,
    ! [P,Q,R] : hAPP_f198738859l_bool(cOMBB_338059395a_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_fun_a_bool_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO,axiom,
    ! [P,Q,R] : hAPP_n215258509l_bool(cOMBB_1146692694ol_nat(P,Q),R) = hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__007,axiom,
    ! [P,Q,R] : hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(cOMBC_1683390479t_bool(P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__008,axiom,
    ! [P,Q,R] : hAPP_n1699378549t_bool(hAPP_f1570313510t_bool(cOMBC_178881787t_bool(P),Q),R) = hAPP_f800510211t_bool(hAPP_n1512601776t_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P,Q,R] : hAPP_f448129468l_bool(hAPP_f284875647l_bool(cOMBC_605892544l_bool(P),Q),R) = hAPP_f448129468l_bool(hAPP_f284875647l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P,Q,R] : hAPP_f54304608l_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_009,axiom,
    ! [P,Q,R] : hAPP_f1721100032l_bool(cOMBB_1974511970t_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f448129468l_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_010,axiom,
    ! [P,Q,R] : hAPP_f1748468828l_bool(cOMBB_444170502t_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f54304608l_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_011,axiom,
    ! [P,Q,R] : hAPP_f1476298914l_bool(cOMBB_2095475776e_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f1664156314l_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
    ! [P,Q,R] : hAPP_f1664156314l_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_f434788991l_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__,axiom,
    ! [P,Q,R] : hAPP_i888906320l_bool(cOMBB_313729068ol_int(P,Q),R) = hAPP_f1631501043l_bool(P,hAPP_int_fun_a_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It___012,axiom,
    ! [P,Q,R] : hAPP_n1414589940l_bool(cOMBB_1823939024ol_nat(P,Q),R) = hAPP_f1631501043l_bool(P,hAPP_nat_fun_a_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P,Q,R] : hAPP_i2112223885l_bool(cOMBB_1761098400ol_int(P,Q),R) = hAPP_f284875647l_bool(P,hAPP_i1948725293t_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__013,axiom,
    ! [P,Q,R] : hAPP_n609083369l_bool(cOMBB_1123824708ol_nat(P,Q),R) = hAPP_f284875647l_bool(P,hAPP_n1082236369t_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
    ! [P,Q,R] : hAPP_i1718399025l_bool(cOMBB_516767136ol_int(P,Q),R) = hAPP_f103356543l_bool(P,hAPP_i418383825t_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__014,axiom,
    ! [P,Q,R] : hAPP_n215258509l_bool(cOMBB_2026977092ol_nat(P,Q),R) = hAPP_f103356543l_bool(P,hAPP_n1699378549t_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
    ! [P,Q,R] : hAPP_i1587699435l_bool(cOMBB_607112096ol_int(P,Q),R) = hAPP_f434788991l_bool(P,hAPP_i7355787e_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It_015,axiom,
    ! [P,Q,R] : hAPP_n850744903l_bool(cOMBB_2117322052ol_nat(P,Q),R) = hAPP_f434788991l_bool(P,hAPP_n1025906991e_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool,axiom,
    ! [P,Q,R] : hAPP_f621171935l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(P),Q),R) = hAPP_f621171935l_bool(hAPP_f1434722111l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__H,axiom,
    ! [P,Q,R] : hAPP_f215623910l_bool(hAPP_f317860125l_bool(cOMBC_348868698l_bool(P),Q),R) = hAPP_f215623910l_bool(hAPP_f317860125l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__H,axiom,
    ! [P,Q,R] : hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(P),Q),R) = hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(P,R),Q) ).

fof(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc_,axiom,
    ! [P,Q,R] : hAPP_f1935102916l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(P),Q),R) = hAPP_f1935102916l_bool(hAPP_f510955609l_bool(P,R),Q) ).

%----Conjectures (7)
fof(conj_0,hypothesis,
    hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,u)) ).

fof(conj_1,hypothesis,
    hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,g),image_pname_a(mgt_call,u))) ).

fof(conj_2,hypothesis,
    hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,na)),hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(mgt_call,u)))) ).

fof(conj_3,hypothesis,
    hAPP_fun_a_bool_nat(finite_card_a,g) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(mgt_call,u))),hAPP_nat_nat(suc,na)) ).

fof(conj_4,hypothesis,
    hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,pn),u)) ).

fof(conj_5,hypothesis,
    ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(mgt_call,pn)),g)) ).

fof(conj_6,conjecture,
    hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_f2050579477a_bool(hAPP_a1206381875a_bool(insert_a,hAPP_pname_a(mgt_call,pn)),g)),image_pname_a(mgt_call,u))) ).

%------------------------------------------------------------------------------