%------------------------------------------------------------------------------
% File     : NUM925+8 : TPTP v8.2.0. Released v5.3.0.
% Domain   : Number Theory
% Problem  : Sum of two squares line 192, 5000 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : s2s_5000_fofpt_l192 [Bla11]

% Status   : Theorem
% Rating   : 0.61 v8.2.0, 0.56 v8.1.0, 0.53 v7.5.0, 0.69 v7.4.0, 0.70 v7.3.0, 0.66 v7.1.0, 0.70 v6.4.0, 0.65 v6.3.0, 0.71 v6.2.0, 0.84 v6.1.0, 0.87 v6.0.0, 0.74 v5.5.0, 0.78 v5.4.0, 0.79 v5.3.0
% Syntax   : Number of formulae    : 5747 (1907 unt;   0 def)
%            Number of atoms       : 14282 (4120 equ)
%            Maximal formula atoms :   17 (   2 avg)
%            Number of connectives : 9557 (1022   ~; 187   |;1082   &)
%                                         ( 907 <=>;6359  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   5 avg)
%            Maximal term depth    :   25 (   2 avg)
%            Number of predicates  :  102 ( 101 usr;   0 prp; 1-2 aty)
%            Number of functors    :  265 ( 265 usr; 149 con; 0-4 aty)
%            Number of variables   : 13369 (13028   !; 341   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 18:25:24
%          : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (281)
fof(tsy_c_Archimedean__Field_Oceiling_res,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ti(fun(X_a,int),archimedean_ceiling(X_a)) = archimedean_ceiling(X_a) ) ).

fof(tsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_res,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ti(fun(X_a,int),archim2085082626_floor(X_a)) = archim2085082626_floor(X_a) ) ).

fof(tsy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_res,axiom,
    ! [X_a,X_c] :
      ( comm_monoid_add(X_c)
     => ti(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),big_co1399186613setsum(X_a,X_c)) = big_co1399186613setsum(X_a,X_c) ) ).

fof(tsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_res,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),big_co302546470etprod(X_a,X_b)) = big_co302546470etprod(X_a,X_b) ) ).

fof(tsy_c_BijectionRel_ObijR_res,axiom,
    ! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool)),bijR(X_a,X_b)) = bijR(X_a,X_b) ).

fof(tsy_c_COMBB_res,axiom,
    ! [X_b,X_c,X_a] : ti(fun(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c))),combb(X_b,X_c,X_a)) = combb(X_b,X_c,X_a) ).

fof(tsy_c_COMBC_res,axiom,
    ! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c))),combc(X_a,X_b,X_c)) = combc(X_a,X_b,X_c) ).

fof(tsy_c_COMBI_res,axiom,
    ! [X_a] : ti(fun(X_a,X_a),combi(X_a)) = combi(X_a) ).

fof(tsy_c_COMBK_res,axiom,
    ! [X_a,X_b] : ti(fun(X_a,fun(X_b,X_a)),combk(X_a,X_b)) = combk(X_a,X_b) ).

fof(tsy_c_COMBS_res,axiom,
    ! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c))),combs(X_a,X_b,X_c)) = combs(X_a,X_b,X_c) ).

fof(tsy_c_Code__Numeral_OSuc__code__numeral_res,axiom,
    ti(fun(code_code_numeral,code_code_numeral),code_S1047413653umeral) = code_S1047413653umeral ).

fof(tsy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__case_res,axiom,
    ! [X_a] : ti(fun(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a))),code_c600343653l_case(X_a)) = code_c600343653l_case(X_a) ).

fof(tsy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__size_res,axiom,
    ti(fun(code_code_numeral,nat),code_c271388182l_size) = code_c271388182l_size ).

fof(tsy_c_Code__Numeral_Odiv__mod__code__numeral_res,axiom,
    ti(fun(code_code_numeral,fun(code_code_numeral,product_prod(code_code_numeral,code_code_numeral))),code_d418564891umeral) = code_d418564891umeral ).

fof(tsy_c_Code__Numeral_Oint__of_res,axiom,
    ti(fun(code_code_numeral,int),code_int_of) = code_int_of ).

fof(tsy_c_Code__Numeral_Onat__of__aux_res,axiom,
    ti(fun(code_code_numeral,fun(nat,nat)),code_nat_of_aux) = code_nat_of_aux ).

fof(tsy_c_Complete__Lattice_OSup__class_OSup_0_res,axiom,
    ti(fun(fun(real,bool),real),complete_Sup_Sup(real)) = complete_Sup_Sup(real) ).

fof(tsy_c_Complete__Lattice_OSup__class_OSup_1_res,axiom,
    ! [X_a] :
      ( comple44059174attice(X_a)
     => ti(fun(fun(X_a,bool),X_a),complete_Sup_Sup(X_a)) = complete_Sup_Sup(X_a) ) ).

fof(tsy_c_Complex_OIm_res,axiom,
    ti(fun(complex,real),im) = im ).

fof(tsy_c_Complex_ORe_res,axiom,
    ti(fun(complex,real),re) = re ).

fof(tsy_c_Complex_Oarg_res,axiom,
    ti(fun(complex,real),arg) = arg ).

fof(tsy_c_Complex_Ocis_res,axiom,
    ti(fun(real,complex),cis) = cis ).

fof(tsy_c_Complex_Ocnj_res,axiom,
    ti(fun(complex,complex),cnj) = cnj ).

fof(tsy_c_Complex_Ocomplex_OComplex_res,axiom,
    ti(fun(real,fun(real,complex)),complex_1) = complex_1 ).

fof(tsy_c_Complex_Ocomplex_Ocomplex__case_res,axiom,
    ! [X_a] : ti(fun(fun(real,fun(real,X_a)),fun(complex,X_a)),complex_case(X_a)) = complex_case(X_a) ).

fof(tsy_c_Complex_Ocomplex_Ocomplex__rec_res,axiom,
    ! [X_a] : ti(fun(fun(real,fun(real,X_a)),fun(complex,X_a)),complex_rec(X_a)) = complex_rec(X_a) ).

fof(tsy_c_Complex_Ocomplex_Ocomplex__size_res,axiom,
    ti(fun(complex,nat),complex_size) = complex_size ).

fof(tsy_c_Complex_Oexpi_res,axiom,
    ti(fun(complex,complex),expi) = expi ).

fof(tsy_c_Complex_Oii_res,axiom,
    ti(complex,ii) = ii ).

fof(tsy_c_Complex_Orcis_res,axiom,
    ti(fun(real,fun(real,complex)),rcis) = rcis ).

fof(tsy_c_Deriv_OBolzano__bisect_res,axiom,
    ti(fun(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real))))),bolzano_bisect) = bolzano_bisect ).

fof(tsy_c_Deriv_Oderiv_res,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ti(fun(fun(X_a,X_a),fun(X_a,fun(X_a,bool))),deriv(X_a)) = deriv(X_a) ) ).

fof(tsy_c_Divides_Oadjust_res,axiom,
    ti(fun(int,fun(product_prod(int,int),product_prod(int,int))),adjust) = adjust ).

fof(tsy_c_Divides_Odiv__class_Odiv_res,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),div_div(X_a)) = div_div(X_a) ) ).

fof(tsy_c_Divides_Odiv__class_Omod_res,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),div_mod(X_a)) = div_mod(X_a) ) ).

fof(tsy_c_Divides_Odivmod__int_res,axiom,
    ti(fun(int,fun(int,product_prod(int,int))),divmod_int) = divmod_int ).

fof(tsy_c_Divides_Odivmod__int__rel_res,axiom,
    ti(fun(int,fun(int,fun(product_prod(int,int),bool))),divmod_int_rel) = divmod_int_rel ).

fof(tsy_c_Divides_Odivmod__nat_res,axiom,
    ti(fun(nat,fun(nat,product_prod(nat,nat))),divmod_nat) = divmod_nat ).

fof(tsy_c_Divides_Odivmod__nat__rel_res,axiom,
    ti(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),divmod_nat_rel) = divmod_nat_rel ).

fof(tsy_c_Divides_OnegDivAlg_res,axiom,
    ti(fun(int,fun(int,product_prod(int,int))),negDivAlg) = negDivAlg ).

fof(tsy_c_Divides_OnegDivAlg__rel_res,axiom,
    ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),negDivAlg_rel) = negDivAlg_rel ).

fof(tsy_c_Divides_OnegateSnd_res,axiom,
    ti(fun(product_prod(int,int),product_prod(int,int)),negateSnd) = negateSnd ).

fof(tsy_c_Divides_Opdivmod_res,axiom,
    ti(fun(int,fun(int,product_prod(int,int))),pdivmod) = pdivmod ).

fof(tsy_c_Divides_OposDivAlg_res,axiom,
    ti(fun(int,fun(int,product_prod(int,int))),posDivAlg) = posDivAlg ).

fof(tsy_c_Divides_OposDivAlg__rel_res,axiom,
    ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),posDivAlg_rel) = posDivAlg_rel ).

fof(tsy_c_EulerFermat_OBnorRset_res,axiom,
    ti(fun(int,fun(int,fun(int,bool))),bnorRset) = bnorRset ).

fof(tsy_c_EulerFermat_ORRset2norRR_res,axiom,
    ti(fun(fun(int,bool),fun(int,fun(int,int))),rRset2norRR) = rRset2norRR ).

fof(tsy_c_EulerFermat_ORsetR_res,axiom,
    ti(fun(int,fun(fun(int,bool),bool)),rsetR) = rsetR ).

fof(tsy_c_EulerFermat_Ois__RRset_res,axiom,
    ti(fun(fun(int,bool),fun(int,bool)),is_RRset) = is_RRset ).

fof(tsy_c_EulerFermat_OnoXRRset_res,axiom,
    ti(fun(int,fun(int,fun(int,bool))),noXRRset) = noXRRset ).

fof(tsy_c_EulerFermat_OnorRRset_res,axiom,
    ti(fun(int,fun(int,bool)),norRRset) = norRRset ).

fof(tsy_c_EulerFermat_Ophi_res,axiom,
    ti(fun(int,nat),phi) = phi ).

fof(tsy_c_EulerFermat_Ozcongm_res,axiom,
    ti(fun(int,fun(int,fun(int,bool))),zcongm) = zcongm ).

fof(tsy_c_Euler_OMultInvPair_res,axiom,
    ti(fun(int,fun(int,fun(int,fun(int,bool)))),multInvPair) = multInvPair ).

fof(tsy_c_Euler_OSetS_res,axiom,
    ti(fun(int,fun(int,fun(fun(int,bool),bool))),setS) = setS ).

fof(tsy_c_EvenOdd_OzEven_res,axiom,
    ti(fun(int,bool),zEven) = zEven ).

fof(tsy_c_EvenOdd_OzOdd_res,axiom,
    ti(fun(int,bool),zOdd) = zOdd ).

fof(tsy_c_Fact_Ofact__class_Ofact_0_res,axiom,
    ti(fun(int,int),fact_fact(int)) = fact_fact(int) ).

fof(tsy_c_Fact_Ofact__class_Ofact_1_res,axiom,
    ti(fun(nat,nat),fact_fact(nat)) = fact_fact(nat) ).

fof(tsy_c_Fields_Oinverse__class_Odivide_0_res,axiom,
    ! [X_a] :
      ( ( inverse(X_a)
        & real_normed_vector(X_a) )
     => ti(fun(X_a,fun(X_a,X_a)),inverse_divide(X_a)) = inverse_divide(X_a) ) ).

fof(tsy_c_Fields_Oinverse__class_Odivide_1_res,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),inverse_divide(X_a)) = inverse_divide(X_a) ) ).

fof(tsy_c_Fields_Oinverse__class_Oinverse_res,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ti(fun(X_a,X_a),inverse_inverse(X_a)) = inverse_inverse(X_a) ) ).

fof(tsy_c_Finite__Set_Ocard_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,bool),nat),finite_card(X_a)) = finite_card(X_a) ).

fof(tsy_c_Finite__Set_Ofinite_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,bool),bool),finite_finite_1(X_a)) = finite_finite_1(X_a) ).

fof(tsy_c_FunDef_Oin__rel_res,axiom,
    ! [X_a,X_b] : ti(fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),in_rel(X_a,X_b)) = in_rel(X_a,X_b) ).

fof(tsy_c_FunDef_Opair__leq_res,axiom,
    ti(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),pair_leq) = pair_leq ).

fof(tsy_c_FunDef_Opair__less_res,axiom,
    ti(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),pair_less) = pair_less ).

fof(tsy_c_GCD_Obezw_res,axiom,
    ti(fun(nat,fun(nat,product_prod(int,int))),bezw) = bezw ).

fof(tsy_c_GCD_Ogcd__class_Ogcd_0_res,axiom,
    ti(fun(int,fun(int,int)),gcd_gcd(int)) = gcd_gcd(int) ).

fof(tsy_c_GCD_Ogcd__class_Ogcd_1_res,axiom,
    ti(fun(nat,fun(nat,nat)),gcd_gcd(nat)) = gcd_gcd(nat) ).

fof(tsy_c_Groups_Oabs__class_Oabs_0_res,axiom,
    ! [X_a] :
      ( abs_if(X_a)
     => ti(fun(X_a,X_a),abs_abs(X_a)) = abs_abs(X_a) ) ).

fof(tsy_c_Groups_Oabs__class_Oabs_1_res,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ti(fun(X_a,X_a),abs_abs(X_a)) = abs_abs(X_a) ) ).

fof(tsy_c_Groups_Ominus__class_Ominus_res,axiom,
    ! [X_a] :
      ( minus(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),minus_minus(X_a)) = minus_minus(X_a) ) ).

fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
    ! [X_a] :
      ( one(X_a)
     => ti(X_a,one_one(X_a)) = one_one(X_a) ) ).

fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
    ! [X_a] :
      ( ( plus(X_a)
        & linorder(X_a)
        & dvd(X_a) )
     => ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).

fof(tsy_c_Groups_Oplus__class_Oplus_1_res,axiom,
    ! [X_a] :
      ( monoid_add(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).

fof(tsy_c_Groups_Oplus__class_Oplus_2_res,axiom,
    ! [X_a] :
      ( ab_semigroup_add(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).

fof(tsy_c_Groups_Oplus__class_Oplus_3_res,axiom,
    ! [X_a] :
      ( cancel_semigroup_add(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).

fof(tsy_c_Groups_Osgn__class_Osgn_0_res,axiom,
    ! [X_a] :
      ( sgn_div_norm(X_a)
     => ti(fun(X_a,X_a),sgn_sgn(X_a)) = sgn_sgn(X_a) ) ).

fof(tsy_c_Groups_Osgn__class_Osgn_1_res,axiom,
    ! [X_a] :
      ( sgn_if(X_a)
     => ti(fun(X_a,X_a),sgn_sgn(X_a)) = sgn_sgn(X_a) ) ).

fof(tsy_c_Groups_Otimes__class_Otimes_0_res,axiom,
    ! [X_a] :
      ( power(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).

fof(tsy_c_Groups_Otimes__class_Otimes_1_res,axiom,
    ! [X_a] :
      ( dvd(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).

fof(tsy_c_Groups_Otimes__class_Otimes_2_res,axiom,
    ! [X_a] :
      ( semiring(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).

fof(tsy_c_Groups_Otimes__class_Otimes_3_res,axiom,
    ! [X_a] :
      ( no_zero_divisors(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).

fof(tsy_c_Groups_Otimes__class_Otimes_4_res,axiom,
    ! [X_a] :
      ( mult_zero(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).

fof(tsy_c_Groups_Otimes__class_Otimes_5_res,axiom,
    ! [X_a] :
      ( ab_semigroup_mult(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).

fof(tsy_c_Groups_Ouminus__class_Ouminus_res,axiom,
    ! [X_b] :
      ( uminus(X_b)
     => ti(fun(X_b,X_b),uminus_uminus(X_b)) = uminus_uminus(X_b) ) ).

fof(tsy_c_Groups_Ozero__class_Ozero_res,axiom,
    ! [X_a] :
      ( zero(X_a)
     => ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(tsy_c_HOL_OThe_0_res,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ti(fun(fun(X_a,bool),X_a),the(X_a)) = the(X_a) ) ).

fof(tsy_c_HOL_OThe_1_res,axiom,
    ti(fun(fun(int,bool),int),the(int)) = the(int) ).

fof(tsy_c_HOL_OThe_2_res,axiom,
    ! [X_a,X_b] : ti(fun(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b)),the(product_prod(X_a,X_b))) = the(product_prod(X_a,X_b)) ).

fof(tsy_c_HOL_OThe_3_res,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ti(fun(fun(X_a,bool),X_a),the(X_a)) = the(X_a) ) ).

fof(tsy_c_HOL_Oundefined_res,axiom,
    ! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).

fof(tsy_c_Hilbert__Choice_OEps_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,bool),X_a),hilbert_Eps(X_a)) = hilbert_Eps(X_a) ).

fof(tsy_c_If_res,axiom,
    ! [X_a] : ti(fun(bool,fun(X_a,fun(X_a,X_a))),if(X_a)) = if(X_a) ).

fof(tsy_c_Int2_OMultInv_res,axiom,
    ti(fun(int,fun(int,int)),multInv) = multInv ).

fof(tsy_c_IntFact_Od22set_res,axiom,
    ti(fun(int,fun(int,bool)),d22set) = d22set ).

fof(tsy_c_IntFact_Ozfact_res,axiom,
    ti(fun(int,int),zfact) = zfact ).

fof(tsy_c_IntPrimes_Oxzgcd_res,axiom,
    ti(fun(int,fun(int,product_prod(int,product_prod(int,int)))),xzgcd) = xzgcd ).

fof(tsy_c_IntPrimes_Oxzgcda_res,axiom,
    ti(fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))))),xzgcda) = xzgcda ).

fof(tsy_c_IntPrimes_Ozcong_res,axiom,
    ti(fun(int,fun(int,fun(int,bool))),zcong) = zcong ).

fof(tsy_c_IntPrimes_Ozprime_res,axiom,
    ti(fun(int,bool),zprime) = zprime ).

fof(tsy_c_Int_OBit0_res,hypothesis,
    ti(fun(int,int),bit0) = bit0 ).

fof(tsy_c_Int_OBit1_res,hypothesis,
    ti(fun(int,int),bit1) = bit1 ).

fof(tsy_c_Int_OMin_res,axiom,
    ti(int,min) = min ).

fof(tsy_c_Int_OPls_res,hypothesis,
    ti(int,pls) = pls ).

fof(tsy_c_Int_Oint__ge__less__than_res,axiom,
    ti(fun(int,fun(product_prod(int,int),bool)),int_ge_less_than) = int_ge_less_than ).

fof(tsy_c_Int_Oint__ge__less__than2_res,axiom,
    ti(fun(int,fun(product_prod(int,int),bool)),int_ge_less_than2) = int_ge_less_than2 ).

fof(tsy_c_Int_Oiszero_res,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ti(fun(X_a,bool),iszero(X_a)) = iszero(X_a) ) ).

fof(tsy_c_Int_Onat_res,axiom,
    ti(fun(int,nat),nat_1) = nat_1 ).

fof(tsy_c_Int_Onat__aux_res,axiom,
    ti(fun(int,fun(nat,nat)),nat_aux) = nat_aux ).

fof(tsy_c_Int_Onumber__class_Onumber__of_res,axiom,
    ! [X_a] :
      ( number(X_a)
     => ti(fun(int,X_a),number_number_of(X_a)) = number_number_of(X_a) ) ).

fof(tsy_c_Int_Opred_res,axiom,
    ti(fun(int,int),pred) = pred ).

fof(tsy_c_Int_Oring__1__class_OInts_res,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ti(fun(X_a,bool),ring_1_Ints(X_a)) = ring_1_Ints(X_a) ) ).

fof(tsy_c_Int_Oring__1__class_Oof__int_res,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ti(fun(int,X_a),ring_1_of_int(X_a)) = ring_1_of_int(X_a) ) ).

fof(tsy_c_Int_Osucc_res,axiom,
    ti(fun(int,int),succ) = succ ).

fof(tsy_c_Lazy__Sequence_Osmall__lazy_H__rel_res,axiom,
    ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),lazy_small_lazy_rel) = lazy_small_lazy_rel ).

fof(tsy_c_Legacy__GCD_Ozgcd_res,axiom,
    ti(fun(int,fun(int,int)),legacy_zgcd) = legacy_zgcd ).

fof(tsy_c_Lim_OisCont_res,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & topological_space(X_b) )
     => ti(fun(fun(X_a,X_b),fun(X_a,bool)),isCont(X_a,X_b)) = isCont(X_a,X_b) ) ).

fof(tsy_c_Limits_Oat_res,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ti(fun(X_a,filter(X_a)),at(X_a)) = at(X_a) ) ).

fof(tsy_c_Limits_Osequentially_res,axiom,
    ti(filter(nat),sequentially) = sequentially ).

fof(tsy_c_Limits_Otendsto_res,axiom,
    ! [X_a,X_b] :
      ( topological_space(X_b)
     => ti(fun(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool))),tendsto(X_a,X_b)) = tendsto(X_a,X_b) ) ).

fof(tsy_c_Limits_Otrivial__limit_res,axiom,
    ! [X_a] : ti(fun(filter(X_a),bool),trivial_limit(X_a)) = trivial_limit(X_a) ).

fof(tsy_c_List_Oupto__rel_res,axiom,
    ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),upto_rel) = upto_rel ).

fof(tsy_c_Log_Olog_res,axiom,
    ti(fun(real,fun(real,real)),log) = log ).

fof(tsy_c_Log_Opowr_res,axiom,
    ti(fun(real,fun(real,real)),powr) = powr ).

fof(tsy_c_Nat_OSuc_res,axiom,
    ti(fun(nat,nat),suc) = suc ).

fof(tsy_c_Nat_Onat_Onat__case_res,axiom,
    ! [X_a] : ti(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),nat_case(X_a)) = nat_case(X_a) ).

fof(tsy_c_Nat_Onat_Onat__rec_res,axiom,
    ! [X_a] : ti(fun(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a))),nat_rec(X_a)) = nat_rec(X_a) ).

fof(tsy_c_Nat_Onat_Onat__size_res,axiom,
    ti(fun(nat,nat),nat_size) = nat_size ).

fof(tsy_c_Nat_Osemiring__1__class_Oof__nat_res,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ti(fun(nat,X_a),semiring_1_of_nat(X_a)) = semiring_1_of_nat(X_a) ) ).

fof(tsy_c_Nat_Osemiring__1__class_Oof__nat__aux_res,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ti(fun(fun(X_a,X_a),fun(nat,fun(X_a,X_a))),semiri532925092at_aux(X_a)) = semiri532925092at_aux(X_a) ) ).

fof(tsy_c_Nat_Osize__class_Osize_0_res,axiom,
    ! [X_a] : ti(fun(list(X_a),nat),size_size(list(X_a))) = size_size(list(X_a)) ).

fof(tsy_c_Nat_Osize__class_Osize_1_res,axiom,
    ti(fun(code_code_numeral,nat),size_size(code_code_numeral)) = size_size(code_code_numeral) ).

fof(tsy_c_Nat_Osize__class_Osize_2_res,axiom,
    ti(fun(nat,nat),size_size(nat)) = size_size(nat) ).

fof(tsy_c_Nat_Osize__class_Osize_3_res,axiom,
    ti(fun(complex,nat),size_size(complex)) = size_size(complex) ).

fof(tsy_c_Nat__Numeral_Oneg_res,axiom,
    ti(fun(int,bool),nat_neg) = nat_neg ).

fof(tsy_c_Nat__Transfer_Ois__nat_res,axiom,
    ti(fun(int,bool),nat_is_nat) = nat_is_nat ).

fof(tsy_c_Nat__Transfer_Onat__set_res,axiom,
    ti(fun(fun(int,bool),bool),nat_nat_set) = nat_nat_set ).

fof(tsy_c_Nat__Transfer_Otransfer__morphism_res,axiom,
    ! [X_a,X_b] : ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),bool)),nat_tr1645093318rphism(X_a,X_b)) = nat_tr1645093318rphism(X_a,X_b) ).

fof(tsy_c_Nat__Transfer_Otsub_res,axiom,
    ti(fun(int,fun(int,int)),nat_tsub) = nat_tsub ).

fof(tsy_c_Nitpick_OAbs__Frac_res,axiom,
    ! [X_a] : ti(fun(product_prod(int,int),X_a),abs_Frac(X_a)) = abs_Frac(X_a) ).

fof(tsy_c_Nitpick_OFrac_res,axiom,
    ti(fun(product_prod(int,int),bool),frac_1) = frac_1 ).

fof(tsy_c_Nitpick_Ofrac_res,axiom,
    ! [X_a] : ti(fun(int,fun(int,X_a)),frac(X_a)) = frac(X_a) ).

fof(tsy_c_Nitpick_Oint__gcd_res,axiom,
    ti(fun(int,fun(int,int)),int_gcd) = int_gcd ).

fof(tsy_c_Nitpick_Oint__lcm_res,axiom,
    ti(fun(int,fun(int,int)),int_lcm) = int_lcm ).

fof(tsy_c_Nitpick_Onat__gcd_res,axiom,
    ti(fun(nat,fun(nat,nat)),nat_gcd) = nat_gcd ).

fof(tsy_c_Nitpick_Onat__gcd__rel_res,axiom,
    ti(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),nat_gcd_rel) = nat_gcd_rel ).

fof(tsy_c_Nitpick_Onat__lcm_res,axiom,
    ti(fun(nat,fun(nat,nat)),nat_lcm) = nat_lcm ).

fof(tsy_c_Nitpick_Onorm__frac_res,axiom,
    ti(fun(int,fun(int,product_prod(int,int))),norm_frac) = norm_frac ).

fof(tsy_c_Nitpick_Onorm__frac__rel_res,axiom,
    ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),norm_frac_rel) = norm_frac_rel ).

fof(tsy_c_Nitpick_Onumber__of__frac_res,axiom,
    ! [X_a] : ti(fun(int,X_a),number_of_frac(X_a)) = number_of_frac(X_a) ).

fof(tsy_c_Nitpick_Oone__frac_res,axiom,
    ! [X_a] : ti(X_a,one_frac(X_a)) = one_frac(X_a) ).

fof(tsy_c_Nitpick_Oprod_res,axiom,
    ! [X_a,X_b] : ti(fun(fun(X_a,bool),fun(fun(X_b,bool),fun(product_prod(X_a,X_b),bool))),prod(X_a,X_b)) = prod(X_a,X_b) ).

fof(tsy_c_Nitpick_Orefl_H_res,axiom,
    ! [X_a] : ti(fun(fun(product_prod(X_a,X_a),bool),bool),refl(X_a)) = refl(X_a) ).

fof(tsy_c_Nitpick_Ozero__frac_res,axiom,
    ! [X_a] : ti(X_a,zero_frac(X_a)) = zero_frac(X_a) ).

fof(tsy_c_NthRoot_Oroot_res,axiom,
    ti(fun(nat,fun(real,real)),root) = root ).

fof(tsy_c_NthRoot_Osqrt_res,axiom,
    ti(fun(real,real),sqrt) = sqrt ).

fof(tsy_c_Orderings_Oord_Omax_res,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),max(X_a)) = max(X_a) ) ).

fof(tsy_c_Orderings_Oord_Omin_res,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),min_1(X_a)) = min_1(X_a) ) ).

fof(tsy_c_Orderings_Oord__class_Oless_res,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)) = ord_less(X_a) ) ).

fof(tsy_c_Orderings_Oord__class_Oless__eq_res,axiom,
    ! [X_b] :
      ( ord(X_b)
     => ti(fun(X_b,fun(X_b,bool)),ord_less_eq(X_b)) = ord_less_eq(X_b) ) ).

fof(tsy_c_Orderings_Oord__class_Omax_res,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),ord_max(X_a)) = ord_max(X_a) ) ).

fof(tsy_c_Orderings_Oord__class_Omin_res,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ti(fun(X_a,fun(X_a,X_a)),ord_min(X_a)) = ord_min(X_a) ) ).

fof(tsy_c_Orderings_Oorder_Omono_res,axiom,
    ! [X_a] :
      ( order(X_a)
     => ti(fun(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool)),mono(nat,X_a)) = mono(nat,X_a) ) ).

fof(tsy_c_Orderings_Oorder_Ostrict__mono_res,axiom,
    ! [X_a] :
      ( order(X_a)
     => ti(fun(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool)),strict_mono(nat,X_a)) = strict_mono(nat,X_a) ) ).

fof(tsy_c_Parity_Oeven__odd__class_Oeven_0_res,axiom,
    ti(fun(nat,bool),even_odd_even(nat)) = even_odd_even(nat) ).

fof(tsy_c_Parity_Oeven__odd__class_Oeven_1_res,axiom,
    ti(fun(int,bool),even_odd_even(int)) = even_odd_even(int) ).

fof(tsy_c_Power_Opower_Opower_res,axiom,
    ! [X_a] : ti(fun(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)))),power_1(X_a)) = power_1(X_a) ).

fof(tsy_c_Power_Opower__class_Opower_res,axiom,
    ! [X_a] :
      ( power(X_a)
     => ti(fun(X_a,fun(nat,X_a)),power_power(X_a)) = power_power(X_a) ) ).

fof(tsy_c_Predicate_OPowp_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),powp(X_a)) = powp(X_a) ).

fof(tsy_c_Primes_Ocoprime_res,axiom,
    ti(fun(nat,fun(nat,bool)),coprime) = coprime ).

fof(tsy_c_Primes_Ofact_res,axiom,
    ti(fun(nat,nat),fact) = fact ).

fof(tsy_c_Primes_Oprime_res,axiom,
    ti(fun(nat,bool),prime) = prime ).

fof(tsy_c_Product__Type_OPair_res,axiom,
    ! [X_a,X_b] : ti(fun(X_a,fun(X_b,product_prod(X_a,X_b))),product_Pair(X_a,X_b)) = product_Pair(X_a,X_b) ).

fof(tsy_c_Product__Type_Oapsnd_res,axiom,
    ! [X_c,X_b,X_a] : ti(fun(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b))),product_apsnd(X_c,X_b,X_a)) = product_apsnd(X_c,X_b,X_a) ).

fof(tsy_c_Product__Type_Ofst_res,axiom,
    ! [X_a,X_b] : ti(fun(product_prod(X_a,X_b),X_a),product_fst(X_a,X_b)) = product_fst(X_a,X_b) ).

fof(tsy_c_Product__Type_Oprod_Oprod__case_res,axiom,
    ! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c)),product_prod_case(X_a,X_b,X_c)) = product_prod_case(X_a,X_b,X_c) ).

fof(tsy_c_Product__Type_Oprod_Oprod__rec_res,axiom,
    ! [X_b,X_c,X_a] : ti(fun(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a)),product_prod_rec(X_b,X_c,X_a)) = product_prod_rec(X_b,X_c,X_a) ).

fof(tsy_c_Product__Type_Oprod_Oprod__size_res,axiom,
    ! [X_a,X_b] : ti(fun(fun(X_a,nat),fun(fun(X_b,nat),fun(product_prod(X_a,X_b),nat))),product_prod_size(X_a,X_b)) = product_prod_size(X_a,X_b) ).

fof(tsy_c_Product__Type_Osnd_res,axiom,
    ! [X_a,X_b] : ti(fun(product_prod(X_a,X_b),X_b),product_snd(X_a,X_b)) = product_snd(X_a,X_b) ).

fof(tsy_c_Quickcheck__Narrowing_Oaround__zero_res,axiom,
    ti(fun(int,list(int)),quickc666637781d_zero) = quickc666637781d_zero ).

fof(tsy_c_Quickcheck__Narrowing_Oaround__zero__rel_res,axiom,
    ti(fun(int,fun(int,bool)),quickc1265749348ro_rel) = quickc1265749348ro_rel ).

fof(tsy_c_Quickcheck__Narrowing_Odiv__mod__code__int_res,axiom,
    ti(fun(quickcheck_code_int,fun(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int))),quickc495462417de_int) = quickc495462417de_int ).

fof(tsy_c_Quickcheck__Narrowing_Oint__of_res,axiom,
    ti(fun(quickcheck_code_int,int),quickcheck_int_of) = quickcheck_int_of ).

fof(tsy_c_Quickcheck__Narrowing_Onat__of_res,axiom,
    ti(fun(quickcheck_code_int,nat),quickcheck_nat_of) = quickcheck_nat_of ).

fof(tsy_c_Quickcheck__Narrowing_Oof__int_res,axiom,
    ti(fun(int,quickcheck_code_int),quickcheck_of_int) = quickcheck_of_int ).

fof(tsy_c_RComplete_Onatceiling_res,axiom,
    ti(fun(real,nat),natceiling) = natceiling ).

fof(tsy_c_RComplete_Onatfloor_res,axiom,
    ti(fun(real,nat),natfloor) = natfloor ).

fof(tsy_c_Rat_OFract_res,axiom,
    ti(fun(int,fun(int,rat)),fract) = fract ).

fof(tsy_c_Rat_OFrct_res,axiom,
    ti(fun(product_prod(int,int),rat),frct) = frct ).

fof(tsy_c_Rat_Ofield__char__0__class_ORats_res,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ti(fun(X_a,bool),field_char_0_Rats(X_a)) = field_char_0_Rats(X_a) ) ).

fof(tsy_c_Rat_Onormalize_res,axiom,
    ti(fun(product_prod(int,int),product_prod(int,int)),normalize) = normalize ).

fof(tsy_c_Rat_Oquotient__of_res,axiom,
    ti(fun(rat,product_prod(int,int)),quotient_of) = quotient_of ).

fof(tsy_c_Rat_Oratrel_res,axiom,
    ti(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),ratrel) = ratrel ).

fof(tsy_c_RealDef_ORatreal_res,axiom,
    ti(fun(rat,real),ratreal) = ratreal ).

fof(tsy_c_RealDef_Oreal_0_res,axiom,
    ti(fun(int,real),real_1(int)) = real_1(int) ).

fof(tsy_c_RealDef_Oreal_1_res,axiom,
    ti(fun(nat,real),real_1(nat)) = real_1(nat) ).

fof(tsy_c_RealDef_Ovanishes_res,axiom,
    ti(fun(fun(nat,rat),bool),vanishes) = vanishes ).

fof(tsy_c_RealVector_Odist__class_Odist_0_res,axiom,
    ! [X_a] :
      ( dist_norm(X_a)
     => ti(fun(X_a,fun(X_a,real)),dist_dist(X_a)) = dist_dist(X_a) ) ).

fof(tsy_c_RealVector_Odist__class_Odist_1_res,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ti(fun(X_a,fun(X_a,real)),dist_dist(X_a)) = dist_dist(X_a) ) ).

fof(tsy_c_RealVector_Onorm__class_Onorm_0_res,axiom,
    ! [X_a] :
      ( sgn_div_norm(X_a)
     => ti(fun(X_a,real),norm_norm(X_a)) = norm_norm(X_a) ) ).

fof(tsy_c_RealVector_Onorm__class_Onorm_1_res,axiom,
    ! [X_a] :
      ( dist_norm(X_a)
     => ti(fun(X_a,real),norm_norm(X_a)) = norm_norm(X_a) ) ).

fof(tsy_c_RealVector_Oof__real_res,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ti(fun(real,X_a),of_real(X_a)) = of_real(X_a) ) ).

fof(tsy_c_RealVector_OscaleR__class_OscaleR_0_res,axiom,
    ! [X_a] :
      ( sgn_div_norm(X_a)
     => ti(fun(real,fun(X_a,X_a)),scaleR_scaleR(X_a)) = scaleR_scaleR(X_a) ) ).

fof(tsy_c_RealVector_OscaleR__class_OscaleR_1_res,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ti(fun(real,fun(X_a,X_a)),scaleR_scaleR(X_a)) = scaleR_scaleR(X_a) ) ).

fof(tsy_c_Residues_OLegendre_res,axiom,
    ti(fun(int,fun(int,int)),legendre) = legendre ).

fof(tsy_c_Residues_OQuadRes_res,axiom,
    ti(fun(int,fun(int,bool)),quadRes) = quadRes ).

fof(tsy_c_Residues_OResSet_res,axiom,
    ti(fun(int,fun(fun(int,bool),bool)),resSet) = resSet ).

fof(tsy_c_Residues_OSR_res,axiom,
    ti(fun(int,fun(int,bool)),sr) = sr ).

fof(tsy_c_Residues_OSRStar_res,axiom,
    ti(fun(int,fun(int,bool)),sRStar) = sRStar ).

fof(tsy_c_Residues_OStandardRes_res,axiom,
    ti(fun(int,fun(int,int)),standardRes) = standardRes ).

fof(tsy_c_Rings_Odvd__class_Odvd_res,axiom,
    ! [X_a] :
      ( dvd(X_a)
     => ti(fun(X_a,fun(X_a,bool)),dvd_dvd(X_a)) = dvd_dvd(X_a) ) ).

fof(tsy_c_SEQ_OBseq_res,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ti(fun(fun(nat,X_a),bool),bseq(X_a)) = bseq(X_a) ) ).

fof(tsy_c_SEQ_OCauchy_res,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ti(fun(fun(nat,X_a),bool),cauchy(X_a)) = cauchy(X_a) ) ).

fof(tsy_c_SEQ_Olim_res,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ti(fun(fun(nat,X_a),X_a),lim(X_a)) = lim(X_a) ) ).

fof(tsy_c_SEQ_Omonoseq_res,axiom,
    ! [X_a] :
      ( order(X_a)
     => ti(fun(fun(nat,X_a),bool),monoseq(X_a)) = monoseq(X_a) ) ).

fof(tsy_c_SMT_Oz3div_res,axiom,
    ti(fun(int,fun(int,int)),z3div) = z3div ).

fof(tsy_c_SMT_Oz3mod_res,axiom,
    ti(fun(int,fun(int,int)),z3mod) = z3mod ).

fof(tsy_c_Series_Osuminf_res,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ti(fun(fun(nat,X_a),X_a),suminf(X_a)) = suminf(X_a) ) ).

fof(tsy_c_Series_Osummable_res,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ti(fun(fun(nat,X_a),bool),summable(X_a)) = summable(X_a) ) ).

fof(tsy_c_Series_Osums_res,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ti(fun(fun(nat,X_a),fun(X_a,bool)),sums(X_a)) = sums(X_a) ) ).

fof(tsy_c_SetInterval_Oord__class_OatLeastAtMost_res,axiom,
    ! [X_a] :
      ( order(X_a)
     => ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_atLeastAtMost(X_a)) = ord_atLeastAtMost(X_a) ) ).

fof(tsy_c_SetInterval_Oord__class_OatLeastLessThan_res,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_atLeastLessThan(X_a)) = ord_atLeastLessThan(X_a) ) ).

fof(tsy_c_SetInterval_Oord__class_OatMost_res,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ti(fun(X_a,fun(X_a,bool)),ord_atMost(X_a)) = ord_atMost(X_a) ) ).

fof(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_0_res,axiom,
    ! [X_a] :
      ( dense_linorder(X_a)
     => ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_gr676655987ssThan(X_a)) = ord_gr676655987ssThan(X_a) ) ).

fof(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_1_res,axiom,
    ti(fun(nat,fun(nat,fun(nat,bool))),ord_gr676655987ssThan(nat)) = ord_gr676655987ssThan(nat) ).

fof(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_2_res,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_gr676655987ssThan(X_a)) = ord_gr676655987ssThan(X_a) ) ).

fof(tsy_c_SetInterval_Oord__class_OlessThan_res,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ti(fun(X_a,fun(X_a,bool)),ord_lessThan(X_a)) = ord_lessThan(X_a) ) ).

fof(tsy_c_Set_OCollect_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,bool),fun(X_a,bool)),collect(X_a)) = collect(X_a) ).

fof(tsy_c_Set_OPow_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),pow(X_a)) = pow(X_a) ).

fof(tsy_c_Set_Oimage_res,axiom,
    ! [X_b,X_a] : ti(fun(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool))),image(X_b,X_a)) = image(X_b,X_a) ).

fof(tsy_c_Sum__Type_OPlus_res,axiom,
    ! [X_a,X_b] : ti(fun(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool))),sum_Plus(X_a,X_b)) = sum_Plus(X_a,X_b) ).

fof(tsy_c_Transcendental_Oarccos_res,axiom,
    ti(fun(real,real),arccos) = arccos ).

fof(tsy_c_Transcendental_Oarcsin_res,axiom,
    ti(fun(real,real),arcsin) = arcsin ).

fof(tsy_c_Transcendental_Oarctan_res,axiom,
    ti(fun(real,real),arctan) = arctan ).

fof(tsy_c_Transcendental_Ocos_res,axiom,
    ti(fun(real,real),cos) = cos ).

fof(tsy_c_Transcendental_Ocos__coeff_res,axiom,
    ti(fun(nat,real),cos_coeff) = cos_coeff ).

fof(tsy_c_Transcendental_Odiffs_res,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ti(fun(fun(nat,X_a),fun(nat,X_a)),diffs(X_a)) = diffs(X_a) ) ).

fof(tsy_c_Transcendental_Oexp_res,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ti(fun(X_a,X_a),exp(X_a)) = exp(X_a) ) ).

fof(tsy_c_Transcendental_Oln_res,axiom,
    ti(fun(real,real),ln) = ln ).

fof(tsy_c_Transcendental_Opi_res,axiom,
    ti(real,pi) = pi ).

fof(tsy_c_Transcendental_Osin_res,axiom,
    ti(fun(real,real),sin) = sin ).

fof(tsy_c_Transcendental_Osin__coeff_res,axiom,
    ti(fun(nat,real),sin_coeff) = sin_coeff ).

fof(tsy_c_Transcendental_Otan_res,axiom,
    ti(fun(real,real),tan) = tan ).

fof(tsy_c_TwoSquares__Mirabelle__gltvpfochi_Ois__sum2sq_res,axiom,
    ti(fun(int,bool),twoSqu1091635170sum2sq) = twoSqu1091635170sum2sq ).

fof(tsy_c_TwoSquares__Mirabelle__gltvpfochi_Osum2sq_res,axiom,
    ti(fun(product_prod(int,int),int),twoSqu1094008024sum2sq) = twoSqu1094008024sum2sq ).

fof(tsy_c_Wellfounded_Oaccp_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,bool)),accp(X_a)) = accp(X_a) ).

fof(tsy_c_Wellfounded_Ofinite__psubset_res,axiom,
    ! [X_a] : ti(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),finite_psubset(X_a)) = finite_psubset(X_a) ).

fof(tsy_c_Wellfounded_Olex__prod_res,axiom,
    ! [X_a,X_b] : ti(fun(fun(product_prod(X_a,X_a),bool),fun(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool))),lex_prod(X_a,X_b)) = lex_prod(X_a,X_b) ).

fof(tsy_c_Wellfounded_Omeasure_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,nat),fun(product_prod(X_a,X_a),bool)),measure(X_a)) = measure(X_a) ).

fof(tsy_c_Wellfounded_Omlex__prod_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool))),mlex_prod(X_a)) = mlex_prod(X_a) ).

fof(tsy_c_Wellfounded_Opred__nat_res,axiom,
    ti(fun(product_prod(nat,nat),bool),pred_nat) = pred_nat ).

fof(tsy_c_WilsonRuss_Oinv_res,axiom,
    ti(fun(int,fun(int,int)),inv) = inv ).

fof(tsy_c_WilsonRuss_Owset_res,axiom,
    ti(fun(int,fun(int,fun(int,bool))),wset) = wset ).

fof(tsy_c_fEx_res,axiom,
    ! [X_a] : ti(fun(fun(X_a,bool),bool),fEx(X_a)) = fEx(X_a) ).

fof(tsy_c_fFalse_res,axiom,
    ti(bool,fFalse) = fFalse ).

fof(tsy_c_fNot_res,axiom,
    ti(fun(bool,bool),fNot) = fNot ).

fof(tsy_c_fTrue_res,axiom,
    ti(bool,fTrue) = fTrue ).

fof(tsy_c_fconj_res,axiom,
    ti(fun(bool,fun(bool,bool)),fconj) = fconj ).

fof(tsy_c_fdisj_res,axiom,
    ti(fun(bool,fun(bool,bool)),fdisj) = fdisj ).

fof(tsy_c_fequal_res,axiom,
    ! [X_a] : ti(fun(X_a,fun(X_a,bool)),fequal(X_a)) = fequal(X_a) ).

fof(tsy_c_fimplies_res,axiom,
    ti(fun(bool,fun(bool,bool)),fimplies) = fimplies ).

fof(tsy_c_hAPP_arg1,axiom,
    ! [X_a,X_c,B_1_1,B_2_1] : hAPP(X_a,X_c,ti(fun(X_a,X_c),B_1_1),B_2_1) = hAPP(X_a,X_c,B_1_1,B_2_1) ).

fof(tsy_c_hAPP_arg2,axiom,
    ! [X_a,X_c,B_1_1,B_2_1] : hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1) ).

fof(tsy_c_hAPP_res,axiom,
    ! [X_c,X_a,B_1_1,B_2_1] : ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1) ).

fof(tsy_c_hBOOL_arg1,axiom,
    ! [B_1_1] :
      ( hBOOL(ti(bool,B_1_1))
    <=> hBOOL(B_1_1) ) ).

fof(tsy_c_member_res,axiom,
    ! [X_a] : ti(fun(X_a,fun(fun(X_a,bool),bool)),member(X_a)) = member(X_a) ).

fof(tsy_v_m_res,axiom,
    ti(int,m) = m ).

fof(tsy_v_m1_____res,axiom,
    ti(int,m1) = m1 ).

fof(tsy_v_n_____res,hypothesis,
    ti(nat,n) = n ).

fof(tsy_v_r_____res,axiom,
    ti(int,r) = r ).

fof(tsy_v_s1_____res,axiom,
    ti(int,s1) = s1 ).

fof(tsy_v_s_____res,axiom,
    ti(int,s) = s ).

fof(tsy_v_sa_____res,axiom,
    ti(int,sa) = sa ).

fof(tsy_v_t_____res,axiom,
    ti(int,t) = t ).

fof(tsy_v_tn_____res,axiom,
    ti(nat,tn) = tn ).

fof(tsy_v_v_____res,axiom,
    ti(int,v) = v ).

fof(tsy_v_w_____res,axiom,
    ti(int,w) = w ).

fof(tsy_v_x_____res,axiom,
    ti(int,x) = x ).

fof(tsy_v_y_____res,axiom,
    ti(int,y) = y ).

%----Relevant facts (4992)
fof(fact_0_n1pos,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) ).

fof(fact_1_t1,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) ).

fof(fact_2_sum__power2__eq__zero__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Ya] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(X_a)
        <=> ( ti(X_a,Xa) = zero_zero(X_a)
            & ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).

fof(fact_3_one__power2,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),one_one(X_a)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(X_a) ) ).

fof(fact_4_zero__power2,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(X_a) ) ).

fof(fact_5_zero__eq__power2,axiom,
    ! [X_a] :
      ( ring_11004092258visors(X_a)
     => ! [A_3] :
          ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(X_a)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_6_add__special_I2_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),W)) ) ).

fof(fact_7_add__special_I3_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,bit1,pls))) ) ).

fof(fact_8_one__add__one__is__two,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ).

fof(fact_9_semiring__one__add__one__is__two,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ).

fof(fact_10_quartic__square__square,axiom,
    ! [X] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_11_power__0__left__number__of,axiom,
    ! [X_a] :
      ( ( power(X_a)
        & semiring_0(X_a) )
     => ! [W] :
          ( ( hAPP(int,nat,number_number_of(nat),W) = zero_zero(nat)
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),W)) = one_one(X_a) )
          & ( hAPP(int,nat,number_number_of(nat),W) != zero_zero(nat)
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),W)) = zero_zero(X_a) ) ) ) ).

fof(fact_12_semiring__norm_I110_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => one_one(X_a) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) ) ).

fof(fact_13_numeral__1__eq__1,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) = one_one(X_a) ) ).

fof(fact_14_n0,axiom,
    hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),n)) ).

fof(fact_15_zless__linear,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),Y))
      | ti(int,X) = ti(int,Y)
      | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X)) ) ).

fof(fact_16_less__number__of__int__code,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),K)),hAPP(int,int,number_number_of(int),L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).

fof(fact_17_plus__numeral__code_I9_J,axiom,
    ! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W)) ).

fof(fact_18_less__number__of,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),hAPP(int,X_a,number_number_of(X_a),Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) ) ) ).

fof(fact_19_zero__is__num__zero,axiom,
    zero_zero(int) = hAPP(int,int,number_number_of(int),pls) ).

fof(fact_20_zpower__int,axiom,
    ! [M,N] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),M)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N)) ).

fof(fact_21_int__power,axiom,
    ! [M,N] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),M)),N) ).

fof(fact_22_zadd__int__left,axiom,
    ! [M,N,Z_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),N)),Z_1)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))),Z_1) ).

fof(fact_23_zadd__int,axiom,
    ! [M,N] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

fof(fact_24_int__1,axiom,
    hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

fof(fact_25_nat__number__of__Pls,axiom,
    hAPP(int,nat,number_number_of(nat),pls) = zero_zero(nat) ).

fof(fact_26_semiring__norm_I113_J,axiom,
    zero_zero(nat) = hAPP(int,nat,number_number_of(nat),pls) ).

fof(fact_27_int__eq__0__conv,axiom,
    ! [Na] :
      ( hAPP(nat,int,semiring_1_of_nat(int),Na) = zero_zero(int)
    <=> Na = zero_zero(nat) ) ).

fof(fact_28_int__0,axiom,
    hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

fof(fact_29_nat__1__add__1,axiom,
    hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ).

fof(fact_30_less__int__code_I16_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit1,K2)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).

fof(fact_31_rel__simps_I17_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit1,L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).

fof(fact_32_rel__simps_I2_J,axiom,
    ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),pls)) ).

fof(fact_33_less__int__code_I13_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit0,K2)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).

fof(fact_34_rel__simps_I14_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit0,L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).

fof(fact_35_zadd__strict__right__mono,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_1),K_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),J_2),K_4))) ) ).

fof(fact_36_add__nat__number__of,axiom,
    ! [V_2,V] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,number_number_of(nat),V_2) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls))
           => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,number_number_of(nat),V) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls))
           => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),V_2)) ) ) ) ) ).

fof(fact_37_one__is__num__one,axiom,
    one_one(int) = hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls)) ).

fof(fact_38_nat__numeral__1__eq__1,axiom,
    hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,pls)) = one_one(nat) ).

fof(fact_39_Numeral1__eq1__nat,axiom,
    one_one(nat) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,pls)) ).

fof(fact_40_eq__number__of,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & ring_char_0(X_a) )
     => ! [Xa,Ya] :
          ( hAPP(int,X_a,number_number_of(X_a),Xa) = hAPP(int,X_a,number_number_of(X_a),Ya)
        <=> ti(int,Xa) = ti(int,Ya) ) ) ).

fof(fact_41_number__of__reorient,axiom,
    ! [X_a] :
      ( number(X_a)
     => ! [Wa,Xa] :
          ( hAPP(int,X_a,number_number_of(X_a),Wa) = ti(X_a,Xa)
        <=> ti(X_a,Xa) = hAPP(int,X_a,number_number_of(X_a),Wa) ) ) ).

fof(fact_42_rel__simps_I51_J,axiom,
    ! [K,L] :
      ( hAPP(int,int,bit1,K) = hAPP(int,int,bit1,L)
    <=> ti(int,K) = ti(int,L) ) ).

fof(fact_43_rel__simps_I48_J,axiom,
    ! [K,L] :
      ( hAPP(int,int,bit0,K) = hAPP(int,int,bit0,L)
    <=> ti(int,K) = ti(int,L) ) ).

fof(fact_44_even__less__0__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)),zero_zero(X_a)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_45_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_46_zadd__left__commute,axiom,
    ! [X,Y,Z_1] : 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_1)) = 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_1)) ).

fof(fact_47_zadd__commute,axiom,
    ! [Z_1,W] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),W) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),Z_1) ).

fof(fact_48_int__int__eq,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,int,semiring_1_of_nat(int),Ma) = hAPP(nat,int,semiring_1_of_nat(int),Na)
    <=> Ma = Na ) ).

fof(fact_49_less__special_I3_J,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),zero_zero(X_a)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),pls)) ) ) ).

fof(fact_50_less__special_I1_J,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Ya)) ) ) ).

fof(fact_51_rel__simps_I12_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),pls))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).

fof(fact_52_less__int__code_I15_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit0,K2)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).

fof(fact_53_rel__simps_I16_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit0,L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).

fof(fact_54_rel__simps_I10_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),pls))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).

fof(fact_55_rel__simps_I4_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),hAPP(int,int,bit0,K)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),K)) ) ).

fof(fact_56_bin__less__0__simps_I4_J,axiom,
    ! [Wa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,Wa)),zero_zero(int)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))) ) ).

fof(fact_57_bin__less__0__simps_I1_J,axiom,
    ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),zero_zero(int))) ).

fof(fact_58_bin__less__0__simps_I3_J,axiom,
    ! [Wa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,Wa)),zero_zero(int)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))) ) ).

fof(fact_59_int__0__less__1,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))) ).

fof(fact_60_zless__add1__eq,axiom,
    ! [Wa,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z))
        | ti(int,Wa) = ti(int,Z) ) ) ).

fof(fact_61_int__less__0__conv,axiom,
    ! [K_4] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),zero_zero(int))) ).

fof(fact_62_less__special_I4_J,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),one_one(X_a)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),hAPP(int,int,bit1,pls))) ) ) ).

fof(fact_63_less__special_I2_J,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),Ya)) ) ) ).

fof(fact_64_odd__less__0,axiom,
    ! [Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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)),Z)),zero_zero(int)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ).

fof(fact_65_double__eq__0__iff,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3) = zero_zero(X_a)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_66_rel__simps_I46_J,axiom,
    ! [K_4] : hAPP(int,int,bit1,K_4) != pls ).

fof(fact_67_rel__simps_I39_J,axiom,
    ! [L_4] : pls != hAPP(int,int,bit1,L_4) ).

fof(fact_68_rel__simps_I50_J,axiom,
    ! [K_4,L_4] : hAPP(int,int,bit1,K_4) != hAPP(int,int,bit0,L_4) ).

fof(fact_69_rel__simps_I49_J,axiom,
    ! [K_4,L_4] : hAPP(int,int,bit0,K_4) != hAPP(int,int,bit1,L_4) ).

fof(fact_70_rel__simps_I44_J,axiom,
    ! [K] :
      ( hAPP(int,int,bit0,K) = pls
    <=> ti(int,K) = pls ) ).

fof(fact_71_rel__simps_I38_J,axiom,
    ! [L] :
      ( pls = hAPP(int,int,bit0,L)
    <=> pls = ti(int,L) ) ).

fof(fact_72_Bit0__Pls,axiom,
    hAPP(int,int,bit0,pls) = pls ).

fof(fact_73_ext,axiom,
    ! [X_a,X_b,F,G] :
      ( ! [X_1] : hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1)
     => ti(fun(X_a,X_b),F) = ti(fun(X_a,X_b),G) ) ).

fof(fact_74_mem__def,axiom,
    ! [X_a,Xa,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
    <=> hBOOL(hAPP(X_a,bool,A_2,Xa)) ) ).

fof(fact_75_Collect__def,axiom,
    ! [X_a,P_1] : hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1) = ti(fun(X_a,bool),P_1) ).

fof(fact_76_Pls__def,axiom,
    pls = zero_zero(int) ).

fof(fact_77_int__0__neq__1,axiom,
    zero_zero(int) != one_one(int) ).

fof(fact_78_add__Pls__right,axiom,
    ! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),pls) = ti(int,K_4) ).

fof(fact_79_add__Pls,axiom,
    ! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),K_4) = ti(int,K_4) ).

fof(fact_80_add__Bit0__Bit0,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4)) ).

fof(fact_81_Bit0__def,axiom,
    ! [K_4] : hAPP(int,int,bit0,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),K_4) ).

fof(fact_82_zadd__0__right,axiom,
    ! [Z_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),zero_zero(int)) = ti(int,Z_1) ).

fof(fact_83_zadd__0,axiom,
    ! [Z_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),zero_zero(int)),Z_1) = ti(int,Z_1) ).

fof(fact_84_semiring__numeral__0__eq__0,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => hAPP(int,X_a,number_number_of(X_a),pls) = zero_zero(X_a) ) ).

fof(fact_85_number__of__Pls,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => hAPP(int,X_a,number_number_of(X_a),pls) = zero_zero(X_a) ) ).

fof(fact_86_semiring__norm_I112_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => zero_zero(X_a) = hAPP(int,X_a,number_number_of(X_a),pls) ) ).

fof(fact_87_add__numeral__0,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),pls)),A_1) = ti(X_a,A_1) ) ).

fof(fact_88_add__numeral__0__right,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),pls)) = ti(X_a,A_1) ) ).

fof(fact_89_power__eq__0__iff__number__of,axiom,
    ! [X_a] :
      ( ( power(X_a)
        & mult_zero(X_a)
        & no_zero_divisors(X_a)
        & zero_neq_one(X_a) )
     => ! [A_3,Wa] :
          ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),hAPP(int,nat,number_number_of(nat),Wa)) = zero_zero(X_a)
        <=> ( ti(X_a,A_3) = zero_zero(X_a)
            & hAPP(int,nat,number_number_of(nat),Wa) != zero_zero(nat) ) ) ) ).

fof(fact_90_add__number__of__left,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),W)),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W))),Z_1) ) ).

fof(fact_91_add__number__of__eq,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W)) ) ).

fof(fact_92_number__of__add,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) ) ).

fof(fact_93_add__Bit1__Bit0,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4)) ).

fof(fact_94_add__Bit0__Bit1,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4)) ).

fof(fact_95_Bit1__def,axiom,
    ! [K_4] : hAPP(int,int,bit1,K_4) = 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_4)),K_4) ).

fof(fact_96_odd__nonzero,axiom,
    ! [Z_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)),Z_1)),Z_1) != zero_zero(int) ).

fof(fact_97_number__of__int,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => ! [N] : hAPP(int,X_a,number_number_of(X_a),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N) ) ).

fof(fact_98_zero__less__power2,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
        <=> ti(X_a,A_3) != zero_zero(X_a) ) ) ).

fof(fact_99_power2__less__0,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),zero_zero(X_a))) ) ).

fof(fact_100_sum__power2__gt__zero__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
        <=> ( ti(X_a,Xa) != zero_zero(X_a)
            | ti(X_a,Ya) != zero_zero(X_a) ) ) ) ).

fof(fact_101_not__sum__power2__lt__zero,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(X_a))) ) ).

fof(fact_102_number__of__Bit0,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W))),hAPP(int,X_a,number_number_of(X_a),W)) ) ).

fof(fact_103_number__of__Bit1,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W))),hAPP(int,X_a,number_number_of(X_a),W)) ) ).

fof(fact_104_semiring__numeral__1__eq__1,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) = one_one(X_a) ) ).

fof(fact_105_mn,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),m1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) ).

fof(fact_106_of__nat__less__two__power,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) ) ).

fof(fact_107_transfer__int__nat__numerals_I3_J,axiom,
    hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_108_transfer__int__nat__numerals_I4_J,axiom,
    hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).

fof(fact_109_realpow__two__sum__zero__iff,axiom,
    ! [Xa,Ya] :
      ( hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real)
    <=> ( Xa = zero_zero(real)
        & Ya = zero_zero(real) ) ) ).

fof(fact_110_of__nat__0__less__iff,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
        <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ) ).

fof(fact_111_one__less__power,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))) ) ) ) ).

fof(fact_112_power__0__left,axiom,
    ! [X_a] :
      ( ( power(X_a)
        & semiring_0(X_a) )
     => ! [N] :
          ( ( N = zero_zero(nat)
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),N) = one_one(X_a) )
          & ( N != zero_zero(nat)
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),N) = zero_zero(X_a) ) ) ) ).

fof(fact_113_power__strict__decreasing,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1,N,N_4] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_4))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))) ) ) ) ) ).

fof(fact_114_zero__less__two,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)))) ) ).

fof(fact_115_int__gr__induct,axiom,
    ! [P_1,K,I] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I))
     => ( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K),one_one(int))))
       => ( ! [I_2] :
              ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I_2))
             => ( hBOOL(hAPP(int,bool,P_1,I_2))
               => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int)))) ) )
         => hBOOL(hAPP(int,bool,P_1,I)) ) ) ) ).

fof(fact_116_transfer__int__nat__numerals_I1_J,axiom,
    zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) ).

fof(fact_117_tn0,axiom,
    hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),tn)) ).

fof(fact_118_less__zeroE,axiom,
    ! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

fof(fact_119_real__zero__not__eq__one,axiom,
    zero_zero(real) != one_one(real) ).

fof(fact_120_less__not__refl,axiom,
    ! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N)) ).

fof(fact_121_not__add__less1,axiom,
    ! [I_1,J_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),I_1)) ).

fof(fact_122_not__add__less2,axiom,
    ! [J_2,I_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),I_1)),I_1)) ).

fof(fact_123_number__of__is__id,axiom,
    ! [K_4] : hAPP(int,int,number_number_of(int),K_4) = ti(int,K_4) ).

fof(fact_124_nat__neq__iff,axiom,
    ! [Ma,Na] :
      ( Ma != Na
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) ) ) ).

fof(fact_125_nat__add__commute,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M) ).

fof(fact_126_nat__add__left__commute,axiom,
    ! [X,Y,Z_1] : 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_1)) = 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_1)) ).

fof(fact_127_nat__add__assoc,axiom,
    ! [M,N,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K_4)) ).

fof(fact_128_nat__add__left__cancel,axiom,
    ! [K,Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Na)
    <=> Ma = Na ) ).

fof(fact_129_nat__add__right__cancel,axiom,
    ! [Ma,K,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K)
    <=> Ma = Na ) ).

fof(fact_130_nat__add__left__cancel__less,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).

fof(fact_131_linorder__neqE__nat,axiom,
    ! [X,Y] :
      ( X != Y
     => ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).

fof(fact_132_less__irrefl__nat,axiom,
    ! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N)) ).

fof(fact_133_less__not__refl2,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
     => M != N ) ).

fof(fact_134_less__not__refl3,axiom,
    ! [S_3,T_5] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S_3),T_5))
     => S_3 != T_5 ) ).

fof(fact_135_trans__less__add1,axiom,
    ! [M,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),M))) ) ).

fof(fact_136_trans__less__add2,axiom,
    ! [M,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),J_2))) ) ).

fof(fact_137_add__less__mono1,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),K_4))) ) ).

fof(fact_138_add__less__mono,axiom,
    ! [K_4,L_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_4),L_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),L_4))) ) ) ).

fof(fact_139_less__add__eq__less,axiom,
    ! [M,N,K_4,L_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_4),L_4))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),L_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N)
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

fof(fact_140_add__lessD1,axiom,
    ! [I_1,J_2,K_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),K_4))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_4)) ) ).

fof(fact_141_nat__less__cases,axiom,
    ! [P_1,Ma,Na] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
     => ( ( Ma = Na
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
       => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma))
           => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) )
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)) ) ) ) ).

fof(fact_142_gr0I,axiom,
    ! [N] :
      ( N != zero_zero(nat)
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

fof(fact_143_gr__implies__not0,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => N != zero_zero(nat) ) ).

fof(fact_144_nat__power__less__imp__less,axiom,
    ! [M,N,I_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I_1))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N)))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

fof(fact_145_less__nat__zero__code,axiom,
    ! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

fof(fact_146_nat__zero__less__power__iff,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa))
        | Na = zero_zero(nat) ) ) ).

fof(fact_147_add__gr__0,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ) ).

fof(fact_148_neq0__conv,axiom,
    ! [Na] :
      ( Na != zero_zero(nat)
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).

fof(fact_149_not__less0,axiom,
    ! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

fof(fact_150_zero__less__power__nat__eq,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Na)))
    <=> ( Na = zero_zero(nat)
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) ) ) ).

fof(fact_151_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) ) ).

fof(fact_152_zero__less__power__nat__eq__number__of,axiom,
    ! [Xa,Wa] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),hAPP(int,nat,number_number_of(nat),Wa))))
    <=> ( hAPP(int,nat,number_number_of(nat),Wa) = zero_zero(nat)
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) ) ) ).

fof(fact_153_zless__int,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).

fof(fact_154_less__nat__number__of,axiom,
    ! [Va,V_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(int,nat,number_number_of(nat),V_1)))
    <=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V_1))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_1)) )
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V_1)) ) ) ).

fof(fact_155_linorder__neqE__linordered__idom,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [X,Y] :
          ( ti(X_a,X) != ti(X_a,Y)
         => ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).

fof(fact_156_add__eq__self__zero,axiom,
    ! [M,N] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = M
     => N = zero_zero(nat) ) ).

fof(fact_157_add__is__0,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na) = zero_zero(nat)
    <=> ( Ma = zero_zero(nat)
        & Na = zero_zero(nat) ) ) ).

fof(fact_158_Nat_Oadd__0__right,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).

fof(fact_159_plus__nat_Oadd__0,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

fof(fact_160_power__one__right,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),one_one(nat)) = ti(X_a,A_1) ) ).

fof(fact_161_of__nat__eq__iff,axiom,
    ! [X_a] :
      ( semiring_char_0(X_a)
     => ! [Ma,Na] :
          ( hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma) = hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)
        <=> Ma = Na ) ) ).

fof(fact_162_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
    ! [Xa,Ya] :
      ( hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),Ya)
    <=> Xa = Ya ) ).

fof(fact_163_int__if__cong,axiom,
    ! [Xa,Ya,P_1] :
      ( ( hBOOL(P_1)
       => hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),P_1),Xa),Ya)) )
      & ( ~ hBOOL(P_1)
       => hAPP(nat,int,semiring_1_of_nat(int),Ya) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),P_1),Xa),Ya)) ) ) ).

fof(fact_164_less__0__number__of,axiom,
    ! [Va] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),Va)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Va)) ) ).

fof(fact_165_zero__less__int__conv,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).

fof(fact_166_one__neq__zero,axiom,
    ! [X_a] :
      ( zero_neq_one(X_a)
     => one_one(X_a) != zero_zero(X_a) ) ).

fof(fact_167_zero__neq__one,axiom,
    ! [X_a] :
      ( zero_neq_one(X_a)
     => zero_zero(X_a) != one_one(X_a) ) ).

fof(fact_168_field__power__not__zero,axiom,
    ! [X_a] :
      ( ring_11004092258visors(X_a)
     => ! [N,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N) != zero_zero(X_a) ) ) ).

fof(fact_169_power__one,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),one_one(X_a)),N) = one_one(X_a) ) ).

fof(fact_170_of__nat__less__iff,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [Ma,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
        <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).

fof(fact_171_less__imp__of__nat__less,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [M,N] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))) ) ) ).

fof(fact_172_of__nat__less__imp__less,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [M,N] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

fof(fact_173_of__nat__add,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [M,N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) ) ).

fof(fact_174_of__nat__1,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => hAPP(nat,X_a,semiring_1_of_nat(X_a),one_one(nat)) = one_one(X_a) ) ).

fof(fact_175_of__nat__power,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [M,N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),N) ) ).

fof(fact_176_transfer__int__nat__numerals_I2_J,axiom,
    one_one(int) = hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) ).

fof(fact_177_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,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),Y)) ).

fof(fact_178_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
    ! [X,N] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),X)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N)) ).

fof(fact_179_pos__add__strict,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [B,C_1,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1))) ) ) ) ).

fof(fact_180_not__one__less__zero,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),zero_zero(X_a))) ) ).

fof(fact_181_zero__less__one,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),one_one(X_a))) ) ).

fof(fact_182_zero__less__power,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))) ) ) ).

fof(fact_183_less__add__one,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a)))) ) ).

fof(fact_184_power__inject__exp,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [Ma,Na,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_3))
         => ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Ma) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na)
          <=> Ma = Na ) ) ) ).

fof(fact_185_power__strict__increasing__iff,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [Xa,Ya,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_2))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B_2),Xa)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B_2),Ya)))
          <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) ) ) ) ).

fof(fact_186_power__less__imp__less__exp,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [M,N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)))
           => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

fof(fact_187_power__strict__increasing,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1,N,N_4] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_4))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4))) ) ) ) ).

fof(fact_188_power__eq__0__iff,axiom,
    ! [X_a] :
      ( ( power(X_a)
        & mult_zero(X_a)
        & no_zero_divisors(X_a)
        & zero_neq_one(X_a) )
     => ! [A_3,Na] :
          ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na) = zero_zero(X_a)
        <=> ( ti(X_a,A_3) = zero_zero(X_a)
            & Na != zero_zero(nat) ) ) ) ).

fof(fact_189_of__nat__less__0__iff,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [M] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),zero_zero(X_a))) ) ).

fof(fact_190_power__0,axiom,
    ! [X_a] :
      ( power(X_a)
     => ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),zero_zero(nat)) = one_one(X_a) ) ).

fof(fact_191_of__nat__0,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => hAPP(nat,X_a,semiring_1_of_nat(X_a),zero_zero(nat)) = zero_zero(X_a) ) ).

fof(fact_192_pos2,axiom,
    hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_193_zero__less__imp__eq__int,axiom,
    ! [K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4))
     => ? [N_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
          & ti(int,K_4) = hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ) ).

fof(fact_194_less__imp__add__positive,axiom,
    ! [I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
     => ? [K_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
          & hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1) = J_2 ) ) ).

fof(fact_195_exp__eq__1,axiom,
    ! [Xa,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Na) = one_one(nat)
    <=> ( Xa = one_one(nat)
        | Na = zero_zero(nat) ) ) ).

fof(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),zero_zero(nat)) = one_one(X_a) ) ).

fof(fact_197_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_198_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)),zero_zero(X_a)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_199_add__pos__pos,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).

fof(fact_200_add__neg__neg,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).

fof(fact_201_tpos,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),t)) ).

fof(fact_202_nat__1__add__number__of,axiom,
    ! [V] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(int,nat,number_number_of(nat),V)) = one_one(nat) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) ) ) ).

fof(fact_203_zle__antisym,axiom,
    ! [Z_1,W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1))
       => ti(int,Z_1) = ti(int,W) ) ) ).

fof(fact_204_zle__trans,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_2))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_2),K_4))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),K_4)) ) ) ).

fof(fact_205_zle__linear,axiom,
    ! [Z_1,W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W))
      | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1)) ) ).

fof(fact_206_zle__refl,axiom,
    ! [W] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),W)) ).

fof(fact_207_le__number__of,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),hAPP(int,X_a,number_number_of(X_a),Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)) ) ) ).

fof(fact_208_of__nat__le__iff,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [Ma,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)))
        <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).

fof(fact_209_add__le__imp__le__left,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).

fof(fact_210_add__le__imp__le__right,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [A_1,C_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).

fof(fact_211_add__mono,axiom,
    ! [X_a] :
      ( ordere779506340up_add(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3))) ) ) ) ).

fof(fact_212_add__left__mono,axiom,
    ! [X_a] :
      ( ordere779506340up_add(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B))) ) ) ).

fof(fact_213_add__right__mono,axiom,
    ! [X_a] :
      ( ordere779506340up_add(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1))) ) ) ).

fof(fact_214_add__le__cancel__left,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [C,A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B_2)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) ) ) ).

fof(fact_215_add__le__cancel__right,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [A_3,C,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),C)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) ) ) ).

fof(fact_216_power__increasing,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1,N,N_4] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_4))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4))) ) ) ) ).

fof(fact_217_less__eq__int__code_I16_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit1,K2)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).

fof(fact_218_rel__simps_I34_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit1,L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).

fof(fact_219_rel__simps_I19_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),pls)) ).

fof(fact_220_less__eq__int__code_I13_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit0,K2)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).

fof(fact_221_rel__simps_I31_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit0,L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).

fof(fact_222_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).

fof(fact_223_zless__le,axiom,
    ! [Z,Wa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),Wa))
    <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),Wa))
        & ti(int,Z) != ti(int,Wa) ) ) ).

fof(fact_224_zadd__left__mono,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_2))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),I_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),J_2))) ) ).

fof(fact_225_less__eq__number__of__int__code,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),K)),hAPP(int,int,number_number_of(int),L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).

fof(fact_226_add__nonpos__nonpos,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).

fof(fact_227_add__increasing2,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1,C_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1))) ) ) ) ).

fof(fact_228_add__increasing,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,C_1,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1))) ) ) ) ).

fof(fact_229_add__nonneg__eq__0__iff,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [Ya,Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Ya))
           => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa),Ya) = zero_zero(X_a)
            <=> ( ti(X_a,Xa) = zero_zero(X_a)
                & ti(X_a,Ya) = zero_zero(X_a) ) ) ) ) ) ).

fof(fact_230_add__nonneg__nonneg,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).

fof(fact_231_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)),zero_zero(X_a)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_232_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_233_add__le__less__mono,axiom,
    ! [X_a] :
      ( ordere223160158up_add(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3))) ) ) ) ).

fof(fact_234_add__less__le__mono,axiom,
    ! [X_a] :
      ( ordere223160158up_add(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3))) ) ) ) ).

fof(fact_235_le__special_I3_J,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),zero_zero(X_a)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),pls)) ) ) ).

fof(fact_236_le__special_I1_J,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Ya)) ) ) ).

fof(fact_237_power__decreasing,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1,N,N_4] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_4))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),one_one(X_a)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))) ) ) ) ) ).

fof(fact_238_power__le__imp__le__exp,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [M,N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)))
           => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

fof(fact_239_power__increasing__iff,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [Xa,Ya,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_2))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B_2),Xa)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B_2),Ya)))
          <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) ) ) ) ).

fof(fact_240_le__special_I4_J,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),one_one(X_a)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),hAPP(int,int,bit1,pls))) ) ) ).

fof(fact_241_le__special_I2_J,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),Ya)) ) ) ).

fof(fact_242_add__nonpos__neg,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).

fof(fact_243_add__neg__nonpos,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).

fof(fact_244_add__strict__increasing2,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,C_1,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1))) ) ) ) ).

fof(fact_245_add__strict__increasing,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,C_1,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1))) ) ) ) ).

fof(fact_246_add__nonneg__pos,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).

fof(fact_247_add__pos__nonneg,axiom,
    ! [X_a] :
      ( ordere216010020id_add(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).

fof(fact_248_zero__le__one,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),one_one(X_a))) ) ).

fof(fact_249_not__one__le__zero,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),zero_zero(X_a))) ) ).

fof(fact_250_le__number__of__eq__not__less,axiom,
    ! [X_a] :
      ( ( number(X_a)
        & linorder(X_a) )
     => ! [Va,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),hAPP(int,X_a,number_number_of(X_a),Wa)))
        <=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),hAPP(int,X_a,number_number_of(X_a),Va))) ) ) ).

fof(fact_251_power__mono,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N))) ) ) ) ).

fof(fact_252_zero__le__power,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))) ) ) ).

fof(fact_253_one__le__power,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))) ) ) ).

fof(fact_254_of__nat__0__le__iff,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))) ) ).

fof(fact_255_zero__le__imp__of__nat,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [M] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M))) ) ).

fof(fact_256_rel__simps_I22_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),hAPP(int,int,bit1,K)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).

fof(fact_257_less__eq__int__code_I14_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit1,K2)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).

fof(fact_258_rel__simps_I32_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit1,L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).

fof(fact_259_rel__simps_I27_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),pls))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),pls)) ) ).

fof(fact_260_rel__simps_I21_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),hAPP(int,int,bit0,K)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).

fof(fact_261_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))) ).

fof(fact_262_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) ) ) ).

fof(fact_263_zadd__zless__mono,axiom,
    ! [Z_4,Z_1,W_2,W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_2),W))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_4),Z_1))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_1))) ) ) ).

fof(fact_264_zero__zle__int,axiom,
    ! [N] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N))) ).

fof(fact_265_transfer__int__nat__quantifiers_I1_J,axiom,
    ! [P_1] :
      ( ! [X_1] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
         => hBOOL(hAPP(int,bool,P_1,X_1)) )
    <=> ! [X_1] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1))) ) ).

fof(fact_266_transfer__int__nat__quantifiers_I2_J,axiom,
    ! [P_1] :
      ( ? [X_1] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
          & hBOOL(hAPP(int,bool,P_1,X_1)) )
    <=> ? [X_1] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1))) ) ).

fof(fact_267_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
    ! [Z_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Z_1))) ).

fof(fact_268_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N))) ) ).

fof(fact_269_zle__iff__zadd,axiom,
    ! [Wa,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z))
    <=> ? [N_1] : ti(int,Z) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),N_1)) ) ).

fof(fact_270_semiring__number__of__add__1,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => ! [V] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,V)) ) ) ).

fof(fact_271_semiring__1__add__number__of,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => ! [V] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),V)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,V)) ) ) ).

fof(fact_272_power__less__imp__less__base,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1,N,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).

fof(fact_273_eq__number__of__0,axiom,
    ! [Va] :
      ( hAPP(int,nat,number_number_of(nat),Va) = zero_zero(nat)
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ).

fof(fact_274_eq__0__number__of,axiom,
    ! [Va] :
      ( zero_zero(nat) = hAPP(int,nat,number_number_of(nat),Va)
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ).

fof(fact_275_rel__simps_I29_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),pls))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).

fof(fact_276_rel__simps_I5_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),hAPP(int,int,bit1,K)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).

fof(fact_277_less__eq__int__code_I15_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit0,K2)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).

fof(fact_278_rel__simps_I33_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit0,L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).

fof(fact_279_less__int__code_I14_J,axiom,
    ! [K1,K2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit1,K2)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).

fof(fact_280_rel__simps_I15_J,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit1,L)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).

fof(fact_281_zero__reorient,axiom,
    ! [X_a] :
      ( zero(X_a)
     => ! [Xa] :
          ( zero_zero(X_a) = ti(X_a,Xa)
        <=> ti(X_a,Xa) = zero_zero(X_a) ) ) ).

fof(fact_282_int__one__le__iff__zero__less,axiom,
    ! [Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

fof(fact_283_int__le__0__conv,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),zero_zero(int)))
    <=> Na = zero_zero(nat) ) ).

fof(fact_284_succ__Pls,axiom,
    hAPP(int,int,succ,pls) = hAPP(int,int,bit1,pls) ).

fof(fact_285_succ__Bit1,axiom,
    ! [K_4] : hAPP(int,int,succ,hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit0,hAPP(int,int,succ,K_4)) ).

fof(fact_286_succ__Bit0,axiom,
    ! [K_4] : hAPP(int,int,succ,hAPP(int,int,bit0,K_4)) = hAPP(int,int,bit1,K_4) ).

fof(fact_287_zless__imp__add1__zle,axiom,
    ! [W,Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z_1))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),one_one(int))),Z_1)) ) ).

fof(fact_288_add1__zle__eq,axiom,
    ! [Wa,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Wa),one_one(int))),Z))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) ) ).

fof(fact_289_zle__add1__eq__le,axiom,
    ! [Wa,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z)) ) ).

fof(fact_290_add__right__imp__eq,axiom,
    ! [X_a] :
      ( cancel_semigroup_add(X_a)
     => ! [B,A_1,C_1] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)
         => ti(X_a,B) = ti(X_a,C_1) ) ) ).

fof(fact_291_add__imp__eq,axiom,
    ! [X_a] :
      ( cancel146912293up_add(X_a)
     => ! [A_1,B,C_1] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)
         => ti(X_a,B) = ti(X_a,C_1) ) ) ).

fof(fact_292_add__left__imp__eq,axiom,
    ! [X_a] :
      ( cancel_semigroup_add(X_a)
     => ! [A_1,B,C_1] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)
         => ti(X_a,B) = ti(X_a,C_1) ) ) ).

fof(fact_293_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B,C_1,D_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3)) ) ).

fof(fact_294_add__right__cancel,axiom,
    ! [X_a] :
      ( cancel_semigroup_add(X_a)
     => ! [B_2,A_3,C] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_3)
        <=> ti(X_a,B_2) = ti(X_a,C) ) ) ).

fof(fact_295_add__left__cancel,axiom,
    ! [X_a] :
      ( cancel_semigroup_add(X_a)
     => ! [A_3,B_2,C] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C)
        <=> ti(X_a,B_2) = ti(X_a,C) ) ) ).

fof(fact_296_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),B) ) ).

fof(fact_297_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [X_a] :
      ( ab_semigroup_add(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1)) ) ).

fof(fact_298_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1)) ) ).

fof(fact_299_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,C_1,D_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),D_3) ) ).

fof(fact_300_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,C_1,D_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),D_3)) ) ).

fof(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1) ) ).

fof(fact_302_one__reorient,axiom,
    ! [X_a] :
      ( one(X_a)
     => ! [Xa] :
          ( one_one(X_a) = ti(X_a,Xa)
        <=> ti(X_a,Xa) = one_one(X_a) ) ) ).

fof(fact_303_succ__def,axiom,
    ! [K_4] : hAPP(int,int,succ,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),one_one(int)) ).

fof(fact_304_power__eq__imp__eq__base,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1,N,B] :
          ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
             => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
               => ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ) ).

fof(fact_305_semiring__add__number__of,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => ! [V_2,V] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),V_2)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),V_2)) ) ) ) ).

fof(fact_306_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).

fof(fact_307_le__imp__0__less,axiom,
    ! [Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z_1))) ) ).

fof(fact_308_number__of__succ,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [K_4] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,K_4)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),K_4)) ) ).

fof(fact_309_add__Bit1__Bit1,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),hAPP(int,int,succ,L_4))) ).

fof(fact_310_power__strict__mono,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
           => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N))) ) ) ) ) ).

fof(fact_311_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_312_power2__ge__self,axiom,
    ! [X] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_313_power2__eq__imp__eq,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [X,Y] :
          ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
             => ti(X_a,X) = ti(X_a,Y) ) ) ) ) ).

fof(fact_314_power2__le__imp__le,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ) ).

fof(fact_315_zero__le__power2,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_316_of__nat__number__of__lemma,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V] :
          ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
           => hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,X_a,number_number_of(X_a),V) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
           => hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(X_a) ) ) ) ).

fof(fact_317_add__0__iff,axiom,
    ! [X_a] :
      ( semiri456707255roduct(X_a)
     => ! [B_2,A_3] :
          ( ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),A_3)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_318_add_Ocomm__neutral,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).

fof(fact_319_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).

fof(fact_320_add__0__right,axiom,
    ! [X_a] :
      ( monoid_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).

fof(fact_321_double__zero__sym,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_322_add__0,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).

fof(fact_323_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).

fof(fact_324_add__0__left,axiom,
    ! [X_a] :
      ( monoid_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).

fof(fact_325_add__less__imp__less__left,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).

fof(fact_326_add__less__imp__less__right,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [A_1,C_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).

fof(fact_327_add__strict__mono,axiom,
    ! [X_a] :
      ( ordere223160158up_add(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3))) ) ) ) ).

fof(fact_328_add__strict__left__mono,axiom,
    ! [X_a] :
      ( ordere223160158up_add(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B))) ) ) ).

fof(fact_329_add__strict__right__mono,axiom,
    ! [X_a] :
      ( ordere223160158up_add(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1))) ) ) ).

fof(fact_330_add__less__cancel__left,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [C,A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B_2)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) ) ) ).

fof(fact_331_add__less__cancel__right,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [A_3,C,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),C)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) ) ) ).

fof(fact_332_power2__less__imp__less,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ) ).

fof(fact_333_sum__power2__le__zero__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(X_a)))
        <=> ( ti(X_a,Xa) = zero_zero(X_a)
            & ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).

fof(fact_334_sum__power2__ge__zero,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).

fof(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),one_one(nat)) = ti(X_a,X) ) ).

fof(fact_336_int__pos__lt__two__imp__zero__or__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
       => ( ti(int,X) = zero_zero(int)
          | ti(int,X) = one_one(int) ) ) ) ).

fof(fact_337_nat__power__eq__0__iff,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ma),Na) = zero_zero(nat)
    <=> ( Na != zero_zero(nat)
        & Ma = zero_zero(nat) ) ) ).

fof(fact_338_nat__number__of__add__1,axiom,
    ! [V] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = one_one(nat) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) ) ) ).

fof(fact_339_number__of1,axiom,
    ! [N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),N)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,N))))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,N)))) ) ) ).

fof(fact_340_realpow__pos__nth__unique,axiom,
    ! [A_1,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
       => ? [X_1] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
            & hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N) = A_1
            & ! [Y_1] :
                ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1))
                  & hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_1),N) = A_1 )
               => Y_1 = X_1 ) ) ) ) ).

fof(fact_341_realpow__pos__nth,axiom,
    ! [A_1,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
       => ? [R_1] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
            & hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_1),N) = A_1 ) ) ) ).

fof(fact_342_number__of2,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),pls))) ).

fof(fact_343_nonneg__eq__int,axiom,
    ! [Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
     => ~ ! [M_2] : ti(int,Z_1) != hAPP(nat,int,semiring_1_of_nat(int),M_2) ) ).

fof(fact_344_nonneg__int__cases,axiom,
    ! [K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
     => ~ ! [N_1] : ti(int,K_4) != hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ).

fof(fact_345_zero__le__imp__eq__int,axiom,
    ! [K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
     => ? [N_1] : ti(int,K_4) = hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ).

fof(fact_346_int__ge__induct,axiom,
    ! [P_1,K,I] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I))
     => ( hBOOL(hAPP(int,bool,P_1,K))
       => ( ! [I_2] :
              ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I_2))
             => ( hBOOL(hAPP(int,bool,P_1,I_2))
               => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int)))) ) )
         => hBOOL(hAPP(int,bool,P_1,I)) ) ) ) ).

fof(fact_347_order__refl,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),X)) ) ).

fof(fact_348_zadd__power3,axiom,
    ! [A_1,B] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),A_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).

fof(fact_349_zadd__power2,axiom,
    ! [A_1,B] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)),B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_350_le0,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

fof(fact_351_real__less__def,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))
    <=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))
        & Xa != Ya ) ) ).

fof(fact_352_less__eq__real__def,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))
    <=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))
        | Xa = Ya ) ) ).

fof(fact_353_le__antisym,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => M = N ) ) ).

fof(fact_354_le__trans,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),K_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),K_4)) ) ) ).

fof(fact_355_eq__imp__le,axiom,
    ! [M,N] :
      ( M = N
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

fof(fact_356_nat__le__linear,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
      | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

fof(fact_357_le__refl,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).

fof(fact_358_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1) ) ).

fof(fact_359_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [Lx,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ry)) ) ).

fof(fact_360_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [Lx,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),Ry) ) ).

fof(fact_361_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [X_a] :
      ( ab_semigroup_mult(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) ) ).

fof(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [Lx,Ly,Rx] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Rx) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),Rx)) ) ).

fof(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [Lx,Ly,Rx] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Rx) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),Ly) ) ).

fof(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry))) ) ).

fof(fact_365_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Ry)) ) ).

fof(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),Ry)) ) ).

fof(fact_367_zmult__commute,axiom,
    ! [Z_1,W] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),W) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z_1) ).

fof(fact_368_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_369_mult__number__of__left,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W)),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W))),Z_1) ) ).

fof(fact_370_arith__simps_I32_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W)) ) ).

fof(fact_371_number__of__mult,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) ) ).

fof(fact_372_of__nat__mult,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [M,N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) ) ).

fof(fact_373_mult__zero__left,axiom,
    ! [X_a] :
      ( mult_zero(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).

fof(fact_374_mult__zero__right,axiom,
    ! [X_a] :
      ( mult_zero(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_375_mult__eq__0__iff,axiom,
    ! [X_a] :
      ( ring_n68954251visors(X_a)
     => ! [A_3,B_2] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2) = zero_zero(X_a)
        <=> ( ti(X_a,A_3) = zero_zero(X_a)
            | ti(X_a,B_2) = zero_zero(X_a) ) ) ) ).

fof(fact_376_no__zero__divisors,axiom,
    ! [X_a] :
      ( no_zero_divisors(X_a)
     => ! [B,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => ( ti(X_a,B) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) != zero_zero(X_a) ) ) ) ).

fof(fact_377_divisors__zero,axiom,
    ! [X_a] :
      ( no_zero_divisors(X_a)
     => ! [A_1,B] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) = zero_zero(X_a)
         => ( ti(X_a,A_1) = zero_zero(X_a)
            | ti(X_a,B) = zero_zero(X_a) ) ) ) ).

fof(fact_378_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).

fof(fact_379_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_380_comm__semiring__class_Odistrib,axiom,
    ! [X_a] :
      ( comm_semiring(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) ) ).

fof(fact_381_combine__common__factor,axiom,
    ! [X_a] :
      ( semiring(X_a)
     => ! [A_1,E_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E_1)),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),E_1)),C_1) ) ).

fof(fact_382_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,Y,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z_1)) ) ).

fof(fact_383_crossproduct__noteq,axiom,
    ! [X_a] :
      ( semiri456707255roduct(X_a)
     => ! [C,D_1,A_3,B_2] :
          ( ( ti(X_a,A_3) != ti(X_a,B_2)
            & ti(X_a,C) != ti(X_a,D_1) )
        <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),D_1)) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),D_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),C)) ) ) ).

fof(fact_384_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) ) ).

fof(fact_385_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,M,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),M)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),M) ) ).

fof(fact_386_crossproduct__eq,axiom,
    ! [X_a] :
      ( semiri456707255roduct(X_a)
     => ! [Wa,Ya,Xa,Z] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Wa),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Wa),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Ya))
        <=> ( ti(X_a,Wa) = ti(X_a,Xa)
            | ti(X_a,Ya) = ti(X_a,Z) ) ) ) ).

fof(fact_387_mult__1__left,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).

fof(fact_388_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).

fof(fact_389_mult__1,axiom,
    ! [X_a] :
      ( comm_monoid_mult(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).

fof(fact_390_mult__1__right,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).

fof(fact_391_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).

fof(fact_392_mult_Ocomm__neutral,axiom,
    ! [X_a] :
      ( comm_monoid_mult(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).

fof(fact_393_power__commutes,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1,N] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) ) ).

fof(fact_394_power__mult__distrib,axiom,
    ! [X_a] :
      ( comm_monoid_mult(X_a)
     => ! [A_1,B,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N)) ) ).

fof(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,Y,Q_3] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),Q_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),Q_3)) ) ).

fof(fact_396_le__0__eq,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),zero_zero(nat)))
    <=> Na = zero_zero(nat) ) ).

fof(fact_397_less__eq__nat_Osimps_I1_J,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

fof(fact_398_mult__Pls,axiom,
    ! [W] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),pls),W) = pls ).

fof(fact_399_mult__Bit0,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,bit0,K_4)),L_4) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),L_4)) ).

fof(fact_400_nat__less__le,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
        & Ma != Na ) ) ).

fof(fact_401_le__eq__less__or__eq,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
        | Ma = Na ) ) ).

fof(fact_402_less__imp__le__nat,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

fof(fact_403_le__neq__implies__less,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( M != N
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

fof(fact_404_less__or__eq__imp__le,axiom,
    ! [M,N] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
        | M = N )
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

fof(fact_405_zmult__1,axiom,
    ! [Z_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),one_one(int)),Z_1) = ti(int,Z_1) ).

fof(fact_406_zmult__1__right,axiom,
    ! [Z_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),one_one(int)) = ti(int,Z_1) ).

fof(fact_407_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_408_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_409_le__add2,axiom,
    ! [N,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) ).

fof(fact_410_le__add1,axiom,
    ! [N,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

fof(fact_411_le__iff__add,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
    <=> ? [K_1] : Na = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K_1) ) ).

fof(fact_412_nat__add__left__cancel__le,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).

fof(fact_413_trans__le__add1,axiom,
    ! [M,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),M))) ) ).

fof(fact_414_trans__le__add2,axiom,
    ! [M,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),J_2))) ) ).

fof(fact_415_add__le__mono1,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),K_4))) ) ).

fof(fact_416_add__le__mono,axiom,
    ! [K_4,L_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),L_4))) ) ) ).

fof(fact_417_add__leD2,axiom,
    ! [M,K_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N)) ) ).

fof(fact_418_add__leD1,axiom,
    ! [M,K_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

fof(fact_419_add__leE,axiom,
    ! [M,K_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),N))
     => ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N)) ) ) ).

fof(fact_420_times__numeral__code_I5_J,axiom,
    ! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W)) ).

fof(fact_421_real__add__left__mono,axiom,
    ! [Z_1,X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z_1),X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z_1),Y))) ) ).

fof(fact_422_semiring__mult__number__of,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => ! [V_2,V] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),V_2)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),V_2)) ) ) ) ).

fof(fact_423_zero__le__square,axiom,
    ! [X_a] :
      ( linordered_ring(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1))) ) ).

fof(fact_424_zero__le__mult__iff,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) ) ) ) ) ).

fof(fact_425_mult__le__0__iff,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),zero_zero(X_a)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) ) ) ) ) ).

fof(fact_426_mult__nonneg__nonneg,axiom,
    ! [X_a] :
      ( ordere453448008miring(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).

fof(fact_427_mult__nonneg__nonpos,axiom,
    ! [X_a] :
      ( ordere453448008miring(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).

fof(fact_428_mult__nonneg__nonpos2,axiom,
    ! [X_a] :
      ( ordere453448008miring(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),zero_zero(X_a))) ) ) ) ).

fof(fact_429_mult__nonpos__nonneg,axiom,
    ! [X_a] :
      ( ordere453448008miring(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).

fof(fact_430_mult__nonpos__nonpos,axiom,
    ! [X_a] :
      ( ordered_ring(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).

fof(fact_431_mult__right__mono,axiom,
    ! [X_a] :
      ( ordered_semiring(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) ) ) ) ).

fof(fact_432_mult__left__mono,axiom,
    ! [X_a] :
      ( ordered_semiring(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).

fof(fact_433_comm__mult__left__mono,axiom,
    ! [X_a] :
      ( ordere1490568538miring(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).

fof(fact_434_mult__right__mono__neg,axiom,
    ! [X_a] :
      ( ordered_ring(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) ) ) ) ).

fof(fact_435_mult__left__mono__neg,axiom,
    ! [X_a] :
      ( ordered_ring(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).

fof(fact_436_mult__mono_H,axiom,
    ! [X_a] :
      ( ordered_semiring(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3))) ) ) ) ) ) ).

fof(fact_437_mult__mono,axiom,
    ! [X_a] :
      ( ordered_semiring(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3))) ) ) ) ) ) ).

fof(fact_438_split__mult__pos__le,axiom,
    ! [X_a] :
      ( ordered_ring(X_a)
     => ! [B,A_1] :
          ( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) ) )
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ).

fof(fact_439_split__mult__neg__le,axiom,
    ! [X_a] :
      ( ordere453448008miring(X_a)
     => ! [B,A_1] :
          ( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) ) )
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ).

fof(fact_440_not__square__less__zero,axiom,
    ! [X_a] :
      ( linordered_ring(X_a)
     => ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)),zero_zero(X_a))) ) ).

fof(fact_441_mult__less__cancel__right__disj,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [A_3,C,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),C)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) ) ) ) ) ).

fof(fact_442_mult__less__cancel__left__disj,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [C,A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) ) ) ) ) ).

fof(fact_443_mult__less__cancel__left__pos,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) ) ) ) ).

fof(fact_444_mult__pos__pos,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).

fof(fact_445_mult__pos__neg,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).

fof(fact_446_mult__pos__neg2,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),zero_zero(X_a))) ) ) ) ).

fof(fact_447_zero__less__mult__pos,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) ) ) ) ).

fof(fact_448_zero__less__mult__pos2,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) ) ) ) ).

fof(fact_449_mult__less__cancel__left__neg,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) ) ) ) ).

fof(fact_450_mult__neg__pos,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).

fof(fact_451_mult__neg__neg,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).

fof(fact_452_mult__strict__right__mono,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) ) ) ) ).

fof(fact_453_mult__strict__left__mono,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).

fof(fact_454_comm__mult__strict__left__mono,axiom,
    ! [X_a] :
      ( linord893533164strict(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).

fof(fact_455_mult__strict__right__mono__neg,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) ) ) ) ).

fof(fact_456_mult__strict__left__mono__neg,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) ) ) ) ).

fof(fact_457_sum__squares__eq__zero__iff,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [Xa,Ya] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),Ya)) = zero_zero(X_a)
        <=> ( ti(X_a,Xa) = zero_zero(X_a)
            & ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).

fof(fact_458_add__scale__eq__noteq,axiom,
    ! [X_a] :
      ( semiri456707255roduct(X_a)
     => ! [C_1,D_3,A_1,B,R_3] :
          ( ti(X_a,R_3) != zero_zero(X_a)
         => ( ( ti(X_a,A_1) = ti(X_a,B)
              & ti(X_a,C_1) != ti(X_a,D_3) )
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_3),C_1)) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_3),D_3)) ) ) ) ).

fof(fact_459_less__1__mult,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,M] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),M))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),N))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),M),N))) ) ) ) ).

fof(fact_460_right__distrib__number__of,axiom,
    ! [X_b] :
      ( ( number(X_b)
        & semiring(X_b) )
     => ! [V,B,C_1] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),B),C_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),B)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),C_1)) ) ).

fof(fact_461_left__distrib__number__of,axiom,
    ! [X_b] :
      ( ( number(X_b)
        & semiring(X_b) )
     => ! [A_1,B,V] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_1),B)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_1),hAPP(int,X_b,number_number_of(X_b),V))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B),hAPP(int,X_b,number_number_of(X_b),V))) ) ).

fof(fact_462_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [M] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M),M) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),M) ) ).

fof(fact_463_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [M,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M) ) ).

fof(fact_464_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,M] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)),M) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M) ) ).

fof(fact_465_power__add,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1,M,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) ) ).

fof(fact_466_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,P_5,Q_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),P_5)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),P_5),Q_3)) ) ).

fof(fact_467_pos__zmult__pos,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) ) ) ).

fof(fact_468_zmult__zless__mono2,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),I_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),J_2))) ) ) ).

fof(fact_469_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) ) ) ).

fof(fact_470_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) ) ).

fof(fact_471_zle__int,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).

fof(fact_472_zpower__zadd__distrib,axiom,
    ! [X,Y,Z_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Z_1)) ).

fof(fact_473_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)) ) ).

fof(fact_474_mult__left__le__imp__le,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ) ).

fof(fact_475_mult__right__le__imp__le,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [A_1,C_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ) ).

fof(fact_476_mult__less__imp__less__left,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).

fof(fact_477_mult__left__less__imp__less,axiom,
    ! [X_a] :
      ( linordered_semiring(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).

fof(fact_478_mult__less__imp__less__right,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [A_1,C_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).

fof(fact_479_mult__right__less__imp__less,axiom,
    ! [X_a] :
      ( linordered_semiring(X_a)
     => ! [A_1,C_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).

fof(fact_480_mult__le__less__imp__less,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3))) ) ) ) ) ) ).

fof(fact_481_mult__less__le__imp__less,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3))) ) ) ) ) ) ).

fof(fact_482_mult__strict__mono_H,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3))) ) ) ) ) ) ).

fof(fact_483_mult__strict__mono,axiom,
    ! [X_a] :
      ( linord20386208strict(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3))) ) ) ) ) ) ).

fof(fact_484_mult__le__cancel__left__neg,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3)) ) ) ) ).

fof(fact_485_mult__le__cancel__left__pos,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) ) ) ) ).

fof(fact_486_sum__squares__ge__zero,axiom,
    ! [X_a] :
      ( linordered_ring(X_a)
     => ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y)))) ) ).

fof(fact_487_sum__squares__le__zero__iff,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),Ya))),zero_zero(X_a)))
        <=> ( ti(X_a,Xa) = zero_zero(X_a)
            & ti(X_a,Ya) = zero_zero(X_a) ) ) ) ).

fof(fact_488_not__sum__squares__lt__zero,axiom,
    ! [X_a] :
      ( linordered_ring(X_a)
     => ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y))),zero_zero(X_a))) ) ).

fof(fact_489_sum__squares__gt__zero__iff,axiom,
    ! [X_a] :
      ( linord581940658strict(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),Ya))))
        <=> ( ti(X_a,Xa) != zero_zero(X_a)
            | ti(X_a,Ya) != zero_zero(X_a) ) ) ) ).

fof(fact_490_mult__right__le__one__le,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),X)) ) ) ) ) ).

fof(fact_491_mult__left__le__one__le,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),X)),X)) ) ) ) ) ).

fof(fact_492_power__gt1__lemma,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)))) ) ) ).

fof(fact_493_power__less__power__Suc,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)))) ) ) ).

fof(fact_494_mult__numeral__1__right,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = ti(X_a,A_1) ) ).

fof(fact_495_mult__numeral__1,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))),A_1) = ti(X_a,A_1) ) ).

fof(fact_496_mult__Bit1,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,bit1,K_4)),L_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),L_4))),L_4) ).

fof(fact_497_pos__zmult__eq__1__iff,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),Na) = one_one(int)
      <=> ( ti(int,Ma) = one_one(int)
          & ti(int,Na) = one_one(int) ) ) ) ).

fof(fact_498_le__nat__number__of,axiom,
    ! [Va,V_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(int,nat,number_number_of(nat),V_1)))
    <=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),V_1))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) ) ) ).

fof(fact_499_linorder__le__cases,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).

fof(fact_500_le__funE,axiom,
    ! [X_a,X_b] :
      ( ord(X_b)
     => ! [Xa,F,G] :
          ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Xa)),hAPP(X_a,X_b,G,Xa))) ) ) ).

fof(fact_501_xt1_I6_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Z_1,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),X)) ) ) ) ).

fof(fact_502_xt1_I5_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
           => ti(X_a,X) = ti(X_a,Y) ) ) ) ).

fof(fact_503_order__trans,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z_1)) ) ) ) ).

fof(fact_504_order__antisym,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
           => ti(X_a,X) = ti(X_a,Y) ) ) ) ).

fof(fact_505_xt1_I4_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
         => ( ti(X_a,B) = ti(X_a,C_1)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_1)) ) ) ) ).

fof(fact_506_ord__le__eq__trans,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( ti(X_a,B) = ti(X_a,C_1)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C_1)) ) ) ) ).

fof(fact_507_xt1_I3_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [C_1,A_1,B] :
          ( ti(X_a,A_1) = ti(X_a,B)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_1)) ) ) ) ).

fof(fact_508_ord__eq__le__trans,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ! [C_1,A_1,B] :
          ( ti(X_a,A_1) = ti(X_a,B)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C_1)) ) ) ) ).

fof(fact_509_order__antisym__conv,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Ya,Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
          <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).

fof(fact_510_le__funD,axiom,
    ! [X_a,X_b] :
      ( ord(X_b)
     => ! [Xa,F,G] :
          ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Xa)),hAPP(X_a,X_b,G,Xa))) ) ) ).

fof(fact_511_order__eq__refl,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [X,Y] :
          ( X = Y
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).

fof(fact_512_order__eq__iff,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Xa,Ya] :
          ( ti(X_a,Xa) = ti(X_a,Ya)
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) ) ) ) ).

fof(fact_513_linorder__linear,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
          | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).

fof(fact_514_le__fun__def,axiom,
    ! [X_a,X_b] :
      ( ord(X_b)
     => ! [F,G] :
          ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
        <=> ! [X_1] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) ) ) ).

fof(fact_515_linorder__cases,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ( ti(X_a,X) != ti(X_a,Y)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).

fof(fact_516_order__less__asym,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).

fof(fact_517_xt1_I10_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Z_1,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X)) ) ) ) ).

fof(fact_518_order__less__trans,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1)) ) ) ) ).

fof(fact_519_xt1_I2_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
         => ( ti(X_a,B) = ti(X_a,C_1)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),A_1)) ) ) ) ).

fof(fact_520_ord__less__eq__trans,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( ti(X_a,B) = ti(X_a,C_1)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C_1)) ) ) ) ).

fof(fact_521_xt1_I1_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [C_1,A_1,B] :
          ( ti(X_a,A_1) = ti(X_a,B)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),A_1)) ) ) ) ).

fof(fact_522_ord__eq__less__trans,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ! [C_1,A_1,B] :
          ( ti(X_a,A_1) = ti(X_a,B)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C_1)) ) ) ) ).

fof(fact_523_xt1_I9_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
         => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).

fof(fact_524_order__less__asym_H,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ).

fof(fact_525_order__less__imp__triv,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [P_1,Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa))
           => hBOOL(P_1) ) ) ) ).

fof(fact_526_order__less__imp__not__eq2,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ti(X_a,Y) != ti(X_a,X) ) ) ).

fof(fact_527_order__less__imp__not__eq,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ti(X_a,X) != ti(X_a,Y) ) ) ).

fof(fact_528_order__less__imp__not__less,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).

fof(fact_529_order__less__not__sym,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).

fof(fact_530_less__imp__neq,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ti(X_a,X) != ti(X_a,Y) ) ) ).

fof(fact_531_linorder__neqE,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] :
          ( ti(X_a,X) != ti(X_a,Y)
         => ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).

fof(fact_532_linorder__antisym__conv3,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Ya,Xa] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa))
         => ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
          <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).

fof(fact_533_linorder__less__linear,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
          | ti(X_a,X) = ti(X_a,Y)
          | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).

fof(fact_534_not__less__iff__gr__or__eq,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa))
            | ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).

fof(fact_535_linorder__neq__iff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( ti(X_a,Xa) != ti(X_a,Ya)
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
            | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) ) ) ) ).

fof(fact_536_order__less__irrefl,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [X] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),X)) ) ).

fof(fact_537_convex__bound__le,axiom,
    ! [X_a] :
      ( linord1278240602ring_1(X_a)
     => ! [V,U_1,Y,X,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),A_1))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V))
               => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),U_1),V) = one_one(X_a)
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),U_1),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)) ) ) ) ) ) ) ).

fof(fact_538_power__Suc__less,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))) ) ) ) ).

fof(fact_539_double__number__of__Bit0,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,W)) ) ).

fof(fact_540_power3__eq__cube,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)),A_1) ) ).

fof(fact_541_zmult__zless__mono2__lemma,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),I_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),J_2))) ) ) ).

fof(fact_542_convex__bound__lt,axiom,
    ! [X_a] :
      ( linord626643107strict(X_a)
     => ! [V,U_1,Y,X,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),A_1))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V))
               => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),U_1),V) = one_one(X_a)
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),U_1),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)) ) ) ) ) ) ) ).

fof(fact_543_mult__2__right,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),Z_1) ) ).

fof(fact_544_semiring__mult__2__right,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => ! [Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),Z_1) ) ).

fof(fact_545_mult__2,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),Z_1) ) ).

fof(fact_546_semiring__mult__2,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => ! [Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),Z_1) ) ).

fof(fact_547_power2__eq__square,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1) ) ).

fof(fact_548_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).

fof(fact_549_cube__square,axiom,
    ! [A_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).

fof(fact_550_two__realpow__ge__one,axiom,
    ! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) ).

fof(fact_551_power2__eq__square__number__of,axiom,
    ! [X_b] :
      ( ( monoid_mult(X_b)
        & number(X_b) )
     => ! [W] : hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),W)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),W)),hAPP(int,X_b,number_number_of(X_b),W)) ) ).

fof(fact_552_xt1_I8_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Z_1,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X)) ) ) ) ).

fof(fact_553_order__le__less__trans,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1)) ) ) ) ).

fof(fact_554_xt1_I7_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Z_1,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X)) ) ) ) ).

fof(fact_555_order__less__le__trans,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1)) ) ) ) ).

fof(fact_556_xt1_I11_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
         => ( ti(X_a,A_1) != ti(X_a,B)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).

fof(fact_557_order__le__neq__trans,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( ti(X_a,A_1) != ti(X_a,B)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).

fof(fact_558_order__le__imp__less__or__eq,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
            | ti(X_a,X) = ti(X_a,Y) ) ) ) ).

fof(fact_559_linorder__antisym__conv2,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
         => ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
          <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).

fof(fact_560_order__less__imp__le,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).

fof(fact_561_leD,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
         => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).

fof(fact_562_xt1_I12_J,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [A_1,B] :
          ( ti(X_a,A_1) != ti(X_a,B)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).

fof(fact_563_order__neq__le__trans,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [A_1,B] :
          ( ti(X_a,A_1) != ti(X_a,B)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).

fof(fact_564_linorder__antisym__conv1,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
          <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).

fof(fact_565_not__leE,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Y,X] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).

fof(fact_566_leI,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).

fof(fact_567_order__le__less,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
            | ti(X_a,Xa) = ti(X_a,Ya) ) ) ) ).

fof(fact_568_less__le__not__le,axiom,
    ! [X_a] :
      ( preorder(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
            & ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) ) ) ) ).

fof(fact_569_order__less__le,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
            & ti(X_a,Xa) != ti(X_a,Ya) ) ) ) ).

fof(fact_570_linorder__le__less__linear,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
          | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).

fof(fact_571_linorder__not__le,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) ) ) ).

fof(fact_572_linorder__not__less,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) ) ) ).

fof(fact_573_conj__le__cong,axiom,
    ! [P_2,P_1,Xa] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
       => ( hBOOL(P_1)
        <=> hBOOL(P_2) ) )
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
          & hBOOL(P_1) )
      <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
          & hBOOL(P_2) ) ) ) ).

fof(fact_574_imp__le__cong,axiom,
    ! [P_2,P_1,Xa] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
       => ( hBOOL(P_1)
        <=> hBOOL(P_2) ) )
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
         => hBOOL(P_1) )
      <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
         => hBOOL(P_2) ) ) ) ).

fof(fact_575_power2__sum,axiom,
    ! [X_a] :
      ( number_semiring(X_a)
     => ! [X,Y] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y)) ) ).

fof(fact_576_self__quotient__aux2,axiom,
    ! [R_3,Q_3,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),R_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_3))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_3),one_one(int))) ) ) ) ).

fof(fact_577_self__quotient__aux1,axiom,
    ! [R_3,Q_3,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),R_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_3))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),A_1))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Q_3)) ) ) ) ).

fof(fact_578_ex__least__nat__less,axiom,
    ! [Na,P_1] :
      ( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
     => ( hBOOL(hAPP(nat,bool,P_1,Na))
       => ? [K_1] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),Na))
            & ! [I_2] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),K_1))
               => ~ hBOOL(hAPP(nat,bool,P_1,I_2)) )
            & hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),one_one(nat)))) ) ) ) ).

fof(fact_579_incr__mult__lemma,axiom,
    ! [K,P_1,D_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
     => ( ! [X_1] :
            ( hBOOL(hAPP(int,bool,P_1,X_1))
           => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D_1))) )
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X_1] :
              ( hBOOL(hAPP(int,bool,P_1,X_1))
             => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),D_1)))) ) ) ) ) ).

fof(fact_580_smaller_I2_J,axiom,
    ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))))
     => ~ hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))))) ) ).

fof(fact_581_q__pos__lemma,axiom,
    ! [B_5,Q_5,R_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_5),Q_5)),R_4)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B_5))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q_5)) ) ) ) ).

fof(fact_582_q__neg__lemma,axiom,
    ! [B_5,Q_5,R_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_5),Q_5)),R_4)),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),zero_zero(int))) ) ) ) ).

fof(fact_583_unique__quotient__lemma,axiom,
    ! [B,Q_5,R_4,Q_3,R_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),Q_5)),R_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),Q_3)) ) ) ) ) ).

fof(fact_584_zdiv__mono2__lemma,axiom,
    ! [B,Q_3,R_3,B_5,Q_5,R_4] :
      ( hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),Q_5)),R_4)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_5),Q_5)),R_4)))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B_5))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5))
             => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_5),B))
               => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_3),Q_5)) ) ) ) ) ) ) ).

fof(fact_585_unique__quotient__lemma__neg,axiom,
    ! [B,Q_5,R_4,Q_3,R_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),Q_5)),R_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int)))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_4))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_3),Q_5)) ) ) ) ) ).

fof(fact_586_nQ1,axiom,
    ~ hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)))))) ).

fof(fact_587_p0,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(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),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ).

fof(fact_588_t__l__p,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),t),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ).

fof(fact_589_qf1pt,axiom,
    hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),t))) ).

fof(fact_590_IH,axiom,
    ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))))
    & hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))))) ) ).

fof(fact_591__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_A,axiom,
    ~ ! [X_1,Y_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))) ).

fof(fact_592_t,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(int)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),t) ).

fof(fact_593_p,axiom,
    hBOOL(hAPP(int,bool,zprime,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ).

fof(fact_594__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_,axiom,
    ( t = one_one(int)
   => ? [X_1,Y_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)) ) ).

fof(fact_595_less__fun__def,axiom,
    ! [X_a,X_b] :
      ( ord(X_b)
     => ! [F,G] :
          ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less(fun(X_a,X_b)),F),G))
        <=> ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
            & ~ hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),G),F)) ) ) ) ).

fof(fact_596_real__le__antisym,axiom,
    ! [Z_1,W] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z_1))
       => Z_1 = W ) ) ).

fof(fact_597_real__le__trans,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),J_2))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),J_2),K_4))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),K_4)) ) ) ).

fof(fact_598_real__le__linear,axiom,
    ! [Z_1,W] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W))
      | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z_1)) ) ).

fof(fact_599_real__le__refl,axiom,
    ! [W] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),W)) ).

fof(fact_600_nat__mult__commute,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),M) ).

fof(fact_601_nat__mult__assoc,axiom,
    ! [M,N,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K_4)) ).

fof(fact_602_mult__0,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

fof(fact_603_mult__0__right,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).

fof(fact_604_mult__is__0,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na) = zero_zero(nat)
    <=> ( Ma = zero_zero(nat)
        | Na = zero_zero(nat) ) ) ).

fof(fact_605_mult__cancel1,axiom,
    ! [K,Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)
    <=> ( Ma = Na
        | K = zero_zero(nat) ) ) ).

fof(fact_606_mult__cancel2,axiom,
    ! [Ma,K,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K)
    <=> ( Ma = Na
        | K = zero_zero(nat) ) ) ).

fof(fact_607_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,P_5,Q_3] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),P_5)),Q_3) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),Q_3)) ) ).

fof(fact_608_power__mult,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1,M,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),N) ) ).

fof(fact_609_real__mult__left__cancel,axiom,
    ! [A_3,B_2,C] :
      ( C != zero_zero(real)
     => ( hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A_3) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B_2)
      <=> A_3 = B_2 ) ) ).

fof(fact_610_real__mult__right__cancel,axiom,
    ! [A_3,B_2,C] :
      ( C != zero_zero(real)
     => ( hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_3),C) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B_2),C)
      <=> A_3 = B_2 ) ) ).

fof(fact_611_add__mult__distrib,axiom,
    ! [M,N,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K_4)) ).

fof(fact_612_add__mult__distrib2,axiom,
    ! [K_4,M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) ).

fof(fact_613_le__square,axiom,
    ! [M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),M))) ).

fof(fact_614_le__cube,axiom,
    ! [M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),M)))) ).

fof(fact_615_mult__le__mono1,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),K_4))) ) ).

fof(fact_616_mult__le__mono2,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),J_2))) ) ).

fof(fact_617_mult__le__mono,axiom,
    ! [K_4,L_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),L_4))) ) ) ).

fof(fact_618_nat__mult__eq__one,axiom,
    ! [Na,Ma] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Ma) = one_one(nat)
    <=> ( Na = one_one(nat)
        & Ma = one_one(nat) ) ) ).

fof(fact_619_nat__mult__eq__1__iff,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na) = one_one(nat)
    <=> ( Ma = one_one(nat)
        & Na = one_one(nat) ) ) ).

fof(fact_620_nat__mult__1__right,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

fof(fact_621_nat__1__eq__mult__iff,axiom,
    ! [Ma,Na] :
      ( one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na)
    <=> ( Ma = one_one(nat)
        & Na = one_one(nat) ) ) ).

fof(fact_622_nat__mult__1,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

fof(fact_623_real__mult__1,axiom,
    ! [Z_1] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),one_one(real)),Z_1) = Z_1 ).

fof(fact_624_real__add__mult__distrib,axiom,
    ! [Z1,Z2,W] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z1),Z2)),W) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),W)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z2),W)) ).

fof(fact_625_zpower__zpower,axiom,
    ! [X,Y,Z_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),Z_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Y),Z_1)) ).

fof(fact_626_mult__less__mono2,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),J_2))) ) ) ).

fof(fact_627_mult__less__mono1,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),K_4))) ) ) ).

fof(fact_628_mult__less__cancel2,axiom,
    ! [Ma,K,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).

fof(fact_629_mult__less__cancel1,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).

fof(fact_630_nat__0__less__mult__iff,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ) ).

fof(fact_631_not__real__square__gt__zero,axiom,
    ! [Xa] :
      ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Xa),Xa)))
    <=> Xa = zero_zero(real) ) ).

fof(fact_632_real__mult__less__mono2,axiom,
    ! [X,Y,Z_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z_1),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z_1),Y))) ) ) ).

fof(fact_633_real__mult__order,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y))) ) ) ).

fof(fact_634_real__mult__less__iff1,axiom,
    ! [Xa,Ya,Z] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Xa),Z)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Ya),Z)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) ) ) ).

fof(fact_635_real__mult__le__cancel__iff2,axiom,
    ! [Xa,Ya,Z] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z),Xa)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z),Ya)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ) ).

fof(fact_636_real__mult__le__cancel__iff1,axiom,
    ! [Xa,Ya,Z] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Xa),Z)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Ya),Z)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ) ).

fof(fact_637_mult__eq__self__implies__10,axiom,
    ! [M,N] :
      ( M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)
     => ( N = one_one(nat)
        | M = zero_zero(nat) ) ) ).

fof(fact_638_real__two__squares__add__zero__iff,axiom,
    ! [Xa,Ya] :
      ( hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Xa),Xa)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Ya),Ya)) = zero_zero(real)
    <=> ( Xa = zero_zero(real)
        & Ya = zero_zero(real) ) ) ).

fof(fact_639_int__mult,axiom,
    ! [M,N] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)) ).

fof(fact_640_zmult__int,axiom,
    ! [M,N] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) ).

fof(fact_641_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,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)) ).

fof(fact_642_is__mult__sum2sq,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,X))
     => ( hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,Y))
       => hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) ) ) ).

fof(fact_643_mult__le__cancel2,axiom,
    ! [Ma,K,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).

fof(fact_644_mult__le__cancel1,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).

fof(fact_645_power__even__eq,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).

fof(fact_646_nat__mult__2,axiom,
    ! [Z_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Z_1),Z_1) ).

fof(fact_647_nat__mult__2__right,axiom,
    ! [Z_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Z_1),Z_1) ).

fof(fact_648_nat__number__of__mult__left,axiom,
    ! [V_2,K_4,V] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = zero_zero(nat) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),V_2))),K_4) ) ) ).

fof(fact_649_mult__nat__number__of,axiom,
    ! [V_2,V] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = zero_zero(nat) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),V_2)) ) ) ).

fof(fact_650_four__x__squared,axiom,
    ! [X] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_651_even__power__le__0__imp__0,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1,K_4] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K_4))),zero_zero(X_a)))
         => ti(X_a,A_1) = zero_zero(X_a) ) ) ).

fof(fact_652_zero__le__even__power_H,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1,N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))) ) ).

fof(fact_653_zdiv__mono2__neg__lemma,axiom,
    ! [B,Q_3,R_3,B_5,Q_5,R_4] :
      ( hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),Q_5)),R_4)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_5),Q_5)),R_4)),zero_zero(int)))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5))
             => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_5),B))
               => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),Q_3)) ) ) ) ) ) ) ).

fof(fact_654_xy,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),x),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))) ).

fof(fact_655_real__sum__squared__expand,axiom,
    ! [X,Y] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y)) ).

fof(fact_656__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_,axiom,
    ~ ! [T_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(int)) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),T_4) ).

fof(fact_657_nat__mult__le__cancel1,axiom,
    ! [Ma,Na,K] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ).

fof(fact_658_mono__nat__linear__lb,axiom,
    ! [Ma,K,F] :
      ( ! [M_2,N_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N_1))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,M_2)),hAPP(nat,nat,F,N_1))) )
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,F,Ma)),K)),hAPP(nat,nat,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K)))) ) ).

fof(fact_659_ex__least__nat__le,axiom,
    ! [Na,P_1] :
      ( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
     => ( hBOOL(hAPP(nat,bool,P_1,Na))
       => ? [K_1] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),Na))
            & ! [I_2] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),K_1))
               => ~ hBOOL(hAPP(nat,bool,P_1,I_2)) )
            & hBOOL(hAPP(nat,bool,P_1,K_1)) ) ) ) ).

fof(fact_660__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),legendre,hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))) = one_one(int) ).

fof(fact_661_mult__left__cancel,axiom,
    ! [N,M,K_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)
       => N = M ) ) ).

fof(fact_662_nat__mult__less__cancel1,axiom,
    ! [Ma,Na,K] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ).

fof(fact_663_real__mult__commute,axiom,
    ! [Z_1,W] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z_1),W) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),W),Z_1) ).

fof(fact_664_real__mult__assoc,axiom,
    ! [Z1,Z2,Z3] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),Z2)),Z3) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z2),Z3)) ).

fof(fact_665_Bit1__Min,axiom,
    hAPP(int,int,bit1,min) = min ).

fof(fact_666_rel__simps_I43_J,axiom,
    ! [L] :
      ( min = hAPP(int,int,bit1,L)
    <=> min = ti(int,L) ) ).

fof(fact_667_rel__simps_I47_J,axiom,
    ! [K] :
      ( hAPP(int,int,bit1,K) = min
    <=> ti(int,K) = min ) ).

fof(fact_668_rel__simps_I40_J,axiom,
    min != pls ).

fof(fact_669_rel__simps_I37_J,axiom,
    pls != min ).

fof(fact_670_rel__simps_I42_J,axiom,
    ! [L_4] : min != hAPP(int,int,bit0,L_4) ).

fof(fact_671_rel__simps_I45_J,axiom,
    ! [K_4] : hAPP(int,int,bit0,K_4) != min ).

fof(fact_672_rel__simps_I7_J,axiom,
    ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),min)) ).

fof(fact_673_rel__simps_I24_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),min)) ).

fof(fact_674_rel__simps_I9_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),hAPP(int,int,bit1,K)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).

fof(fact_675_rel__simps_I13_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),min))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)) ) ).

fof(fact_676_rel__simps_I6_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),pls)) ).

fof(fact_677_rel__simps_I3_J,axiom,
    ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),min)) ).

fof(fact_678_rel__simps_I8_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),hAPP(int,int,bit0,K)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).

fof(fact_679_rel__simps_I30_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),min))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).

fof(fact_680_rel__simps_I26_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),hAPP(int,int,bit1,K)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),K)) ) ).

fof(fact_681_bin__less__0__simps_I2_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),zero_zero(int))) ).

fof(fact_682_rel__simps_I20_J,axiom,
    ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),min)) ).

fof(fact_683_rel__simps_I23_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),pls)) ).

fof(fact_684_rel__simps_I28_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),min))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).

fof(fact_685_eq__number__of__Pls__Min,axiom,
    hAPP(int,int,number_number_of(int),pls) != hAPP(int,int,number_number_of(int),min) ).

fof(fact_686_succ__Min,axiom,
    hAPP(int,int,succ,min) = pls ).

fof(fact_687_Legendre__1mod4,axiom,
    ! [M] :
      ( hBOOL(hAPP(int,bool,zprime,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),M)),one_one(int))))
     => hAPP(int,int,hAPP(int,fun(int,int),legendre,hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),M)),one_one(int))) = one_one(int) ) ).

fof(fact_688_rel__simps_I11_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),min))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).

fof(fact_689_rel__simps_I25_J,axiom,
    ! [K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),hAPP(int,int,bit0,K)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).

fof(fact_690_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M,N] :
      ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N) = one_one(int)
     => ( ti(int,M) = one_one(int)
        | ti(int,M) = hAPP(int,int,number_number_of(int),min) ) ) ).

fof(fact_691_zmult__eq__1__iff,axiom,
    ! [Ma,Na] :
      ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),Na) = one_one(int)
    <=> ( ( ti(int,Ma) = one_one(int)
          & ti(int,Na) = one_one(int) )
        | ( ti(int,Ma) = hAPP(int,int,number_number_of(int),min)
          & ti(int,Na) = hAPP(int,int,number_number_of(int),min) ) ) ) ).

fof(fact_692_zprime__2,axiom,
    hBOOL(hAPP(int,bool,zprime,hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_693_nat__mult__eq__cancel__disj,axiom,
    ! [K,Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)
    <=> ( K = zero_zero(nat)
        | Ma = Na ) ) ).

fof(fact_694_left__add__mult__distrib,axiom,
    ! [I_1,U_1,J_2,K_4] : 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_2),U_1)),K_4)) = 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_2)),U_1)),K_4) ).

fof(fact_695_power__m1__even,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = one_one(X_a) ) ).

fof(fact_696_nat__mult__eq__cancel1,axiom,
    ! [Ma,Na,K] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)
      <=> Ma = Na ) ) ).

fof(fact_697__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N,axiom,
    ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,number_number_of(int),min)))
   => hAPP(int,int,hAPP(int,fun(int,int),legendre,hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))) != one_one(int) ) ).

fof(fact_698__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,number_number_of(int),min))) ).

fof(fact_699_s,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ).

fof(fact_700__0964_A_K_Am_A_L_A1_Advd_As_____A_094_A2_A_L_A1_096,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(int)))) ).

fof(fact_701__096s_____A_094_A2_A_N_A_N1_A_061_As_____A_094_A2_A_L_A1_096,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,number_number_of(int),min)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(int)) ).

fof(fact_702_s0p,axiom,
    ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),s))
    & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),s),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))))
    & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,s1),s),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ) ).

fof(fact_703_s1,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ).

fof(fact_704_prime__g__5,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( ti(int,P_5) != hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))
       => ( ti(int,P_5) != hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_5)) ) ) ) ).

fof(fact_705__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_,axiom,
    ~ ! [S1] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),S1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ).

fof(fact_706_dvd__0__right,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),zero_zero(X_a))) ) ).

fof(fact_707__0964_A_K_Am_A_L_A1_Advd_As_____A_094_A2_A_N_A_N1_096,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,number_number_of(int),min)))) ).

fof(fact_708__096_091s_____A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ).

fof(fact_709__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_,axiom,
    ~ ! [S_1] :
        ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),S_1))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),S_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,s1),S_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) ) ).

fof(fact_710__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1,axiom,
    ? [X_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
      & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))))
      & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,s1),X_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))))
      & ! [Y_1] :
          ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,s1),Y_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) )
         => ti(int,Y_1) = ti(int,X_1) ) ) ).

fof(fact_711_dvd__diff,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Z_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z_1))) ) ) ) ).

fof(fact_712_dvd__trans,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C_1)) ) ) ) ).

fof(fact_713_dvd__refl,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),A_1)) ) ).

fof(fact_714_zdvd__zdiffD,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),N)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M)) ) ) ).

fof(fact_715_diff__eq__diff__eq,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C),D_1)
         => ( ti(X_a,A_3) = ti(X_a,B_2)
          <=> ti(X_a,C) = ti(X_a,D_1) ) ) ) ).

fof(fact_716_number__of__diff,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) ) ).

fof(fact_717_inv__not__p__minus__1__aux,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))),one_one(int)),P_4))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),P_4)) ) ).

fof(fact_718_inf__period_I4_J,axiom,
    ! [X_a] :
      ( ( comm_ring(X_a)
        & dvd(X_a) )
     => ! [Ta,D_1,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),D))
         => ! [X_1,K_1] :
              ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Ta)))
            <=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_1),D))),Ta))) ) ) ) ).

fof(fact_719_inf__period_I3_J,axiom,
    ! [X_a] :
      ( ( comm_ring(X_a)
        & dvd(X_a) )
     => ! [Ta,D_1,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),D))
         => ! [X_1,K_1] :
              ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Ta)))
            <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_1),D))),Ta))) ) ) ) ).

fof(fact_720_of__nat__diff,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [N,M] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) ) ) ).

fof(fact_721_dvd__0__left,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),zero_zero(X_a)),A_1))
         => ti(X_a,A_1) = zero_zero(X_a) ) ) ).

fof(fact_722_dvd__mult__right,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B,C_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C_1)) ) ) ).

fof(fact_723_dvd__mult__left,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B,C_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C_1)) ) ) ).

fof(fact_724_dvdI,axiom,
    ! [X_a] :
      ( dvd(X_a)
     => ! [A_1,B,K_4] :
          ( ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),K_4)
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),A_1)) ) ) ).

fof(fact_725_mult__dvd__mono,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [C_1,D_3,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C_1),D_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3))) ) ) ) ).

fof(fact_726_dvd__mult,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [B,A_1,C_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) ) ) ).

fof(fact_727_dvd__mult2,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) ) ) ).

fof(fact_728_dvd__triv__right,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1))) ) ).

fof(fact_729_dvd__triv__left,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ).

fof(fact_730_dvd__add,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1))) ) ) ) ).

fof(fact_731_one__dvd,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),one_one(X_a)),A_1)) ) ).

fof(fact_732_dvd__power__same,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [N,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N))) ) ) ).

fof(fact_733_right__minus__eq,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_3,B_2] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) = zero_zero(X_a)
        <=> ti(X_a,A_3) = ti(X_a,B_2) ) ) ).

fof(fact_734_eq__iff__diff__eq__0,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [A_3,B_2] :
          ( ti(X_a,A_3) = ti(X_a,B_2)
        <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) = zero_zero(X_a) ) ) ).

fof(fact_735_diff__self,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),A_1) = zero_zero(X_a) ) ).

fof(fact_736_diff__0__right,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).

fof(fact_737_diff__eq__diff__less__eq,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C),D_1)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1)) ) ) ) ).

fof(fact_738_diff__eq__diff__less,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C),D_1)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1)) ) ) ) ).

fof(fact_739_add__diff__cancel,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B) = ti(X_a,A_1) ) ).

fof(fact_740_diff__add__cancel,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),B) = ti(X_a,A_1) ) ).

fof(fact_741_add__diff__add,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [A_1,C_1,B,D_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C_1),D_3)) ) ).

fof(fact_742_diff__bin__simps_I1_J,axiom,
    ! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),pls) = ti(int,K_4) ).

fof(fact_743_diff__bin__simps_I7_J,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4)) ).

fof(fact_744_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_745_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_746_dvd__mult__cancel__right,axiom,
    ! [X_a] :
      ( idom(X_a)
     => ! [A_3,C,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),C)))
        <=> ( ti(X_a,C) = zero_zero(X_a)
            | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) ) ) ) ).

fof(fact_747_dvd__mult__cancel__left,axiom,
    ! [X_a] :
      ( idom(X_a)
     => ! [C,A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2)))
        <=> ( ti(X_a,C) = zero_zero(X_a)
            | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) ) ) ) ).

fof(fact_748_power__le__dvd,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [M,A_1,N,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),B))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),B)) ) ) ) ).

fof(fact_749_dvd__power__le,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [N,M,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),M))) ) ) ) ).

fof(fact_750_le__imp__power__dvd,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_1,M,N] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))) ) ) ).

fof(fact_751_le__iff__diff__le__0,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),zero_zero(X_a))) ) ) ).

fof(fact_752_less__iff__diff__less__0,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),zero_zero(X_a))) ) ) ).

fof(fact_753_eq__add__iff1,axiom,
    ! [X_a] :
      ( ring(X_a)
     => ! [A_3,E_2,C,B_2,D_1] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1)
        <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),E_2)),C) = ti(X_a,D_1) ) ) ).

fof(fact_754_eq__add__iff2,axiom,
    ! [X_a] :
      ( ring(X_a)
     => ! [A_3,E_2,C,B_2,D_1] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1)
        <=> ti(X_a,C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_3)),E_2)),D_1) ) ) ).

fof(fact_755_mult__diff__mult,axiom,
    ! [X_a] :
      ( ring(X_a)
     => ! [X,Y,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),B)) ) ).

fof(fact_756_right__diff__distrib__number__of,axiom,
    ! [X_b] :
      ( ( number(X_b)
        & ring(X_b) )
     => ! [V,B,C_1] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),B),C_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),B)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),C_1)) ) ).

fof(fact_757_left__diff__distrib__number__of,axiom,
    ! [X_b] :
      ( ( number(X_b)
        & ring(X_b) )
     => ! [A_1,B,V] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_1),B)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_1),hAPP(int,X_b,number_number_of(X_b),V))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B),hAPP(int,X_b,number_number_of(X_b),V))) ) ).

fof(fact_758_zdvd__not__zless,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M),N))
       => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M)) ) ) ).

fof(fact_759_zdvd__mult__cancel,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)))
     => ( ti(int,K_4) != zero_zero(int)
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N)) ) ) ).

fof(fact_760_zdvd__mono,axiom,
    ! [Ma,Ta,K] :
      ( ti(int,K) != zero_zero(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Ta))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),Ma)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),Ta))) ) ) ).

fof(fact_761_zdvd__antisym__nonneg,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M))
           => ti(int,M) = ti(int,N) ) ) ) ) ).

fof(fact_762_zcong__zpower__zmult,axiom,
    ! [Z_1,X,Y,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),one_one(int)),P_5))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Y),Z_1))),one_one(int)),P_5)) ) ).

fof(fact_763_zdvd__reduce,axiom,
    ! [K,Na,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Na),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),Ma))))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Na)) ) ).

fof(fact_764_zdvd__period,axiom,
    ! [C,Xa,Ta,A_3,D_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),D_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),Ta)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D_1))),Ta))) ) ) ).

fof(fact_765_diff__bin__simps_I10_J,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4)) ).

fof(fact_766_diff__bin__simps_I9_J,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4)) ).

fof(fact_767_diff__bin__simps_I3_J,axiom,
    ! [L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),L_4)) ).

fof(fact_768_less__bin__lemma,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K),L)),zero_zero(int))) ) ).

fof(fact_769_zprime__zdvd__power,axiom,
    ! [A_1,N,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),A_1)) ) ) ).

fof(fact_770_diff__bin__simps_I2_J,axiom,
    ! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),min) = hAPP(int,int,succ,K_4) ).

fof(fact_771_unity__coeff__ex,axiom,
    ! [X_a] :
      ( ( dvd(X_a)
        & semiring_0(X_a) )
     => ! [P_1,L] :
          ( ? [X_1] : hBOOL(hAPP(X_a,bool,P_1,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),L),X_1)))
        <=> ? [X_1] :
              ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),zero_zero(X_a))))
              & hBOOL(hAPP(X_a,bool,P_1,X_1)) ) ) ) ).

fof(fact_772_le__add__iff2,axiom,
    ! [X_a] :
      ( ordered_ring(X_a)
     => ! [A_3,E_2,C,B_2,D_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_3)),E_2)),D_1))) ) ) ).

fof(fact_773_le__add__iff1,axiom,
    ! [X_a] :
      ( ordered_ring(X_a)
     => ! [A_3,E_2,C,B_2,D_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),E_2)),C)),D_1)) ) ) ).

fof(fact_774_less__add__iff2,axiom,
    ! [X_a] :
      ( ordered_ring(X_a)
     => ! [A_3,E_2,C,B_2,D_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_3)),E_2)),D_1))) ) ) ).

fof(fact_775_less__add__iff1,axiom,
    ! [X_a] :
      ( ordered_ring(X_a)
     => ! [A_3,E_2,C,B_2,D_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),E_2)),C)),D_1)) ) ) ).

fof(fact_776_real__squared__diff__one__factored,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),one_one(X_a)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a))) ) ).

fof(fact_777_add__number__of__diff1,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),W)),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W))),C_1) ) ).

fof(fact_778_zdvd__imp__le,axiom,
    ! [Z_1,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),N))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),N)) ) ) ).

fof(fact_779_zle__diff1__eq,axiom,
    ! [Wa,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),one_one(int))))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) ) ).

fof(fact_780_diff__bin__simps_I4_J,axiom,
    ! [L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4)) ).

fof(fact_781_diff__bin__simps_I6_J,axiom,
    ! [L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4)) ).

fof(fact_782_diff__bin__simps_I5_J,axiom,
    ! [L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4)) ).

fof(fact_783_zprime__power__zdvd__cancel__left,axiom,
    ! [N,B,A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_5),N)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_5),N)),B)) ) ) ) ).

fof(fact_784_zprime__power__zdvd__cancel__right,axiom,
    ! [N,A_1,B,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),B))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_5),N)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_5),N)),A_1)) ) ) ) ).

fof(fact_785_dvd__power,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,N] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            | ti(X_a,X) = one_one(X_a) )
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))) ) ) ).

fof(fact_786_zspecial__product,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_787_power2__diff,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [X,Y] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y)) ) ).

fof(fact_788_zdiff__power3,axiom,
    ! [A_1,B] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),A_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).

fof(fact_789_zdiff__power2,axiom,
    ! [A_1,B] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)),B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_790_neg__one__power__eq__mod__m,axiom,
    ! [J_2,K_4,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),J_2)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),K_4)),M))
       => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),J_2) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),K_4) ) ) ).

fof(fact_791_Euler_Oaux____1,axiom,
    ! [Y,X,P_5] :
      ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X),P_5))
       => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),Y)) ) ) ).

fof(fact_792_zcong__neg__1__impl__ne__1,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,number_number_of(int),min)),P_5))
       => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),one_one(int)),P_5)) ) ) ).

fof(fact_793_one__not__neg__one__mod__m,axiom,
    ! [M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M))
     => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,one_one(int)),hAPP(int,int,number_number_of(int),min)),M)) ) ).

fof(fact_794_Legendre__def,axiom,
    ! [A_1,P_5] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5))
       => hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_5) = zero_zero(int) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5))
       => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),A_1))
           => hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_5) = one_one(int) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),A_1))
           => hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_5) = hAPP(int,int,number_number_of(int),min) ) ) ) ) ).

fof(fact_795_QuadRes__def,axiom,
    ! [Ma,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,Ma),Xa))
    <=> ? [Y_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),Ma)) ) ).

fof(fact_796_zcong__square__zless,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_5))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),A_1)),one_one(int)),P_5))
           => ( ti(int,A_1) = one_one(int)
              | ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)) ) ) ) ) ) ).

fof(fact_797_zcong__square,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),A_1)),one_one(int)),P_5))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),one_one(int)),P_5))
            | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),P_5)) ) ) ) ) ).

fof(fact_798_zprime__def,axiom,
    ! [P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
    <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),P_4))
        & ! [M_2] :
            ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_2))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_2),P_4)) )
           => ( ti(int,M_2) = one_one(int)
              | ti(int,M_2) = ti(int,P_4) ) ) ) ) ).

fof(fact_799_tn,axiom,
    tn = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,t)),one_one(nat)) ).

fof(fact_800_dvd__diff__nat,axiom,
    ! [N,K_4,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).

fof(fact_801_diff__commute,axiom,
    ! [I_1,J_2,K_4] : 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_2)),K_4) = 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_4)),J_2) ).

fof(fact_802_dvd__diffD,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M)) ) ) ) ).

fof(fact_803_dvd__diffD1,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N)) ) ) ) ).

fof(fact_804_diffs0__imp__equal,axiom,
    ! [M,N] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat)
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M) = zero_zero(nat)
       => M = N ) ) ).

fof(fact_805_diff__self__eq__0,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).

fof(fact_806_minus__nat_Odiff__0,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).

fof(fact_807_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_808_less__imp__diff__less,axiom,
    ! [N,J_2,K_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_2),K_4))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),N)),K_4)) ) ).

fof(fact_809_diff__less__mono2,axiom,
    ! [L_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),L_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),M))) ) ) ).

fof(fact_810_diff__add__inverse2,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = M ).

fof(fact_811_diff__add__inverse,axiom,
    ! [N,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M)),N) = M ).

fof(fact_812_diff__diff__left,axiom,
    ! [I_1,J_2,K_4] : 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_2)),K_4) = 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_2),K_4)) ).

fof(fact_813_Nat_Odiff__cancel,axiom,
    ! [K_4,M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) ).

fof(fact_814_diff__cancel2,axiom,
    ! [M,K_4,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) ).

fof(fact_815_le__diff__iff,axiom,
    ! [Na,K,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Na))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),K)))
        <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ) ) ).

fof(fact_816_Nat_Odiff__diff__eq,axiom,
    ! [N,K_4,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ).

fof(fact_817_eq__diff__iff,axiom,
    ! [Na,K,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Na))
       => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),K)
        <=> Ma = Na ) ) ) ).

fof(fact_818_diff__diff__cancel,axiom,
    ! [I_1,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_819_diff__le__mono,axiom,
    ! [L_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),L_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),L_4))) ) ).

fof(fact_820_diff__le__mono2,axiom,
    ! [L_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),M))) ) ).

fof(fact_821_Nat_Odiff__le__self,axiom,
    ! [M,N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ).

fof(fact_822_diff__mult__distrib,axiom,
    ! [M,N,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K_4)) ).

fof(fact_823_diff__mult__distrib2,axiom,
    ! [K_4,M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) ).

fof(fact_824_dvd__reduce,axiom,
    ! [K,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Na)) ) ).

fof(fact_825_divides__add__revr,axiom,
    ! [B,D_3,A_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),A_1))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B)))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),B)) ) ) ).

fof(fact_826_divides__mul__l,axiom,
    ! [C_1,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C_1),A_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C_1),B))) ) ).

fof(fact_827_divides__mul__r,axiom,
    ! [C_1,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1))) ) ).

fof(fact_828_nat__dvd__1__iff__1,axiom,
    ! [Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),one_one(nat)))
    <=> Ma = one_one(nat) ) ).

fof(fact_829_divides__exp,axiom,
    ! [N,X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),N))) ) ).

fof(fact_830_real__le__eq__diff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),Ya)),zero_zero(real))) ) ).

fof(fact_831_zero__less__diff,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),Ma)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).

fof(fact_832_diff__less,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ) ) ).

fof(fact_833_diff__add__0,axiom,
    ! [N,M] : 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)) = zero_zero(nat) ).

fof(fact_834_nat__dvd__not__less,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
       => ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ) ).

fof(fact_835_diff__is__0__eq,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),Na) = zero_zero(nat)
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).

fof(fact_836_diff__is__0__eq_H,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) ) ).

fof(fact_837_less__diff__conv,axiom,
    ! [I,J,K] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ).

fof(fact_838_add__diff__inverse,axiom,
    ! [M,N] :
      ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),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),N)) = M ) ).

fof(fact_839_divides__ge,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
     => ( B = zero_zero(nat)
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_1),B)) ) ) ).

fof(fact_840_diff__less__mono,axiom,
    ! [C_1,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_1),B))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C_1),A_1))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),C_1))) ) ) ).

fof(fact_841_less__diff__iff,axiom,
    ! [Na,K,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Na))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),K)))
        <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ) ) ).

fof(fact_842_nat__mult__dvd__cancel__disj,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))
    <=> ( K = zero_zero(nat)
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) ) ) ).

fof(fact_843_nat__mult__dvd__cancel__disj_H,axiom,
    ! [Ma,K,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K)))
    <=> ( K = zero_zero(nat)
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) ) ) ).

fof(fact_844_diff__diff__right,axiom,
    ! [I_1,K_4,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
     => 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_2),K_4)) = 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_4)),J_2) ) ).

fof(fact_845_le__diff__conv,axiom,
    ! [J,K,I] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)),I))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ).

fof(fact_846_le__add__diff,axiom,
    ! [M,K_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M)),K_4))) ) ).

fof(fact_847_le__add__diff__inverse,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) = M ) ).

fof(fact_848_add__diff__assoc,axiom,
    ! [I_1,K_4,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
     => 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_2),K_4)) = 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_2)),K_4) ) ).

fof(fact_849_le__diff__conv2,axiom,
    ! [I,K,J] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ) ).

fof(fact_850_le__add__diff__inverse2,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = M ) ).

fof(fact_851_le__imp__diff__is__add,axiom,
    ! [K,I,J] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),I) = K
      <=> J = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ).

fof(fact_852_diff__add__assoc,axiom,
    ! [I_1,K_4,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
     => 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_2)),K_4) = 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_2),K_4)) ) ).

fof(fact_853_add__diff__assoc2,axiom,
    ! [I_1,K_4,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)),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_2),I_1)),K_4) ) ).

fof(fact_854_diff__add__assoc2,axiom,
    ! [I_1,K_4,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),I_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)),I_1) ) ).

fof(fact_855_divides__rev,axiom,
    ! [A_1,N,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N)))
     => ( N != zero_zero(nat)
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) ) ) ).

fof(fact_856_divides__exp2,axiom,
    ! [X,Y,N] :
      ( N != zero_zero(nat)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N)),Y))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).

fof(fact_857_zdvd__int,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))) ) ).

fof(fact_858_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) ) ).

fof(fact_859_nat__diff__split__asm,axiom,
    ! [P_1,A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_3),B_2)))
    <=> ~ ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2))
            & ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
          | ? [D_2] :
              ( A_3 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B_2),D_2)
              & ~ hBOOL(hAPP(nat,bool,P_1,D_2)) ) ) ) ).

fof(fact_860_nat__diff__split,axiom,
    ! [P_1,A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_3),B_2)))
    <=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2))
         => hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
        & ! [D_2] :
            ( A_3 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B_2),D_2)
           => hBOOL(hAPP(nat,bool,P_1,D_2)) ) ) ) ).

fof(fact_861_dvd__imp__le,axiom,
    ! [K_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N)) ) ) ).

fof(fact_862_nat__mult__dvd__cancel1,axiom,
    ! [Ma,Na,K] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) ) ) ).

fof(fact_863_dvd__mult__cancel,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

fof(fact_864_nat__eq__add__iff2,axiom,
    ! [U,Ma,Na,I,J] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U)),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na)
      <=> Ma = 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)),U)),Na) ) ) ).

fof(fact_865_nat__diff__add__eq2,axiom,
    ! [U_1,M,N,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
     => 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)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),U_1)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(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_2),I_1)),U_1)),N)) ) ).

fof(fact_866_nat__le__add__iff2,axiom,
    ! [U,Ma,Na,I,J] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),U)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),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)),U)),Na))) ) ) ).

fof(fact_867_nat__eq__add__iff1,axiom,
    ! [U,Ma,Na,J,I] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U)),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),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),J)),U)),Ma) = Na ) ) ).

fof(fact_868_nat__diff__add__eq1,axiom,
    ! [U_1,M,N,J_2,I_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),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)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),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_2)),U_1)),M)),N) ) ).

fof(fact_869_nat__le__add__iff1,axiom,
    ! [U,Ma,Na,J,I] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),U)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),J)),U)),Ma)),Na)) ) ) ).

fof(fact_870_zdiff__int,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) ) ).

fof(fact_871_zcong__trans,axiom,
    ! [C_1,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),C_1),M))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),C_1),M)) ) ) ).

fof(fact_872_zcong__eq__trans,axiom,
    ! [D_3,C_1,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => ( ti(int,B) = ti(int,C_1)
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),D_3),M))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),D_3),M)) ) ) ) ).

fof(fact_873_zcong__refl,axiom,
    ! [K_4,M] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),K_4),M)) ).

fof(fact_874_zcong__sym,axiom,
    ! [A_3,B_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B_2),A_3),Ma)) ) ).

fof(fact_875_divides__div__not,axiom,
    ! [X,Q_3,N,R_3] :
      ( X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Q_3),N)),R_3)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),R_3))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_3),N))
         => ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),X)) ) ) ) ).

fof(fact_876_mult__eq__if,axiom,
    ! [N,M] :
      ( ( M = zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
      & ( M != zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),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),one_one(nat))),N)) ) ) ).

fof(fact_877_nat__less__add__iff1,axiom,
    ! [U,Ma,Na,J,I] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),U)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),J)),U)),Ma)),Na)) ) ) ).

fof(fact_878_nat__less__add__iff2,axiom,
    ! [U,Ma,Na,I,J] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),U)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),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)),U)),Na))) ) ) ).

fof(fact_879_dvd__mult__cancel1,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na)),Ma))
      <=> Na = one_one(nat) ) ) ).

fof(fact_880_dvd__mult__cancel2,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Ma)),Ma))
      <=> Na = one_one(nat) ) ) ).

fof(fact_881_power__eq__if,axiom,
    ! [P_5,M] :
      ( ( M = zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),M) = one_one(nat) )
      & ( M != zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ).

fof(fact_882_power__dvd__imp__le,axiom,
    ! [I_1,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N)))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I_1))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

fof(fact_883_realpow__minus__mult,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [X,N] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),X) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N) ) ) ).

fof(fact_884_zdiff__int__split,axiom,
    ! [P_1,Xa,Ya] :
      ( hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Xa),Ya))))
    <=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ya),Xa))
         => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya)))) )
        & ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya))
         => hBOOL(hAPP(int,bool,P_1,zero_zero(int))) ) ) ) ).

fof(fact_885_order__le__neq__implies__less,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => ( ti(X_a,X) != ti(X_a,Y)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ) ).

fof(fact_886_Euler_Oaux2,axiom,
    ! [B,A_1,C_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),C_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),B))
          | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A_1)) ) ) ) ).

fof(fact_887_divides__cases,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M))
     => ( M = zero_zero(nat)
        | M = N
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)),M)) ) ) ).

fof(fact_888_diff__square,axiom,
    ! [X,Y] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),Y)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X),Y)) ).

fof(fact_889_zcong__id,axiom,
    ! [M] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,M),zero_zero(int)),M)) ).

fof(fact_890_IntPrimes_Ozcong__zero,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),zero_zero(int)))
    <=> ti(int,A_3) = ti(int,B_2) ) ).

fof(fact_891_Int2_Oaux1,axiom,
    ! [A_1,B,C_1] :
      ( hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B) = ti(int,C_1)
     => ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C_1),B) ) ).

fof(fact_892_zcong__1,axiom,
    ! [A_1,B] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),one_one(int))) ).

fof(fact_893_zcong__zmult,axiom,
    ! [C_1,D_3,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),D_3),M))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D_3)),M)) ) ) ).

fof(fact_894_zcong__scalar2,axiom,
    ! [K_4,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),A_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),B)),M)) ) ).

fof(fact_895_zcong__scalar,axiom,
    ! [K_4,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),K_4)),M)) ) ).

fof(fact_896_zcong__zmult__prop2,axiom,
    ! [C,D_1,A_3,B_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),A_3)),Ma))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),B_2)),Ma)) ) ) ).

fof(fact_897_zcong__zmult__prop1,axiom,
    ! [C,D_1,A_3,B_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),D_1)),Ma))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1)),Ma)) ) ) ).

fof(fact_898_zcong__zmult__self,axiom,
    ! [A_1,M,B] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),M)),M)) ).

fof(fact_899_zcong__zadd,axiom,
    ! [C_1,D_3,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),D_3),M))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),D_3)),M)) ) ) ).

fof(fact_900_zcong__shift,axiom,
    ! [C_1,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C_1)),M)) ) ).

fof(fact_901_zcong__zdiff,axiom,
    ! [C_1,D_3,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),D_3),M))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),D_3)),M)) ) ) ).

fof(fact_902_zcong__zpower,axiom,
    ! [Z_1,X,Y,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),M))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Z_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),Z_1)),M)) ) ).

fof(fact_903_zcong__less__eq,axiom,
    ! [M,Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),M))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M))
             => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),M))
               => ti(int,X) = ti(int,Y) ) ) ) ) ) ) ).

fof(fact_904_zcong__not__zero,axiom,
    ! [M,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M))
       => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),M)) ) ) ).

fof(fact_905_zcong__not,axiom,
    ! [B,M,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1))
           => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) ) ) ) ) ).

fof(fact_906_xzgcda__linear__aux1,axiom,
    ! [A_1,R_3,B,M,C_1,D_3,N] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_3),B))),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_3),D_3))),N)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),N))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_3),N)))) ).

fof(fact_907_zdvd__bounds,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),zero_zero(int)))
        | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),M)) ) ) ).

fof(fact_908_zcong__iff__lin,axiom,
    ! [A_3,B_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))
    <=> ? [K_1] : ti(int,B_2) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),K_1)) ) ).

fof(fact_909_zcong__eq__zdvd__prop,axiom,
    ! [Xa,P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),P_4))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),Xa)) ) ).

fof(fact_910_zcong__zero__equiv__div,axiom,
    ! [A_3,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),Ma))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),A_3)) ) ).

fof(fact_911_zcong__def,axiom,
    ! [A_3,B_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_3),B_2))) ) ).

fof(fact_912_zprime__zdvd__zmult__better,axiom,
    ! [M,N,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),M))
          | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),N)) ) ) ) ).

fof(fact_913_Euler_Oaux1,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),A_1))
       => ( ti(int,X) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),one_one(int))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),one_one(int)))) ) ) ) ).

fof(fact_914_Int2_Ozcong__zero,axiom,
    ! [M,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),M))
         => ti(int,X) = zero_zero(int) ) ) ) ).

fof(fact_915_zcong__zless__0,axiom,
    ! [M,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),M))
         => ti(int,A_1) = zero_zero(int) ) ) ) ).

fof(fact_916_zcong__zless__imp__eq,axiom,
    ! [B,M,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),M))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
             => ti(int,A_1) = ti(int,B) ) ) ) ) ) ).

fof(fact_917_zpower__zdvd__prop1,axiom,
    ! [P_5,Y,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),Y))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),N))) ) ) ).

fof(fact_918_neg__one__power,axiom,
    ! [N] :
      ( hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),N) = one_one(int)
      | hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),N) = hAPP(int,int,number_number_of(int),min) ) ).

fof(fact_919_zcong__zmult__prop3,axiom,
    ! [Y,X,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y),zero_zero(int)),P_5))
         => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),zero_zero(int)),P_5)) ) ) ) ).

fof(fact_920_zcong__zprime__prod__zero__contra,axiom,
    ! [B,A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
       => ( ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5))
            & ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),zero_zero(int)),P_5)) )
         => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),zero_zero(int)),P_5)) ) ) ) ).

fof(fact_921_zcong__zprime__prod__zero,axiom,
    ! [B,A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),zero_zero(int)),P_5))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5))
            | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),zero_zero(int)),P_5)) ) ) ) ) ).

fof(fact_922_zprime__zdvd__zmult,axiom,
    ! [N,P_5,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),M))
            | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),N)) ) ) ) ) ).

fof(fact_923_zpower__zdvd__prop2,axiom,
    ! [Y,N,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),N)))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),Y)) ) ) ) ).

fof(fact_924_realpow__num__eq__if,axiom,
    ! [X_a] :
      ( power(X_a)
     => ! [M,N] :
          ( ( N = zero_zero(nat)
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M),N) = one_one(X_a) )
          & ( N != zero_zero(nat)
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),M),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).

fof(fact_925__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An_,axiom,
    ~ ! [R_1,V_3] :
        ~ ( ti(int,V_3) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),x),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),V_3))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) ) ).

fof(fact_926__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An_,axiom,
    ~ ! [S_1,W_1] :
        ~ ( ti(int,W_1) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),y),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),W_1))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) ) ).

fof(fact_927_decr__mult__lemma,axiom,
    ! [K,P_1,D_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
     => ( ! [X_1] :
            ( hBOOL(hAPP(int,bool,P_1,X_1))
           => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D_1))) )
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X_1] :
              ( hBOOL(hAPP(int,bool,P_1,X_1))
             => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),D_1)))) ) ) ) ) ).

fof(fact_928_zprime__factor__exists,axiom,
    ! [A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
     => ? [P_6] :
          ( hBOOL(hAPP(int,bool,zprime,P_6))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_6),A_1)) ) ) ).

fof(fact_929_zcong__zless__unique,axiom,
    ! [A_1,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ? [X_1] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),M))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),X_1),M))
          & ! [Y_1] :
              ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
                & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),M))
                & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),Y_1),M)) )
             => ti(int,Y_1) = ti(int,X_1) ) ) ) ).

fof(fact_930__096sum2sq_A_Is_____M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096,axiom,
    hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),s),one_one(int))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),t) ).

fof(fact_931_norR__mem__unique__aux,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),one_one(int))))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) ) ).

fof(fact_932_dvd_Oorder__refl,axiom,
    ! [X] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X)) ).

fof(fact_933__096_B_Bthesis_O_A_I_B_Btn_O_A_091_124_Atn_A_061_Anat_At_A_N_A1_059_A0_,axiom,
    hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,t)),one_one(nat)))) ).

fof(fact_934_dvd_Oless__asym,axiom,
    ! [X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).

fof(fact_935_dvd_Oless__trans,axiom,
    ! [Z_1,X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y)) )
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X)) ) ) ) ).

fof(fact_936_dvd_Oless__asym_H,axiom,
    ! [A_1,B] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) )
     => ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) ) ) ).

fof(fact_937_dvd_Oless__le__trans,axiom,
    ! [Z_1,X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_1))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X)) ) ) ) ).

fof(fact_938_dvd_Oord__less__eq__trans,axiom,
    ! [C_1,A_1,B] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) )
     => ( B = C_1
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C_1),A_1)) ) ) ) ).

fof(fact_939_dvd_Oless__imp__triv,axiom,
    ! [P_1,Xa,Ya] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
     => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) )
       => hBOOL(P_1) ) ) ).

fof(fact_940_dvd_Oless__imp__not__eq2,axiom,
    ! [X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => Y != X ) ).

fof(fact_941_dvd_Oless__imp__not__eq,axiom,
    ! [X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => X != Y ) ).

fof(fact_942_dvd_Oless__imp__not__less,axiom,
    ! [X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).

fof(fact_943_dvd_Oless__imp__le,axiom,
    ! [X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ).

fof(fact_944_dvd_Oless__not__sym,axiom,
    ! [X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).

fof(fact_945_dvd_Oless__imp__neq,axiom,
    ! [X,Y] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
     => X != Y ) ).

fof(fact_946_dvd_Ole__less__trans,axiom,
    ! [Z_1,X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
     => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y)) )
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X)) ) ) ) ).

fof(fact_947_dvd_Oord__eq__less__trans,axiom,
    ! [C_1,A_1,B] :
      ( A_1 = B
     => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C_1),B)) )
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C_1),A_1)) ) ) ) ).

fof(fact_948_dvd_Oorder__trans,axiom,
    ! [Z_1,X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_1))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1)) ) ) ).

fof(fact_949_dvd_Oantisym,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
       => X = Y ) ) ).

fof(fact_950_dvd__antisym,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M))
       => M = N ) ) ).

fof(fact_951_dvd_Oord__le__eq__trans,axiom,
    ! [C_1,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
     => ( B = C_1
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1)) ) ) ).

fof(fact_952_dvd_Oord__eq__le__trans,axiom,
    ! [C_1,A_1,B] :
      ( A_1 = B
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C_1))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1)) ) ) ).

fof(fact_953_dvd_Ole__neq__trans,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
     => ( A_1 != B
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) ) ) ) ).

fof(fact_954_dvd_Ole__imp__less__or__eq,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
     => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
        | X = Y ) ) ).

fof(fact_955_dvd_Oantisym__conv,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
      <=> Xa = Ya ) ) ).

fof(fact_956_dvd_Oeq__refl,axiom,
    ! [X,Y] :
      ( X = Y
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ).

fof(fact_957_dvd_Oneq__le__trans,axiom,
    ! [A_1,B] :
      ( A_1 != B
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) ) ) ) ).

fof(fact_958_dvd_Oless__le__not__le,axiom,
    ! [Xa,Ya] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) ) ) ).

fof(fact_959_dvd_Oless__le,axiom,
    ! [Xa,Ya] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
        & Xa != Ya ) ) ).

fof(fact_960_dvd_Ole__less,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
    <=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
        | Xa = Ya ) ) ).

fof(fact_961_dvd_Oeq__iff,axiom,
    ! [Xa,Ya] :
      ( Xa = Ya
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) ) ) ).

fof(fact_962_dvd_Oless__irrefl,axiom,
    ! [X] :
      ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X)) ) ).

fof(fact_963_divides__antisym,axiom,
    ! [Xa,Ya] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
    <=> Xa = Ya ) ).

fof(fact_964_nat__if__cong,axiom,
    ! [Xa,Ya,P_1] :
      ( ( hBOOL(P_1)
       => hAPP(int,nat,nat_1,Xa) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),P_1),Xa),Ya)) )
      & ( ~ hBOOL(P_1)
       => hAPP(int,nat,nat_1,Ya) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),P_1),Xa),Ya)) ) ) ).

fof(fact_965_abs__idempotent,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) = hAPP(X_a,X_a,abs_abs(X_a),A_1) ) ).

fof(fact_966_nat__abs__mult__distrib,axiom,
    ! [W,Z_1] : hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),W))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z_1))) ).

fof(fact_967_int__dvd__iff,axiom,
    ! [Ma,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z)))) ) ).

fof(fact_968_dvd__int__iff,axiom,
    ! [Z,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z),hAPP(nat,int,semiring_1_of_nat(int),Ma)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z))),Ma)) ) ).

fof(fact_969_abs__eq__0,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_3] :
          ( hAPP(X_a,X_a,abs_abs(X_a),A_3) = zero_zero(X_a)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_970_abs__zero,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => hAPP(X_a,X_a,abs_abs(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_971_abs__ge__self,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),hAPP(X_a,X_a,abs_abs(X_a),A_1))) ) ).

fof(fact_972_abs__le__D1,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).

fof(fact_973_abs__mult__self,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),A_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1) ) ).

fof(fact_974_abs__mult,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ).

fof(fact_975_abs__add__abs,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ).

fof(fact_976_abs__one,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => hAPP(X_a,X_a,abs_abs(X_a),one_one(X_a)) = one_one(X_a) ) ).

fof(fact_977_abs__minus__commute,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)) ) ).

fof(fact_978_power__abs,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1,N] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),N) ) ).

fof(fact_979_dvd__if__abs__eq,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [L_4,K_4] :
          ( hAPP(X_a,X_a,abs_abs(X_a),L_4) = hAPP(X_a,X_a,abs_abs(X_a),K_4)
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L_4),K_4)) ) ) ).

fof(fact_980_abs__dvd__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Ma,K] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,abs_abs(X_a),Ma)),K))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)) ) ) ).

fof(fact_981_dvd__abs__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Ma,K] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),hAPP(X_a,X_a,abs_abs(X_a),K)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)) ) ) ).

fof(fact_982_abs__of__nat,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [N] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N) ) ).

fof(fact_983_abs__int__eq,axiom,
    ! [M] : hAPP(int,int,abs_abs(int),hAPP(nat,int,semiring_1_of_nat(int),M)) = hAPP(nat,int,semiring_1_of_nat(int),M) ).

fof(fact_984_nat__int,axiom,
    ! [N] : hAPP(int,nat,nat_1,hAPP(nat,int,semiring_1_of_nat(int),N)) = N ).

fof(fact_985_zdvd__antisym__abs,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),A_1))
       => hAPP(int,int,abs_abs(int),A_1) = hAPP(int,int,abs_abs(int),B) ) ) ).

fof(fact_986_abs__ge__zero,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,abs_abs(X_a),A_1))) ) ).

fof(fact_987_abs__le__zero__iff,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_3)),zero_zero(X_a)))
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_988_abs__of__nonneg,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
         => hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1) ) ) ).

fof(fact_989_abs__of__pos,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1) ) ) ).

fof(fact_990_zero__less__abs__iff,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,abs_abs(X_a),A_3)))
        <=> ti(X_a,A_3) != zero_zero(X_a) ) ) ).

fof(fact_991_abs__not__less__zero,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),zero_zero(X_a))) ) ).

fof(fact_992_abs__mult__less,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [B,D_3,A_1,C_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),C_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),B)),D_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),D_3))) ) ) ) ).

fof(fact_993_abs__triangle__ineq,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))) ) ).

fof(fact_994_abs__triangle__ineq2__sym,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)))) ) ).

fof(fact_995_abs__triangle__ineq2,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).

fof(fact_996_abs__triangle__ineq3,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).

fof(fact_997_nat__0,axiom,
    hAPP(int,nat,nat_1,zero_zero(int)) = zero_zero(nat) ).

fof(fact_998_transfer__nat__int__numerals_I1_J,axiom,
    zero_zero(nat) = hAPP(int,nat,nat_1,zero_zero(int)) ).

fof(fact_999_abs__zmult__eq__1,axiom,
    ! [M,N] :
      ( hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)) = one_one(int)
     => hAPP(int,int,abs_abs(int),M) = one_one(int) ) ).

fof(fact_1000_eq__nat__nat__iff,axiom,
    ! [Z_3,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_3))
       => ( hAPP(int,nat,nat_1,Z) = hAPP(int,nat,nat_1,Z_3)
        <=> ti(int,Z) = ti(int,Z_3) ) ) ) ).

fof(fact_1001_transfer__nat__int__relations_I1_J,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
       => ( hAPP(int,nat,nat_1,Xa) = hAPP(int,nat,nat_1,Ya)
        <=> ti(int,Xa) = ti(int,Ya) ) ) ) ).

fof(fact_1002_all__nat,axiom,
    ! [P_1] :
      ( ! [X1] : hBOOL(hAPP(nat,bool,P_1,X1))
    <=> ! [X_1] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
         => hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,X_1))) ) ) ).

fof(fact_1003_ex__nat,axiom,
    ! [P_1] :
      ( ? [X1] : hBOOL(hAPP(nat,bool,P_1,X1))
    <=> ? [X_1] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
          & hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,X_1))) ) ) ).

fof(fact_1004_nat__number__of__def,axiom,
    ! [V] : hAPP(int,nat,number_number_of(nat),V) = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),V)) ).

fof(fact_1005_nat__number__of,axiom,
    ! [W] : hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),W)) = hAPP(int,nat,number_number_of(nat),W) ).

fof(fact_1006_transfer__nat__int__numerals_I2_J,axiom,
    one_one(nat) = hAPP(int,nat,nat_1,one_one(int)) ).

fof(fact_1007_abs__eq__1__iff,axiom,
    ! [Z] :
      ( hAPP(int,int,abs_abs(int),Z) = one_one(int)
    <=> ( ti(int,Z) = one_one(int)
        | ti(int,Z) = hAPP(int,int,number_number_of(int),min) ) ) ).

fof(fact_1008_zdvd1__eq,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),one_one(int)))
    <=> hAPP(int,int,abs_abs(int),Xa) = one_one(int) ) ).

fof(fact_1009_mult__sum2sq,axiom,
    ! [A_1,B,P_5,Q_3] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_1),B))),hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3))) = hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),P_5)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),P_5)))) ).

fof(fact_1010_abs__mult__pos,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),Y)),X) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),X)) ) ) ).

fof(fact_1011_abs__eq__mult,axiom,
    ! [X_a] :
      ( ordered_ring_abs(X_a)
     => ! [B,A_1] :
          ( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
              | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) )
            & ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
              | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) ) )
         => hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ) ).

fof(fact_1012_zero__le__power__abs,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1,N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),N))) ) ).

fof(fact_1013_abs__triangle__ineq4,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))) ) ).

fof(fact_1014_abs__diff__triangle__ineq,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B,C_1,D_3] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C_1))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D_3))))) ) ).

fof(fact_1015_abs__minus__one,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),min)) = one_one(X_a) ) ).

fof(fact_1016_nat__0__iff,axiom,
    ! [I] :
      ( hAPP(int,nat,nat_1,I) = zero_zero(nat)
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int))) ) ).

fof(fact_1017_nat__le__0,axiom,
    ! [Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),zero_zero(int)))
     => hAPP(int,nat,nat_1,Z_1) = zero_zero(nat) ) ).

fof(fact_1018_zabs__less__one__iff,axiom,
    ! [Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,abs_abs(int),Z)),one_one(int)))
    <=> ti(int,Z) = zero_zero(int) ) ).

fof(fact_1019_zless__nat__conj,axiom,
    ! [Wa,Z] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Wa)),hAPP(int,nat,nat_1,Z)))
    <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) ) ) ).

fof(fact_1020_nat__mono__iff,axiom,
    ! [Wa,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Wa)),hAPP(int,nat,nat_1,Z)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) ) ) ).

fof(fact_1021_transfer__nat__int__relations_I3_J,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,Xa)),hAPP(int,nat,nat_1,Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)) ) ) ) ).

fof(fact_1022_nat__0__le,axiom,
    ! [Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
     => hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z_1)) = ti(int,Z_1) ) ).

fof(fact_1023_int__eq__iff,axiom,
    ! [Ma,Z] :
      ( hAPP(nat,int,semiring_1_of_nat(int),Ma) = ti(int,Z)
    <=> ( Ma = hAPP(int,nat,nat_1,Z)
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

fof(fact_1024_int__nat__eq,axiom,
    ! [Z_1] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
       => hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z_1)) = ti(int,Z_1) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
       => hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z_1)) = zero_zero(int) ) ) ).

fof(fact_1025_zero__le__zpower__abs,axiom,
    ! [X,N] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),X)),N))) ).

fof(fact_1026_zless__nat__eq__int__zless,axiom,
    ! [Ma,Z] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(int,nat,nat_1,Z)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z)) ) ).

fof(fact_1027_dvd__imp__le__int,axiom,
    ! [D_3,I_1] :
      ( ti(int,I_1) != zero_zero(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),I_1))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),D_3)),hAPP(int,int,abs_abs(int),I_1))) ) ) ).

fof(fact_1028_transfer__nat__int__relations_I4_J,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,Xa)),hAPP(int,nat,nat_1,Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),Ya)) ) ) ) ).

fof(fact_1029_abs__power__minus__one,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [N] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),N)) = one_one(X_a) ) ).

fof(fact_1030_zero__less__nat__eq,axiom,
    ! [Z] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,nat_1,Z)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

fof(fact_1031_transfer__nat__int__numerals_I4_J,axiom,
    hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).

fof(fact_1032_nat__less__eq__zless,axiom,
    ! [Z,Wa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Wa)),hAPP(int,nat,nat_1,Z)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) ) ) ).

fof(fact_1033_transfer__nat__int__relations_I2_J,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Xa)),hAPP(int,nat,nat_1,Ya)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) ) ) ) ).

fof(fact_1034_abs__power3__distrib,axiom,
    ! [X] : hAPP(int,int,abs_abs(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).

fof(fact_1035_zero__less__zpower__abs__iff,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),Xa)),Na)))
    <=> ( ti(int,Xa) != zero_zero(int)
        | Na = zero_zero(nat) ) ) ).

fof(fact_1036_split__nat,axiom,
    ! [P_1,I] :
      ( hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,I)))
    <=> ( ! [N_1] :
            ( ti(int,I) = hAPP(nat,int,semiring_1_of_nat(int),N_1)
           => hBOOL(hAPP(nat,bool,P_1,N_1)) )
        & ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
         => hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) ) ) ) ).

fof(fact_1037_nat__eq__iff,axiom,
    ! [Wa,Ma] :
      ( hAPP(int,nat,nat_1,Wa) = Ma
    <=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
         => ti(int,Wa) = hAPP(nat,int,semiring_1_of_nat(int),Ma) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
         => Ma = zero_zero(nat) ) ) ) ).

fof(fact_1038_nat__eq__iff2,axiom,
    ! [Ma,Wa] :
      ( Ma = hAPP(int,nat,nat_1,Wa)
    <=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
         => ti(int,Wa) = hAPP(nat,int,semiring_1_of_nat(int),Ma) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
         => Ma = zero_zero(nat) ) ) ) ).

fof(fact_1039_nat__le__eq__zle,axiom,
    ! [Z,Wa] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Wa))
        | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) )
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,Wa)),hAPP(int,nat,nat_1,Z)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z)) ) ) ).

fof(fact_1040_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)) ) ) ).

fof(fact_1041_nat__mult__distrib,axiom,
    ! [Z_4,Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
     => hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,Z_1)),hAPP(int,nat,nat_1,Z_4)) ) ).

fof(fact_1042_nat__add__distrib,axiom,
    ! [Z_4,Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_4))
       => hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,Z_1)),hAPP(int,nat,nat_1,Z_4)) ) ) ).

fof(fact_1043_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)) ) ) ).

fof(fact_1044_zdvd__mult__cancel1,axiom,
    ! [Na,Ma] :
      ( ti(int,Ma) != zero_zero(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),Na)),Ma))
      <=> hAPP(int,int,abs_abs(int),Na) = one_one(int) ) ) ).

fof(fact_1045_int__eq__iff__number__of,axiom,
    ! [Ma,Va] :
      ( hAPP(nat,int,semiring_1_of_nat(int),Ma) = hAPP(int,int,number_number_of(int),Va)
    <=> ( Ma = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),Va))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),Va))) ) ) ).

fof(fact_1046_nat__diff__distrib,axiom,
    ! [Z_1,Z_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_4),Z_1))
       => hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Z_1)),hAPP(int,nat,nat_1,Z_4)) ) ) ).

fof(fact_1047_nat__power__eq,axiom,
    ! [N,Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
     => hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,nat_1,Z_1)),N) ) ).

fof(fact_1048_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,nat_1,X)),N) = hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N)) ) ).

fof(fact_1049_Little__Fermat,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),X))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))),one_one(int)),P_5)) ) ) ).

fof(fact_1050_power2__abs,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).

fof(fact_1051_abs__power2,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).

fof(fact_1052_transfer__nat__int__numerals_I3_J,axiom,
    hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_1053_power2__eq__iff__abs__eq,axiom,
    ! [A_3,B_2] :
      ( hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))
    <=> hAPP(int,int,abs_abs(int),A_3) = hAPP(int,int,abs_abs(int),B_2) ) ).

fof(fact_1054_abs__power2__distrib,axiom,
    ! [A_1] : hAPP(int,int,abs_abs(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_1055_nat__less__iff,axiom,
    ! [Ma,Wa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Wa)),Ma))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),Ma))) ) ) ).

fof(fact_1056_nat__dvd__iff,axiom,
    ! [Z,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,Z)),Ma))
    <=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z),hAPP(nat,int,semiring_1_of_nat(int),Ma))) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => Ma = zero_zero(nat) ) ) ) ).

fof(fact_1057_power2__eq1__iff,axiom,
    ! [A_1] :
      ( hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(int)
     => hAPP(int,int,abs_abs(int),A_1) = one_one(int) ) ).

fof(fact_1058_decr__lemma,axiom,
    ! [X,Z_1,D_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_3))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Z_1))),one_one(int))),D_3))),Z_1)) ) ).

fof(fact_1059_incr__lemma,axiom,
    ! [Z_1,X,D_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_3))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),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),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Z_1))),one_one(int))),D_3)))) ) ).

fof(fact_1060_Euler_Oaux__1,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_5))
     => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,P_5)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,P_5)),one_one(nat)))) ) ).

fof(fact_1061_Int2_Oaux__2,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).

fof(fact_1062_Int2_Oaux__1,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,P_5)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ).

fof(fact_1063_of__nat__double,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [X] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
         => hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,X))),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,X))) ) ) ).

fof(fact_1064_best__division__abs,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ? [N_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Y),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),X))))),X)) ) ).

fof(fact_1065_nat0__intermed__int__val,axiom,
    ! [K,F,Na] :
      ( ! [I_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),Na))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,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_2),one_one(nat)))),hAPP(nat,int,F,I_2)))),one_one(int))) )
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na)))
         => ? [I_2] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),Na))
              & hAPP(nat,int,F,I_2) = ti(int,K) ) ) ) ) ).

fof(fact_1066_int__val__lemma,axiom,
    ! [K,F,Na] :
      ( ! [I_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),Na))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,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_2),one_one(nat)))),hAPP(nat,int,F,I_2)))),one_one(int))) )
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na)))
         => ? [I_2] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),Na))
              & hAPP(nat,int,F,I_2) = ti(int,K) ) ) ) ) ).

fof(fact_1067_is__sum2sq__def,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,Xa))
    <=> ? [A_4,B_4] : hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_4),B_4)) = ti(int,Xa) ) ).

fof(fact_1068_nat__intermed__int__val,axiom,
    ! [K,F,Na,Ma] :
      ( ! [I_2] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),I_2))
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),Na)) )
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,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_2),one_one(nat)))),hAPP(nat,int,F,I_2)))),one_one(int))) )
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,Ma)),K))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na)))
           => ? [I_2] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),I_2))
                & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),Na))
                & hAPP(nat,int,F,I_2) = ti(int,K) ) ) ) ) ) ).

fof(fact_1069_nat__aux__def,axiom,
    ! [I_1,N] : hAPP(nat,nat,hAPP(int,fun(nat,nat),nat_aux,I_1),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,I_1)),N) ).

fof(fact_1070_Wilson__Russ,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))),hAPP(int,int,number_number_of(int),min)),P_5)) ) ).

fof(fact_1071_Euler__part1,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),X))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(int,int,number_number_of(int),min)),P_5)) ) ) ) ) ).

fof(fact_1072_zdiv__zero,axiom,
    ! [B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),zero_zero(int)),B) = zero_zero(int) ).

fof(fact_1073_div__0,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).

fof(fact_1074_div__by__0,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_1075_div__by__1,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).

fof(fact_1076_div__dvd__div,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [C,A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),C))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B_2),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),C),A_3)))
            <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B_2),C)) ) ) ) ) ).

fof(fact_1077_zdiv__self,axiom,
    ! [A_1] :
      ( ti(int,A_1) != zero_zero(int)
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),A_1) = one_one(int) ) ).

fof(fact_1078_pos__imp__zdiv__neg__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2)),zero_zero(int)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),zero_zero(int))) ) ) ).

fof(fact_1079_neg__imp__zdiv__neg__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2)),zero_zero(int)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_3)) ) ) ).

fof(fact_1080_div__neg__pos__less0,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int))) ) ) ).

fof(fact_1081_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y))) ) ) ).

fof(fact_1082_zdiv__number__of__Bit0,axiom,
    ! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)) ).

fof(fact_1083_zdvd__mult__div__cancel,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M))
     => hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M),N)) = ti(int,M) ) ).

fof(fact_1084_abs__div,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Y),X))
     => hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,abs_abs(int),X)),hAPP(int,int,abs_abs(int),Y)) ) ).

fof(fact_1085_div__mult__mult1,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] :
          ( ti(X_a,C_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B) ) ) ).

fof(fact_1086_div__mult__mult2,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] :
          ( ti(X_a,C_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B) ) ) ).

fof(fact_1087_div__mult__self1__is__id,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),B) = ti(X_a,A_1) ) ) ).

fof(fact_1088_div__mult__self2__is__id,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),B) = ti(X_a,A_1) ) ) ).

fof(fact_1089_div__mult__mult1__if,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] :
          ( ( ti(X_a,C_1) = zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)) = zero_zero(X_a) )
          & ( ti(X_a,C_1) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B) ) ) ) ).

fof(fact_1090_div__self,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),A_1) = one_one(X_a) ) ) ).

fof(fact_1091_dvd__mult__div__cancel,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)) = ti(X_a,B) ) ) ).

fof(fact_1092_div__mult__swap,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,C_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C_1),B))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) ) ) ).

fof(fact_1093_dvd__div__mult__self,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)),A_1) = ti(X_a,B) ) ) ).

fof(fact_1094_dvd__div__mult,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)),A_1) ) ) ).

fof(fact_1095_div__mult__div__if__dvd,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [Z_1,W,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z_1),W))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),W)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z_1)) ) ) ) ).

fof(fact_1096_div__add,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [Y,Z_1,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z_1),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z_1),Y))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),Y),Z_1)) ) ) ) ).

fof(fact_1097_div__power,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [N,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N)) ) ) ).

fof(fact_1098_int__div__less__self,axiom,
    ! [K_4,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),K_4))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),K_4)),X)) ) ) ).

fof(fact_1099_div__prop1,axiom,
    ! [X,Y,Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Y),Z_1)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Z_1)),Y)) ) ) ).

fof(fact_1100_zdiv__zmult2__eq,axiom,
    ! [A_1,B,C_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),C_1) ) ).

fof(fact_1101_zdiv__mono1__neg,axiom,
    ! [B,A_1,A_6] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),A_6))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_6),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))) ) ) ).

fof(fact_1102_zdiv__mono1,axiom,
    ! [B,A_1,A_6] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),A_6))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_6),B))) ) ) ).

fof(fact_1103_div__neg__neg__trivial,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = zero_zero(int) ) ) ).

fof(fact_1104_zdiv__mono2__neg,axiom,
    ! [B,B_5,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_5),B))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B_5)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))) ) ) ) ).

fof(fact_1105_div__nonpos__pos__le0,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int))) ) ) ).

fof(fact_1106_neg__imp__zdiv__nonneg__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_3),zero_zero(int))) ) ) ).

fof(fact_1107_div__pos__pos__trivial,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = zero_zero(int) ) ) ).

fof(fact_1108_div__nonneg__neg__le0,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int))) ) ) ).

fof(fact_1109_zdiv__mono2,axiom,
    ! [B,B_5,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_5),B))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B_5))) ) ) ) ).

fof(fact_1110_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [B_2,A_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_3))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2)))
      <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) ) ) ) ).

fof(fact_1111_pos__imp__zdiv__pos__iff,axiom,
    ! [I,K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),I),K)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I)) ) ) ).

fof(fact_1112_pos__imp__zdiv__nonneg__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_3)) ) ) ).

fof(fact_1113_zdiv__eq__0__iff,axiom,
    ! [I,K] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_div(int),I),K) = zero_zero(int)
    <=> ( ti(int,K) = zero_zero(int)
        | ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K)) )
        | ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int)))
          & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I)) ) ) ) ).

fof(fact_1114_abs__add__one__not__less__self,axiom,
    ! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),X)),one_one(real))),X)) ).

fof(fact_1115_div__mult__self2,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,C_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)) ) ) ).

fof(fact_1116_div__mult__self1,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,C_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)) ) ) ).

fof(fact_1117_div__add__self2,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),one_one(X_a)) ) ) ).

fof(fact_1118_div__add__self1,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),one_one(X_a)) ) ) ).

fof(fact_1119_dvd__div__eq__mult,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [C,B_2,A_3] :
          ( ti(X_a,A_3) != zero_zero(X_a)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2))
           => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B_2),A_3) = ti(X_a,C)
            <=> ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3) ) ) ) ) ).

fof(fact_1120_dvd__div__div__eq__mult,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [D_1,B_2,C,A_3] :
          ( ti(X_a,A_3) != zero_zero(X_a)
         => ( ti(X_a,C) != zero_zero(X_a)
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),D_1))
               => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B_2),A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),D_1),C)
                <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),D_1) ) ) ) ) ) ) ).

fof(fact_1121_zdiv__leq__prop,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Y),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y))),X)) ) ).

fof(fact_1122_div__eq__minus1,axiom,
    ! [B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),min)),B) = hAPP(int,int,number_number_of(int),min) ) ).

fof(fact_1123_abs__add__one__gt__zero,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,abs_abs(real),X)))) ).

fof(fact_1124_div__prop2,axiom,
    ! [X,Y,Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Y),Z_1)),Z_1)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Z_1)),Y)) ) ) ).

fof(fact_1125_divmod__int__rel__div__eq,axiom,
    ! [A_9,B_8,Y,R_5] :
      ( ti(int,A_9) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_8),Y)),R_5)
     => ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_8))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_5))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_5),B_8)) ) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_8))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_8),R_5))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_5),zero_zero(int))) ) ) )
       => ( ti(int,B_8) != zero_zero(int)
         => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_9),B_8) = ti(int,Y) ) ) ) ).

fof(fact_1126_split__zdiv,axiom,
    ! [P_1,Na,K] :
      ( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Na),K)))
    <=> ( ( ti(int,K) = zero_zero(int)
         => hBOOL(hAPP(int,bool,P_1,zero_zero(int))) )
        & ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I_2,J_1] :
              ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1))
                & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K))
                & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
             => hBOOL(hAPP(int,bool,P_1,I_2)) ) )
        & ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I_2,J_1] :
              ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1))
                & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int)))
                & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
             => hBOOL(hAPP(int,bool,P_1,I_2)) ) ) ) ) ).

fof(fact_1127_div__pos__neg__trivial,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),zero_zero(int)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(int,int,number_number_of(int),min) ) ) ).

fof(fact_1128_zdiv__number__of__Bit1,axiom,
    ! [V,W] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),one_one(int))),hAPP(int,int,number_number_of(int),W)) ) ) ).

fof(fact_1129_less__one__imp__sqr__less__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),one_one(real)))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(real))) ) ).

fof(fact_1130_pos__zdiv__mult__2,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A_1) ) ).

fof(fact_1131_neg__zdiv__mult__2,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),one_one(int))),A_1) ) ).

fof(fact_1132_zfact__prop,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),A_1))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_5)) ) ) ) ) ).

fof(fact_1133_Euler_Oaux____2,axiom,
    ! [P_5] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_1134_Euler__part2,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_5)) ) ) ) ).

fof(fact_1135_Euler__Criterion,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_5)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_5)) ) ) ).

fof(fact_1136_Euler__part3,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),X))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int)),P_5)) ) ) ) ) ).

fof(fact_1137_zfact_Osimps,axiom,
    ! [N] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),zero_zero(int)))
       => hAPP(int,int,zfact,N) = one_one(int) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),zero_zero(int)))
       => hAPP(int,int,zfact,N) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N),one_one(int)))) ) ) ).

fof(fact_1138_abs__diff__less__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,A_3,Ra] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Xa),A_3))),Ra))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),Ra)),Xa))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),Ra))) ) ) ) ).

fof(fact_1139_rv,axiom,
    ( v = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),x),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))))
    & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),v))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) ) ).

fof(fact_1140_sw,axiom,
    ( w = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),y),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))))
    & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),w))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) ) ).

fof(fact_1141_inv__inv,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_5))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_5))
           => hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1)) = ti(int,A_1) ) ) ) ) ).

fof(fact_1142_MultInvPair__distinct,axiom,
    ! [J_2,A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_5))
           => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),A_1))
             => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),P_5)) ) ) ) ) ) ).

fof(fact_1143_int__le__induct,axiom,
    ! [P_1,I,K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K))
     => ( hBOOL(hAPP(int,bool,P_1,K))
       => ( ! [I_2] :
              ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K))
             => ( hBOOL(hAPP(int,bool,P_1,I_2))
               => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int)))) ) )
         => hBOOL(hAPP(int,bool,P_1,I)) ) ) ) ).

fof(fact_1144_m1,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),m1) ).

fof(fact_1145__0961_A_L_Aint_An_Advd_Av_A_094_A2_A_L_Aw_A_094_A2_096,axiom,
    hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).

fof(fact_1146__096_Ix_A_094_A2_A_L_Ay_A_094_A2_J_A_K_A_Iv_A_094_A2_A_L_Aw_A_094_A2_J,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),x),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),x),v)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),y),w))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),x),w)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),y),v))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_1147__096_I_Ir_A_K_A_I1_A_L_Aint_An_J_A_L_Av_J_A_K_Av_A_L_A_Is_A_K_A_I1_A_L,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),v)),v)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),w)),w))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),v)),w)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),w)),v))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),v))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),w)))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),w))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),v)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_1148__096_I_I1_A_L_Aint_An_J_A_K_A_Ir_A_K_Av_J_A_L_A_I1_A_L_Aint_An_J_A_K_A,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),v))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),w)))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),w))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),v)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),v))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),w)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),m1))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),w))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),v)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_1149__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),x),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_1150__096_Ix_A_K_Av_A_L_Ay_A_K_Aw_J_A_094_A2_A_L_A_Ix_A_K_Aw_A_N_Ay_A_K_Av_,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),x),v)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),y),w))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),x),w)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),y),v))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),v)),v)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),w)),w))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),v)),w)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),w)),v))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_1151__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),v))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),w)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),m1))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),w))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),v)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_1152__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),v)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),w))),m1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),w)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),v))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_1153__096_I1_A_L_Aint_An_J_A_094_A2_A_K_A_I_I4_A_K_Am_A_L_A1_J_A_K_Am1_J_A_,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),m1)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),v)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),w))),m1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),w)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),v))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_1154_calculation,axiom,
    hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),m1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),v)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),w))),m1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),r),w)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),sa),v))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = zero_zero(int) ).

fof(fact_1155__096_B_Bthesis_O_A_I_B_Bm1_O_Av_A_094_A2_A_L_Aw_A_094_A2_A_061_A_I1_A_,axiom,
    ~ ! [M1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),M1) ).

fof(fact_1156_div__le__mono,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),K_4))) ) ).

fof(fact_1157_div__le__dividend,axiom,
    ! [M,N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),M)) ).

fof(fact_1158_div__mult2__eq,axiom,
    ! [A_1,B,C_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B)),C_1) ).

fof(fact_1159_div__less,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = zero_zero(nat) ) ).

fof(fact_1160_nat__mult__div__cancel__disj,axiom,
    ! [M,N,K_4] :
      ( ( K_4 = zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) = zero_zero(nat) )
      & ( K_4 != zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) ) ) ).

fof(fact_1161_zdiv__int,axiom,
    ! [A_1,B] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,semiring_1_of_nat(int),A_1)),hAPP(nat,int,semiring_1_of_nat(int),B)) ).

fof(fact_1162_Divides_Otransfer__int__nat__functions_I1_J,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y)) ).

fof(fact_1163_aux______3,axiom,
    ! [J_2,K_4,A_1,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),A_1),P_5))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2)),J_2)),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2)),A_1)),P_5)) ) ).

fof(fact_1164_aux______1,axiom,
    ! [J_2,A_1,P_5,K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),P_5))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),K_4)),P_5)) ) ).

fof(fact_1165_div__le__mono2,axiom,
    ! [K_4,N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),K_4),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),K_4),M))) ) ) ).

fof(fact_1166_div__mult__self1__is__m,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),M)),N) = M ) ).

fof(fact_1167_div__mult__self__is__m,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),N) = M ) ).

fof(fact_1168_nat__mult__div__cancel1,axiom,
    ! [M,N,K_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) ) ).

fof(fact_1169_div__less__dividend,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),M)) ) ) ).

fof(fact_1170_split__div,axiom,
    ! [P_1,Na,K] :
      ( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Na),K)))
    <=> ( ( K = zero_zero(nat)
         => hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
        & ( K != zero_zero(nat)
         => ! [I_2,J_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K))
             => ( Na = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),I_2)),J_1)
               => hBOOL(hAPP(nat,bool,P_1,I_2)) ) ) ) ) ) ).

fof(fact_1171_Divides_Otransfer__nat__int__functions_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)) ) ) ).

fof(fact_1172_nat__div__distrib,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) ) ).

fof(fact_1173_add__self__div__2,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),M)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = M ).

fof(fact_1174_div__2__gt__zero,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_1175_inv__less__p__minus__1,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))) ) ) ) ).

fof(fact_1176_inv__g__1,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1))) ) ) ) ).

fof(fact_1177_inv__not__p__minus__1,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))
         => hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)) ) ) ) ).

fof(fact_1178_inv__not__1,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))
         => hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) != one_one(int) ) ) ) ).

fof(fact_1179_inv__distinct,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))
         => ti(int,A_1) != hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) ) ) ) ).

fof(fact_1180_MultInv__prop1,axiom,
    ! [X,Y,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),P_5))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),Y)),P_5)) ) ) ).

fof(fact_1181_Suc__n__div__2__gt__zero,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_1182_inv__not__0,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))
         => hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) != zero_zero(int) ) ) ) ).

fof(fact_1183_MultInv__zcong__prop1,axiom,
    ! [A_1,J_2,K_4,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),K_4),P_5))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),P_5)) ) ) ).

fof(fact_1184_MultInv__def,axiom,
    ! [P_5,X] : hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_1185_MultInv__prop5,axiom,
    ! [Y,X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y),zero_zero(int)),P_5))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),Y)),P_5))
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),P_5)) ) ) ) ) ) ).

fof(fact_1186_MultInv__prop4,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),X),P_5)) ) ) ) ).

fof(fact_1187_MultInv__prop3,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)),zero_zero(int)),P_5)) ) ) ) ).

fof(fact_1188_inv__is__inv,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1))),one_one(int)),P_5)) ) ) ) ).

fof(fact_1189_aux______4,axiom,
    ! [K_4,A_1,J_2,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_5))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2)),J_2)),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2)),A_1)),P_5))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),P_5)) ) ) ) ) ).

fof(fact_1190_aux______2,axiom,
    ! [J_2,A_1,K_4,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_5))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),K_4)),P_5))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),A_1),P_5)) ) ) ) ) ).

fof(fact_1191_MultInv__zcong__prop2,axiom,
    ! [A_1,J_2,K_4,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_5))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_5))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),P_5))
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),P_5)) ) ) ) ) ) ).

fof(fact_1192_MultInv__zcong__prop3,axiom,
    ! [J_2,K_4,A_1,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_5))
           => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_5))
             => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),P_5))
               => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),K_4),P_5)) ) ) ) ) ) ) ).

fof(fact_1193_Int2_Oaux____2,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)))),X),P_5)) ) ) ) ).

fof(fact_1194_Int2_Oaux____1,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)))),P_5)) ) ) ) ).

fof(fact_1195_MultInv__prop2a,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)),X)),one_one(int)),P_5)) ) ) ) ).

fof(fact_1196_MultInv__prop2,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5))
     => ( hBOOL(hAPP(int,bool,zprime,P_5))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),one_one(int)),P_5)) ) ) ) ).

fof(fact_1197_sin__bound__lemma,axiom,
    ! [U_1,V,X,Y] :
      ( X = Y
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),U_1)),V))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),U_1)),Y))),V)) ) ) ).

fof(fact_1198_wset__mem__inv__mem,axiom,
    ! [B_2,A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_4))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
         => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))
           => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),B_2)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) ) ) ) ) ).

fof(fact_1199_wset__inv__mem__mem,axiom,
    ! [B_2,A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_4))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
             => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),B_2)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))
               => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) ) ) ) ) ) ) ).

fof(fact_1200_rabs__ratiotest__lemma,axiom,
    ! [X,Y,C_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C_1),zero_zero(real)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C_1),hAPP(real,real,abs_abs(real),Y))))
       => X = zero_zero(real) ) ) ).

fof(fact_1201_int__less__induct,axiom,
    ! [P_1,I,K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K))
     => ( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K),one_one(int))))
       => ( ! [I_2] :
              ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),K))
             => ( hBOOL(hAPP(int,bool,P_1,I_2))
               => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int)))) ) )
         => hBOOL(hAPP(int,bool,P_1,I)) ) ) ) ).

fof(fact_1202_d22set__induct__old,axiom,
    ! [Xa,P_1] :
      ( ! [A_4] :
          ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_4))
           => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_4),one_one(int)))) )
         => hBOOL(hAPP(int,bool,P_1,A_4)) )
     => hBOOL(hAPP(int,bool,P_1,Xa)) ) ).

fof(fact_1203_minf_I10_J,axiom,
    ! [X_a] :
      ( ( plus(X_a)
        & linorder(X_a)
        & dvd(X_a) )
     => ! [D_1,Sb] :
        ? [Z_2] :
        ! [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_2))
         => ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb)))
          <=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb))) ) ) ) ).

fof(fact_1204_wset__mem__mem,axiom,
    ! [P_4,A_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3))
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) ) ).

fof(fact_1205_wset__subset,axiom,
    ! [B_2,P_4,A_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_3),one_one(int))),P_4)))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) ) ) ).

fof(fact_1206_wset__g__1,axiom,
    ! [B_2,A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) ) ) ) ).

fof(fact_1207_wset__less,axiom,
    ! [B_2,A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) ) ) ) ).

fof(fact_1208_wset__mem__imp__or,axiom,
    ! [B_2,P_4,A_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3))
     => ( ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_3),one_one(int))),P_4)))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))
         => ( ti(int,B_2) = ti(int,A_3)
            | ti(int,B_2) = hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_3) ) ) ) ) ).

fof(fact_1209_le__Suc__ex__iff,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),L))
    <=> ? [N_1] : L = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N_1) ) ).

fof(fact_1210_wset__mem,axiom,
    ! [B_2,A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3))
           => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) ) ) ) ) ).

fof(fact_1211_eq__diff__eq_H,axiom,
    ! [Xa,Ya,Z] :
      ( Xa = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Ya),Z)
    <=> Ya = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Z) ) ).

fof(fact_1212_d22set__eq__wset,axiom,
    ! [P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => hAPP(int,fun(int,bool),d22set,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_4) ) ).

fof(fact_1213_pinf_I9_J,axiom,
    ! [X_a] :
      ( ( plus(X_a)
        & linorder(X_a)
        & dvd(X_a) )
     => ! [D_1,Sb] :
        ? [Z_2] :
        ! [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb))) ) ) ) ).

fof(fact_1214_minf_I9_J,axiom,
    ! [X_a] :
      ( ( plus(X_a)
        & linorder(X_a)
        & dvd(X_a) )
     => ! [D_1,Sb] :
        ? [Z_2] :
        ! [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_2))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb))) ) ) ) ).

fof(fact_1215_pinf_I10_J,axiom,
    ! [X_a] :
      ( ( plus(X_a)
        & linorder(X_a)
        & dvd(X_a) )
     => ! [D_1,Sb] :
        ? [Z_2] :
        ! [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_1))
         => ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb)))
          <=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb))) ) ) ) ).

fof(fact_1216_phi__prime,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => hAPP(int,nat,phi,P_5) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))) ) ).

fof(fact_1217_Euler_Oaux__2,axiom,
    ! [P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),P_4),zOdd))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_1218_int__nat__abs,axiom,
    ! [X] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))) = hAPP(int,int,abs_abs(int),X) ).

fof(fact_1219_EvenOdd_Oodd__times__odd,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zOdd))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),zOdd)) ) ) ).

fof(fact_1220_odd__mult__odd__prop,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),zOdd))
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) ) ).

fof(fact_1221_d22set__le__swap,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2))
     => ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),d22set,A_3))) ) ).

fof(fact_1222_d22set__le,axiom,
    ! [B_2,A_3] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),d22set,A_3)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) ) ).

fof(fact_1223_d22set__g__1,axiom,
    ! [B_2,A_3] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),d22set,A_3)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) ) ).

fof(fact_1224_power__preserves__odd,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Xa),Na)),zOdd))
      <=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) ) ) ).

fof(fact_1225_d22set__mem,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),d22set,A_3))) ) ) ).

fof(fact_1226_zdvd__self__abs1,axiom,
    ! [D_3] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),hAPP(int,int,abs_abs(int),D_3))) ).

fof(fact_1227_zdvd__self__abs2,axiom,
    ! [D_3] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,abs_abs(int),D_3)),D_3)) ).

fof(fact_1228_zprime__zOdd__eq__grt__2,axiom,
    ! [P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),P_4),zOdd))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) ) ) ).

fof(fact_1229_EvenOdd_Oneg__one__odd__power,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
       => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Xa)) = hAPP(int,int,number_number_of(int),min) ) ) ).

fof(fact_1230_zOddI,axiom,
    ! [Xa,K] :
      ( ti(int,Xa) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K)),one_one(int))
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) ) ).

fof(fact_1231_divides__le,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        | N = zero_zero(nat) ) ) ).

fof(fact_1232_best__odd__division__abs,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Xa))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
       => ? [N_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ya),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),Xa))))),Xa)) ) ) ).

fof(fact_1233_zOddE,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
     => ~ ! [K_1] : ti(int,Xa) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K_1)),one_one(int)) ) ).

fof(fact_1234_pow__divides__eq__int,axiom,
    ! [A_3,B_2,Na] :
      ( Na != zero_zero(nat)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),Na)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B_2),Na)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),B_2)) ) ) ).

fof(fact_1235_pow__divides__pow__int,axiom,
    ! [A_1,N,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N)))
     => ( N != zero_zero(nat)
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B)) ) ) ).

fof(fact_1236_pow__divides__eq__nat,axiom,
    ! [A_3,B_2,Na] :
      ( Na != zero_zero(nat)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_3),Na)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B_2),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),B_2)) ) ) ).

fof(fact_1237_dvd__pos__nat,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) ) ) ).

fof(fact_1238_mult_Oprod__diff__prod,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X,Y,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))) ) ).

fof(fact_1239_mult_Ozero__left,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),B) = zero_zero(X_a) ) ).

fof(fact_1240_mult__left_Ozero,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),Y) = zero_zero(X_a) ) ).

fof(fact_1241_mult_Ozero__right,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_1242_mult__right_Ozero,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_1243_mult_Oadd__right,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [A_1,B,B_5] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),B_5)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B_5)) ) ).

fof(fact_1244_mult__right_Oadd,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa_2,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),Y)) ) ).

fof(fact_1245_mult_Oadd__left,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [A_1,A_6,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),A_6)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_6),B)) ) ).

fof(fact_1246_mult__left_Oadd,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X,Y,Ya_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Ya_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Ya_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Ya_1)) ) ).

fof(fact_1247_mult__left_Odiff,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X,Y,Ya_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),Ya_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Ya_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Ya_1)) ) ).

fof(fact_1248_mult_Odiff__left,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [A_1,A_6,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),A_6)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_6),B)) ) ).

fof(fact_1249_mult__right_Odiff,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa_2,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),Y)) ) ).

fof(fact_1250_mult_Odiff__right,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [A_1,B,B_5] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),B_5)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B_5)) ) ).

fof(fact_1251_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
    ! [A_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1))
     => A_1 = zero_zero(nat) ) ).

fof(fact_1252_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_3))
    <=> A_3 = zero_zero(nat) ) ).

fof(fact_1253_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
    ! [A_3] :
      ( A_3 != zero_zero(nat)
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),zero_zero(nat)))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_3)) ) ) ).

fof(fact_1254_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
    ! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat))) ).

fof(fact_1255_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
    ! [A_1] :
      ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat))) ) ).

fof(fact_1256_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
    ! [A_1] :
      ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat)))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1)) ) ).

fof(fact_1257_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
    ! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1)) ).

fof(fact_1258_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
    ! [A_3] :
      ( A_3 != one_one(nat)
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_3))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat))) ) ) ).

fof(fact_1259_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat)))
    <=> A_3 = one_one(nat) ) ).

fof(fact_1260_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
    ! [A_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat)))
     => A_1 = one_one(nat) ) ).

fof(fact_1261_SR__def,axiom,
    ! [P_4] : hAPP(int,fun(int,bool),sr,P_4) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),P_4))) ).

fof(fact_1262_zmod__number__of__Bit1,axiom,
    ! [V,W] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)))),one_one(int)) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),one_one(int))),hAPP(int,int,number_number_of(int),W)))),one_one(int)) ) ) ).

fof(fact_1263_neg__zmod__mult__2,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),one_one(int))),A_1))),one_one(int)) ) ).

fof(fact_1264_transfer__morphism__nat__int,axiom,
    hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,nat),fun(fun(int,bool),bool),nat_tr1645093318rphism(int,nat),nat_1),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))) ).

fof(fact_1265_StandardRes__prop4,axiom,
    ! [X,Y,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X)),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),Y))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),M)) ) ).

fof(fact_1266_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y)) ) ) ).

fof(fact_1267_inv__inv__aux,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_5))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))))) ) ).

fof(fact_1268_transfer__morphismI,axiom,
    ! [X_a,X_b,F,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),bool),nat_tr1645093318rphism(X_a,X_b),F),A_2)) ).

fof(fact_1269_Suc__mono,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N))) ) ).

fof(fact_1270_lessI,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,N))) ).

fof(fact_1271_zero__less__Suc,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,suc,N))) ).

fof(fact_1272_dvd__1__left,axiom,
    ! [K_4] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_4)) ).

fof(fact_1273_StandardRes__SR__prop,axiom,
    ! [Xa,P_4] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sr,P_4)))
     => hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa) = ti(int,Xa) ) ).

fof(fact_1274_StandardRes__def,axiom,
    ! [M,X] : hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M) ).

fof(fact_1275_Suc__inject,axiom,
    ! [X,Y] :
      ( hAPP(nat,nat,suc,X) = hAPP(nat,nat,suc,Y)
     => X = Y ) ).

fof(fact_1276_nat_Oinject,axiom,
    ! [Nat,Nat_5] :
      ( hAPP(nat,nat,suc,Nat) = hAPP(nat,nat,suc,Nat_5)
    <=> Nat = Nat_5 ) ).

fof(fact_1277_Suc__n__not__n,axiom,
    ! [N] : hAPP(nat,nat,suc,N) != N ).

fof(fact_1278_n__not__Suc__n,axiom,
    ! [N] : N != hAPP(nat,nat,suc,N) ).

fof(fact_1279_mod__mod__trivial,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).

fof(fact_1280_mod__self,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),A_1) = zero_zero(X_a) ) ).

fof(fact_1281_mod__by__0,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).

fof(fact_1282_mod__0,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).

fof(fact_1283_mod__mult__right__eq,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) ) ).

fof(fact_1284_mod__mult__left__eq,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1) ) ).

fof(fact_1285_mod__mult__eq,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) ) ).

fof(fact_1286_mod__mult__mult1,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [C_1,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) ) ).

fof(fact_1287_mod__mult__mult2,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,C_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),C_1) ) ).

fof(fact_1288_zmod__simps_I4_J,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,C_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) ) ).

fof(fact_1289_mod__mult__cong,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [B,B_5,A_1,C_1,A_6] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_6),C_1)
         => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_5),C_1)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_6),B_5)),C_1) ) ) ) ).

fof(fact_1290_mod__add__cong,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [B,B_5,A_1,C_1,A_6] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_6),C_1)
         => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_5),C_1)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_6),B_5)),C_1) ) ) ) ).

fof(fact_1291_zmod__simps_I1_J,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,C_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) ) ).

fof(fact_1292_zmod__simps_I2_J,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) ) ).

fof(fact_1293_mod__add__eq,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) ) ).

fof(fact_1294_mod__add__left__eq,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1) ) ).

fof(fact_1295_mod__add__right__eq,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) ) ).

fof(fact_1296_mod__add__self1,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [B,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).

fof(fact_1297_mod__add__self2,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).

fof(fact_1298_mod__diff__cong,axiom,
    ! [X_a] :
      ( ring_div(X_a)
     => ! [B,B_5,A_1,C_1,A_6] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_6),C_1)
         => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_5),C_1)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_6),B_5)),C_1) ) ) ) ).

fof(fact_1299_mod__diff__eq,axiom,
    ! [X_a] :
      ( ring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) ) ).

fof(fact_1300_mod__diff__left__eq,axiom,
    ! [X_a] :
      ( ring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1) ) ).

fof(fact_1301_mod__diff__right__eq,axiom,
    ! [X_a] :
      ( ring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) ) ).

fof(fact_1302_dvd__mod__imp__dvd,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [K_4,M,N] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),M),N)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),N))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),M)) ) ) ) ).

fof(fact_1303_dvd__mod,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [N,K_4,M] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),M))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),N))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),M),N))) ) ) ) ).

fof(fact_1304_mod__mod__cancel,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,C_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C_1),B))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1) ) ) ).

fof(fact_1305_dvd__mod__iff,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [Ma,K,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),Na))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),Ma),Na)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),Ma)) ) ) ) ).

fof(fact_1306_zmod__self,axiom,
    ! [A_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),A_1) = zero_zero(int) ).

fof(fact_1307_zmod__zero,axiom,
    ! [B] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),zero_zero(int)),B) = zero_zero(int) ).

fof(fact_1308_zmod__zmult1__eq,axiom,
    ! [A_1,B,C_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C_1))),C_1) ).

fof(fact_1309_zmod__simps_I3_J,axiom,
    ! [A_1,B,C_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C_1))),C_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1) ).

fof(fact_1310_zdiff__zmod__left,axiom,
    ! [X,M,Y] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M)),Y)),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y)),M) ).

fof(fact_1311_zdiff__zmod__right,axiom,
    ! [X,Y,M] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Y),M))),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y)),M) ).

fof(fact_1312_Zero__not__Suc,axiom,
    ! [M] : zero_zero(nat) != hAPP(nat,nat,suc,M) ).

fof(fact_1313_nat_Osimps_I2_J,axiom,
    ! [Nat_4] : zero_zero(nat) != hAPP(nat,nat,suc,Nat_4) ).

fof(fact_1314_Suc__not__Zero,axiom,
    ! [M] : hAPP(nat,nat,suc,M) != zero_zero(nat) ).

fof(fact_1315_nat_Osimps_I3_J,axiom,
    ! [Nat_3] : hAPP(nat,nat,suc,Nat_3) != zero_zero(nat) ).

fof(fact_1316_Zero__neq__Suc,axiom,
    ! [M] : zero_zero(nat) != hAPP(nat,nat,suc,M) ).

fof(fact_1317_Suc__neq__Zero,axiom,
    ! [M] : hAPP(nat,nat,suc,M) != zero_zero(nat) ).

fof(fact_1318_zpower__zmod,axiom,
    ! [X,M,Y] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M)),Y)),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),M) ).

fof(fact_1319_Residues_Oaux,axiom,
    ! [X,M,Y] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Y),M)
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),M)) ) ).

fof(fact_1320_mod__mod__is__mod,axiom,
    ! [X,M] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M)),M)) ).

fof(fact_1321_zcong__zmod,axiom,
    ! [A_3,B_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_3),Ma)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B_2),Ma)),Ma)) ) ).

fof(fact_1322_zdvd__zmod__imp__zdvd,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M)) ) ) ).

fof(fact_1323_zdvd__zmod,axiom,
    ! [N,F_4,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),N))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N))) ) ) ).

fof(fact_1324_Suc__less__SucD,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N)))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

fof(fact_1325_Suc__lessD,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

fof(fact_1326_less__SucE,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N)))
     => ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
       => M = N ) ) ).

fof(fact_1327_less__trans__Suc,axiom,
    ! [K_4,I_1,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_2),K_4))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I_1)),K_4)) ) ) ).

fof(fact_1328_Suc__lessI,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( hAPP(nat,nat,suc,M) != N
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),N)) ) ) ).

fof(fact_1329_less__SucI,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N))) ) ).

fof(fact_1330_less__antisym,axiom,
    ! [N,M] :
      ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M)))
       => M = N ) ) ).

fof(fact_1331_not__less__less__Suc__eq,axiom,
    ! [Na,Ma] :
      ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(nat,nat,suc,Ma)))
      <=> Na = Ma ) ) ).

fof(fact_1332_Suc__less__eq,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Ma)),hAPP(nat,nat,suc,Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).

fof(fact_1333_less__Suc__eq,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,suc,Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
        | Ma = Na ) ) ).

fof(fact_1334_not__less__eq,axiom,
    ! [Ma,Na] :
      ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(nat,nat,suc,Ma))) ) ).

fof(fact_1335_transfer__morphism__def,axiom,
    ! [X_a,X_b,F,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),bool),nat_tr1645093318rphism(X_a,X_b),F),A_2)) ).

fof(fact_1336_add__Suc__shift,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,suc,N)) ).

fof(fact_1337_add__Suc,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

fof(fact_1338_add__Suc__right,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

fof(fact_1339_Suc__n__not__le__n,axiom,
    ! [N] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),N)) ).

fof(fact_1340_not__less__eq__eq,axiom,
    ! [Ma,Na] :
      ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Na)),Ma)) ) ).

fof(fact_1341_le__Suc__eq,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),hAPP(nat,nat,suc,Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
        | Ma = hAPP(nat,nat,suc,Na) ) ) ).

fof(fact_1342_Suc__le__mono,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Na)),hAPP(nat,nat,suc,Ma)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),Ma)) ) ).

fof(fact_1343_le__SucI,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,suc,N))) ) ).

fof(fact_1344_le__SucE,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,suc,N)))
     => ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => M = hAPP(nat,nat,suc,N) ) ) ).

fof(fact_1345_Suc__leD,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

fof(fact_1346_Suc__mult__cancel1,axiom,
    ! [K,Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Na)
    <=> Ma = Na ) ).

fof(fact_1347_diff__Suc__Suc,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) ).

fof(fact_1348_Suc__diff__diff,axiom,
    ! [M,N,K_4] : 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)),N)),hAPP(nat,nat,suc,K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),K_4) ).

fof(fact_1349_exp__mono__eq,axiom,
    ! [Xa,Na,Ya] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ya),hAPP(nat,nat,suc,Na))
    <=> Xa = Ya ) ).

fof(fact_1350_StandardRes__prop1,axiom,
    ! [X,M] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X)),M)) ).

fof(fact_1351_mod__mult__self2__is__0,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),B) = zero_zero(X_a) ) ).

fof(fact_1352_mod__mult__self1__is__0,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [B,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),B) = zero_zero(X_a) ) ).

fof(fact_1353_mod__by__1,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),one_one(X_a)) = zero_zero(X_a) ) ).

fof(fact_1354_mod__mult__self2,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).

fof(fact_1355_mod__mult__self1,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,C_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).

fof(fact_1356_dvd__imp__mod__0,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),A_1) = zero_zero(X_a) ) ) ).

fof(fact_1357_dvd__eq__mod__eq__0,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2))
        <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_2),A_3) = zero_zero(X_a) ) ) ).

fof(fact_1358_mod__div__trivial,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),B) = zero_zero(X_a) ) ).

fof(fact_1359_neg__mod__bound,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))) ) ).

fof(fact_1360_pos__mod__bound,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B)) ) ).

fof(fact_1361_zmod__eq__0__iff,axiom,
    ! [Ma,D_1] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Ma),D_1) = zero_zero(int)
    <=> ? [Q_4] : ti(int,Ma) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),Q_4) ) ).

fof(fact_1362_zmod__le__nonneg__dividend,axiom,
    ! [K_4,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),K_4)),M)) ) ).

fof(fact_1363_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y))) ) ) ).

fof(fact_1364_power__0__Suc,axiom,
    ! [X_a] :
      ( ( power(X_a)
        & semiring_0(X_a) )
     => ! [N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(nat,nat,suc,N)) = zero_zero(X_a) ) ).

fof(fact_1365_zmod__zdiv__trivial,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B) = zero_zero(int) ).

fof(fact_1366_DIVISION__BY__ZERO,axiom,
    ! [A_1] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),zero_zero(int)) = zero_zero(int)
      & hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),zero_zero(int)) = ti(int,A_1) ) ).

fof(fact_1367_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,Q_3] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3)) ) ).

fof(fact_1368_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,Q_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_3)) ) ).

fof(fact_1369_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,Q_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3)),X) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_3)) ) ).

fof(fact_1370_power__Suc2,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),A_1) ) ).

fof(fact_1371_power__Suc,axiom,
    ! [X_a] :
      ( power(X_a)
     => ! [A_1,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) ) ).

fof(fact_1372_zdiv__zadd1__eq,axiom,
    ! [A_1,B,C_1] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),C_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C_1))),C_1)) ).

fof(fact_1373_nat__lt__two__imp__zero__or__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))
     => ( X = zero_zero(nat)
        | X = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).

fof(fact_1374_less__Suc__eq__0__disj,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,suc,Na)))
    <=> ( Ma = zero_zero(nat)
        | ? [J_1] :
            ( Ma = hAPP(nat,nat,suc,J_1)
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),Na)) ) ) ) ).

fof(fact_1375_less__Suc0,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))
    <=> Na = zero_zero(nat) ) ).

fof(fact_1376_gr0__conv__Suc,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
    <=> ? [M_2] : Na = hAPP(nat,nat,suc,M_2) ) ).

fof(fact_1377_dvd__1__iff__1,axiom,
    ! [Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),hAPP(nat,nat,suc,zero_zero(nat))))
    <=> Ma = hAPP(nat,nat,suc,zero_zero(nat)) ) ).

fof(fact_1378_zmod__eq__dvd__iff,axiom,
    ! [Xa,Na,Ya] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),Na) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Ya),Na)
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Na),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya))) ) ).

fof(fact_1379_one__is__add,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na)
    <=> ( ( Ma = hAPP(nat,nat,suc,zero_zero(nat))
          & Na = zero_zero(nat) )
        | ( Ma = zero_zero(nat)
          & Na = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ) ).

fof(fact_1380_add__is__1,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na) = hAPP(nat,nat,suc,zero_zero(nat))
    <=> ( ( Ma = hAPP(nat,nat,suc,zero_zero(nat))
          & Na = zero_zero(nat) )
        | ( Ma = zero_zero(nat)
          & Na = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ) ).

fof(fact_1381_mult__eq__1__iff,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na) = hAPP(nat,nat,suc,zero_zero(nat))
    <=> ( Ma = hAPP(nat,nat,suc,zero_zero(nat))
        & Na = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).

fof(fact_1382_One__nat__def,axiom,
    one_one(nat) = hAPP(nat,nat,suc,zero_zero(nat)) ).

fof(fact_1383_less__iff__Suc__add,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
    <=> ? [K_1] : Na = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K_1)) ) ).

fof(fact_1384_less__add__Suc2,axiom,
    ! [I_1,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),I_1)))) ).

fof(fact_1385_less__add__Suc1,axiom,
    ! [I_1,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),M)))) ).

fof(fact_1386_Suc__le__lessD,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

fof(fact_1387_le__less__Suc__eq,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(nat,nat,suc,Ma)))
      <=> Na = Ma ) ) ).

fof(fact_1388_Suc__leI,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N)) ) ).

fof(fact_1389_le__imp__less__Suc,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N))) ) ).

fof(fact_1390_Suc__le__eq,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Ma)),Na))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).

fof(fact_1391_less__Suc__eq__le,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,suc,Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).

fof(fact_1392_less__eq__Suc__le,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Na)),Ma)) ) ).

fof(fact_1393_Suc__mult__less__cancel1,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) ) ).

fof(fact_1394_diff__less__Suc,axiom,
    ! [M,N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),hAPP(nat,nat,suc,M))) ).

fof(fact_1395_mult__Suc,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,M)),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),N)) ).

fof(fact_1396_mult__Suc__right,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) ).

fof(fact_1397_Suc__mult__le__cancel1,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) ) ).

fof(fact_1398_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_1399_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_1400_Suc__diff__le,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) ) ).

fof(fact_1401_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_1402_diff__Suc__eq__diff__pred,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),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),one_one(nat))),N) ).

fof(fact_1403_div__1,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),hAPP(nat,nat,suc,zero_zero(nat))) = M ).

fof(fact_1404_power__Suc__0,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,suc,zero_zero(nat))),N) = hAPP(nat,nat,suc,zero_zero(nat)) ).

fof(fact_1405_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa,Ma] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Ma) = hAPP(nat,nat,suc,zero_zero(nat))
    <=> ( Ma = zero_zero(nat)
        | Xa = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).

fof(fact_1406_exp__mono__lt,axiom,
    ! [Xa,Na,Ya] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),hAPP(nat,nat,suc,Na))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ya),hAPP(nat,nat,suc,Na))))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) ) ).

fof(fact_1407_divides__rexp,axiom,
    ! [N,X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),hAPP(nat,nat,suc,N)))) ) ).

fof(fact_1408_exp__mono__le,axiom,
    ! [Xa,Na,Ya] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),hAPP(nat,nat,suc,Na))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ya),hAPP(nat,nat,suc,Na))))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) ) ).

fof(fact_1409_dvd__eq__mod__eq__0__number__of,axiom,
    ! [X_a] :
      ( ( semiring_div(X_a)
        & number(X_a) )
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),hAPP(int,X_a,number_number_of(X_a),Ya)))
        <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(int,X_a,number_number_of(X_a),Ya)),hAPP(int,X_a,number_number_of(X_a),Xa)) = zero_zero(X_a) ) ) ).

fof(fact_1410_semiring__div__class_Omod__div__equality_H,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)) = ti(X_a,A_1) ) ).

fof(fact_1411_mod__div__equality2,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [B,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) = ti(X_a,A_1) ) ).

fof(fact_1412_mod__div__equality,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) = ti(X_a,A_1) ) ).

fof(fact_1413_div__mod__equality2,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [B,A_1,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1) ) ).

fof(fact_1414_div__mod__equality,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1) ) ).

fof(fact_1415_StandardRes__ubound,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_5))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_5),X)),P_5)) ) ).

fof(fact_1416_StandardRes__prop3,axiom,
    ! [Xa,P_4] :
      ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),P_4))
    <=> hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa) != zero_zero(int) ) ).

fof(fact_1417_StandardRes__eq__zcong,axiom,
    ! [Ma,Xa] :
      ( hAPP(int,int,hAPP(int,fun(int,int),standardRes,Ma),Xa) = zero_zero(int)
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),Ma)) ) ).

fof(fact_1418_pos__mod__sign,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))) ) ).

fof(fact_1419_pos__mod__conj,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B)) ) ) ).

fof(fact_1420_mod__pos__pos__trivial,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = ti(int,A_1) ) ) ).

fof(fact_1421_neg__mod__sign,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),zero_zero(int))) ) ).

fof(fact_1422_neg__mod__conj,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),zero_zero(int)))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))) ) ) ).

fof(fact_1423_mod__neg__neg__trivial,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = ti(int,A_1) ) ) ).

fof(fact_1424_power__inject__base,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1,N,B] :
          ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),hAPP(nat,nat,suc,N))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
             => ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ).

fof(fact_1425_power__le__imp__le__base,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [A_1,N,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),hAPP(nat,nat,suc,N))))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ) ).

fof(fact_1426_zcong__zmod__eq,axiom,
    ! [A_3,B_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))
      <=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_3),Ma) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B_2),Ma) ) ) ).

fof(fact_1427_zmod__zdvd__zmod,axiom,
    ! [A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),B))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),M) ) ) ).

fof(fact_1428_zmod__minus1__right,axiom,
    ! [A_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,number_number_of(int),min)) = zero_zero(int) ).

fof(fact_1429_zdvd__iff__zmod__eq__0__number__of,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,number_number_of(int),Xa)),hAPP(int,int,number_number_of(int),Ya)))
    <=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),Ya)),hAPP(int,int,number_number_of(int),Xa)) = zero_zero(int) ) ).

fof(fact_1430_power__gt1,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N)))) ) ) ).

fof(fact_1431_zmod__zdiv__equality,axiom,
    ! [A_1,B] : ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) ).

fof(fact_1432_zdiv__zmult1__eq,axiom,
    ! [A_1,B,C_1] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C_1))),C_1)) ).

fof(fact_1433_zdiv__zmod__equality,axiom,
    ! [B,A_1,K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))),K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),K_4) ).

fof(fact_1434_zdiv__zmod__equality2,axiom,
    ! [A_1,B,K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))),K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),K_4) ).

fof(fact_1435_zmult__div__cancel,axiom,
    ! [N,M] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M),N)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N)) ).

fof(fact_1436_zmod__zdiv__equality_H,axiom,
    ! [M,N] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M),N)),N)) ).

fof(fact_1437_of__nat__Suc,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [M] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,M)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)) ) ).

fof(fact_1438_one__less__mult,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

fof(fact_1439_n__less__n__mult__m,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),M))) ) ) ).

fof(fact_1440_n__less__m__mult__n,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

fof(fact_1441_mn__eq__m__one,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = M
       => N = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).

fof(fact_1442_prod__mn__less__k,axiom,
    ! [M,K_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M))
         => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = K_4
           => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),K_4)) ) ) ) ) ).

fof(fact_1443_one__less__k,axiom,
    ! [M,K_4] :
      ( M != hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_4)) ) ) ).

fof(fact_1444_one__less__m,axiom,
    ! [M,K_4] :
      ( M != hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)
     => ( M != hAPP(nat,nat,suc,zero_zero(nat))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M)) ) ) ).

fof(fact_1445_diff__Suc__less,axiom,
    ! [I_1,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_1446_Suc__pred,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_1447_one__le__mult__iff,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),Ma))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),Na)) ) ) ).

fof(fact_1448_int__Suc0__eq__1,axiom,
    hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,zero_zero(nat))) = one_one(int) ).

fof(fact_1449_nat__1,axiom,
    hAPP(int,nat,nat_1,one_one(int)) = hAPP(nat,nat,suc,zero_zero(nat)) ).

fof(fact_1450_int__Suc,axiom,
    ! [M] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,M)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),M)) ).

fof(fact_1451_zless__iff__Suc__zadd,axiom,
    ! [Wa,Z] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z))
    <=> ? [N_1] : ti(int,Z) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1))) ) ).

fof(fact_1452_diff__Suc__diff__eq2,axiom,
    ! [M,K_4,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
     => 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_2),K_4))),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,J_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),M)) ) ).

fof(fact_1453_diff__Suc__diff__eq1,axiom,
    ! [M,K_4,J_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),hAPP(nat,nat,suc,J_2)) ) ).

fof(fact_1454_nat__one__le__power,axiom,
    ! [N,I_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),I_1))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N))) ) ).

fof(fact_1455_mod__pos__neg__trivial,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),zero_zero(int)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B) ) ) ).

fof(fact_1456_realpow__Suc__le__self,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,R_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),R_3))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),R_3),one_one(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),R_3),hAPP(nat,nat,suc,N))),R_3)) ) ) ) ).

fof(fact_1457_power__Suc__less__one,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N))),one_one(X_a))) ) ) ) ).

fof(fact_1458_xzgcda__linear__aux2,axiom,
    ! [R_3,S_3,T_5,R_4,S_4,M,T_6,N] :
      ( ti(int,R_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_6),N))
     => ( ti(int,R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_3),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_5),N))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),R_4),R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),S_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_4),R_3)),S_3))),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),T_6),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_4),R_3)),T_5))),N)) ) ) ).

fof(fact_1459_zcong__zmod__aux,axiom,
    ! [A_1,B,M] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),M)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),M)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),M)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),M))) ).

fof(fact_1460_numeral__3__eq__3,axiom,
    hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) ).

fof(fact_1461_numeral__1__eq__Suc__0,axiom,
    hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,pls)) = hAPP(nat,nat,suc,zero_zero(nat)) ).

fof(fact_1462_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),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N) ).

fof(fact_1463_lemma__realpow__diff,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [Y,P_5,N] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_5),N))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,N)),P_5)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),P_5))),Y) ) ) ).

fof(fact_1464_Suc__diff__1,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_1465_Suc__pred_H,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_1466_add__eq__if,axiom,
    ! [N,M] :
      ( ( M = zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = N )
      & ( M != zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),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),one_one(nat))),N)) ) ) ).

fof(fact_1467_StandardRes__lbound,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_5))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_5),X))) ) ).

fof(fact_1468_div__if,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
         => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = zero_zero(nat) )
        & ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
         => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).

fof(fact_1469_div__geq,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ).

fof(fact_1470_StandardRes__prop2,axiom,
    ! [X1_2,X2_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => ( hAPP(int,int,hAPP(int,fun(int,int),standardRes,Ma),X1_2) = hAPP(int,int,hAPP(int,fun(int,int),standardRes,Ma),X2_2)
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X1_2),X2_2),Ma)) ) ) ).

fof(fact_1471_zmod__number__of__Bit0,axiom,
    ! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W))) ).

fof(fact_1472_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y))) ) ) ).

fof(fact_1473_split__zmod,axiom,
    ! [P_1,Na,K] :
      ( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Na),K)))
    <=> ( ( ti(int,K) = zero_zero(int)
         => hBOOL(hAPP(int,bool,P_1,Na)) )
        & ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I_2,J_1] :
              ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1))
                & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K))
                & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
             => hBOOL(hAPP(int,bool,P_1,J_1)) ) )
        & ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I_2,J_1] :
              ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1))
                & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int)))
                & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
             => hBOOL(hAPP(int,bool,P_1,J_1)) ) ) ) ) ).

fof(fact_1474_zmult2__lemma__aux3,axiom,
    ! [Q_3,B,R_3,C_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3))) ) ) ) ).

fof(fact_1475_zmult2__lemma__aux4,axiom,
    ! [Q_3,B,R_3,C_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1))) ) ) ) ).

fof(fact_1476_zmult2__lemma__aux1,axiom,
    ! [Q_3,B,R_3,C_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int)))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3))) ) ) ) ).

fof(fact_1477_zmult2__lemma__aux2,axiom,
    ! [Q_3,B,R_3,C_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int)))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3)),zero_zero(int))) ) ) ) ).

fof(fact_1478_divmod__int__rel__mod__eq,axiom,
    ! [A_9,B_8,Q_6,Y] :
      ( ti(int,A_9) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_8),Q_6)),Y)
     => ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_8))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),B_8)) ) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_8))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_8),Y))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int))) ) ) )
       => ( ti(int,B_8) != zero_zero(int)
         => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_9),B_8) = ti(int,Y) ) ) ) ).

fof(fact_1479_zmod__zmult2__eq,axiom,
    ! [A_1,B,C_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1))
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) ) ).

fof(fact_1480_tsub__eq,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_1481_realpow__two__diff,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) ) ).

fof(fact_1482_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X),Y)) ).

fof(fact_1483_numeral__2__eq__2,axiom,
    hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))) ).

fof(fact_1484_semiring__norm_I115_J,axiom,
    hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ).

fof(fact_1485_add__2__eq__Suc,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,N)) ).

fof(fact_1486_add__2__eq__Suc_H,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,N)) ).

fof(fact_1487_Suc__diff__eq__diff__pred,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),pls)),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,pls)))) ) ).

fof(fact_1488_one__less__nat__eq,axiom,
    ! [Z] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,nat_1,Z)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ).

fof(fact_1489_div2__Suc__Suc,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,M))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_1490_expand__Suc,axiom,
    ! [V] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),V)))
     => hAPP(int,nat,number_number_of(nat),V) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat))) ) ).

fof(fact_1491_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
     => hAPP(nat,nat,suc,hAPP(int,nat,nat_1,Z_1)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z_1)) ) ).

fof(fact_1492_Suc__div__eq__add3__div__number__of,axiom,
    ! [M,V] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M)))),hAPP(int,nat,number_number_of(nat),V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M)),hAPP(int,nat,number_number_of(nat),V)) ).

fof(fact_1493_Suc__div__eq__add3__div,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M)))),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M)),N) ).

fof(fact_1494_div__Suc__eq__div__add3,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,N)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N)) ).

fof(fact_1495_split__div__lemma,axiom,
    ! [Q_2,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Q_2)),Ma))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),hAPP(nat,nat,suc,Q_2)))) )
      <=> Q_2 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Ma),Na) ) ) ).

fof(fact_1496_split__div_H,axiom,
    ! [P_1,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Ma),Na)))
    <=> ( ( Na = zero_zero(nat)
          & hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
        | ? [Q_4] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Q_4)),Ma))
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),hAPP(nat,nat,suc,Q_4))))
            & hBOOL(hAPP(nat,bool,P_1,Q_4)) ) ) ) ).

fof(fact_1497_le__div__geq,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ).

fof(fact_1498_neq__one__mod__two,axiom,
    ! [Xa] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(int)
    <=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(int) ) ).

fof(fact_1499_split__pos__lemma,axiom,
    ! [P_1,Na,K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Na),K)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Na),K)))
      <=> ! [I_2,J_1] :
            ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K))
              & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,I_2),J_1)) ) ) ) ).

fof(fact_1500_split__neg__lemma,axiom,
    ! [P_1,Na,K] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Na),K)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Na),K)))
      <=> ! [I_2,J_1] :
            ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int)))
              & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,I_2),J_1)) ) ) ) ).

fof(fact_1501_zmod__minus1,axiom,
    ! [B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),min)),B) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),one_one(int)) ) ).

fof(fact_1502_less__2__cases,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
     => ( N = zero_zero(nat)
        | N = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).

fof(fact_1503_nat__2,axiom,
    hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))) ).

fof(fact_1504_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [X,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))) ) ).

fof(fact_1505_power__odd__eq,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [A_1,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ).

fof(fact_1506_tsub__def,axiom,
    ! [Y,X] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(int,fun(int,bool),ord_less_eq(int),Y),X))
       => hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y) = zero_zero(int) ) ) ).

fof(fact_1507_lemma__Suc__Suc__4n__diff__2,axiom,
    ! [N] :
      ( N != zero_zero(nat)
     => hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N) ) ).

fof(fact_1508_Suc__Suc__mult__two__diff__two,axiom,
    ! [N] :
      ( N != zero_zero(nat)
     => hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N) ) ).

fof(fact_1509_int__power__div__base,axiom,
    ! [K_4,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_4),M)),K_4) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,suc,zero_zero(nat)))) ) ) ).

fof(fact_1510_one__div__nat__number__of,axiom,
    ! [V_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),one_one(int)),hAPP(int,int,number_number_of(int),V_2))) ).

fof(fact_1511_odd__0__le__power__imp__0__le,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1,N] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) ) ) ).

fof(fact_1512_odd__power__less__zero,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [N,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))),zero_zero(X_a))) ) ) ).

fof(fact_1513_power__m1__odd,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) = hAPP(int,X_a,number_number_of(X_a),min) ) ).

fof(fact_1514_Suc__mult__two__diff__one,axiom,
    ! [N] :
      ( N != zero_zero(nat)
     => hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)),one_one(nat))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N) ) ).

fof(fact_1515_WilsonRuss_Oinv__def,axiom,
    ! [P_5,A_1] : hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_5) ).

fof(fact_1516_inv__is__inv__aux,axiom,
    ! [M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
     => hAPP(nat,nat,suc,hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),one_one(int))) ) ).

fof(fact_1517_pos__zmod__mult__2,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),A_1))) ) ).

fof(fact_1518_SRStar__mult__prop2,axiom,
    ! [Xa,A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4))
         => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4)))
           => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),Xa)))),hAPP(int,fun(int,bool),sRStar,P_4))) ) ) ) ) ).

fof(fact_1519_realpow__pos__nth2,axiom,
    ! [N,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ? [R_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
          & hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_1),hAPP(nat,nat,suc,N)) = A_1 ) ) ).

fof(fact_1520_mod__div__decomp,axiom,
    ! [X_a] :
      ( semiring_div(X_a)
     => ! [A_1,B] : ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) ) ).

fof(fact_1521_zmod__eq__0D,axiom,
    ! [M_5,D_4] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_5),D_4) = zero_zero(int)
     => ? [Q_4] : ti(int,M_5) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_4),Q_4) ) ).

fof(fact_1522_dvd_Olift__Suc__mono__le,axiom,
    ! [Na,N_6,F] :
      ( ! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_1)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1))))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),N_6))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,Na)),hAPP(nat,nat,F,N_6))) ) ) ).

fof(fact_1523_inc__induct,axiom,
    ! [P_1,I,J] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( hBOOL(hAPP(nat,bool,P_1,J))
       => ( ! [I_2] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J))
             => ( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,I_2)))
               => hBOOL(hAPP(nat,bool,P_1,I_2)) ) )
         => hBOOL(hAPP(nat,bool,P_1,I)) ) ) ) ).

fof(fact_1524_less__imp__Suc__add,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => ? [K_1] : N = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_1)) ) ).

fof(fact_1525_mod__Suc__eq__Suc__mod,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))),N) ).

fof(fact_1526_mod__less,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) = M ) ).

fof(fact_1527_mod__less__eq__dividend,axiom,
    ! [M,N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)),M)) ).

fof(fact_1528_mod__mult__distrib2,axiom,
    ! [K_4,M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) ).

fof(fact_1529_mod__mult__distrib,axiom,
    ! [M,N,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K_4)) ).

fof(fact_1530_StandardRes__SRStar__prop3,axiom,
    ! [Xa,P_4] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4)))
     => hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa) = ti(int,Xa) ) ).

fof(fact_1531_mod__1,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

fof(fact_1532_mod__Suc,axiom,
    ! [M,N] :
      ( ( hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) = N
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M)),N) = zero_zero(nat) )
      & ( hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) != N
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) ) ) ).

fof(fact_1533_mod__less__divisor,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)),N)) ) ).

fof(fact_1534_mod__eq__0__iff,axiom,
    ! [Ma,D_1] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Ma),D_1) = zero_zero(nat)
    <=> ? [Q_4] : Ma = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),D_1),Q_4) ) ).

fof(fact_1535_mod__geq,axiom,
    ! [M,N] :
      ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N) ) ).

fof(fact_1536_mod__if,axiom,
    ! [M,N] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) = M )
      & ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N) ) ) ).

fof(fact_1537_mod__mult__self3,axiom,
    ! [K_4,N,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)),M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) ).

fof(fact_1538_le__mod__geq,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N) ) ).

fof(fact_1539_div__add1__eq,axiom,
    ! [A_1,B,C_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B)),C_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C_1))),C_1)) ).

fof(fact_1540_Divides_Otransfer__int__nat__functions_I2_J,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ).

fof(fact_1541_zmod__int,axiom,
    ! [A_1,B] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,semiring_1_of_nat(int),A_1)),hAPP(nat,int,semiring_1_of_nat(int),B)) ).

fof(fact_1542_SRStar__SR__prop,axiom,
    ! [Xa,P_4] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4)))
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sr,P_4))) ) ).

fof(fact_1543_mod__le__divisor,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)),N)) ) ).

fof(fact_1544_mod__mult__self4,axiom,
    ! [K_4,N,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)),M))),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M)),N) ).

fof(fact_1545_SRStar__def,axiom,
    ! [P_4] : hAPP(int,fun(int,bool),sRStar,P_4) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),P_4))) ).

fof(fact_1546_mod__mult2__eq,axiom,
    ! [A_1,B,C_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B)),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),B)) ).

fof(fact_1547_div__mult1__eq,axiom,
    ! [A_1,B,C_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)),C_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C_1))),C_1)) ).

fof(fact_1548_div__mod__equality_H,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) ).

fof(fact_1549_mult__div__cancel,axiom,
    ! [N,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) ).

fof(fact_1550_Divides_Omod__div__equality_H,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),N)) ).

fof(fact_1551_StandardRes__SRStar__prop1a,axiom,
    ! [Xa,P_4] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4)))
     => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),P_4)) ) ).

fof(fact_1552_split__mod,axiom,
    ! [P_1,Na,K] :
      ( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Na),K)))
    <=> ( ( K = zero_zero(nat)
         => hBOOL(hAPP(nat,bool,P_1,Na)) )
        & ( K != zero_zero(nat)
         => ! [I_2,J_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K))
             => ( Na = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),I_2)),J_1)
               => hBOOL(hAPP(nat,bool,P_1,J_1)) ) ) ) ) ) ).

fof(fact_1553_mod__lemma,axiom,
    ! [Q_3,R_3,B,C_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C_1))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_3),B))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Q_3),C_1))),R_3)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1))) ) ) ).

fof(fact_1554_Suc__times__mod__eq,axiom,
    ! [M,K_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),K_4))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M))),K_4) = one_one(nat) ) ).

fof(fact_1555_div__Suc,axiom,
    ! [A_1,C_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,A_1)),C_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,zero_zero(nat))),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),C_1))),C_1)) ).

fof(fact_1556_nat__mod__distrib,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) ) ) ).

fof(fact_1557_Divides_Otransfer__nat__int__functions_I2_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)) ) ) ).

fof(fact_1558_mod2__Suc__Suc,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,M))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_1559_mod__Suc__eq__mod__add3,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,N)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N)) ).

fof(fact_1560_Suc__mod__eq__add3__mod,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M)))),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M)),N) ).

fof(fact_1561_Suc__mod__eq__add3__mod__number__of,axiom,
    ! [M,V] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M)))),hAPP(int,nat,number_number_of(nat),V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M)),hAPP(int,nat,number_number_of(nat),V)) ).

fof(fact_1562_mod__exhaust__less__4,axiom,
    ! [M] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(nat)
      | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = one_one(nat)
      | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))
      | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) ) ).

fof(fact_1563_mod2__gr__0,axiom,
    ! [Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Ma),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
    <=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Ma),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(nat) ) ).

fof(fact_1564_StandardRes__SRStar__prop4,axiom,
    ! [Xa,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4)))
         => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa)),hAPP(int,fun(int,bool),sRStar,P_4))) ) ) ) ).

fof(fact_1565_StandardRes__SRStar__prop1,axiom,
    ! [Xa,P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa)),hAPP(int,fun(int,bool),sRStar,P_4)))
      <=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),P_4)) ) ) ).

fof(fact_1566_SRStar__mult__prop1,axiom,
    ! [Ya,Xa,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4)))
         => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),hAPP(int,fun(int,bool),sRStar,P_4)))
           => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya))),hAPP(int,fun(int,bool),sRStar,P_4))) ) ) ) ) ).

fof(fact_1567_StandardRes__SRStar__prop2,axiom,
    ! [Xa,P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
     => ( hBOOL(hAPP(int,bool,zprime,P_4))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4)))
         => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),Xa))),hAPP(int,fun(int,bool),sRStar,P_4))) ) ) ) ).

fof(fact_1568_nat__mod__eq__lemma,axiom,
    ! [X,N,Y] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Y),N)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
       => ? [Q_4] : X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),Q_4)) ) ) ).

fof(fact_1569_dvd_Olift__Suc__mono__less__iff,axiom,
    ! [Na,Ma,F] :
      ( ! [N_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_1)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1))))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1))),hAPP(nat,nat,F,N_1))) )
     => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,Na)),hAPP(nat,nat,F,Ma)))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,Ma)),hAPP(nat,nat,F,Na))) )
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) ) ) ).

fof(fact_1570_dvd_Olift__Suc__mono__less,axiom,
    ! [Na,N_6,F] :
      ( ! [N_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_1)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1))))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1))),hAPP(nat,nat,F,N_1))) )
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),N_6))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,Na)),hAPP(nat,nat,F,N_6)))
          & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_6)),hAPP(nat,nat,F,Na))) ) ) ) ).

fof(fact_1571_mod__induct__0,axiom,
    ! [I,P_1,P_4] :
      ( ! [I_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),P_4))
         => ( hBOOL(hAPP(nat,bool,P_1,I_2))
           => hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,I_2)),P_4))) ) )
     => ( hBOOL(hAPP(nat,bool,P_1,I))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),P_4))
         => hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) ) ) ) ).

fof(fact_1572_gr0__implies__Suc,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ? [M_2] : N = hAPP(nat,nat,suc,M_2) ) ).

fof(fact_1573_lift__Suc__mono__le,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Na,N_6,F] :
          ( ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,N_1)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1))))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),N_6))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,Na)),hAPP(nat,X_a,F,N_6))) ) ) ) ).

fof(fact_1574_mod__eqD,axiom,
    ! [M,D_3,R_3] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),D_3) = R_3
     => ? [Q_4] : M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),R_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Q_4),D_3)) ) ).

fof(fact_1575_nat__mod__eq__iff,axiom,
    ! [Xa,Na,Ya] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Xa),Na) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Ya),Na)
    <=> ? [Q1,Q2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Xa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Q1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ya),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Q2)) ) ).

fof(fact_1576_mod__eq__0D,axiom,
    ! [M_5,D_4] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_5),D_4) = zero_zero(nat)
     => ? [Q_4] : M_5 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),D_4),Q_4) ) ).

fof(fact_1577_mod__induct,axiom,
    ! [J,I,P_1,P_4] :
      ( ! [I_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),P_4))
         => ( hBOOL(hAPP(nat,bool,P_1,I_2))
           => hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,I_2)),P_4))) ) )
     => ( hBOOL(hAPP(nat,bool,P_1,I))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),P_4))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),P_4))
           => hBOOL(hAPP(nat,bool,P_1,J)) ) ) ) ) ).

fof(fact_1578_lift__Suc__mono__less__iff,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Na,Ma,F] :
          ( ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N_1)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1))))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,Na)),hAPP(nat,X_a,F,Ma)))
          <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) ) ) ) ).

fof(fact_1579_lift__Suc__mono__less,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Na,N_6,F] :
          ( ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N_1)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1))))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),N_6))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,Na)),hAPP(nat,X_a,F,N_6))) ) ) ) ).

fof(fact_1580_pdivmod__def,axiom,
    ! [K_4,L_4] : hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4))) ).

fof(fact_1581_zmult2__lemma,axiom,
    ! [C_1,A_1,B,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))
     => ( ti(int,B) != zero_zero(int)
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1))
         => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Q_3),C_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3)))) ) ) ) ).

fof(fact_1582_one__mod__nat__number__of,axiom,
    ! [V_2] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(nat,nat,suc,zero_zero(nat)) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),V_2))) ) ) ).

fof(fact_1583_two__times__odd__div__two__plus__one,axiom,
    ! [X] :
      ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(int)) = ti(int,X) ) ).

fof(fact_1584_lemma__interval,axiom,
    ! [B,A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),B))
       => ? [D_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
            & ! [Y_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y_1))),D_2))
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_1),Y_1))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),B)) ) ) ) ) ) ).

fof(fact_1585_negDivAlg__eqn__1__number__of,axiom,
    ! [W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),W))))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),W))) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),W))))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),W)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,number_number_of(int),W)))) ) ) ) ).

fof(fact_1586_not__neg__0,axiom,
    ~ hBOOL(hAPP(int,bool,nat_neg,zero_zero(int))) ).

fof(fact_1587_even__zero__int,axiom,
    hBOOL(hAPP(int,bool,even_odd_even(int),zero_zero(int))) ).

fof(fact_1588_not__neg__1,axiom,
    ~ hBOOL(hAPP(int,bool,nat_neg,one_one(int))) ).

fof(fact_1589_odd__one__int,axiom,
    ~ hBOOL(hAPP(int,bool,even_odd_even(int),one_one(int))) ).

fof(fact_1590_Parity_Oodd__times__odd,axiom,
    ! [Y,X] :
      ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),Y))
       => ~ hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) ) ) ).

fof(fact_1591_anything__times__even,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),Y))
     => hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) ) ).

fof(fact_1592_even__times__anything,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) ) ).

fof(fact_1593_Parity_Oeven__product,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)))
    <=> ( hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
        | hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) ) ) ).

fof(fact_1594_Parity_Oodd__plus__odd,axiom,
    ! [Y,X] :
      ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),Y))
       => hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) ) ) ).

fof(fact_1595_odd__plus__even,axiom,
    ! [Y,X] :
      ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => ( hBOOL(hAPP(int,bool,even_odd_even(int),Y))
       => ~ hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) ) ) ).

fof(fact_1596_Parity_Oeven__plus__odd,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),Y))
       => ~ hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) ) ) ).

fof(fact_1597_Parity_Oeven__plus__even,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => ( hBOOL(hAPP(int,bool,even_odd_even(int),Y))
       => hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) ) ) ).

fof(fact_1598_even__sum,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),Ya)))
    <=> ( ( hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
          & hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) )
        | ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
          & ~ hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) ) ) ) ).

fof(fact_1599_negDivAlg__correct,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
       => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B))) ) ) ).

fof(fact_1600_not__neg__int,axiom,
    ! [N] : ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(nat,int,semiring_1_of_nat(int),N))) ).

fof(fact_1601_even__difference,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)))
    <=> ( ( hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
          & hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) )
        | ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
          & ~ hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) ) ) ) ).

fof(fact_1602_odd__pow,axiom,
    ! [N,X] :
      ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => ~ hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N))) ) ).

fof(fact_1603_neg__def,axiom,
    ! [Z_5] :
      ( hBOOL(hAPP(int,bool,nat_neg,Z_5))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_5),zero_zero(int))) ) ).

fof(fact_1604_neg__number__of__Bit1,axiom,
    ! [Wa] :
      ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,Wa))))
    <=> hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Wa))) ) ).

fof(fact_1605_not__neg__eq__ge__0,axiom,
    ! [Xa] :
      ( ~ hBOOL(hAPP(int,bool,nat_neg,Xa))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) ) ).

fof(fact_1606_not__neg__number__of__Pls,axiom,
    ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),pls))) ).

fof(fact_1607_neg__number__of__Bit0,axiom,
    ! [Wa] :
      ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,Wa))))
    <=> hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Wa))) ) ).

fof(fact_1608_neg__nat,axiom,
    ! [Z_1] :
      ( hBOOL(hAPP(int,bool,nat_neg,Z_1))
     => hAPP(int,nat,nat_1,Z_1) = zero_zero(nat) ) ).

fof(fact_1609_even__power,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Xa),Na)))
    <=> ( hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
        & Na != zero_zero(nat) ) ) ).

fof(fact_1610_neg__number__of__Min,axiom,
    hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),min))) ).

fof(fact_1611_not__neg__nat,axiom,
    ! [Z_1] :
      ( ~ hBOOL(hAPP(int,bool,nat_neg,Z_1))
     => hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z_1)) = ti(int,Z_1) ) ).

fof(fact_1612_unique__remainder,axiom,
    ! [Q_5,R_4,A_1,B,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))
     => ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_5),R_4)))
       => ( ti(int,B) != zero_zero(int)
         => ti(int,R_3) = ti(int,R_4) ) ) ) ).

fof(fact_1613_unique__quotient,axiom,
    ! [Q_5,R_4,A_1,B,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))
     => ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_5),R_4)))
       => ( ti(int,B) != zero_zero(int)
         => ti(int,Q_3) = ti(int,Q_5) ) ) ) ).

fof(fact_1614_self__remainder,axiom,
    ! [A_1,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),A_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))
     => ( ti(int,A_1) != zero_zero(int)
       => ti(int,R_3) = zero_zero(int) ) ) ).

fof(fact_1615_divmod__int__rel__0,axiom,
    ! [B] :
      ( ti(int,B) != zero_zero(int)
     => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,zero_zero(int)),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)))) ) ).

fof(fact_1616_neg__imp__number__of__eq__0,axiom,
    ! [V] :
      ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
     => hAPP(int,nat,number_number_of(nat),V) = zero_zero(nat) ) ).

fof(fact_1617_self__quotient,axiom,
    ! [A_1,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),A_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))
     => ( ti(int,A_1) != zero_zero(int)
       => ti(int,Q_3) = one_one(int) ) ) ).

fof(fact_1618_divmod__int__rel__div,axiom,
    ! [A_1,B,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))
     => ( ti(int,B) != zero_zero(int)
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = ti(int,Q_3) ) ) ).

fof(fact_1619_divmod__int__rel__mod,axiom,
    ! [A_1,B,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))
     => ( ti(int,B) != zero_zero(int)
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = ti(int,R_3) ) ) ).

fof(fact_1620_eq__nat__number__of,axiom,
    ! [Va,V_1] :
      ( hAPP(int,nat,number_number_of(nat),Va) = hAPP(int,nat,number_number_of(nat),V_1)
    <=> ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Va)))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),V_1)),zero_zero(int))) )
        & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Va)))
         => ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_1)))
             => hAPP(int,int,number_number_of(int),Va) = zero_zero(int) )
            & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_1)))
             => ti(int,Va) = ti(int,V_1) ) ) ) ) ) ).

fof(fact_1621_nat__number__of__add__left,axiom,
    ! [V_2,K_4,V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2)))
           => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),K_4) )
          & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2)))
           => hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),V_2))),K_4) ) ) ) ) ).

fof(fact_1622_int__nat__number__of,axiom,
    ! [V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(int) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,int,number_number_of(int),V) ) ) ).

fof(fact_1623_divmod__int__rel__div__mod,axiom,
    ! [A_1,B] :
      ( ti(int,B) != zero_zero(int)
     => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))) ) ).

fof(fact_1624_of__nat__number__of__eq,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V] :
          ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
           => hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(X_a) )
          & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
           => hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,X_a,number_number_of(X_a),V) ) ) ) ).

fof(fact_1625_even__equiv__def,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
    <=> ? [Y_1] : ti(int,Xa) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Y_1) ) ).

fof(fact_1626_int__even__iff__2__dvd,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Xa)) ) ).

fof(fact_1627_Suc__nat__number__of,axiom,
    ! [V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,suc,hAPP(int,nat,number_number_of(nat),V)) = one_one(nat) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,suc,hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) ) ) ).

fof(fact_1628_zadd1__lemma,axiom,
    ! [B,Bq,Br,A_1,C_1,Aq,Ar] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Aq),Ar)))
     => ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,B),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Bq),Br)))
       => ( ti(int,C_1) != zero_zero(int)
         => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aq),Bq)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ar),Br)),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ar),Br)),C_1)))) ) ) ) ).

fof(fact_1629_even__def,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
    <=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(int) ) ).

fof(fact_1630_two__times__even__div__two,axiom,
    ! [X] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = ti(int,X) ) ).

fof(fact_1631_power__nat__number__of,axiom,
    ! [N,V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),N) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),N) = hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),V)),N)) ) ) ).

fof(fact_1632_power__nat__number__of__number__of,axiom,
    ! [W,V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),W)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),W)) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),W)) = hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,nat,number_number_of(nat),W))) ) ) ).

fof(fact_1633_div__nat__number__of,axiom,
    ! [V_2,V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = zero_zero(nat) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),V_2))) ) ) ).

fof(fact_1634_mod__nat__number__of,axiom,
    ! [V_2,V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = zero_zero(nat) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2)))
           => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,number_number_of(nat),V) )
          & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2)))
           => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),V_2))) ) ) ) ) ).

fof(fact_1635_Suc__nat__number__of__add,axiom,
    ! [N,V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,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,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V))),N) ) ) ).

fof(fact_1636_negDivAlg__minus1,axiom,
    ! [B] : hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),min)),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),one_one(int))) ).

fof(fact_1637_negDivAlg__div__mod,axiom,
    ! [L_4,K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),L_4))
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4)) ) ) ).

fof(fact_1638_negDivAlg_Osimps,axiom,
    ! [A_1,B] :
      ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))
          | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) )
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)) )
      & ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))
            | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) )
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) ) ) ).

fof(fact_1639_odd__equiv__def,axiom,
    ! [Xa] :
      ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
    <=> ? [Y_1] : ti(int,Xa) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Y_1)),one_one(int)) ) ).

fof(fact_1640_divmod__int__relI,axiom,
    ! [A_1,B,Q_3,R_3] :
      ( ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3)
     => ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B)) ) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int))) ) ) )
       => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) ) ) ).

fof(fact_1641_odd__plus__one__div__two,axiom,
    ! [X] :
      ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(int)) ) ).

fof(fact_1642_even__plus__one__div__two,axiom,
    ! [X] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),X))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).

fof(fact_1643_zmult1__lemma,axiom,
    ! [A_1,B,C_1,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,B),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))
     => ( ti(int,C_1) != zero_zero(int)
       => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_3)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),R_3)),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),R_3)),C_1)))) ) ) ).

fof(fact_1644_negDivAlg__eqn__number__of,axiom,
    ! [W,V] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V))))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V))) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V))))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),V)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,number_number_of(int),V)))) ) ) ) ).

fof(fact_1645_negDivAlg__eqn,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) ) ) ) ).

fof(fact_1646_lemma__interval__lt,axiom,
    ! [B,A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),B))
       => ? [D_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
            & ! [Y_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y_1))),D_2))
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),Y_1))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),B)) ) ) ) ) ) ).

fof(fact_1647_posDivAlg_Osimps,axiom,
    ! [A_1,B] :
      ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
          | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) )
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A_1) )
      & ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
            | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) )
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) ) ) ).

fof(fact_1648_posDivAlg__eqn__1__number__of,axiom,
    ! [W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),W)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,number_number_of(int),W)))) ) ) ) ).

fof(fact_1649_posDivAlg__eqn__number__of,axiom,
    ! [W,V] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),V)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,number_number_of(int),V)))) ) ) ) ).

fof(fact_1650_posDivAlg__eqn,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A_1) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) ) ) ) ).

fof(fact_1651_lemma__f__mono__add,axiom,
    ! [Ma,No_1,F] :
      ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),hAPP(nat,real,F,hAPP(nat,nat,suc,N_1))))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,Ma)),hAPP(nat,real,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),No_1)))) ) ).

fof(fact_1652_posDivAlg__0,axiom,
    ! [B] : hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,zero_zero(int)),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) ).

fof(fact_1653_posDivAlg__correct,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
       => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B))) ) ) ).

fof(fact_1654_pdivmod__posDivAlg,axiom,
    ! [K_4,L_4] :
      ( ( ti(int,L_4) = zero_zero(int)
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),hAPP(int,int,abs_abs(int),K_4)) )
      & ( ti(int,L_4) != zero_zero(int)
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4)) ) ) ).

fof(fact_1655_posDivAlg__div__mod,axiom,
    ! [L_4,K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4)) ) ) ).

fof(fact_1656_natceiling__add__number__of,axiom,
    ! [X,N] :
      ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),N)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,number_number_of(real),N))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),hAPP(int,nat,number_number_of(nat),N)) ) ) ).

fof(fact_1657_negDivAlg_Opsimps,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_2)))
     => ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)))
            | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),zero_zero(int))) )
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_3),B_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)) )
        & ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)))
              | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),zero_zero(int))) )
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_3),B_2) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B_2))) ) ) ) ).

fof(fact_1658_natfloor__add__number__of,axiom,
    ! [X,N] :
      ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),N)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,number_number_of(real),N))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),hAPP(int,nat,number_number_of(nat),N)) ) ) ).

fof(fact_1659_natceiling__le__eq__number__of,axiom,
    ! [Xa,Na] :
      ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Na)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,Xa)),hAPP(int,nat,number_number_of(nat),Na)))
        <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(int,real,number_number_of(real),Na))) ) ) ) ).

fof(fact_1660_posDivAlg_Opsimps,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_2)))
     => ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2))
            | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),zero_zero(int))) )
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_3),B_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A_3) )
        & ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2))
              | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),zero_zero(int))) )
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_3),B_2) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B_2))) ) ) ) ).

fof(fact_1661_SRStar__card,axiom,
    ! [P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
     => hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),sRStar,P_4))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)) ) ).

fof(fact_1662_natfloor__zero,axiom,
    hAPP(real,nat,natfloor,zero_zero(real)) = zero_zero(nat) ).

fof(fact_1663_zero__le__natfloor,axiom,
    ! [X] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(real,nat,natfloor,X))) ).

fof(fact_1664_natfloor__mono,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natfloor,X)),hAPP(real,nat,natfloor,Y))) ) ).

fof(fact_1665_natfloor__number__of__eq,axiom,
    ! [N] : hAPP(real,nat,natfloor,hAPP(int,real,number_number_of(real),N)) = hAPP(int,nat,number_number_of(nat),N) ).

fof(fact_1666_natfloor__one,axiom,
    hAPP(real,nat,natfloor,one_one(real)) = one_one(nat) ).

fof(fact_1667_natceiling__zero,axiom,
    hAPP(real,nat,natceiling,zero_zero(real)) = zero_zero(nat) ).

fof(fact_1668_zero__le__natceiling,axiom,
    ! [X] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(real,nat,natceiling,X))) ).

fof(fact_1669_natceiling__mono,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X)),hAPP(real,nat,natceiling,Y))) ) ).

fof(fact_1670_natceiling__number__of__eq,axiom,
    ! [N] : hAPP(real,nat,natceiling,hAPP(int,real,number_number_of(real),N)) = hAPP(int,nat,number_number_of(nat),N) ).

fof(fact_1671_natceiling__one,axiom,
    hAPP(real,nat,natceiling,one_one(real)) = one_one(nat) ).

fof(fact_1672_natfloor__neg,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => hAPP(real,nat,natfloor,X) = zero_zero(nat) ) ).

fof(fact_1673_natceiling__neg,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => hAPP(real,nat,natceiling,X) = zero_zero(nat) ) ).

fof(fact_1674_le__natfloor__eq__one,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(real,nat,natfloor,Xa)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Xa)) ) ).

fof(fact_1675_natceiling__le__eq__one,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,Xa)),one_one(nat)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real))) ) ).

fof(fact_1676_le__mult__natfloor,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(real,nat,natfloor,A_1)),hAPP(real,nat,natfloor,B))),hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)))) ) ) ).

fof(fact_1677_natfloor__add__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),one_one(nat)) ) ).

fof(fact_1678_natceiling__add__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),one_one(nat)) ) ).

fof(fact_1679_le__natfloor__eq__number__of,axiom,
    ! [Xa,Na] :
      ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Na)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),Na)),hAPP(real,nat,natfloor,Xa)))
        <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),Na)),Xa)) ) ) ) ).

fof(fact_1680_negDivAlg_Opinduct,axiom,
    ! [P_1,A0,A1] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
     => ( ! [A_4,B_4] :
            ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_4),B_4)))
           => ( ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_4),B_4)))
                    | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_4),zero_zero(int))) )
               => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B_4))) )
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),B_4)) ) )
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).

fof(fact_1681_posDivAlg_Opinduct,axiom,
    ! [P_1,A0,A1] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
     => ( ! [A_4,B_4] :
            ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_4),B_4)))
           => ( ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_4),B_4))
                    | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_4),zero_zero(int))) )
               => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B_4))) )
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),B_4)) ) )
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).

fof(fact_1682_MultInvPair__card__two,axiom,
    ! [J,A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3))
           => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J),zero_zero(int)),P_4))
             => hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),hAPP(int,fun(int,fun(int,fun(int,bool))),multInvPair,A_3),P_4),J)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ) ) ) ) ).

fof(fact_1683_SetS__elems__card,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3))
           => ! [X_1] :
                ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)))
               => hAPP(fun(int,bool),nat,finite_card(int),X_1) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ) ) ) ) ).

fof(fact_1684_Code__Numeral_Oint__of__code,axiom,
    ! [K_4] :
      ( ( K_4 = zero_zero(code_code_numeral)
       => hAPP(code_code_numeral,int,code_int_of,K_4) = zero_zero(int) )
      & ( K_4 != zero_zero(code_code_numeral)
       => ( ( hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(code_code_numeral)
           => hAPP(code_code_numeral,int,code_int_of,K_4) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(code_code_numeral,int,code_int_of,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) )
          & ( hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(code_code_numeral)
           => hAPP(code_code_numeral,int,code_int_of,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(code_code_numeral,int,code_int_of,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int)) ) ) ) ) ).

fof(fact_1685_int__diff__cases,axiom,
    ! [Z_1] :
      ~ ! [M_2,N_1] : ti(int,Z_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M_2)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) ).

fof(fact_1686_code__numeral__zero__minus__one,axiom,
    hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),zero_zero(code_code_numeral)),one_one(code_code_numeral)) = zero_zero(code_code_numeral) ).

fof(fact_1687_zero__code__numeral__code,axiom,
    zero_zero(code_code_numeral) = hAPP(int,code_code_numeral,number_number_of(code_code_numeral),pls) ).

fof(fact_1688_one__code__numeral__code,axiom,
    one_one(code_code_numeral) = hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit1,pls)) ).

fof(fact_1689_SetS__card,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3))
           => hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(fun(int,bool),bool),nat,finite_card(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ) ) ) ).

fof(fact_1690_MultInvPair__prop2,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4))
         => hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)) = hAPP(int,fun(int,bool),sRStar,P_4) ) ) ) ).

fof(fact_1691_EvenOdd_Oneg__one__even__power,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
       => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Xa)) = one_one(int) ) ) ).

fof(fact_1692_le__Suc__ex,axiom,
    ! [K_4,L_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4))
     => ? [N_1] : L_4 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N_1) ) ).

fof(fact_1693_zero__induct__lemma,axiom,
    ! [I,P_1,K] :
      ( hBOOL(hAPP(nat,bool,P_1,K))
     => ( ! [N_1] :
            ( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_1)))
           => hBOOL(hAPP(nat,bool,P_1,N_1)) )
       => hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),K),I))) ) ) ).

fof(fact_1694_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_1695_one__not__even,axiom,
    ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),one_one(int)),zEven)) ).

fof(fact_1696_even__times__either,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),zEven)) ) ).

fof(fact_1697_EvenOdd_Oeven__product,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),zEven))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
        | hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) ) ) ).

fof(fact_1698_EvenOdd_Oeven__plus__even,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),Ya)),zEven)) ) ) ).

fof(fact_1699_even__diff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zEven))
    <=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
      <=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) ) ) ).

fof(fact_1700_even__minus__even,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zEven)) ) ) ).

fof(fact_1701_not__odd__impl__even,axiom,
    ! [Xa] :
      ( ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) ) ).

fof(fact_1702_even__odd__disj,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
      | hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) ) ).

fof(fact_1703_odd__iff__not__even,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
    <=> ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) ) ).

fof(fact_1704_even__odd__conj,axiom,
    ! [Xa] :
      ~ ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
        & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) ) ).

fof(fact_1705_ln__less__cancel__iff,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,Xa)),hAPP(real,real,ln,Ya)))
        <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) ) ) ) ).

fof(fact_1706_ln__inj__iff,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya))
       => ( hAPP(real,real,ln,Xa) = hAPP(real,real,ln,Ya)
        <=> Xa = Ya ) ) ) ).

fof(fact_1707_ln__less__self,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,X)),X)) ) ).

fof(fact_1708_ln__one,axiom,
    hAPP(real,real,ln,one_one(real)) = zero_zero(real) ).

fof(fact_1709_even__plus__odd__prop2,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)),zOdd))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),zEven))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),zOdd)) ) ) ).

fof(fact_1710_even__plus__odd__prop1,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)),zOdd))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),zOdd))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),zEven)) ) ) ).

fof(fact_1711_IntNatAux_Oeven__plus__odd,axiom,
    ! [B_2,A_3] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),zEven))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),zOdd))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)),zOdd)) ) ) ).

fof(fact_1712_IntNatAux_Oodd__plus__odd,axiom,
    ! [B_2,A_3] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),zOdd))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),zOdd))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)),zEven)) ) ) ).

fof(fact_1713_even__minus__odd,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zOdd))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zOdd)) ) ) ).

fof(fact_1714_odd__minus__even,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zOdd)) ) ) ).

fof(fact_1715_odd__minus__odd,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zOdd))
       => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zEven)) ) ) ).

fof(fact_1716_ln__eq__minus__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hAPP(real,real,ln,X) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real))
       => X = one_one(real) ) ) ).

fof(fact_1717_ln__le__cancel__iff,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,Xa)),hAPP(real,real,ln,Ya)))
        <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ) ) ).

fof(fact_1718_ln__ge__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,X))) ) ).

fof(fact_1719_ln__gt__zero__imp__gt__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,X)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ).

fof(fact_1720_ln__less__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,X)),zero_zero(real))) ) ) ).

fof(fact_1721_ln__less__zero__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,Xa)),zero_zero(real)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) ) ) ).

fof(fact_1722_ln__eq__zero__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => ( hAPP(real,real,ln,Xa) = zero_zero(real)
      <=> Xa = one_one(real) ) ) ).

fof(fact_1723_ln__gt__zero__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,Xa)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa)) ) ) ).

fof(fact_1724_ln__gt__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,X))) ) ).

fof(fact_1725_ln__le__minus__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real)))) ) ).

fof(fact_1726_power__preserves__even,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Xa),Na)),zEven))
      <=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) ) ) ).

fof(fact_1727_odd__minus__one__even,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),one_one(int))),zEven)) ) ).

fof(fact_1728_ln__ge__zero__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,Xa)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Xa)) ) ) ).

fof(fact_1729_ln__ge__zero__imp__ge__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,X)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ).

fof(fact_1730_ln__mult,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,Y)) ) ) ).

fof(fact_1731_ln__add__one__self__le__self,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

fof(fact_1732_ln__add__one__self__le__self2,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

fof(fact_1733_zEvenI,axiom,
    ! [Xa,K] :
      ( ti(int,Xa) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K)
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) ) ).

fof(fact_1734_even__div__2__prop1,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(int) ) ).

fof(fact_1735_even__div__2__l,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Ya),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_1736_even__prod__div__2,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Ya) ) ).

fof(fact_1737_even__div__2__prop2,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = ti(int,Xa) ) ).

fof(fact_1738_even__sum__div__2,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),Ya)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Ya),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ).

fof(fact_1739_neg__one__power__parity,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
       => ( ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
          <=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) )
         => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Xa)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Ya)) ) ) ) ).

fof(fact_1740_ln__one__plus__pos__lower__bound,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)))) ) ) ).

fof(fact_1741_zEvenE,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => ~ ! [K_1] : ti(int,Xa) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K_1) ) ).

fof(fact_1742_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).

fof(fact_1743_nat__of__aux__code,axiom,
    ! [N,I_1] :
      ( ( I_1 = zero_zero(code_code_numeral)
       => hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,I_1),N) = N )
      & ( I_1 != zero_zero(code_code_numeral)
       => hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,I_1),N) = hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),I_1),one_one(code_code_numeral))),hAPP(nat,nat,suc,N)) ) ) ).

fof(fact_1744_ln__bound,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X)),X)) ) ).

fof(fact_1745_aux4,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))) ) ) ).

fof(fact_1746_UnionI,axiom,
    ! [X_a,A_2,X_2,C_2] :
      ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_2),C_2))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),X_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),C_2))) ) ) ).

fof(fact_1747_exp__less__mono,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X)),hAPP(real,real,exp(real),Y))) ) ).

fof(fact_1748_exp__diff,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [X,Y] : hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y)) ) ).

fof(fact_1749_exp__inj__iff,axiom,
    ! [Xa,Ya] :
      ( hAPP(real,real,exp(real),Xa) = hAPP(real,real,exp(real),Ya)
    <=> Xa = Ya ) ).

fof(fact_1750_divide_Ozero,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),zero_zero(X_a)),Y) = zero_zero(X_a) ) ).

fof(fact_1751_divide_Oadd,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [X,Y,Ya_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Ya_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Ya_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Ya_1)) ) ).

fof(fact_1752_divide_Odiff,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [X,Y,Ya_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),Ya_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Ya_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Ya_1)) ) ).

fof(fact_1753_power__divide,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_1,B,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N)) ) ).

fof(fact_1754_exp__le__cancel__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),Xa)),hAPP(real,real,exp(real),Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ).

fof(fact_1755_exp__less__cancel,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X)),hAPP(real,real,exp(real),Y)))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y)) ) ).

fof(fact_1756_exp__less__cancel__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),Xa)),hAPP(real,real,exp(real),Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) ) ).

fof(fact_1757_real__divide__square__eq,axiom,
    ! [R_3,A_1] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),R_3)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A_1),R_3) ).

fof(fact_1758_abs__exp__cancel,axiom,
    ! [X] : hAPP(real,real,abs_abs(real),hAPP(real,real,exp(real),X)) = hAPP(real,real,exp(real),X) ).

fof(fact_1759_ln__exp,axiom,
    ! [X] : hAPP(real,real,ln,hAPP(real,real,exp(real),X)) = X ).

fof(fact_1760_exp__ln__eq,axiom,
    ! [U_1,X] :
      ( hAPP(real,real,exp(real),U_1) = X
     => hAPP(real,real,ln,X) = U_1 ) ).

fof(fact_1761_ln__unique,axiom,
    ! [Y,X] :
      ( hAPP(real,real,exp(real),Y) = X
     => hAPP(real,real,ln,X) = Y ) ).

fof(fact_1762_ln__x__over__x__mono,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),one_one(real))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,Y)),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,X)),X))) ) ) ).

fof(fact_1763_exp__not__eq__zero,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [X] : hAPP(X_a,X_a,exp(X_a),X) != zero_zero(X_a) ) ).

fof(fact_1764_nonzero__power__divide,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [A_1,N,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N)) ) ) ).

fof(fact_1765_power__one__over,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_1,N] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1)),N) ) ).

fof(fact_1766_not__exp__le__zero,axiom,
    ! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),X)),zero_zero(real))) ).

fof(fact_1767_exp__ge__zero,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,exp(real),X))) ).

fof(fact_1768_not__exp__less__zero,axiom,
    ! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X)),zero_zero(real))) ).

fof(fact_1769_exp__gt__zero,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,exp(real),X))) ).

fof(fact_1770_real__0__le__divide__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Xa),Ya)))
    <=> ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),zero_zero(real)))
          | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Ya)) )
        & ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
          | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),zero_zero(real))) ) ) ) ).

fof(fact_1771_exp__eq__one__iff,axiom,
    ! [Xa] :
      ( hAPP(real,real,exp(real),Xa) = one_one(real)
    <=> Xa = zero_zero(real) ) ).

fof(fact_1772_lemma__MVT,axiom,
    ! [F,A_3,B_2] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,A_3)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3))),A_3)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3))),B_2)) ).

fof(fact_1773_exp__zero,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => hAPP(X_a,X_a,exp(X_a),zero_zero(X_a)) = one_one(X_a) ) ).

fof(fact_1774_mult__exp__exp,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y)) = hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) ) ).

fof(fact_1775_exp__add,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [X,Y] : hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y)) ) ).

fof(fact_1776_eq__divide__eq__number__of1,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & number(X_a) )
     => ! [A_3,B_2,Wa] :
          ( ti(X_a,A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa))
        <=> ( ( hAPP(int,X_a,number_number_of(X_a),Wa) != zero_zero(X_a)
             => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa)) = ti(X_a,B_2) )
            & ( hAPP(int,X_a,number_number_of(X_a),Wa) = zero_zero(X_a)
             => ti(X_a,A_3) = zero_zero(X_a) ) ) ) ) ).

fof(fact_1777_eq__divide__eq__number__of,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & number(X_a) )
     => ! [Wa,B_2,C] :
          ( hAPP(int,X_a,number_number_of(X_a),Wa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)
        <=> ( ( ti(X_a,C) != zero_zero(X_a)
             => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C) = ti(X_a,B_2) )
            & ( ti(X_a,C) = zero_zero(X_a)
             => hAPP(int,X_a,number_number_of(X_a),Wa) = zero_zero(X_a) ) ) ) ) ).

fof(fact_1778_divide__eq__eq__number__of,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & number(X_a) )
     => ! [B_2,C,Wa] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C) = hAPP(int,X_a,number_number_of(X_a),Wa)
        <=> ( ( ti(X_a,C) != zero_zero(X_a)
             => ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C) )
            & ( ti(X_a,C) = zero_zero(X_a)
             => hAPP(int,X_a,number_number_of(X_a),Wa) = zero_zero(X_a) ) ) ) ) ).

fof(fact_1779_divide__eq__eq__number__of1,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & number(X_a) )
     => ! [B_2,Wa,A_3] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa)) = ti(X_a,A_3)
        <=> ( ( hAPP(int,X_a,number_number_of(X_a),Wa) != zero_zero(X_a)
             => ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa)) )
            & ( hAPP(int,X_a,number_number_of(X_a),Wa) = zero_zero(X_a)
             => ti(X_a,A_3) = zero_zero(X_a) ) ) ) ) ).

fof(fact_1780_DERIV__mult__lemma,axiom,
    ! [X_a] :
      ( real_field(X_a)
     => ! [A_1,B,C_1,D_3,H_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),D_3))),H_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D_3)),H_2))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C_1)),H_2)),D_3)) ) ).

fof(fact_1781_divide__Numeral0,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & number_ring(X_a) )
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),pls)) = zero_zero(X_a) ) ).

fof(fact_1782_divide__numeral__1,axiom,
    ! [X_a] :
      ( ( field(X_a)
        & number_ring(X_a) )
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = ti(X_a,A_1) ) ).

fof(fact_1783_divide__Numeral1,axiom,
    ! [X_a] :
      ( ( field(X_a)
        & number_ring(X_a) )
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = ti(X_a,X) ) ).

fof(fact_1784_exp__le__one__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),Xa)),one_one(real)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),zero_zero(real))) ) ).

fof(fact_1785_one__le__exp__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,exp(real),Xa)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) ) ).

fof(fact_1786_one__less__exp__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,exp(real),Xa)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) ) ).

fof(fact_1787_exp__less__one__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),Xa)),one_one(real)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) ) ).

fof(fact_1788_exp__gt__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,exp(real),X))) ) ).

fof(fact_1789_exp__ge__add__one__self,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(real,real,exp(real),X))) ).

fof(fact_1790_exp__ln,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,exp(real),hAPP(real,real,ln,X)) = X ) ).

fof(fact_1791_exp__ln__iff,axiom,
    ! [Xa] :
      ( hAPP(real,real,exp(real),hAPP(real,real,ln,Xa)) = Xa
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) ) ).

fof(fact_1792_less__divide__eq__number__of1,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number(X_a) )
     => ! [A_3,B_2,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa))))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))),B_2)) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa)))) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ) ) ) ) ).

fof(fact_1793_less__divide__eq__number__of,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number(X_a) )
     => ! [Wa,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)),B_2)) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C))) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) ) ) ) ) ) ) ).

fof(fact_1794_divide__less__eq__number__of,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number(X_a) )
     => ! [B_2,C,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),hAPP(int,X_a,number_number_of(X_a),Wa)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C))) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)),B_2)) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) ) ) ) ) ) ) ).

fof(fact_1795_divide__less__eq__number__of1,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number(X_a) )
     => ! [B_2,Wa,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa))),A_3))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa)))) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))),B_2)) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) ) ) ) ) ) ) ).

fof(fact_1796_Union__upper,axiom,
    ! [X_a,B_1,A_2] :
      ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),B_1),A_2))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2))) ) ).

fof(fact_1797_Union__mono,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),A_2),B_1))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),B_1))) ) ).

fof(fact_1798_power__diff,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [N,M,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) ) ) ) ).

fof(fact_1799_exp__ge__add__one__self__aux,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(real,real,exp(real),X))) ) ).

fof(fact_1800_ln__div,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,Y)) ) ) ).

fof(fact_1801_divide__le__eq__number__of1,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number(X_a) )
     => ! [B_2,Wa,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa))),A_3))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa)))) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))),B_2)) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) ) ) ) ) ) ) ).

fof(fact_1802_divide__le__eq__number__of,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number(X_a) )
     => ! [B_2,C,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),hAPP(int,X_a,number_number_of(X_a),Wa)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C))) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)),B_2)) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) ) ) ) ) ) ) ).

fof(fact_1803_le__divide__eq__number__of,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number(X_a) )
     => ! [Wa,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)),B_2)) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C))) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) ) ) ) ) ) ) ).

fof(fact_1804_le__divide__eq__number__of1,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number(X_a) )
     => ! [A_3,B_2,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa))))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))),B_2)) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa)))) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) ) ) ) ) ) ) ).

fof(fact_1805_eq__divide__2__times__iff,axiom,
    ! [Xa,Ya,Z] :
      ( Xa = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Ya),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z))
    <=> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Xa) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Ya),Z) ) ).

fof(fact_1806_real__sum__of__halves,axiom,
    ! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = X ).

fof(fact_1807_half__gt__zero__iff,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number_ring(X_a) )
     => ! [Ra] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ra),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Ra)) ) ) ).

fof(fact_1808_half__gt__zero,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number_ring(X_a) )
     => ! [R_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),R_3))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),R_3),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_1809_real__less__half__sum,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_1810_real__gt__half__sum,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Y)) ) ).

fof(fact_1811_real__average__minus__first,axiom,
    ! [A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),A_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_1812_real__average__minus__second,axiom,
    ! [B,A_1] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),A_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_1813_half,axiom,
    ! [X_a] :
      ( ( linord1117847801e_zero(X_a)
        & number_ring(X_a) )
     => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_1814_Sup__le__iff,axiom,
    ! [X_a] :
      ( comple44059174attice(X_a)
     => ! [A_2,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),B_2))
        <=> ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),B_2)) ) ) ) ).

fof(fact_1815_complete__lattice__class_OSup__upper,axiom,
    ! [X_a] :
      ( comple44059174attice(X_a)
     => ! [Xa,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2))) ) ) ).

fof(fact_1816_Sup__subset__mono,axiom,
    ! [X_a] :
      ( comple44059174attice(X_a)
     => ! [A_2,B_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),B_1))) ) ) ).

fof(fact_1817_complete__lattice__class_OSup__upper2,axiom,
    ! [X_a] :
      ( comple44059174attice(X_a)
     => ! [Va,U,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),U),A_2))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Va),U))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Va),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2))) ) ) ) ).

fof(fact_1818_less__Sup__iff,axiom,
    ! [X_a] :
      ( comple1803745165norder(X_a)
     => ! [A_3,S] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),S)))
        <=> ? [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),X_1)) ) ) ) ).

fof(fact_1819_exp__bound,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ) ).

fof(fact_1820_lemma__exp__total,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y))
     => ? [X_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y),one_one(real))))
          & hAPP(real,real,exp(real),X_1) = Y ) ) ).

fof(fact_1821_le__divide__eq,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) ) ) ) ) ) ) ).

fof(fact_1822_divide__le__eq,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [B_2,C,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) ) ) ) ) ) ) ).

fof(fact_1823_pos__le__divide__eq,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) ) ) ) ).

fof(fact_1824_pos__divide__le__eq,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [B_2,A_3,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) ) ) ) ).

fof(fact_1825_divide__zero,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_1826_divide__zero__left,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).

fof(fact_1827_times__divide__eq__right,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) ) ).

fof(fact_1828_times__divide__times__eq,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [X,Y,Z_1,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Z_1),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),W)) ) ).

fof(fact_1829_add__divide__distrib,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)) ) ).

fof(fact_1830_divide__1,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).

fof(fact_1831_diff__divide__distrib,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)) ) ).

fof(fact_1832_abs__divide,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ).

fof(fact_1833_zero__le__divide__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) ) ) ) ) ).

fof(fact_1834_divide__le__0__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),zero_zero(X_a)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) ) ) ) ) ).

fof(fact_1835_divide__right__mono,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1))) ) ) ) ).

fof(fact_1836_divide__right__mono__neg,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1))) ) ) ) ).

fof(fact_1837_divide__strict__right__mono__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1))) ) ) ) ).

fof(fact_1838_divide__strict__right__mono,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1))) ) ) ) ).

fof(fact_1839_divide__neg__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).

fof(fact_1840_divide__neg__pos,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a))) ) ) ) ).

fof(fact_1841_divide__pos__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a))) ) ) ) ).

fof(fact_1842_divide__pos__pos,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).

fof(fact_1843_divide__less__cancel,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,C,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) )
            & ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) )
            & ti(X_a,C) != zero_zero(X_a) ) ) ) ).

fof(fact_1844_divide__less__0__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),zero_zero(X_a)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),zero_zero(X_a))) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_2)) ) ) ) ) ).

fof(fact_1845_zero__less__divide__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_2)) )
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),zero_zero(X_a))) ) ) ) ) ).

fof(fact_1846_eq__divide__eq,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_3,B_2,C] :
          ( ti(X_a,A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)
        <=> ( ( ti(X_a,C) != zero_zero(X_a)
             => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C) = ti(X_a,B_2) )
            & ( ti(X_a,C) = zero_zero(X_a)
             => ti(X_a,A_3) = zero_zero(X_a) ) ) ) ) ).

fof(fact_1847_divide__eq__eq,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [B_2,C,A_3] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C) = ti(X_a,A_3)
        <=> ( ( ti(X_a,C) != zero_zero(X_a)
             => ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C) )
            & ( ti(X_a,C) = zero_zero(X_a)
             => ti(X_a,A_3) = zero_zero(X_a) ) ) ) ) ).

fof(fact_1848_nonzero__eq__divide__eq,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_3,B_2,C] :
          ( ti(X_a,C) != zero_zero(X_a)
         => ( ti(X_a,A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)
          <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C) = ti(X_a,B_2) ) ) ) ).

fof(fact_1849_nonzero__divide__eq__eq,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [B_2,A_3,C] :
          ( ti(X_a,C) != zero_zero(X_a)
         => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C) = ti(X_a,A_3)
          <=> ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C) ) ) ) ).

fof(fact_1850_mult__divide__mult__cancel__right,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_1,B,C_1] :
          ( ti(X_a,C_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) ) ) ).

fof(fact_1851_mult__divide__mult__cancel__left,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_1,B,C_1] :
          ( ti(X_a,C_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) ) ) ).

fof(fact_1852_frac__eq__eq,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [Xa,Wa,Z,Ya] :
          ( ti(X_a,Ya) != zero_zero(X_a)
         => ( ti(X_a,Z) != zero_zero(X_a)
           => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Xa),Ya) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Wa),Z)
            <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Wa),Ya) ) ) ) ) ).

fof(fact_1853_divide__eq__imp,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [B,A_1,C_1] :
          ( ti(X_a,C_1) != zero_zero(X_a)
         => ( ti(X_a,B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1) = ti(X_a,A_1) ) ) ) ).

fof(fact_1854_eq__divide__imp,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B,C_1] :
          ( ti(X_a,C_1) != zero_zero(X_a)
         => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1) = ti(X_a,B)
           => ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1) ) ) ) ).

fof(fact_1855_right__inverse__eq,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_3,B_2] :
          ( ti(X_a,B_2) != zero_zero(X_a)
         => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2) = one_one(X_a)
          <=> ti(X_a,A_3) = ti(X_a,B_2) ) ) ) ).

fof(fact_1856_divide__self,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1) = one_one(X_a) ) ) ).

fof(fact_1857_divide__self__if,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => ! [A_1] :
          ( ( ti(X_a,A_1) = zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1) = zero_zero(X_a) )
          & ( ti(X_a,A_1) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1) = one_one(X_a) ) ) ) ).

fof(fact_1858_nonzero__abs__divide,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ) ).

fof(fact_1859_divide__le__cancel,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,C,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) )
            & ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3)) ) ) ) ) ).

fof(fact_1860_divide__nonneg__pos,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).

fof(fact_1861_divide__nonneg__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a))) ) ) ) ).

fof(fact_1862_frac__less2,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Z_1,W,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W),Z_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W))) ) ) ) ) ) ).

fof(fact_1863_frac__less,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Z_1,W,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W),Z_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W))) ) ) ) ) ) ).

fof(fact_1864_frac__le,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Z_1,W,Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W),Z_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W))) ) ) ) ) ) ).

fof(fact_1865_divide__nonpos__pos,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a))) ) ) ) ).

fof(fact_1866_divide__nonpos__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).

fof(fact_1867_less__divide__eq,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ) ) ) ) ).

fof(fact_1868_divide__less__eq,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [B_2,C,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) )
            & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
             => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) )
                & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) ) ) ) ) ) ) ).

fof(fact_1869_pos__less__divide__eq,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) ) ) ) ).

fof(fact_1870_pos__divide__less__eq,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [B_2,A_3,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) ) ) ) ).

fof(fact_1871_mult__imp__div__pos__less,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [X,Z_1,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z_1)) ) ) ) ).

fof(fact_1872_mult__imp__less__div__pos,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y)),X))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).

fof(fact_1873_neg__less__divide__eq,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) ) ) ) ).

fof(fact_1874_neg__divide__less__eq,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [B_2,A_3,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) ) ) ) ).

fof(fact_1875_divide__strict__left__mono,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),B))) ) ) ) ) ).

fof(fact_1876_divide__strict__left__mono__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),B))) ) ) ) ) ).

fof(fact_1877_add__frac__eq,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [X,W,Z_1,Y] :
          ( ti(X_a,Y) != zero_zero(X_a)
         => ( ti(X_a,Z_1) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W),Y))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z_1)) ) ) ) ).

fof(fact_1878_divide__add__eq__iff,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [X,Y,Z_1] :
          ( ti(X_a,Z_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))),Z_1) ) ) ).

fof(fact_1879_add__frac__num,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [X,Z_1,Y] :
          ( ti(X_a,Y) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))),Y) ) ) ).

fof(fact_1880_add__divide__eq__iff,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [X,Y,Z_1] :
          ( ti(X_a,Z_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),X)),Y)),Z_1) ) ) ).

fof(fact_1881_add__num__frac,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [Z_1,X,Y] :
          ( ti(X_a,Y) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))),Y) ) ) ).

fof(fact_1882_diff__divide__eq__iff,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [X,Y,Z_1] :
          ( ti(X_a,Z_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),X)),Y)),Z_1) ) ) ).

fof(fact_1883_divide__diff__eq__iff,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [X,Y,Z_1] :
          ( ti(X_a,Z_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))),Z_1) ) ) ).

fof(fact_1884_diff__frac__eq,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [X,W,Z_1,Y] :
          ( ti(X_a,Y) != zero_zero(X_a)
         => ( ti(X_a,Z_1) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W),Y))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z_1)) ) ) ) ).

fof(fact_1885_gt__half__sum,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)))),B)) ) ) ).

fof(fact_1886_less__half__sum,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))))) ) ) ).

fof(fact_1887_abs__div__pos,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),Y) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) ) ) ).

fof(fact_1888_divide__left__mono__neg,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [C_1,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),zero_zero(X_a)))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),B))) ) ) ) ) ).

fof(fact_1889_divide__left__mono,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [C_1,B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),B))) ) ) ) ) ).

fof(fact_1890_neg__divide__le__eq,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [B_2,A_3,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) ) ) ) ).

fof(fact_1891_neg__le__divide__eq,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_3,B_2,C] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) ) ) ) ).

fof(fact_1892_mult__imp__le__div__pos,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y)),X))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).

fof(fact_1893_mult__imp__div__pos__le,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [X,Z_1,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z_1)) ) ) ) ).

fof(fact_1894_field__le__mult__one__interval,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [X,Y] :
          ( ! [Z_2] :
              ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Z_2))
             => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),one_one(X_a)))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_2),X)),Y)) ) )
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).

fof(fact_1895_lemma__termdiff3,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [N,Z_1,K_5,H_2] :
          ( ti(X_a,H_2) != zero_zero(X_a)
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),Z_1)),K_5))
           => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),H_2))),K_5))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),H_2)),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),N))),H_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),N)),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),K_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(X_a,real,norm_norm(X_a),H_2)))) ) ) ) ) ).

fof(fact_1896_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ) ).

fof(fact_1897_field__le__epsilon,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [X,Y] :
          ( ! [E] :
              ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),E))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),E))) )
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).

fof(fact_1898_log__base__10__eq1,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,hAPP(real,real,exp(real),one_one(real)))),hAPP(real,real,ln,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(real,real,ln,X)) ) ).

fof(fact_1899_le__imp__neg__le,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B)),hAPP(X_a,X_a,uminus_uminus(X_a),A_1))) ) ) ).

fof(fact_1900_neg__le__iff__le,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) ) ) ).

fof(fact_1901_minus__le__iff,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),B_2))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)),A_3)) ) ) ).

fof(fact_1902_le__minus__iff,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) ) ) ).

fof(fact_1903_complex__mod__minus__le__complex__mod,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,norm_norm(complex),X))),hAPP(complex,real,norm_norm(complex),X))) ).

fof(fact_1904_square__eq__iff,axiom,
    ! [X_a] :
      ( idom(X_a)
     => ! [A_3,B_2] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),B_2)
        <=> ( ti(X_a,A_3) = ti(X_a,B_2)
            | ti(X_a,A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),B_2) ) ) ) ).

fof(fact_1905_minus__mult__minus,axiom,
    ! [X_a] :
      ( ring(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) ) ).

fof(fact_1906_minus__mult__commute,axiom,
    ! [X_a] :
      ( ring(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).

fof(fact_1907_minus__mult__left,axiom,
    ! [X_a] :
      ( ring(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) ) ).

fof(fact_1908_minus__mult__right,axiom,
    ! [X_a] :
      ( ring(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).

fof(fact_1909_mult_Ominus__right,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) ) ).

fof(fact_1910_mult__right_Ominus,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa_2,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),X)) ) ).

fof(fact_1911_mult_Ominus__left,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) ) ).

fof(fact_1912_mult__left_Ominus,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) ) ).

fof(fact_1913_minus__add__cancel,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)) = ti(X_a,B) ) ).

fof(fact_1914_add__minus__cancel,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)) = ti(X_a,B) ) ).

fof(fact_1915_minus__add,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B)),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) ) ).

fof(fact_1916_minus__add__distrib,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).

fof(fact_1917_minus__diff__minus,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) ) ).

fof(fact_1918_minus__diff__eq,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1) ) ).

fof(fact_1919_minus__dvd__iff,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Xa)),Ya))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Xa),Ya)) ) ) ).

fof(fact_1920_dvd__minus__iff,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),Ya)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Xa),Ya)) ) ) ).

fof(fact_1921_minus__minus,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = ti(X_a,A_1) ) ).

fof(fact_1922_equation__minus__iff,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_3,B_2] :
          ( ti(X_a,A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),B_2)
        <=> ti(X_a,B_2) = hAPP(X_a,X_a,uminus_uminus(X_a),A_3) ) ) ).

fof(fact_1923_minus__equation__iff,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_3,B_2] :
          ( hAPP(X_a,X_a,uminus_uminus(X_a),A_3) = ti(X_a,B_2)
        <=> hAPP(X_a,X_a,uminus_uminus(X_a),B_2) = ti(X_a,A_3) ) ) ).

fof(fact_1924_neg__equal__iff__equal,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_3,B_2] :
          ( hAPP(X_a,X_a,uminus_uminus(X_a),A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),B_2)
        <=> ti(X_a,A_3) = ti(X_a,B_2) ) ) ).

fof(fact_1925_norm__minus__cancel,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,real,norm_norm(X_a),X) ) ).

fof(fact_1926_minus__number__of__mult,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W))),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W))),Z_1) ) ).

fof(fact_1927_arith__simps_I30_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W)) ) ).

fof(fact_1928_number__of__minus,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W)) ) ).

fof(fact_1929_norm__minus__commute,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)) ) ).

fof(fact_1930_less__minus__iff,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) ) ) ).

fof(fact_1931_minus__less__iff,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),B_2))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)),A_3)) ) ) ).

fof(fact_1932_neg__less__iff__less,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) ) ) ).

fof(fact_1933_neg__equal__zero,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hAPP(X_a,X_a,uminus_uminus(X_a),A_3) = ti(X_a,A_3)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_1934_neg__equal__0__iff__equal,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_3] :
          ( hAPP(X_a,X_a,uminus_uminus(X_a),A_3) = zero_zero(X_a)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_1935_equal__neg__zero,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( ti(X_a,A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),A_3)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_1936_neg__0__equal__iff__equal,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_3] :
          ( zero_zero(X_a) = hAPP(X_a,X_a,uminus_uminus(X_a),A_3)
        <=> zero_zero(X_a) = ti(X_a,A_3) ) ) ).

fof(fact_1937_minus__zero,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => hAPP(X_a,X_a,uminus_uminus(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_1938_norm__of__nat,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => ! [N] : hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) = hAPP(nat,real,semiring_1_of_nat(real),N) ) ).

fof(fact_1939_divide_Ominus,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) ) ).

fof(fact_1940_minus__divide__right,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).

fof(fact_1941_minus__divide__left,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) ) ).

fof(fact_1942_minus__divide__divide,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) ) ).

fof(fact_1943_abs__minus__cancel,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = hAPP(X_a,X_a,abs_abs(X_a),A_1) ) ).

fof(fact_1944_mod__minus__eq,axiom,
    ! [X_a] :
      ( ring_div(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),B) ) ).

fof(fact_1945_mod__minus__cong,axiom,
    ! [X_a] :
      ( ring_div(X_a)
     => ! [A_1,B,A_6] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_6),B)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_6)),B) ) ) ).

fof(fact_1946_abs__norm__cancel,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1] : hAPP(real,real,abs_abs(real),hAPP(X_a,real,norm_norm(X_a),A_1)) = hAPP(X_a,real,norm_norm(X_a),A_1) ) ).

fof(fact_1947_real__norm__def,axiom,
    ! [R_3] : hAPP(real,real,norm_norm(real),R_3) = hAPP(real,real,abs_abs(real),R_3) ).

fof(fact_1948_minus__le__self__iff,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),A_3))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_1949_neg__le__0__iff__le,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),zero_zero(X_a)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_1950_le__minus__self__iff,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_1951_neg__0__le__iff__le,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_1952_less__minus__self__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_1953_neg__0__less__iff__less,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_1954_neg__less__0__iff__less,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),zero_zero(X_a)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_1955_neg__less__nonneg,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),A_3))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_1956_add__eq__0__iff,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [Xa,Ya] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa),Ya) = zero_zero(X_a)
        <=> ti(X_a,Ya) = hAPP(X_a,X_a,uminus_uminus(X_a),Xa) ) ) ).

fof(fact_1957_right__minus,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = zero_zero(X_a) ) ).

fof(fact_1958_eq__neg__iff__add__eq__0,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_3,B_2] :
          ( ti(X_a,A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),B_2)
        <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2) = zero_zero(X_a) ) ) ).

fof(fact_1959_left__minus,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),A_1) = zero_zero(X_a) ) ).

fof(fact_1960_ab__left__minus,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),A_1) = zero_zero(X_a) ) ).

fof(fact_1961_minus__unique,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1,B] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = zero_zero(X_a)
         => hAPP(X_a,X_a,uminus_uminus(X_a),A_1) = ti(X_a,B) ) ) ).

fof(fact_1962_norm__zero,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => hAPP(X_a,real,norm_norm(X_a),zero_zero(X_a)) = zero_zero(real) ) ).

fof(fact_1963_norm__eq__zero,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa] :
          ( hAPP(X_a,real,norm_norm(X_a),Xa) = zero_zero(real)
        <=> ti(X_a,Xa) = zero_zero(X_a) ) ) ).

fof(fact_1964_diff__0,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),zero_zero(X_a)),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1) ) ).

fof(fact_1965_square__eq__1__iff,axiom,
    ! [X_a] :
      ( ring_11004092258visors(X_a)
     => ! [Xa] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa) = one_one(X_a)
        <=> ( ti(X_a,Xa) = one_one(X_a)
            | ti(X_a,Xa) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ) ).

fof(fact_1966_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X] : hAPP(X_a,X_a,uminus_uminus(X_a),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) ) ).

fof(fact_1967_Groups_Odiff__def,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).

fof(fact_1968_ab__diff__minus,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).

fof(fact_1969_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).

fof(fact_1970_diff__minus__eq__add,axiom,
    ! [X_a] :
      ( group_add(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) ) ).

fof(fact_1971_norm__mult,axiom,
    ! [X_a] :
      ( real_n1866405975lgebra(X_a)
     => ! [X,Y] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y)) ) ).

fof(fact_1972_nonzero__minus__divide__right,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ) ).

fof(fact_1973_nonzero__minus__divide__divide,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) ) ) ).

fof(fact_1974_norm__ge__zero,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,norm_norm(X_a),X))) ) ).

fof(fact_1975_norm__not__less__zero,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),zero_zero(real))) ) ).

fof(fact_1976_abs__ge__minus__self,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),A_1))) ) ).

fof(fact_1977_abs__le__iff,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_3)),B_2))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),B_2)) ) ) ) ).

fof(fact_1978_abs__leI,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B)) ) ) ) ).

fof(fact_1979_abs__le__D2,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)) ) ) ).

fof(fact_1980_norm__one,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => hAPP(X_a,real,norm_norm(X_a),one_one(X_a)) = one_one(real) ) ).

fof(fact_1981_abs__less__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_3)),B_2))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),B_2)) ) ) ) ).

fof(fact_1982_norm__power,axiom,
    ! [X_a] :
      ( real_n1866405975lgebra(X_a)
     => ! [X,N] : hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(X_a,real,norm_norm(X_a),X)),N) ) ).

fof(fact_1983_abs__power__minus,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1,N] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),N)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) ) ).

fof(fact_1984_norm__divide,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & real_normed_field(X_a) )
     => ! [A_1,B] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)) ) ).

fof(fact_1985_dvd__neg__div,axiom,
    ! [X_a] :
      ( ring_div(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)) ) ) ).

fof(fact_1986_dvd__div__neg,axiom,
    ! [X_a] :
      ( ring_div(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)) ) ) ).

fof(fact_1987_complex__mod__triangle__ineq2,axiom,
    ! [B,A_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),A_1))),hAPP(complex,real,norm_norm(complex),B))),hAPP(complex,real,norm_norm(complex),A_1))) ).

fof(fact_1988_real__minus__mult__self__le,axiom,
    ! [U_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),U_1),U_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X))) ).

fof(fact_1989_real__add__eq__0__iff,axiom,
    ! [Xa,Ya] :
      ( hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya) = zero_zero(real)
    <=> Ya = hAPP(real,real,uminus_uminus(real),Xa) ) ).

fof(fact_1990_real__add__minus__iff,axiom,
    ! [Xa,A_3] :
      ( hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),hAPP(real,real,uminus_uminus(real),A_3)) = zero_zero(real)
    <=> Xa = A_3 ) ).

fof(fact_1991_abs__le__interval__iff,axiom,
    ! [Xa,Ra] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),Ra))
    <=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),Ra)),Xa))
        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ra)) ) ) ).

fof(fact_1992_real__diff__def,axiom,
    ! [R_3,S_3] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_3),S_3) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_3),hAPP(real,real,uminus_uminus(real),S_3)) ).

fof(fact_1993_minus__real__def,axiom,
    ! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),Y)) ).

fof(fact_1994_abs__minus__add__cancel,axiom,
    ! [X,Y] : hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),Y))) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Y),hAPP(real,real,uminus_uminus(real),X))) ).

fof(fact_1995_log__one,axiom,
    ! [A_1] : hAPP(real,real,hAPP(real,fun(real,real),log,A_1),one_one(real)) = zero_zero(real) ).

fof(fact_1996_Log_Olog__def,axiom,
    ! [A_1,X] : hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,A_1)) ).

fof(fact_1997_norm__le__zero__iff,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),Xa)),zero_zero(real)))
        <=> ti(X_a,Xa) = zero_zero(X_a) ) ) ).

fof(fact_1998_zero__less__norm__iff,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,norm_norm(X_a),Xa)))
        <=> ti(X_a,Xa) != zero_zero(X_a) ) ) ).

fof(fact_1999_power__minus,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [A_1,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) ) ).

fof(fact_2000_abs__of__nonpos,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
         => hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1) ) ) ).

fof(fact_2001_abs__minus__le__zero,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1))),zero_zero(X_a))) ) ).

fof(fact_2002_abs__of__neg,axiom,
    ! [X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
         => hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1) ) ) ).

fof(fact_2003_abs__if,axiom,
    ! [X_a] :
      ( abs_if(X_a)
     => ! [A_1] :
          ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
           => hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1) )
          & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
           => hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1) ) ) ) ).

fof(fact_2004_norm__mult__ineq,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y)))) ) ).

fof(fact_2005_norm__mult__less,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Y,S_3,X,R_3] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),R_3))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Y)),S_3))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),S_3))) ) ) ) ).

fof(fact_2006_nonzero__norm__divide,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [A_1,B] :
          ( ti(X_a,B) != zero_zero(X_a)
         => hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)) ) ) ).

fof(fact_2007_norm__triangle__ineq,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y)))) ) ).

fof(fact_2008_norm__add__less,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Y,S_3,X,R_3] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),R_3))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Y)),S_3))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_3),S_3))) ) ) ) ).

fof(fact_2009_realpow__two__disj,axiom,
    ! [X_a] :
      ( idom(X_a)
     => ! [Xa,Ya] :
          ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))
        <=> ( ti(X_a,Xa) = ti(X_a,Ya)
            | ti(X_a,Xa) = hAPP(X_a,X_a,uminus_uminus(X_a),Ya) ) ) ) ).

fof(fact_2010_norm__triangle__ineq4,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)))) ) ).

fof(fact_2011_norm__power__ineq,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => ! [X,N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(X_a,real,norm_norm(X_a),X)),N))) ) ).

fof(fact_2012_norm__diff__ineq,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)))) ) ).

fof(fact_2013_mult__minus1,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),min)),Z_1) = hAPP(X_a,X_a,uminus_uminus(X_a),Z_1) ) ).

fof(fact_2014_mult__minus1__right,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),hAPP(int,X_a,number_number_of(X_a),min)) = hAPP(X_a,X_a,uminus_uminus(X_a),Z_1) ) ).

fof(fact_2015_norm__triangle__ineq2,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).

fof(fact_2016_arith__simps_I31_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),min) ) ).

fof(fact_2017_number__of__Min,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => hAPP(int,X_a,number_number_of(X_a),min) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ).

fof(fact_2018_divide__minus1,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & number_ring(X_a) )
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),min)) = hAPP(X_a,X_a,uminus_uminus(X_a),X) ) ).

fof(fact_2019_norm__number__of,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & real_n2089651433ebra_1(X_a) )
     => ! [W] : hAPP(X_a,real,norm_norm(X_a),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),W)) ) ).

fof(fact_2020_real__0__le__add__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),Xa)),Ya)) ) ).

fof(fact_2021_real__add__le__0__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya)),zero_zero(real)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),hAPP(real,real,uminus_uminus(real),Xa))) ) ).

fof(fact_2022_real__0__less__add__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),Xa)),Ya)) ) ).

fof(fact_2023_real__add__less__0__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya)),zero_zero(real)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),hAPP(real,real,uminus_uminus(real),Xa))) ) ).

fof(fact_2024_abs__real__def,axiom,
    ! [A_1] :
      ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real)))
       => hAPP(real,real,abs_abs(real),A_1) = hAPP(real,real,uminus_uminus(real),A_1) )
      & ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real)))
       => hAPP(real,real,abs_abs(real),A_1) = A_1 ) ) ).

fof(fact_2025_real__abs__def,axiom,
    ! [R_3] :
      ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),zero_zero(real)))
       => hAPP(real,real,abs_abs(real),R_3) = hAPP(real,real,uminus_uminus(real),R_3) )
      & ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),zero_zero(real)))
       => hAPP(real,real,abs_abs(real),R_3) = R_3 ) ) ).

fof(fact_2026_abs__sum__triangle__ineq,axiom,
    ! [X,Y,L_4,M] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,uminus_uminus(real),L_4)),hAPP(real,real,uminus_uminus(real),M))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),L_4)))),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Y),hAPP(real,real,uminus_uminus(real),M)))))) ).

fof(fact_2027_abs__number__of,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & linordered_idom(X_a) )
     => ! [X] :
          ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X)),zero_zero(X_a)))
           => hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),X)) )
          & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X)),zero_zero(X_a)))
           => hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),X)) = hAPP(int,X_a,number_number_of(X_a),X) ) ) ) ).

fof(fact_2028_norm__ratiotest__lemma,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X,Y,C_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C_1),zero_zero(real)))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C_1),hAPP(X_a,real,norm_norm(X_a),Y))))
           => ti(X_a,X) = zero_zero(X_a) ) ) ) ).

fof(fact_2029_norm__diff__triangle__ineq,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B,C_1,D_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C_1))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D_3))))) ) ).

fof(fact_2030_log__eq__one,axiom,
    ! [A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ( A_1 != one_one(real)
       => hAPP(real,real,hAPP(real,fun(real,real),log,A_1),A_1) = one_one(real) ) ) ).

fof(fact_2031_log__less__cancel__iff,axiom,
    ! [Ya,Xa,A_3] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A_3))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_3),Xa)),hAPP(real,real,hAPP(real,fun(real,real),log,A_3),Ya)))
          <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) ) ) ) ) ).

fof(fact_2032_minus1__divide,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & number_ring(X_a) )
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),X)) ) ).

fof(fact_2033_norm__triangle__ineq3,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).

fof(fact_2034_log__ln,axiom,
    ! [X] : hAPP(real,real,ln,X) = hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(real,real,exp(real),one_one(real))),X) ).

fof(fact_2035_power2__minus,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).

fof(fact_2036_log__le__cancel__iff,axiom,
    ! [Ya,Xa,A_3] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A_3))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_3),Xa)),hAPP(real,real,hAPP(real,fun(real,real),log,A_3),Ya)))
          <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ) ) ) ).

fof(fact_2037_power2__eq__1__iff,axiom,
    ! [X_a] :
      ( ring_11004092258visors(X_a)
     => ! [A_3] :
          ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(X_a)
        <=> ( ti(X_a,A_3) = one_one(X_a)
            | ti(X_a,A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ) ).

fof(fact_2038_ring__1__class_Opower__minus__even,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [A_1,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) ) ).

fof(fact_2039_realpow__square__minus__le,axiom,
    ! [U_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),U_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2040_ln__one__minus__pos__upper__bound,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X))),hAPP(real,real,uminus_uminus(real),X))) ) ) ).

fof(fact_2041_aux5,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real)))
     => hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X))))) ) ).

fof(fact_2042_log__mult,axiom,
    ! [Y,X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ( A_1 != one_one(real)
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
           => hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),Y)) ) ) ) ) ).

fof(fact_2043_log__divide,axiom,
    ! [Y,X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ( A_1 != one_one(real)
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
           => hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),Y)) ) ) ) ) ).

fof(fact_2044_power__minus1__even,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = one_one(X_a) ) ).

fof(fact_2045_power__minus1__odd,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ).

fof(fact_2046_log__eq__div__ln__mult__log,axiom,
    ! [X,B,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ( A_1 != one_one(real)
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B))
         => ( B != one_one(real)
           => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,B)),hAPP(real,real,ln,A_1))),hAPP(real,real,hAPP(real,fun(real,real),log,B),X)) ) ) ) ) ) ).

fof(fact_2047_log__base__10__eq2,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,exp(real),one_one(real)))),hAPP(real,real,ln,X)) ) ).

fof(fact_2048_ln__one__minus__pos__lower__bound,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,uminus_uminus(real),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X)))) ) ) ).

fof(fact_2049_divide_Opos__bounded,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ) ).

fof(fact_2050_mult_Opos__bounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
          & ! [A_4,B_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2))) ) ) ).

fof(fact_2051_mult__right_Opos__bounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ) ).

fof(fact_2052_mult__left_Opos__bounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Y] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ) ).

fof(fact_2053_divide_Ononneg__bounded,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ) ).

fof(fact_2054_negative__zle,axiom,
    ! [N,M] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N))),hAPP(nat,int,semiring_1_of_nat(int),M))) ).

fof(fact_2055_negative__zless,axiom,
    ! [N,M] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N)))),hAPP(nat,int,semiring_1_of_nat(int),M))) ).

fof(fact_2056_complex__diff__def,axiom,
    ! [X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),hAPP(complex,complex,uminus_uminus(complex),Y)) ).

fof(fact_2057_zminus__zminus,axiom,
    ! [Z_1] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,uminus_uminus(int),Z_1)) = ti(int,Z_1) ).

fof(fact_2058_minus__Pls,axiom,
    hAPP(int,int,uminus_uminus(int),pls) = pls ).

fof(fact_2059_minus__Bit0,axiom,
    ! [K_4] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit0,K_4)) = hAPP(int,int,bit0,hAPP(int,int,uminus_uminus(int),K_4)) ).

fof(fact_2060_zminus__0,axiom,
    hAPP(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

fof(fact_2061_zmult__zminus,axiom,
    ! [Z_1,W] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,uminus_uminus(int),Z_1)),W) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),W)) ).

fof(fact_2062_zminus__zadd__distrib,axiom,
    ! [Z_1,W] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),W)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),Z_1)),hAPP(int,int,uminus_uminus(int),W)) ).

fof(fact_2063_minus__numeral__code_I5_J,axiom,
    ! [W] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),W)) = hAPP(int,int,number_number_of(int),hAPP(int,int,uminus_uminus(int),W)) ).

fof(fact_2064_zcong__zminus,axiom,
    ! [A_3,B_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),hAPP(int,int,uminus_uminus(int),Ma)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) ) ).

fof(fact_2065_uminus__dvd__conv_I1_J,axiom,
    ! [D_1,Ta] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),Ta))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,uminus_uminus(int),D_1)),Ta)) ) ).

fof(fact_2066_uminus__dvd__conv_I2_J,axiom,
    ! [D_1,Ta] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),Ta))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,uminus_uminus(int),Ta))) ) ).

fof(fact_2067_zdiv__zminus__zminus,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) ).

fof(fact_2068_zdiv__zminus2,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B) ).

fof(fact_2069_zminus__zmod,axiom,
    ! [X,M] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M))),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),X)),M) ).

fof(fact_2070_zmod__zminus__zminus,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) ).

fof(fact_2071_zmod__zminus2,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B)) ).

fof(fact_2072_even__neg,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,uminus_uminus(int),Xa)))
    <=> hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) ) ).

fof(fact_2073_zadd__zminus__inverse2,axiom,
    ! [Z_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),Z_1)),Z_1) = zero_zero(int) ).

fof(fact_2074_negative__eq__positive,axiom,
    ! [Na,Ma] :
      ( hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),Na)) = hAPP(nat,int,semiring_1_of_nat(int),Ma)
    <=> ( Na = zero_zero(nat)
        & Ma = zero_zero(nat) ) ) ).

fof(fact_2075_not__int__zless__negative,axiom,
    ! [N,M] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),N)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),M)))) ).

fof(fact_2076_diff__int__def__symmetric,axiom,
    ! [Z_1,W] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),hAPP(int,int,uminus_uminus(int),W)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),W) ).

fof(fact_2077_diff__int__def,axiom,
    ! [Z_1,W] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),W) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),hAPP(int,int,uminus_uminus(int),W)) ).

fof(fact_2078_Int_OMin__def,axiom,
    min = hAPP(int,int,uminus_uminus(int),one_one(int)) ).

fof(fact_2079_mult__Min,axiom,
    ! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),min),K_4) = hAPP(int,int,uminus_uminus(int),K_4) ).

fof(fact_2080_zmod__zminus1__not__zero,axiom,
    ! [K_4,L_4] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),K_4)),L_4) != zero_zero(int)
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4) != zero_zero(int) ) ).

fof(fact_2081_zmod__zminus2__not__zero,axiom,
    ! [K_4,L_4] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),hAPP(int,int,uminus_uminus(int),L_4)) != zero_zero(int)
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4) != zero_zero(int) ) ).

fof(fact_2082_diff__number__of__eq,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),W))) ) ).

fof(fact_2083_minus__Min,axiom,
    hAPP(int,int,uminus_uminus(int),min) = hAPP(int,int,bit1,pls) ).

fof(fact_2084_int__zle__neg,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),Ma))))
    <=> ( Na = zero_zero(nat)
        & Ma = zero_zero(nat) ) ) ).

fof(fact_2085_negative__zle__0,axiom,
    ! [N] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N))),zero_zero(int))) ).

fof(fact_2086_nat__zminus__int,axiom,
    ! [N] : hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

fof(fact_2087_minus__numeral__code_I6_J,axiom,
    ! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),W))) ).

fof(fact_2088_zabs__def,axiom,
    ! [I_1] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int)))
       => hAPP(int,int,abs_abs(int),I_1) = hAPP(int,int,uminus_uminus(int),I_1) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int)))
       => hAPP(int,int,abs_abs(int),I_1) = ti(int,I_1) ) ) ).

fof(fact_2089_zmod__zminus1__eq__if,axiom,
    ! [A_1,B] :
      ( ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int)
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = zero_zero(int) )
      & ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int)
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) ) ) ).

fof(fact_2090_zmod__zminus2__eq__if,axiom,
    ! [A_1,B] :
      ( ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int)
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = zero_zero(int) )
      & ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int)
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B) ) ) ).

fof(fact_2091_zdiv__minus1__right,axiom,
    ! [A_1] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,number_number_of(int),min)) = hAPP(int,int,uminus_uminus(int),A_1) ).

fof(fact_2092_neg__zminus__int,axiom,
    ! [N] : hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N))))) ).

fof(fact_2093_add__number__of__diff2,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V,C_1,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C_1),hAPP(int,X_a,number_number_of(X_a),W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),W)))),C_1) ) ).

fof(fact_2094_power3__minus,axiom,
    ! [A_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(int,int,uminus_uminus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).

fof(fact_2095_negative__zless__0,axiom,
    ! [N] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N)))),zero_zero(int))) ).

fof(fact_2096_not__zle__0__negative,axiom,
    ! [N] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N))))) ).

fof(fact_2097_nat__mult__distrib__neg,axiom,
    ! [Z_4,Z_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),zero_zero(int)))
     => hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),Z_1))),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),Z_4))) ) ).

fof(fact_2098_zdiv__zminus2__eq__if,axiom,
    ! [A_1,B] :
      ( ti(int,B) != zero_zero(int)
     => ( ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int)
         => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)) )
        & ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int)
         => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),one_one(int)) ) ) ) ).

fof(fact_2099_zdiv__zminus1__eq__if,axiom,
    ! [A_1,B] :
      ( ti(int,B) != zero_zero(int)
     => ( ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int)
         => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)) )
        & ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int)
         => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),one_one(int)) ) ) ) ).

fof(fact_2100_neg__even__power,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
       => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),A_3)),hAPP(int,nat,nat_1,Xa)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),hAPP(int,nat,nat_1,Xa)) ) ) ).

fof(fact_2101_neg__odd__power,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
       => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),A_3)),hAPP(int,nat,nat_1,Xa)) = hAPP(int,int,uminus_uminus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),hAPP(int,nat,nat_1,Xa))) ) ) ).

fof(fact_2102_zminus1__lemma,axiom,
    ! [A_3,B_2,Q_2,Ra] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_3),B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),Ra)))
     => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,uminus_uminus(int),A_3)),B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),Ra),zero_zero(int))),hAPP(int,int,uminus_uminus(int),Q_2)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,uminus_uminus(int),Q_2)),one_one(int)))),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),Ra),zero_zero(int))),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B_2),Ra))))) ) ).

fof(fact_2103_diff__special_I2_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))) ) ).

fof(fact_2104_diff__special_I1_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),W))) ) ).

fof(fact_2105_negD,axiom,
    ! [X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
     => ? [N_1] : ti(int,X) = hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1))) ) ).

fof(fact_2106_mult__left_Ononneg__bounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Y] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ) ).

fof(fact_2107_mult__right_Ononneg__bounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ) ).

fof(fact_2108_mult_Ononneg__bounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
          & ! [A_4,B_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2))) ) ) ).

fof(fact_2109_divide_Obounded,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y] :
        ? [K_2] :
        ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ).

fof(fact_2110_mult__left_Obounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Y] :
        ? [K_2] :
        ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ).

fof(fact_2111_mult__right_Obounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [X] :
        ? [K_2] :
        ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ).

fof(fact_2112_mult_Obounded,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ? [K_2] :
        ! [A_4,B_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2))) ) ).

fof(fact_2113_UnionE,axiom,
    ! [X_a,A_2,C_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),C_2)))
     => ~ ! [X_3] :
            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),X_3))
           => ~ hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_3),C_2)) ) ) ).

fof(fact_2114_exp__total,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ? [X_1] : hAPP(real,real,exp(real),X_1) = Y ) ).

fof(fact_2115_lemmaCauchy,axiom,
    ! [X_b,X_a] :
      ( ( ord(X_a)
        & real_normed_vector(X_b) )
     => ! [X_2,M_1] :
          ( ! [N_1] :
              ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),M_1),N_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,X_2,M_1)),hAPP(X_a,X_b,X_2,N_1)))),one_one(real))) )
         => ! [N_1] :
              ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),M_1),N_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,M_1))))) ) ) ) ).

fof(fact_2116_lemma__sqrt__hcomplex__capprox,axiom,
    ! [Y,X,U_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),U_1))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
           => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),U_1)) ) ) ) ) ) ).

fof(fact_2117_arctan__add,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),one_one(real)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),Y)),one_one(real)))
       => hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y)) = hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)))) ) ) ).

fof(fact_2118_eq__special_I4_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Xa] :
          ( hAPP(int,X_a,number_number_of(X_a),Xa) = one_one(X_a)
        <=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_2119_eq__special_I2_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Ya] :
          ( one_one(X_a) = hAPP(int,X_a,number_number_of(X_a),Ya)
        <=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),Ya))))) ) ) ).

fof(fact_2120_real__sqrt__divide,axiom,
    ! [X,Y] : hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y)) ).

fof(fact_2121_real__sqrt__le__mono,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y))) ) ).

fof(fact_2122_real__sqrt__le__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,Xa)),hAPP(real,real,sqrt,Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ).

fof(fact_2123_real__sqrt__eq__iff,axiom,
    ! [Xa,Ya] :
      ( hAPP(real,real,sqrt,Xa) = hAPP(real,real,sqrt,Ya)
    <=> Xa = Ya ) ).

fof(fact_2124_real__sqrt__less__mono,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y))) ) ).

fof(fact_2125_real__sqrt__less__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,Xa)),hAPP(real,real,sqrt,Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) ) ).

fof(fact_2126_real__sqrt__eq__0__iff,axiom,
    ! [Xa] :
      ( hAPP(real,real,sqrt,Xa) = zero_zero(real)
    <=> Xa = zero_zero(real) ) ).

fof(fact_2127_real__sqrt__zero,axiom,
    hAPP(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

fof(fact_2128_real__sqrt__one,axiom,
    hAPP(real,real,sqrt,one_one(real)) = one_one(real) ).

fof(fact_2129_real__sqrt__eq__1__iff,axiom,
    ! [Xa] :
      ( hAPP(real,real,sqrt,Xa) = one_one(real)
    <=> Xa = one_one(real) ) ).

fof(fact_2130_real__sqrt__power,axiom,
    ! [X,K_4] : hAPP(real,real,sqrt,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),K_4)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,X)),K_4) ).

fof(fact_2131_real__sqrt__mult,axiom,
    ! [X,Y] : hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y)) ).

fof(fact_2132_real__sqrt__minus,axiom,
    ! [X] : hAPP(real,real,sqrt,hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sqrt,X)) ).

fof(fact_2133_arctan__monotone_H,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y))) ) ).

fof(fact_2134_arctan__zero__zero,axiom,
    hAPP(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

fof(fact_2135_arctan__monotone,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y))) ) ).

fof(fact_2136_arctan__minus,axiom,
    ! [X] : hAPP(real,real,arctan,hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,arctan,X)) ).

fof(fact_2137_real__sqrt__eq__zero__cancel,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hAPP(real,real,sqrt,X) = zero_zero(real)
       => X = zero_zero(real) ) ) ).

fof(fact_2138_real__sqrt__ge__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,X))) ) ).

fof(fact_2139_real__sqrt__le__0__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,Xa)),zero_zero(real)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),zero_zero(real))) ) ).

fof(fact_2140_real__sqrt__ge__0__iff,axiom,
    ! [Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Ya)) ) ).

fof(fact_2141_real__sqrt__gt__0__iff,axiom,
    ! [Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya)) ) ).

fof(fact_2142_real__sqrt__lt__0__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,Xa)),zero_zero(real)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) ) ).

fof(fact_2143_real__sqrt__not__eq__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,sqrt,X) != zero_zero(real) ) ).

fof(fact_2144_real__sqrt__gt__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,X))) ) ).

fof(fact_2145_iszero__0,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => hBOOL(hAPP(X_a,bool,iszero(X_a),zero_zero(X_a))) ) ).

fof(fact_2146_iszero__def,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [Z] :
          ( hBOOL(hAPP(X_a,bool,iszero(X_a),Z))
        <=> ti(X_a,Z) = zero_zero(X_a) ) ) ).

fof(fact_2147_real__sqrt__ge__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,sqrt,X))) ) ).

fof(fact_2148_real__sqrt__le__1__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,Xa)),one_one(real)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real))) ) ).

fof(fact_2149_real__sqrt__ge__1__iff,axiom,
    ! [Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,sqrt,Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Ya)) ) ).

fof(fact_2150_real__sqrt__lt__1__iff,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,Xa)),one_one(real)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) ) ).

fof(fact_2151_real__sqrt__gt__1__iff,axiom,
    ! [Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,sqrt,Ya)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Ya)) ) ).

fof(fact_2152_not__iszero__1,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ~ hBOOL(hAPP(X_a,bool,iszero(X_a),one_one(X_a))) ) ).

fof(fact_2153_real__sqrt__abs2,axiom,
    ! [X] : hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)) = hAPP(real,real,abs_abs(real),X) ).

fof(fact_2154_le__real__sqrt__sumsq,axiom,
    ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),Y))))) ).

fof(fact_2155_iszero__number__of__Bit1,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & ring_char_0(X_a) )
     => ! [W] : ~ hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,W)))) ) ).

fof(fact_2156_iszero__Numeral0,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),pls))) ) ).

fof(fact_2157_iszero__number__of__Bit0,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & ring_char_0(X_a) )
     => ! [Wa] :
          ( hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,Wa))))
        <=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),Wa))) ) ) ).

fof(fact_2158_nonzero__number__of__Min,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ~ hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),min))) ) ).

fof(fact_2159_real__sqrt__mult__self__sum__ge__zero,axiom,
    ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),Y))))) ).

fof(fact_2160_not__iszero__Numeral1,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ~ hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)))) ) ).

fof(fact_2161_arctan__half,axiom,
    ! [X] : hAPP(real,real,arctan,X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) ).

fof(fact_2162_eq__number__of__eq,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Xa,Ya] :
          ( hAPP(int,X_a,number_number_of(X_a),Xa) = hAPP(int,X_a,number_number_of(X_a),Ya)
        <=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),hAPP(int,int,uminus_uminus(int),Ya))))) ) ) ).

fof(fact_2163_real__sqrt__two__ge__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2164_real__sqrt__two__gt__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2165_real__sqrt__two__gt__one,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2166_real__sqrt__abs,axiom,
    ! [X] : hAPP(real,real,sqrt,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(real,real,abs_abs(real),X) ).

fof(fact_2167_real__sqrt__pow2__iff,axiom,
    ! [Xa] :
      ( hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,Xa)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = Xa
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) ) ).

fof(fact_2168_real__sqrt__pow2,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = X ) ).

fof(fact_2169_real__sqrt__unique,axiom,
    ! [Y,X] :
      ( hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = X
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => hAPP(real,real,sqrt,X) = Y ) ) ).

fof(fact_2170_real__sqrt__pow2__gt__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_2171_lemma__real__divide__sqrt__less,axiom,
    ! [U_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),U_1))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),U_1)) ) ).

fof(fact_2172_real__sqrt__sum__squares__ge1,axiom,
    ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).

fof(fact_2173_real__sqrt__sum__squares__ge2,axiom,
    ! [Y,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).

fof(fact_2174_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A_1,C_1,B,D_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),C_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D_3)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),D_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ).

fof(fact_2175_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X,Y] :
      ( hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = X
     => Y = zero_zero(real) ) ).

fof(fact_2176_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X,Y] :
      ( hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = Y
     => X = zero_zero(real) ) ).

fof(fact_2177_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X,Y,Xa_2,Ya_1] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2178_real__sqrt__sum__squares__ge__zero,axiom,
    ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).

fof(fact_2179_real__sqrt__ge__abs1,axiom,
    ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).

fof(fact_2180_real__sqrt__ge__abs2,axiom,
    ! [Y,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Y)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).

fof(fact_2181_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X,Y,Xa_2,Ya_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ).

fof(fact_2182_eq__special_I1_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Ya] :
          ( zero_zero(X_a) = hAPP(int,X_a,number_number_of(X_a),Ya)
        <=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),Ya))))) ) ) ).

fof(fact_2183_eq__special_I3_J,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Xa] :
          ( hAPP(int,X_a,number_number_of(X_a),Xa) = zero_zero(X_a)
        <=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),hAPP(int,int,uminus_uminus(int),pls))))) ) ) ).

fof(fact_2184_cos__x__y__le__one,axiom,
    ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),one_one(real))) ).

fof(fact_2185_real__sqrt__sum__squares__less,axiom,
    ! [Y,X,U_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),U_1)) ) ) ).

fof(fact_2186_machin,axiom,
    hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,hAPP(int,int,bit1,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))))))))))) ).

fof(fact_2187_z3mod__def,axiom,
    ! [K_4,L_4] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
       => hAPP(int,int,hAPP(int,fun(int,int),z3mod,K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
       => hAPP(int,int,hAPP(int,fun(int,int),z3mod,K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),hAPP(int,int,uminus_uminus(int),L_4)) ) ) ).

fof(fact_2188_z3div__def,axiom,
    ! [K_4,L_4] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
       => hAPP(int,int,hAPP(int,fun(int,int),z3div,K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
       => hAPP(int,int,hAPP(int,fun(int,int),z3div,K_4),L_4) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),hAPP(int,int,uminus_uminus(int),L_4))) ) ) ).

fof(fact_2189_subsetD,axiom,
    ! [X_a,C,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1)) ) ) ).

fof(fact_2190_equalityI,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
       => ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ).

fof(fact_2191_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

fof(fact_2192_pi__ge__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),pi)) ).

fof(fact_2193_pi__gt__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),pi)) ).

fof(fact_2194_pi__not__less__zero,axiom,
    ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),pi),zero_zero(real))) ).

fof(fact_2195_Compl__anti__mono,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) ) ).

fof(fact_2196_Compl__subset__Compl__iff,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1)))
    <=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ).

fof(fact_2197_psubset__eq,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
    <=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
        & ti(fun(X_a,bool),A_2) != ti(fun(X_a,bool),B_1) ) ) ).

fof(fact_2198_subset__iff__psubset__eq,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
    <=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
        | ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ).

fof(fact_2199_psubset__imp__subset,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).

fof(fact_2200_psubset__subset__trans,axiom,
    ! [X_a,C_2,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2)) ) ) ).

fof(fact_2201_subset__psubset__trans,axiom,
    ! [X_a,C_2,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_1),C_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2)) ) ) ).

fof(fact_2202_equalityE,axiom,
    ! [X_a,A_2,B_1] :
      ( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
     => ~ ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
         => ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ) ).

fof(fact_2203_subset__trans,axiom,
    ! [X_a,C_2,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),C_2)) ) ) ).

fof(fact_2204_set__mp,axiom,
    ! [X_a,Xa,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),B_1)) ) ) ).

fof(fact_2205_set__rev__mp,axiom,
    ! [X_a,B_1,Xa,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),B_1)) ) ) ).

fof(fact_2206_in__mono,axiom,
    ! [X_a,Xa,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),B_1)) ) ) ).

fof(fact_2207_equalityD2,axiom,
    ! [X_a,A_2,B_1] :
      ( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ).

fof(fact_2208_equalityD1,axiom,
    ! [X_a,A_2,B_1] :
      ( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).

fof(fact_2209_set__eq__subset,axiom,
    ! [X_a,A_2,B_1] :
      ( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
    <=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
        & hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ) ).

fof(fact_2210_subset__refl,axiom,
    ! [X_a,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),A_2)) ).

fof(fact_2211_double__diff,axiom,
    ! [X_a,C_2,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2))
       => hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C_2),A_2)) = ti(fun(X_a,bool),A_2) ) ) ).

fof(fact_2212_Diff__mono,axiom,
    ! [X_a,D,B_1,A_2,C_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),C_2))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),D),B_1))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C_2),D))) ) ) ).

fof(fact_2213_Diff__subset,axiom,
    ! [X_a,A_2,B_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),A_2)) ).

fof(fact_2214_pi__ge__two,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi)) ).

fof(fact_2215_pi__less__4,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2216_pi__half__neq__two,axiom,
    hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ).

fof(fact_2217_pi__half__le__two,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_2218_pi__half__neq__zero,axiom,
    hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(real) ).

fof(fact_2219_pi__half__less__two,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_2220_pi__half__ge__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2221_pi__half__gt__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2222_m2pi__less__pi,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))),pi)) ).

fof(fact_2223_arctan__ubound,axiom,
    ! [Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2224_arctan1__eq__pi4,axiom,
    hAPP(real,real,arctan,one_one(real)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_2225_minus__pi__half__less__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(real))) ).

fof(fact_2226_arctan__lbound,axiom,
    ! [Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y))) ).

fof(fact_2227_arctan__bounded,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y)))
      & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_2228_arcsin__lt__bounded,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),one_one(real)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ) ).

fof(fact_2229_arcsin__lbound,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y))) ) ) ).

fof(fact_2230_arcsin__bounded,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ) ).

fof(fact_2231_arcsin__ubound,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_2232_DiffE,axiom,
    ! [X_a,C,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
     => ~ ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1)) ) ) ).

fof(fact_2233_DiffI,axiom,
    ! [X_a,B_1,C,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2))
     => ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) ) ) ).

fof(fact_2234_ComplI,axiom,
    ! [X_a,C,A_2] :
      ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) ) ).

fof(fact_2235_ComplE,axiom,
    ! [X_a,C,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)))
     => ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) ) ).

fof(fact_2236_ComplD,axiom,
    ! [X_a,C,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)))
     => ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) ) ).

fof(fact_2237_Compl__eq__Compl__iff,axiom,
    ! [X_a,A_2,B_1] :
      ( hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1)
    <=> ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ).

fof(fact_2238_Compl__iff,axiom,
    ! [X_a,C,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)))
    <=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) ) ).

fof(fact_2239_double__complement,axiom,
    ! [X_a,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)) = ti(fun(X_a,bool),A_2) ).

fof(fact_2240_Compl__eq,axiom,
    ! [X_a,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))) ).

fof(fact_2241_DiffD2,axiom,
    ! [X_a,C,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
     => ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1)) ) ).

fof(fact_2242_DiffD1,axiom,
    ! [X_a,C,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) ) ).

fof(fact_2243_Diff__idemp,axiom,
    ! [X_a,A_2,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) ).

fof(fact_2244_Diff__iff,axiom,
    ! [X_a,C,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
    <=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2))
        & ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1)) ) ) ).

fof(fact_2245_set__diff__eq,axiom,
    ! [X_a,A_2,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1)))) ).

fof(fact_2246_psubsetD,axiom,
    ! [X_a,C,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1)) ) ) ).

fof(fact_2247_psubset__trans,axiom,
    ! [X_a,C_2,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_1),C_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2)) ) ) ).

fof(fact_2248_arctan__inverse,axiom,
    ! [X] :
      ( X != zero_zero(real)
     => hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),X)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sgn_sgn(real),X)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(real,real,arctan,X)) ) ).

fof(fact_2249_subsetI,axiom,
    ! [X_a,B_1,A_2] :
      ( ! [X_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1)) )
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).

fof(fact_2250_cos__arcsin,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => hAPP(real,real,cos,hAPP(real,real,arcsin,X)) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_2251_arcsin,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
          & hAPP(real,real,sin,hAPP(real,real,arcsin,Y)) = Y ) ) ) ).

fof(fact_2252_sin__diff,axiom,
    ! [X,Y] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,sin,Y))) ).

fof(fact_2253_sin__diff2,axiom,
    ! [X,Y] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Y)),hAPP(real,real,sin,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,Y)),hAPP(real,real,cos,X))) ).

fof(fact_2254_sin__minus,axiom,
    ! [X] : hAPP(real,real,sin,hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X)) ).

fof(fact_2255_cos__minus,axiom,
    ! [X] : hAPP(real,real,cos,hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,cos,X) ).

fof(fact_2256_sin__zero,axiom,
    hAPP(real,real,sin,zero_zero(real)) = zero_zero(real) ).

fof(fact_2257_sgn__sgn,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),A_1)) = hAPP(X_a,X_a,sgn_sgn(X_a),A_1) ) ).

fof(fact_2258_cos__one__sin__zero,axiom,
    ! [X] :
      ( hAPP(real,real,cos,X) = one_one(real)
     => hAPP(real,real,sin,X) = zero_zero(real) ) ).

fof(fact_2259_sin__add,axiom,
    ! [X,Y] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,sin,Y))) ).

fof(fact_2260_sgn0,axiom,
    ! [X_a] :
      ( sgn_if(X_a)
     => hAPP(X_a,X_a,sgn_sgn(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_2261_sgn__0__0,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hAPP(X_a,X_a,sgn_sgn(X_a),A_3) = zero_zero(X_a)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_2262_sgn__zero,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => hAPP(X_a,X_a,sgn_sgn(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_2263_sgn__zero__iff,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa] :
          ( hAPP(X_a,X_a,sgn_sgn(X_a),Xa) = zero_zero(X_a)
        <=> ti(X_a,Xa) = zero_zero(X_a) ) ) ).

fof(fact_2264_sgn__times,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),A_1)),hAPP(X_a,X_a,sgn_sgn(X_a),B)) ) ).

fof(fact_2265_sgn__mult,axiom,
    ! [X_a] :
      ( real_n1866405975lgebra(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)),hAPP(X_a,X_a,sgn_sgn(X_a),Y)) ) ).

fof(fact_2266_sgn__one,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => hAPP(X_a,X_a,sgn_sgn(X_a),one_one(X_a)) = one_one(X_a) ) ).

fof(fact_2267_sin__cos__squared__add3,axiom,
    ! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,X))) = one_one(real) ).

fof(fact_2268_sin__zero__abs__cos__one,axiom,
    ! [X] :
      ( hAPP(real,real,sin,X) = zero_zero(real)
     => hAPP(real,real,abs_abs(real),hAPP(real,real,cos,X)) = one_one(real) ) ).

fof(fact_2269_cos__diff2,axiom,
    ! [X,Y] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,Y)),hAPP(real,real,sin,X))) ).

fof(fact_2270_cos__diff,axiom,
    ! [X,Y] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,Y))) ).

fof(fact_2271_cos__add,axiom,
    ! [X,Y] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,Y))) ).

fof(fact_2272_sgn__minus,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)) ) ).

fof(fact_2273_sin__le__one,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sin,X)),one_one(real))) ).

fof(fact_2274_cos__le__one,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,X)),one_one(real))) ).

fof(fact_2275_cos__zero,axiom,
    hAPP(real,real,cos,zero_zero(real)) = one_one(real) ).

fof(fact_2276_sin__pi,axiom,
    hAPP(real,real,sin,pi) = zero_zero(real) ).

fof(fact_2277_cos__arctan__not__zero,axiom,
    ! [X] : hAPP(real,real,cos,hAPP(real,real,arctan,X)) != zero_zero(real) ).

fof(fact_2278_sin__double,axiom,
    ! [X] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,sin,X))),hAPP(real,real,cos,X)) ).

fof(fact_2279_sgn__less,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),A_3)),zero_zero(X_a)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_2280_sgn__greater,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,sgn_sgn(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_2281_real__sgn__eq,axiom,
    ! [X] : hAPP(real,real,sgn_sgn(real),X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,abs_abs(real),X)) ).

fof(fact_2282_mult__sgn__abs,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)),hAPP(X_a,X_a,abs_abs(X_a),X)) = ti(X_a,X) ) ).

fof(fact_2283_abs__sgn,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [K_4] : hAPP(X_a,X_a,abs_abs(X_a),K_4) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_4),hAPP(X_a,X_a,sgn_sgn(X_a),K_4)) ) ).

fof(fact_2284_sin__ge__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sin,X))) ) ) ).

fof(fact_2285_sin__gt__zero__pi,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),pi))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X))) ) ) ).

fof(fact_2286_sin__ge__minus__one,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),hAPP(real,real,sin,X))) ).

fof(fact_2287_cos__monotone__0__pi_H,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) ) ) ) ).

fof(fact_2288_abs__sin__le__one,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,sin,X))),one_one(real))) ).

fof(fact_2289_cos__ge__minus__one,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),hAPP(real,real,cos,X))) ).

fof(fact_2290_abs__cos__le__one,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,cos,X))),one_one(real))) ).

fof(fact_2291_sin__periodic__pi2,axiom,
    ! [X] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),pi),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X)) ).

fof(fact_2292_sin__periodic__pi,axiom,
    ! [X] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X)) ).

fof(fact_2293_cos__pi,axiom,
    hAPP(real,real,cos,pi) = hAPP(int,real,number_number_of(real),min) ).

fof(fact_2294_cos__periodic__pi,axiom,
    ! [X] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,cos,X)) ).

fof(fact_2295_sin__cos__squared__add2__mult,axiom,
    ! [R_3,A_1] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(real,real,cos,A_1))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(real,real,sin,A_1))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2296_sin__cos__squared__add,axiom,
    ! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = one_one(real) ).

fof(fact_2297_sin__cos__squared__add2,axiom,
    ! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = one_one(real) ).

fof(fact_2298_sin__squared__eq,axiom,
    ! [X] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_2299_cos__squared__eq,axiom,
    ! [X] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_2300_cos__sin__eq,axiom,
    ! [X] : hAPP(real,real,cos,X) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X)) ).

fof(fact_2301_sin__cos__eq,axiom,
    ! [X] : hAPP(real,real,sin,X) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X)) ).

fof(fact_2302_real__sgn__pos,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,sgn_sgn(real),X) = one_one(real) ) ).

fof(fact_2303_sgn__pos,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => hAPP(X_a,X_a,sgn_sgn(X_a),A_1) = one_one(X_a) ) ) ).

fof(fact_2304_sgn__1__pos,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hAPP(X_a,X_a,sgn_sgn(X_a),A_3) = one_one(X_a)
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_2305_cos__double,axiom,
    ! [X] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_2306_minus__sin__cos__eq,axiom,
    ! [X] : hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X)) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2307_cos__monotone__0__pi,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) ) ) ) ).

fof(fact_2308_cos__monotone__minus__pi__0_H,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),pi)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))) ) ) ) ).

fof(fact_2309_sin__cos__add,axiom,
    ! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,sin,Y))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,Y))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real) ).

fof(fact_2310_sin__cos__minus,axiom,
    ! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,sin,hAPP(real,real,uminus_uminus(real),X))),hAPP(real,real,sin,X))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,cos,hAPP(real,real,uminus_uminus(real),X))),hAPP(real,real,cos,X))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real) ).

fof(fact_2311_real__sgn__neg,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
     => hAPP(real,real,sgn_sgn(real),X) = hAPP(int,real,number_number_of(real),min) ) ).

fof(fact_2312_real__sgn__def,axiom,
    ! [X] :
      ( ( X = zero_zero(real)
       => hAPP(real,real,sgn_sgn(real),X) = zero_zero(real) )
      & ( X != zero_zero(real)
       => ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
           => hAPP(real,real,sgn_sgn(real),X) = one_one(real) )
          & ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
           => hAPP(real,real,sgn_sgn(real),X) = hAPP(real,real,uminus_uminus(real),one_one(real)) ) ) ) ) ).

fof(fact_2313_sgn__real__def,axiom,
    ! [A_1] :
      ( ( A_1 = zero_zero(real)
       => hAPP(real,real,sgn_sgn(real),A_1) = zero_zero(real) )
      & ( A_1 != zero_zero(real)
       => ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
           => hAPP(real,real,sgn_sgn(real),A_1) = one_one(real) )
          & ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
           => hAPP(real,real,sgn_sgn(real),A_1) = hAPP(real,real,uminus_uminus(real),one_one(real)) ) ) ) ) ).

fof(fact_2314_cos__two__neq__zero,axiom,
    hAPP(real,real,cos,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(real) ).

fof(fact_2315_sgn__if,axiom,
    ! [X_a] :
      ( sgn_if(X_a)
     => ! [X] :
          ( ( ti(X_a,X) = zero_zero(X_a)
           => hAPP(X_a,X_a,sgn_sgn(X_a),X) = zero_zero(X_a) )
          & ( ti(X_a,X) != zero_zero(X_a)
           => ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
               => hAPP(X_a,X_a,sgn_sgn(X_a),X) = one_one(X_a) )
              & ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
               => hAPP(X_a,X_a,sgn_sgn(X_a),X) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ) ) ) ).

fof(fact_2316_sgn__neg,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
         => hAPP(X_a,X_a,sgn_sgn(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ).

fof(fact_2317_sgn__1__neg,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hAPP(X_a,X_a,sgn_sgn(X_a),A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_2318_cos__monotone__minus__pi__0,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),pi)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))) ) ) ) ).

fof(fact_2319_norm__sgn,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] :
          ( ( ti(X_a,X) = zero_zero(X_a)
           => hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)) = zero_zero(real) )
          & ( ti(X_a,X) != zero_zero(X_a)
           => hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)) = one_one(real) ) ) ) ).

fof(fact_2320_sin__arcsin,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => hAPP(real,real,sin,hAPP(real,real,arcsin,Y)) = Y ) ) ).

fof(fact_2321_sin__gt__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X))) ) ) ).

fof(fact_2322_cos__two__le__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),zero_zero(real))) ).

fof(fact_2323_cos__two__less__zero,axiom,
    hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),zero_zero(real))) ).

fof(fact_2324_sin__two__pi,axiom,
    hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi)) = zero_zero(real) ).

fof(fact_2325_sin__periodic,axiom,
    ! [X] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,sin,X) ).

fof(fact_2326_sin__pi__half,axiom,
    hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = one_one(real) ).

fof(fact_2327_sin__30,axiom,
    hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2328_cos__pi__half,axiom,
    hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real) ).

fof(fact_2329_sin__45,axiom,
    hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2330_sin__60,axiom,
    hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2331_cos__two__pi,axiom,
    hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi)) = one_one(real) ).

fof(fact_2332_cos__periodic,axiom,
    ! [X] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,cos,X) ).

fof(fact_2333_cos__60,axiom,
    hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2334_cos__30,axiom,
    hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2335_cos__45,axiom,
    hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2336_sin__gt__zero2,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X))) ) ) ).

fof(fact_2337_cos__double__less__one,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),one_one(real))) ) ) ).

fof(fact_2338_sin__monotone__2pi_H,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sin,Y)),hAPP(real,real,sin,X))) ) ) ) ).

fof(fact_2339_cos__gt__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,cos,X))) ) ) ).

fof(fact_2340_cos__3over2__pi,axiom,
    hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),pi)) = zero_zero(real) ).

fof(fact_2341_sin__less__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sin,X)),zero_zero(real))) ) ) ).

fof(fact_2342_sin__3over2__pi,axiom,
    hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),pi)) = hAPP(real,real,uminus_uminus(real),one_one(real)) ).

fof(fact_2343_cos__ge__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,cos,X))) ) ) ).

fof(fact_2344_cos__gt__zero__pi,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,cos,X))) ) ) ).

fof(fact_2345_arcsin__sin,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hAPP(real,real,arcsin,hAPP(real,real,sin,X)) = X ) ) ).

fof(fact_2346_arcsin__pi,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),pi))
          & hAPP(real,real,sin,hAPP(real,real,arcsin,Y)) = Y ) ) ) ).

fof(fact_2347_sin__total,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ? [X_1] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
            & hAPP(real,real,sin,X_1) = Y
            & ! [Y_1] :
                ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y_1))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
                  & hAPP(real,real,sin,Y_1) = Y )
               => Y_1 = X_1 ) ) ) ) ).

fof(fact_2348_cos__total,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ? [X_1] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),pi))
            & hAPP(real,real,cos,X_1) = Y
            & ! [Y_1] :
                ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),pi))
                  & hAPP(real,real,cos,Y_1) = Y )
               => Y_1 = X_1 ) ) ) ) ).

fof(fact_2349_cos__is__zero,axiom,
    ? [X_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
      & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
      & hAPP(real,real,cos,X_1) = zero_zero(real)
      & ! [Y_1] :
          ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
            & hAPP(real,real,cos,Y_1) = zero_zero(real) )
         => Y_1 = X_1 ) ) ).

fof(fact_2350_sin__arccos,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => hAPP(real,real,sin,hAPP(real,real,arccos,X)) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_2351_zsgn__def,axiom,
    ! [I_1] :
      ( ( ti(int,I_1) = zero_zero(int)
       => hAPP(int,int,sgn_sgn(int),I_1) = zero_zero(int) )
      & ( ti(int,I_1) != zero_zero(int)
       => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),I_1))
           => hAPP(int,int,sgn_sgn(int),I_1) = one_one(int) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),I_1))
           => hAPP(int,int,sgn_sgn(int),I_1) = hAPP(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

fof(fact_2352_arccos__cos,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi))
       => hAPP(real,real,arccos,hAPP(real,real,cos,X)) = X ) ) ).

fof(fact_2353_cos__arccos__abs,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Y)),one_one(real)))
     => hAPP(real,real,cos,hAPP(real,real,arccos,Y)) = Y ) ).

fof(fact_2354_arccos__lbound,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y))) ) ) ).

fof(fact_2355_cos__arccos,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => hAPP(real,real,cos,hAPP(real,real,arccos,Y)) = Y ) ) ).

fof(fact_2356_arccos__ubound,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi)) ) ) ).

fof(fact_2357_arccos__cos2,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),pi)),X))
       => hAPP(real,real,arccos,hAPP(real,real,cos,X)) = hAPP(real,real,uminus_uminus(real),X) ) ) ).

fof(fact_2358_arccos__bounded,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi)) ) ) ) ).

fof(fact_2359_arccos__lt__bounded,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),one_one(real)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,arccos,Y)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arccos,Y)),pi)) ) ) ) ).

fof(fact_2360_arccos,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi))
          & hAPP(real,real,cos,hAPP(real,real,arccos,Y)) = Y ) ) ) ).

fof(fact_2361_cos__arccos__lemma1,axiom,
    ! [X_4,Y_3] : hAPP(real,real,cos,hAPP(real,real,arccos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_4),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_4),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).

fof(fact_2362_sin__arccos__abs,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Y)),one_one(real)))
     => hAPP(real,real,sin,hAPP(real,real,arccos,Y)) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_2363_sin__arccos__lemma1,axiom,
    ! [X_4,Y_3] : hAPP(real,real,sin,hAPP(real,real,arccos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_4),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_4),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2364_tan__half,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hAPP(real,real,tan,X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),one_one(real))) ) ) ).

fof(fact_2365_tan__double,axiom,
    ! [X] :
      ( hAPP(real,real,cos,X) != zero_zero(real)
     => ( hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) != zero_zero(real)
       => hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,tan,X))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,tan,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).

fof(fact_2366_sin__cos__npi,axiom,
    ! [N] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N) ).

fof(fact_2367_cos__expansion__lemma,axiom,
    ! [X,M] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,M))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).

fof(fact_2368_real__of__nat__ge__zero,axiom,
    ! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,real_1(nat),N))) ).

fof(fact_2369_tan__minus,axiom,
    ! [X] : hAPP(real,real,tan,hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,tan,X)) ).

fof(fact_2370_natceiling__real__of__nat,axiom,
    ! [N] : hAPP(real,nat,natceiling,hAPP(nat,real,real_1(nat),N)) = N ).

fof(fact_2371_real__of__nat__inject,axiom,
    ! [Na,Ma] :
      ( hAPP(nat,real,real_1(nat),Na) = hAPP(nat,real,real_1(nat),Ma)
    <=> Na = Ma ) ).

fof(fact_2372_tan__zero,axiom,
    hAPP(real,real,tan,zero_zero(real)) = zero_zero(real) ).

fof(fact_2373_abs__real__of__nat__cancel,axiom,
    ! [X] : hAPP(real,real,abs_abs(real),hAPP(nat,real,real_1(nat),X)) = hAPP(nat,real,real_1(nat),X) ).

fof(fact_2374_natfloor__real__of__nat,axiom,
    ! [N] : hAPP(real,nat,natfloor,hAPP(nat,real,real_1(nat),N)) = N ).

fof(fact_2375_tan__arctan,axiom,
    ! [Y] : hAPP(real,real,tan,hAPP(real,real,arctan,Y)) = Y ).

fof(fact_2376_tan__npi,axiom,
    ! [N] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),pi)) = zero_zero(real) ).

fof(fact_2377_tan__periodic__nat,axiom,
    ! [X,N] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),pi))) = hAPP(real,real,tan,X) ).

fof(fact_2378_real__of__nat__def,axiom,
    real_1(nat) = semiring_1_of_nat(real) ).

fof(fact_2379_real__eq__of__nat,axiom,
    real_1(nat) = semiring_1_of_nat(real) ).

fof(fact_2380_real__of__nat__zero,axiom,
    hAPP(nat,real,real_1(nat),zero_zero(nat)) = zero_zero(real) ).

fof(fact_2381_real__of__nat__zero__iff,axiom,
    ! [Na] :
      ( hAPP(nat,real,real_1(nat),Na) = zero_zero(real)
    <=> Na = zero_zero(nat) ) ).

fof(fact_2382_not__real__of__nat__less__zero,axiom,
    ! [N] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),zero_zero(real))) ).

fof(fact_2383_real__of__nat__less__iff,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,real_1(nat),Ma)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) ) ).

fof(fact_2384_real__of__nat__le__iff,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,real_1(nat),Ma)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),Ma)) ) ).

fof(fact_2385_real__of__nat__mult,axiom,
    ! [M,N] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M)),hAPP(nat,real,real_1(nat),N)) ).

fof(fact_2386_real__of__nat__add,axiom,
    ! [M,N] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),M)),hAPP(nat,real,real_1(nat),N)) ).

fof(fact_2387_real__of__nat__1,axiom,
    hAPP(nat,real,real_1(nat),one_one(nat)) = one_one(real) ).

fof(fact_2388_power__real__of__nat,axiom,
    ! [M,N] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,real_1(nat),M)),N) = hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N)) ).

fof(fact_2389_real__of__nat__power,axiom,
    ! [M,N] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,real_1(nat),M)),N) ).

fof(fact_2390_real__natceiling__ge,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(nat,real,real_1(nat),hAPP(real,nat,natceiling,X)))) ).

fof(fact_2391_tan__pi,axiom,
    hAPP(real,real,tan,pi) = zero_zero(real) ).

fof(fact_2392_tan__periodic__pi,axiom,
    ! [X] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,tan,X) ).

fof(fact_2393_real__of__nat__le__zero__cancel__iff,axiom,
    ! [Na] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),Na)),zero_zero(real)))
    <=> Na = zero_zero(nat) ) ).

fof(fact_2394_real__of__nat__Suc__gt__zero,axiom,
    ! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N)))) ).

fof(fact_2395_real__of__nat__one,axiom,
    hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,zero_zero(nat))) = one_one(real) ).

fof(fact_2396_real__of__nat__Suc,axiom,
    ! [N] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N)),one_one(real)) ).

fof(fact_2397_real__of__nat__diff,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,real_1(nat),M)),hAPP(nat,real,real_1(nat),N)) ) ).

fof(fact_2398_real__of__nat__div4,axiom,
    ! [N,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),X))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),X)))) ).

fof(fact_2399_real__natfloor__le,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))),X)) ) ).

fof(fact_2400_le__natfloor,axiom,
    ! [X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),X)),A_1))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(real,nat,natfloor,A_1))) ) ).

fof(fact_2401_exp__real__of__nat__mult,axiom,
    ! [N,X] : hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),X)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,exp(real),X)),N) ).

fof(fact_2402_natceiling__le,axiom,
    ! [X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(nat,real,real_1(nat),A_1)))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X)),A_1)) ) ).

fof(fact_2403_natfloor__power,axiom,
    ! [N,X] :
      ( X = hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))
     => hAPP(real,nat,natfloor,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(real,nat,natfloor,X)),N) ) ).

fof(fact_2404_tan__def,axiom,
    ! [X] : hAPP(real,real,tan,X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,X)) ).

fof(fact_2405_real__of__nat__gt__zero__cancel__iff,axiom,
    ! [Na] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),Na)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).

fof(fact_2406_nat__less__real__le,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),Na)),one_one(real))),hAPP(nat,real,real_1(nat),Ma))) ) ).

fof(fact_2407_nat__le__real__less,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),Ma))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),Na)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),Ma)),one_one(real)))) ) ).

fof(fact_2408_sin__npi,axiom,
    ! [N] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),pi)) = zero_zero(real) ).

fof(fact_2409_sin__npi2,axiom,
    ! [N] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(nat),N))) = zero_zero(real) ).

fof(fact_2410_le__natfloor__eq,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_3),hAPP(real,nat,natfloor,Xa)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),A_3)),Xa)) ) ) ).

fof(fact_2411_natceiling__le__eq,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,Xa)),A_3))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(nat,real,real_1(nat),A_3))) ) ) ).

fof(fact_2412_real__natfloor__add__one__gt,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))),one_one(real)))) ).

fof(fact_2413_natfloor__subtract,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),A_1)),X))
     => hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(real,nat,natfloor,X)),A_1) ) ).

fof(fact_2414_real__natfloor__gt__diff__one,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real))),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X)))) ).

fof(fact_2415_natceiling__subtract,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),A_1)),X))
     => hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(real,nat,natceiling,X)),A_1) ) ).

fof(fact_2416_tan__periodic__n,axiom,
    ! [X,N] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),N)),pi))) = hAPP(real,real,tan,X) ).

fof(fact_2417_lemma__NBseq__def,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [X_2] :
          ( ? [K_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
              & ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),K_2)) )
        <=> ? [N_2] :
            ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2)))) ) ) ).

fof(fact_2418_lemma__NBseq__def2,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [X_2] :
          ( ? [K_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
              & ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),K_2)) )
        <=> ? [N_2] :
            ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2)))) ) ) ).

fof(fact_2419_real__of__nat__div,axiom,
    ! [N,D_3] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),D_3))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),N))
       => hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),D_3)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),D_3)) ) ) ).

fof(fact_2420_ln__realpow,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,ln,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(real,real,ln,X)) ) ).

fof(fact_2421_less__natfloor,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),N)))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(real,nat,natfloor,X)),N)) ) ) ).

fof(fact_2422_real__of__nat__div2,axiom,
    ! [N,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),X))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),X))))) ).

fof(fact_2423_real__of__nat__number__of,axiom,
    ! [V] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,real,real_1(nat),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(real) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
       => hAPP(nat,real,real_1(nat),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,real,number_number_of(real),V) ) ) ).

fof(fact_2424_real__of__nat__div3,axiom,
    ! [N,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),X))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),X)))),one_one(real))) ).

fof(fact_2425_natfloor__add,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),A_1) ) ).

fof(fact_2426_ge__natfloor__plus__one__imp__gt,axiom,
    ! [Z_1,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,Z_1)),one_one(nat))),N))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z_1),hAPP(nat,real,real_1(nat),N))) ) ).

fof(fact_2427_log__nat__power,axiom,
    ! [B,N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,hAPP(real,fun(real,real),log,B),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(real,real,hAPP(real,fun(real,real),log,B),X)) ) ).

fof(fact_2428_natfloor__eq,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N)),one_one(real))))
       => hAPP(real,nat,natfloor,X) = N ) ) ).

fof(fact_2429_natceiling__add,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),A_1) ) ).

fof(fact_2430_two__realpow__gt,axiom,
    ! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) ).

fof(fact_2431_LIMSEQ__inverse__realpow__zero__lemma,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),X)),one_one(real))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))),N))) ) ).

fof(fact_2432_real__of__nat__div__aux,axiom,
    ! [X,D_3] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),D_3))
     => hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),X)),hAPP(nat,real,real_1(nat),D_3)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),D_3))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),D_3))),hAPP(nat,real,real_1(nat),D_3))) ) ).

fof(fact_2433_cos__npi,axiom,
    ! [N] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),pi)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N) ).

fof(fact_2434_cos__npi2,axiom,
    ! [N] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(nat),N))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N) ).

fof(fact_2435_tan__60,axiom,
    hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).

fof(fact_2436_add__tan__eq,axiom,
    ! [Y,X] :
      ( hAPP(real,real,cos,X) != zero_zero(real)
     => ( hAPP(real,real,cos,Y) != zero_zero(real)
       => hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) ) ) ).

fof(fact_2437_fact__lemma,axiom,
    ! [N] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N)) ).

fof(fact_2438_natfloor__div__nat,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
       => hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(nat,real,real_1(nat),Y))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(real,nat,natfloor,X)),Y) ) ) ).

fof(fact_2439_natceiling__eq,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N)),one_one(real))))
       => hAPP(real,nat,natceiling,X) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ) ) ).

fof(fact_2440_tan__periodic,axiom,
    ! [X] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,tan,X) ).

fof(fact_2441_tan__45,axiom,
    hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = one_one(real) ).

fof(fact_2442_tan__add,axiom,
    ! [Y,X] :
      ( hAPP(real,real,cos,X) != zero_zero(real)
     => ( hAPP(real,real,cos,Y) != zero_zero(real)
       => ( hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) != zero_zero(real)
         => hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y)))) ) ) ) ).

fof(fact_2443_lemma__tan__add1,axiom,
    ! [Y,X] :
      ( hAPP(real,real,cos,X) != zero_zero(real)
     => ( hAPP(real,real,cos,Y) != zero_zero(real)
       => hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) ) ) ).

fof(fact_2444_sin__2npi,axiom,
    ! [N] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,real_1(nat),N))),pi)) = zero_zero(real) ).

fof(fact_2445_cos__2npi,axiom,
    ! [N] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,real_1(nat),N))),pi)) = one_one(real) ).

fof(fact_2446_tan__gt__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,tan,X))) ) ) ).

fof(fact_2447_tan__monotone_H,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Ya))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Xa))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
           => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),Xa))
            <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,Ya)),hAPP(real,real,tan,Xa))) ) ) ) ) ) ).

fof(fact_2448_tan__monotone,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,Y)),hAPP(real,real,tan,X))) ) ) ) ).

fof(fact_2449_tan__inverse,axiom,
    ! [Y] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,tan,Y)) = hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Y)) ).

fof(fact_2450_tan__30,axiom,
    hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).

fof(fact_2451_tan__less__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,X)),zero_zero(real))) ) ) ).

fof(fact_2452_arctan,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y)))
      & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
      & hAPP(real,real,tan,hAPP(real,real,arctan,Y)) = Y ) ).

fof(fact_2453_arctan__tan,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hAPP(real,real,arctan,hAPP(real,real,tan,X)) = X ) ) ).

fof(fact_2454_sin__expansion__lemma,axiom,
    ! [X,M] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,M))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2455_cos__pi__eq__zero,axiom,
    ! [M] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real) ).

fof(fact_2456_tan__total__pi4,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),one_one(real)))
     => ? [Z_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),Z_2))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
          & hAPP(real,real,tan,Z_2) = X ) ) ).

fof(fact_2457_tan__total__pos,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ? [X_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
          & hAPP(real,real,tan,X_1) = Y ) ) ).

fof(fact_2458_lemma__tan__total1,axiom,
    ! [Y] :
    ? [X_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1))
      & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
      & hAPP(real,real,tan,X_1) = Y ) ).

fof(fact_2459_tan__total,axiom,
    ! [Y] :
    ? [X_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1))
      & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
      & hAPP(real,real,tan,X_1) = Y
      & ! [Y_1] :
          ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y_1))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
            & hAPP(real,real,tan,Y_1) = Y )
         => Y_1 = X_1 ) ) ).

fof(fact_2460_lemma__tan__total,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ? [X_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),hAPP(real,real,tan,X_1))) ) ) ).

fof(fact_2461_reals__Archimedean6,axiom,
    ! [R_3] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_3))
     => ? [N_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat)))),R_3))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(nat,real,real_1(nat),N_1))) ) ) ).

fof(fact_2462_reals__Archimedean4,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ? [N_1] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),Y)),X))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1))),Y))) ) ) ) ).

fof(fact_2463_reals__Archimedean6a,axiom,
    ! [R_3] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_3))
     => ? [N_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N_1)),R_3))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1)))) ) ) ).

fof(fact_2464_reals__Archimedean3,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ! [Y_1] :
        ? [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),X))) ) ).

fof(fact_2465_lemma__STAR__cos,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),N)) = zero_zero(real) ) ).

fof(fact_2466_sin__zero__iff,axiom,
    ! [Xa] :
      ( hAPP(real,real,sin,Xa) = zero_zero(real)
    <=> ( ? [N_1] :
            ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
            & Xa = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) )
        | ? [N_1] :
            ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
            & Xa = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ) ).

fof(fact_2467_cos__zero__iff,axiom,
    ! [Xa] :
      ( hAPP(real,real,cos,Xa) = zero_zero(real)
    <=> ( ? [N_1] :
            ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
            & Xa = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) )
        | ? [N_1] :
            ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
            & Xa = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ) ).

fof(fact_2468_odd__add,axiom,
    ! [Ma,Na] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na)))
    <=> ~ ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Ma))
        <=> ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) ) ) ).

fof(fact_2469_even__add,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na)))
    <=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Ma))
      <=> hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) ) ) ).

fof(fact_2470_even__sum__nat,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Xa),Ya)))
    <=> ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
          & hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) )
        | ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
          & ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) ) ) ) ).

fof(fact_2471_even__zero__nat,axiom,
    hBOOL(hAPP(nat,bool,even_odd_even(nat),zero_zero(nat))) ).

fof(fact_2472_even__Suc,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,suc,Xa)))
    <=> ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) ) ).

fof(fact_2473_odd__1__nat,axiom,
    ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),one_one(nat))) ).

fof(fact_2474_even__product__nat,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Xa),Ya)))
    <=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
        | hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) ) ) ).

fof(fact_2475_real__of__nat__fact__not__zero,axiom,
    ! [N] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)) != zero_zero(real) ).

fof(fact_2476_power__mono__odd,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [X,Y,N] :
          ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N))) ) ) ) ).

fof(fact_2477_odd__pos,axiom,
    ! [N] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

fof(fact_2478_power__minus__odd,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X,N] :
          ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)) ) ) ).

fof(fact_2479_Parity_Opower__minus__even,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X,N] :
          ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N) ) ) ).

fof(fact_2480_neg__power__if,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X,N] :
          ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N) )
          & ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)) ) ) ) ).

fof(fact_2481_power__even__abs,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [X,N] :
          ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),N) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N) ) ) ).

fof(fact_2482_even__difference__nat,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Xa),Ya)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya))
        | ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
          & hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) )
        | ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
          & ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) ) ) ) ).

fof(fact_2483_even__nat__def,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
    <=> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,semiring_1_of_nat(int),Xa))) ) ).

fof(fact_2484_Parity_Otransfer__int__nat__relations,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)))
    <=> hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) ) ).

fof(fact_2485_real__of__nat__fact__ge__zero,axiom,
    ! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))) ).

fof(fact_2486_real__of__nat__fact__gt__zero,axiom,
    ! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))) ).

fof(fact_2487_zero__le__power__eq,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)))
        <=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
            | ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa)) ) ) ) ) ).

fof(fact_2488_zero__le__power__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3))
            | hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) ) ) ) ).

fof(fact_2489_zero__le__even__power,axiom,
    ! [X_a] :
      ( ( monoid_mult(X_a)
        & linordered_ring(X_a) )
     => ! [X,N] :
          ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))) ) ) ).

fof(fact_2490_zero__le__odd__power,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Na] :
          ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa)) ) ) ) ).

fof(fact_2491_power__less__zero__eq,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)),zero_zero(X_a)))
        <=> ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),zero_zero(X_a))) ) ) ) ).

fof(fact_2492_minus__one__odd__power,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X] :
          ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ).

fof(fact_2493_minus__one__even__power,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X] :
          ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = one_one(X_a) ) ) ).

fof(fact_2494_minus__one__even__odd__power,axiom,
    ! [X_a] :
      ( comm_ring_1(X_a)
     => ! [X] :
          ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = one_one(X_a) )
          & ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ) ).

fof(fact_2495_power__mono__even,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [X,Y,N] :
          ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),hAPP(X_a,X_a,abs_abs(X_a),Y)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N))) ) ) ) ).

fof(fact_2496_even__nat__equiv__def2,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
    <=> ? [Y_1] : Xa = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),Y_1) ) ).

fof(fact_2497_odd__nat__equiv__def2,axiom,
    ! [Xa] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
    <=> ? [Y_1] : Xa = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),Y_1)) ) ).

fof(fact_2498_Parity_Oneg__one__odd__power,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [X] :
          ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = hAPP(int,X_a,number_number_of(X_a),min) ) ) ).

fof(fact_2499_power__even__abs__number__of,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [X,W] :
          ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W)))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),hAPP(int,nat,number_number_of(nat),W)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),W)) ) ) ).

fof(fact_2500_odd__nat__plus__one__div__two,axiom,
    ! [X] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,X)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))) ) ).

fof(fact_2501_even__nat__plus__one__div__two,axiom,
    ! [X] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,X)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) ) ).

fof(fact_2502_odd__nat__mod__two__eq__one,axiom,
    ! [X] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,nat,suc,zero_zero(nat)) ) ).

fof(fact_2503_even__nat__mod__two__eq__zero,axiom,
    ! [X] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = zero_zero(nat) ) ).

fof(fact_2504_odd__nat__equiv__def,axiom,
    ! [Xa] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
    <=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Xa),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,nat,suc,zero_zero(nat)) ) ).

fof(fact_2505_even__nat__equiv__def,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
    <=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Xa),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = zero_zero(nat) ) ).

fof(fact_2506_even__power__nat,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Ya)))
    <=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ya)) ) ) ).

fof(fact_2507_fact__diff__Suc,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M)))
     => hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ).

fof(fact_2508_power__le__zero__eq,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)),zero_zero(X_a)))
        <=> ( Na != zero_zero(nat)
            & ( ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
                & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a))) )
              | ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
                & ti(X_a,Xa) = zero_zero(X_a) ) ) ) ) ) ).

fof(fact_2509_zero__less__power__eq,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)))
        <=> ( Na = zero_zero(nat)
            | ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
              & ti(X_a,Xa) != zero_zero(X_a) )
            | ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)) ) ) ) ) ).

fof(fact_2510_zero__le__power__eq__number__of,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),Wa))))
        <=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa)))
            | ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa)) ) ) ) ) ).

fof(fact_2511_power__less__zero__eq__number__of,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),Wa))),zero_zero(X_a)))
        <=> ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa)))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),zero_zero(X_a))) ) ) ) ).

fof(fact_2512_neg__one__even__odd__power,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [X] :
          ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = one_one(X_a) )
          & ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = hAPP(int,X_a,number_number_of(X_a),min) ) ) ) ).

fof(fact_2513_Parity_Oneg__one__even__power,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [X] :
          ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
         => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = one_one(X_a) ) ) ).

fof(fact_2514_even__num__iff,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
      <=> ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))) ) ) ).

fof(fact_2515_odd__nat__div__two__times__two__plus__one,axiom,
    ! [X] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
     => hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))) = X ) ).

fof(fact_2516_even__nat__div__two__times__two,axiom,
    ! [X] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))) = X ) ).

fof(fact_2517_pos__int__even__equiv__nat__even,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
     => ( hBOOL(hAPP(int,bool,even_odd_even(int),Xa))
      <=> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,nat_1,Xa))) ) ) ).

fof(fact_2518_power__le__zero__eq__number__of,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),Wa))),zero_zero(X_a)))
        <=> ( hAPP(int,nat,number_number_of(nat),Wa) != zero_zero(nat)
            & ( ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa)))
                & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a))) )
              | ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa)))
                & ti(X_a,Xa) = zero_zero(X_a) ) ) ) ) ) ).

fof(fact_2519_zero__less__power__eq__number__of,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Xa,Wa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),Wa))))
        <=> ( hAPP(int,nat,number_number_of(nat),Wa) = zero_zero(nat)
            | ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa)))
              & ti(X_a,Xa) != zero_zero(X_a) )
            | ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa)))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)) ) ) ) ) ).

fof(fact_2520_nat__even__iff__2__dvd,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Xa)) ) ).

fof(fact_2521_even__dvd,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)) ) ).

fof(fact_2522_even__mult__two__ex,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
    <=> ? [M_2] : Na = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_2) ) ).

fof(fact_2523_even__even__mod__4__iff,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
    <=> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Na),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).

fof(fact_2524_odd__Suc__mult__two__ex,axiom,
    ! [Na] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
    <=> ? [M_2] : Na = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_2)) ) ).

fof(fact_2525_lemma__even__div2,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).

fof(fact_2526_lemma__STAR__sin,axiom,
    ! [Na] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),hAPP(bool,fun(real,fun(real,real)),if(real),hAPP(nat,bool,even_odd_even(nat),Na)),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),Na)) = zero_zero(real) ).

fof(fact_2527_lemma__not__even__div2,axiom,
    ! [N] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ).

fof(fact_2528_lemma__even__mod__4__div__2,axiom,
    ! [N] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = one_one(nat)
     => hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_2529_lemma__odd__mod__4__div__2,axiom,
    ! [N] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))
     => ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_2530_sin__zero__lemma,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hAPP(real,real,sin,X) = zero_zero(real)
       => ? [N_1] :
            ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
            & X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ) ).

fof(fact_2531_cos__zero__lemma,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hAPP(real,real,cos,X) = zero_zero(real)
       => ? [N_1] :
            ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
            & X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ) ).

fof(fact_2532_odd__square,axiom,
    ! [N] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
     => ? [X_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_1)),one_one(nat)) ) ).

fof(fact_2533_sin__coeff__def,axiom,
    ! [X_1] :
      ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
       => hAPP(nat,real,sin_coeff,X_1) = zero_zero(real) )
      & ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
       => hAPP(nat,real,sin_coeff,X_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X_1),hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),X_1))) ) ) ).

fof(fact_2534_even__square,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
     => ? [X_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_1) ) ).

fof(fact_2535_cos__coeff__def,axiom,
    ! [X_1] :
      ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
       => hAPP(nat,real,cos_coeff,X_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),X_1))) )
      & ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
       => hAPP(nat,real,cos_coeff,X_1) = zero_zero(real) ) ) ).

fof(fact_2536_fact__add__num__eq__if__nat,axiom,
    ! [M,N] :
      ( ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat)
       => hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = one_one(nat) )
      & ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) != zero_zero(nat)
       => hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),one_one(nat)))) ) ) ).

fof(fact_2537_fact__add__num__eq__if2__nat,axiom,
    ! [N,M] :
      ( ( M = zero_zero(nat)
       => hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(nat,nat,fact_fact(nat),N) )
      & ( M != zero_zero(nat)
       => hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N))) ) ) ).

fof(fact_2538_fact__nonzero__nat,axiom,
    ! [N] : hAPP(nat,nat,fact_fact(nat),N) != zero_zero(nat) ).

fof(fact_2539_fact__mono__nat,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,fact_fact(nat),M)),hAPP(nat,nat,fact_fact(nat),N))) ) ).

fof(fact_2540_fact__1__nat,axiom,
    hAPP(nat,nat,fact_fact(nat),one_one(nat)) = one_one(nat) ).

fof(fact_2541_of__nat__fact__not__zero,axiom,
    ! [X_a] :
      ( semiring_char_0(X_a)
     => ! [N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N)) != zero_zero(X_a) ) ).

fof(fact_2542_Fact_Ofact__0__nat,axiom,
    hAPP(nat,nat,fact_fact(nat),zero_zero(nat)) = hAPP(nat,nat,suc,zero_zero(nat)) ).

fof(fact_2543_fact__Suc__0__nat,axiom,
    hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,suc,zero_zero(nat))) = hAPP(nat,nat,suc,zero_zero(nat)) ).

fof(fact_2544_fact__gt__zero__nat,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,fact_fact(nat),N))) ).

fof(fact_2545_fact__less__mono__nat,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,fact_fact(nat),M)),hAPP(nat,nat,fact_fact(nat),N))) ) ) ).

fof(fact_2546_Fact_Ofact__Suc,axiom,
    ! [X] : hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,suc,X)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,X)),hAPP(nat,nat,fact_fact(nat),X)) ).

fof(fact_2547_fact__dvd,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,fact_fact(nat),N)),hAPP(nat,nat,fact_fact(nat),M))) ) ).

fof(fact_2548_fact__ge__one__nat,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(nat,nat,fact_fact(nat),N))) ).

fof(fact_2549_of__nat__fact__ge__zero,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N)))) ) ).

fof(fact_2550_of__nat__fact__gt__zero,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N)))) ) ).

fof(fact_2551_fact__ge__Suc__0__nat,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,fact_fact(nat),N))) ).

fof(fact_2552_dvd__fact__nat,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),hAPP(nat,nat,fact_fact(nat),N))) ) ) ).

fof(fact_2553_fact__plus__one__nat,axiom,
    ! [N] : hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(nat,nat,fact_fact(nat),N)) ).

fof(fact_2554_fact__mod,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,fact_fact(nat),N)),hAPP(nat,nat,fact_fact(nat),M)) = zero_zero(nat) ) ).

fof(fact_2555_fact__num__eq__if__nat,axiom,
    ! [M] :
      ( ( M = zero_zero(nat)
       => hAPP(nat,nat,fact_fact(nat),M) = one_one(nat) )
      & ( M != zero_zero(nat)
       => hAPP(nat,nat,fact_fact(nat),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ).

fof(fact_2556_fact__reduce__nat,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(nat,nat,fact_fact(nat),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ).

fof(fact_2557_Ln_Oaux1,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N)))) ) ) ).

fof(fact_2558_tan__sec,axiom,
    ! [X] :
      ( hAPP(real,real,cos,X) != zero_zero(real)
     => hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,tan,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,cos,X))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).

fof(fact_2559_polar__ex2,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),zero_zero(real)))
     => ? [R_1,A_4] :
          ( X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,cos,A_4))
          & Y = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,sin,A_4)) ) ) ).

fof(fact_2560_polar__ex1,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ? [R_1,A_4] :
          ( X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,cos,A_4))
          & Y = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,sin,A_4)) ) ) ).

fof(fact_2561_inverse__minus__eq,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => ! [A_1] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) ) ).

fof(fact_2562_real__sqrt__inverse,axiom,
    ! [X] : hAPP(real,real,sqrt,hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X)) ).

fof(fact_2563_power__inverse,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => ! [A_1,N] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),N) ) ).

fof(fact_2564_INVERSE__ZERO,axiom,
    hAPP(real,real,inverse_inverse(real),zero_zero(real)) = zero_zero(real) ).

fof(fact_2565_inverse__1,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => hAPP(X_a,X_a,inverse_inverse(X_a),one_one(X_a)) = one_one(X_a) ) ).

fof(fact_2566_inverse__eq__1__iff,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [Xa] :
          ( hAPP(X_a,X_a,inverse_inverse(X_a),Xa) = one_one(X_a)
        <=> ti(X_a,Xa) = one_one(X_a) ) ) ).

fof(fact_2567_abs__inverse,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) ) ).

fof(fact_2568_inverse__eq__imp__eq,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => ! [A_1,B] :
          ( hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),B)
         => ti(X_a,A_1) = ti(X_a,B) ) ) ).

fof(fact_2569_inverse__eq__iff__eq,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => ! [A_3,B_2] :
          ( hAPP(X_a,X_a,inverse_inverse(X_a),A_3) = hAPP(X_a,X_a,inverse_inverse(X_a),B_2)
        <=> ti(X_a,A_3) = ti(X_a,B_2) ) ) ).

fof(fact_2570_inverse__inverse__eq,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => ! [A_1] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = ti(X_a,A_1) ) ).

fof(fact_2571_inverse__mult__distrib,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ).

fof(fact_2572_field__inverse__zero,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => hAPP(X_a,X_a,inverse_inverse(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_2573_inverse__zero,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => hAPP(X_a,X_a,inverse_inverse(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_2574_inverse__nonzero__iff__nonzero,axiom,
    ! [X_a] :
      ( divisi14063676e_zero(X_a)
     => ! [A_3] :
          ( hAPP(X_a,X_a,inverse_inverse(X_a),A_3) = zero_zero(X_a)
        <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ).

fof(fact_2575_nonzero__imp__inverse__nonzero,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,inverse_inverse(X_a),A_1) != zero_zero(X_a) ) ) ).

fof(fact_2576_nonzero__inverse__inverse__eq,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = ti(X_a,A_1) ) ) ).

fof(fact_2577_inverse__zero__imp__zero,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] :
          ( hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = zero_zero(X_a)
         => ti(X_a,A_1) = zero_zero(X_a) ) ) ).

fof(fact_2578_nonzero__inverse__eq__imp__eq,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B] :
          ( hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),B)
         => ( ti(X_a,A_1) != zero_zero(X_a)
           => ( ti(X_a,B) != zero_zero(X_a)
             => ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ).

fof(fact_2579_inverse__divide,axiom,
    ! [X_a] :
      ( field_inverse_zero(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),A_1) ) ).

fof(fact_2580_nonzero__norm__inverse,axiom,
    ! [X_a] :
      ( real_n1866405975lgebra(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),A_1)) ) ) ).

fof(fact_2581_norm__inverse,axiom,
    ! [X_a] :
      ( ( divisi14063676e_zero(X_a)
        & real_n1866405975lgebra(X_a) )
     => ! [A_1] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),A_1)) ) ).

fof(fact_2582_Bseq__inverse__lemma,axiom,
    ! [X_a] :
      ( real_n1866405975lgebra(X_a)
     => ! [R_3,X] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_3),hAPP(X_a,real,norm_norm(X_a),X)))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_3))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),X))),hAPP(real,real,inverse_inverse(real),R_3))) ) ) ) ).

fof(fact_2583_inverse__nonnegative__iff__nonnegative,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_2584_inverse__nonpositive__iff__nonpositive,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),zero_zero(X_a)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_2585_inverse__less__imp__less__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).

fof(fact_2586_inverse__less__imp__less,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).

fof(fact_2587_inverse__negative__imp__negative,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),zero_zero(X_a)))
         => ( ti(X_a,A_1) != zero_zero(X_a)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) ) ) ) ).

fof(fact_2588_less__imp__inverse__less__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).

fof(fact_2589_less__imp__inverse__less,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).

fof(fact_2590_negative__imp__inverse__negative,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),zero_zero(X_a))) ) ) ).

fof(fact_2591_inverse__positive__imp__positive,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))
         => ( ti(X_a,A_1) != zero_zero(X_a)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) ) ) ) ).

fof(fact_2592_positive__imp__inverse__positive,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ).

fof(fact_2593_inverse__negative__iff__negative,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),zero_zero(X_a)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ).

fof(fact_2594_inverse__positive__iff__positive,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) ) ) ).

fof(fact_2595_nonzero__inverse__mult__distrib,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [B,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => ( ti(X_a,B) != zero_zero(X_a)
           => hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) ) ) ) ).

fof(fact_2596_inverse__unique,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B] :
          ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) = one_one(X_a)
         => hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = ti(X_a,B) ) ) ).

fof(fact_2597_nonzero__power__inverse,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [N,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),N) ) ) ).

fof(fact_2598_nonzero__inverse__minus__eq,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) ) ) ).

fof(fact_2599_divide__inverse,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ).

fof(fact_2600_field__divide__inverse,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ).

fof(fact_2601_nonzero__abs__inverse,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) ) ) ).

fof(fact_2602_inverse__eq__divide,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] : hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1) ) ).

fof(fact_2603_field__class_Onormalizing__field__rules_I2_J,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [A_1] : hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1) ) ).

fof(fact_2604_real__divide__def,axiom,
    ! [R,S_5] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),R),S_5) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R),hAPP(real,real,inverse_inverse(real),S_5)) ).

fof(fact_2605_divide__real__def,axiom,
    ! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),hAPP(real,real,inverse_inverse(real),Y)) ).

fof(fact_2606_exp__minus,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [X] : hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,exp(X_a),X)) ) ).

fof(fact_2607_le__imp__inverse__le,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).

fof(fact_2608_le__imp__inverse__le__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).

fof(fact_2609_inverse__le__imp__le,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) ) ) ) ).

fof(fact_2610_inverse__le__imp__le__neg,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) ) ) ) ).

fof(fact_2611_inverse__le__1__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Xa)),one_one(X_a)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a)))
            | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),Xa)) ) ) ) ).

fof(fact_2612_inverse__add,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [B,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => ( ti(X_a,B) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ) ) ).

fof(fact_2613_division__ring__inverse__add,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [B,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => ( ti(X_a,B) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ) ) ).

fof(fact_2614_one__less__inverse,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).

fof(fact_2615_one__less__inverse__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),Xa)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),one_one(X_a))) ) ) ) ).

fof(fact_2616_left__inverse,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),A_1) = one_one(X_a) ) ) ).

fof(fact_2617_field__inverse,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),A_1) = one_one(X_a) ) ) ).

fof(fact_2618_right__inverse,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = one_one(X_a) ) ) ).

fof(fact_2619_division__ring__inverse__diff,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [B,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => ( ti(X_a,B) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1))),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ) ) ).

fof(fact_2620_nonzero__inverse__eq__divide,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1) ) ) ).

fof(fact_2621_inverse__numeral__1,axiom,
    ! [X_a] :
      ( ( field(X_a)
        & number_ring(X_a) )
     => hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) ) ).

fof(fact_2622_inverse__eq__divide__number__of,axiom,
    ! [X_b] :
      ( ( division_ring(X_b)
        & number(X_b) )
     => ! [W] : hAPP(X_b,X_b,inverse_inverse(X_b),hAPP(int,X_b,number_number_of(X_b),W)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),one_one(X_b)),hAPP(int,X_b,number_number_of(X_b),W)) ) ).

fof(fact_2623_real__mult__inverse__cancel2,axiom,
    ! [Y,U_1,X1_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X1_1))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),U_1)))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,inverse_inverse(real),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),U_1),hAPP(real,real,inverse_inverse(real),X1_1)))) ) ) ) ).

fof(fact_2624_real__mult__inverse__cancel,axiom,
    ! [Y,U_1,X1_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X1_1))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),U_1)))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X)),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X1_1)),U_1))) ) ) ) ).

fof(fact_2625_real__mult__inverse__left,axiom,
    ! [X] :
      ( X != zero_zero(real)
     => hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X)),X) = one_one(real) ) ).

fof(fact_2626_inverse__less__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),hAPP(X_a,X_a,inverse_inverse(X_a),B_2)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) )
            & ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),zero_zero(X_a)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) ) ) ) ) ).

fof(fact_2627_inverse__le__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),hAPP(X_a,X_a,inverse_inverse(X_a),B_2)))
        <=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3)) )
            & ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),zero_zero(X_a)))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) ) ) ) ) ).

fof(fact_2628_one__le__inverse,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),one_one(X_a)))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).

fof(fact_2629_inverse__less__1__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Xa)),one_one(X_a)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a)))
            | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),Xa)) ) ) ) ).

fof(fact_2630_one__le__inverse__iff,axiom,
    ! [X_a] :
      ( linord1117847801e_zero(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),Xa)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),one_one(X_a))) ) ) ) ).

fof(fact_2631_inverse__diff__inverse,axiom,
    ! [X_a] :
      ( division_ring(X_a)
     => ! [B,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => ( ti(X_a,B) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(X_a,X_a,inverse_inverse(X_a),B))) ) ) ) ).

fof(fact_2632_inv__of__nat__fact__ge__zero,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N))))) ) ).

fof(fact_2633_inv__of__nat__fact__gt__zero,axiom,
    ! [X_a] :
      ( linordered_field(X_a)
     => ! [N] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N))))) ) ).

fof(fact_2634_inv__real__of__nat__fact__ge__zero,axiom,
    ! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N))))) ).

fof(fact_2635_sqrt__divide__self__eq,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,X)),X) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X)) ) ).

fof(fact_2636_inv__real__of__nat__fact__gt__zero,axiom,
    ! [N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N))))) ).

fof(fact_2637_ln__inverse,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,ln,hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,ln,X)) ) ).

fof(fact_2638_DERIV__inverse__lemma,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [H_2,B,A_1] :
          ( ti(X_a,A_1) != zero_zero(X_a)
         => ( ti(X_a,B) != zero_zero(X_a)
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))),H_2) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),H_2))),hAPP(X_a,X_a,inverse_inverse(X_a),B))) ) ) ) ).

fof(fact_2639_log__inverse,axiom,
    ! [X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ( A_1 != one_one(real)
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)) ) ) ) ).

fof(fact_2640_real__inv__sqrt__pow2,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,inverse_inverse(real),X) ) ).

fof(fact_2641_reals__Archimedean,axiom,
    ! [X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ? [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1)))),X)) ) ).

fof(fact_2642_DERIV__arccos,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arccos),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) ) ) ).

fof(fact_2643_DERIV__arcsin,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arcsin),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ) ) ).

fof(fact_2644_ex__inverse__of__nat__less,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
         => ? [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))),X)) ) ) ) ).

fof(fact_2645_DERIV__exp,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [Xa] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),exp(X_a)),Xa),hAPP(X_a,X_a,exp(X_a),Xa))) ) ).

fof(fact_2646_Deriv_Olemma__DERIV__subst,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [E_3,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( ti(X_a,D) = ti(X_a,E_3)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),E_3)) ) ) ) ).

fof(fact_2647_DERIV__cong,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y_2,F,Xa,X_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),X_2))
         => ( ti(X_a,X_2) = ti(X_a,Y_2)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),Y_2)) ) ) ) ).

fof(fact_2648_DERIV__unique,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [E_3,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),E_3))
           => ti(X_a,D) = ti(X_a,E_3) ) ) ) ).

fof(fact_2649_DERIV__cmult__Id,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [C,Xa] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),Xa),C)) ) ).

fof(fact_2650_DERIV__sin,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sin),Xa),hAPP(real,real,cos,Xa))) ).

fof(fact_2651_DERIV__cos,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),cos),Xa),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa)))) ).

fof(fact_2652_DERIV__ln,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),ln),Xa),hAPP(real,real,inverse_inverse(real),Xa))) ) ).

fof(fact_2653_DERIV__ln__divide,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),ln),Xa),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),Xa))) ) ).

fof(fact_2654_DERIV__real__sqrt,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sqrt),Xa),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,Xa))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).

fof(fact_2655_DERIV__tan,axiom,
    ! [Xa] :
      ( hAPP(real,real,cos,Xa) != zero_zero(real)
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),tan),Xa),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,Xa)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).

fof(fact_2656_DERIV__arctan,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arctan),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).

fof(fact_2657_DERIV__real__sqrt__generic,axiom,
    ! [D,Xa] :
      ( Xa != zero_zero(real)
     => ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
         => D = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,Xa))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) )
       => ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real)))
           => D = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,Xa)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) )
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sqrt),Xa),D)) ) ) ) ).

fof(fact_2658_DERIV__const__average,axiom,
    ! [Va,K,A_3,B_2] :
      ( A_3 != B_2
     => ( ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),Va),X_1),K))
       => hAPP(real,real,Va,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_3),B_2)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,Va,A_3)),hAPP(real,real,Va,B_2))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ) ).

fof(fact_2659_DERIV__local__max,axiom,
    ! [D_1,F,Xa,L] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
       => ( ! [Y_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),Y_1))),D_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,Y_1)),hAPP(real,real,F,Xa))) )
         => L = zero_zero(real) ) ) ) ).

fof(fact_2660_DERIV__local__min,axiom,
    ! [D_1,F,Xa,L] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
       => ( ! [Y_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),Y_1))),D_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,Xa)),hAPP(real,real,F,Y_1))) )
         => L = zero_zero(real) ) ) ) ).

fof(fact_2661_DERIV__local__const,axiom,
    ! [D_1,F,Xa,L] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
       => ( ! [Y_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),Y_1))),D_1))
             => hAPP(real,real,F,Xa) = hAPP(real,real,F,Y_1) )
         => L = zero_zero(real) ) ) ) ).

fof(fact_2662_ex__inverse__of__nat__Suc__less,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
         => ? [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,N_1)))),X)) ) ) ).

fof(fact_2663_DERIV__neg__dec__left,axiom,
    ! [F,Xa,L] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
            & ! [H_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1))
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_2))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Xa)),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),H_1)))) ) ) ) ) ) ).

fof(fact_2664_DERIV__pos__inc__left,axiom,
    ! [F,Xa,L] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
            & ! [H_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1))
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_2))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),H_1))),hAPP(real,real,F,Xa))) ) ) ) ) ) ).

fof(fact_2665_DERIV__neg__dec__right,axiom,
    ! [F,Xa,L] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
            & ! [H_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1))
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_2))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),H_1))),hAPP(real,real,F,Xa))) ) ) ) ) ) ).

fof(fact_2666_DERIV__pos__inc__right,axiom,
    ! [F,Xa,L] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
            & ! [H_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1))
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_2))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Xa)),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),H_1)))) ) ) ) ) ) ).

fof(fact_2667_ex__less__of__nat__mult,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
         => ? [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)),X))) ) ) ).

fof(fact_2668_DERIV__const__ratio__const2,axiom,
    ! [F,K,A_3,B_2] :
      ( A_3 != B_2
     => ( ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),K))
       => hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3)) = K ) ) ).

fof(fact_2669_DERIV__const__ratio__const,axiom,
    ! [F,K,A_3,B_2] :
      ( A_3 != B_2
     => ( ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),K))
       => hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3)),K) ) ) ).

fof(fact_2670_DERIV__real__root__generic,axiom,
    ! [D,Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( Xa != zero_zero(real)
       => ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
           => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
             => D = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))) ) )
         => ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
             => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real)))
               => D = hAPP(real,real,uminus_uminus(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))))) ) )
           => ( ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
               => D = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))) )
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,Na)),Xa),D)) ) ) ) ) ) ).

fof(fact_2671_DERIV__even__real__root,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real)))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,Na)),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,real_1(nat),Na))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))) ) ) ) ).

fof(fact_2672_DERIV__real__root,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,Na)),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))) ) ) ).

fof(fact_2673_MVT2,axiom,
    ! [F,F_2,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2))
     => ( ! [X_1] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),hAPP(real,real,F_2,X_1))) )
       => ? [Z_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),Z_2))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z_2),B_2))
            & hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3)),hAPP(real,real,F_2,Z_2)) ) ) ) ).

fof(fact_2674_real__root__zero,axiom,
    ! [N] : hAPP(real,real,hAPP(nat,fun(real,real),root,N),zero_zero(real)) = zero_zero(real) ).

fof(fact_2675_real__root__Suc__0,axiom,
    ! [X] : hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,suc,zero_zero(nat))),X) = X ).

fof(fact_2676_real__root__eq__iff,axiom,
    ! [Xa,Ya,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)
      <=> Xa = Ya ) ) ).

fof(fact_2677_real__root__commute,axiom,
    ! [X,N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => hAPP(real,real,hAPP(nat,fun(real,real),root,M),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) = hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,hAPP(nat,fun(real,real),root,M),X)) ) ) ).

fof(fact_2678_odd__real__root__unique,axiom,
    ! [Y,X,N] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
     => ( hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),N) = X
       => hAPP(real,real,hAPP(nat,fun(real,real),root,N),X) = Y ) ) ).

fof(fact_2679_odd__real__root__pow,axiom,
    ! [X,N] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
     => hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),N) = X ) ).

fof(fact_2680_odd__real__root__power__cancel,axiom,
    ! [X,N] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = X ) ).

fof(fact_2681_real__root__le__iff,axiom,
    ! [Xa,Ya,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ) ).

fof(fact_2682_real__root__le__mono,axiom,
    ! [X,Y,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y))) ) ) ).

fof(fact_2683_real__root__eq__0__iff,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = zero_zero(real)
      <=> Xa = zero_zero(real) ) ) ).

fof(fact_2684_real__root__less__iff,axiom,
    ! [Xa,Ya,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) ) ) ).

fof(fact_2685_real__root__less__mono,axiom,
    ! [X,Y,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y))) ) ) ).

fof(fact_2686_real__root__mult,axiom,
    ! [X,Y,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y)) ) ).

fof(fact_2687_real__root__mult__exp,axiom,
    ! [X,N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),X) = hAPP(real,real,hAPP(nat,fun(real,real),root,M),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) ) ) ).

fof(fact_2688_real__root__power,axiom,
    ! [X,K_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),K_4)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),K_4) ) ).

fof(fact_2689_real__root__one,axiom,
    ! [N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,N),one_one(real)) = one_one(real) ) ).

fof(fact_2690_real__root__eq__1__iff,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = one_one(real)
      <=> Xa = one_one(real) ) ) ).

fof(fact_2691_real__root__divide,axiom,
    ! [X,Y,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y)) ) ).

fof(fact_2692_real__root__minus,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) ) ).

fof(fact_2693_real__root__abs,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,abs_abs(real),X)) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) ) ).

fof(fact_2694_real__root__inverse,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) ) ).

fof(fact_2695_real__root__ge__0__iff,axiom,
    ! [Ya,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Ya)) ) ) ).

fof(fact_2696_real__root__le__0__iff,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),zero_zero(real)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),zero_zero(real))) ) ) ).

fof(fact_2697_real__root__ge__zero,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))) ) ) ).

fof(fact_2698_real__root__gt__0__iff,axiom,
    ! [Ya,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya)) ) ) ).

fof(fact_2699_real__root__lt__0__iff,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),zero_zero(real)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) ) ) ).

fof(fact_2700_real__root__gt__zero,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))) ) ) ).

fof(fact_2701_real__root__ge__1__iff,axiom,
    ! [Ya,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Ya)) ) ) ).

fof(fact_2702_real__root__le__1__iff,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),one_one(real)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real))) ) ) ).

fof(fact_2703_real__root__decreasing,axiom,
    ! [X,N_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_4))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_4),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))) ) ) ) ).

fof(fact_2704_real__root__gt__1__iff,axiom,
    ! [Ya,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Ya)) ) ) ).

fof(fact_2705_real__root__lt__1__iff,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),one_one(real)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) ) ) ).

fof(fact_2706_real__root__strict__decreasing,axiom,
    ! [X,N_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_4))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_4),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))) ) ) ) ).

fof(fact_2707_real__root__pos,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,suc,N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(nat,nat,suc,N))) = X ) ).

fof(fact_2708_real__root__pos__pos,axiom,
    ! [X_4,N_5] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_5))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_4))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_5),X_4))) ) ) ).

fof(fact_2709_real__root__less__mono__lemma,axiom,
    ! [Y,X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y))) ) ) ) ).

fof(fact_2710_real__root__mult__lemma,axiom,
    ! [Y,X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
         => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y)) ) ) ) ).

fof(fact_2711_real__root__pos__mult__exp,axiom,
    ! [X,N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),X) = hAPP(real,real,hAPP(nat,fun(real,real),root,M),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) ) ) ) ).

fof(fact_2712_real__root__pos2,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = X ) ) ).

fof(fact_2713_real__root__pow__pos2,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),N) = X ) ) ).

fof(fact_2714_real__root__pos__unique,axiom,
    ! [X,Y,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),N) = X
         => hAPP(real,real,hAPP(nat,fun(real,real),root,N),X) = Y ) ) ) ).

fof(fact_2715_real__root__pow__pos,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),N) = X ) ) ).

fof(fact_2716_real__root__increasing,axiom,
    ! [X,N_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_4))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_4),X))) ) ) ) ) ).

fof(fact_2717_real__root__strict__increasing,axiom,
    ! [X,N_4,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_4))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real)))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_4),X))) ) ) ) ) ).

fof(fact_2718_sqrt__def,axiom,
    sqrt = hAPP(nat,fun(real,real),root,hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2719_real__root__inverse__lemma,axiom,
    ! [X,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) ) ) ).

fof(fact_2720_DERIV__odd__real__root,axiom,
    ! [Xa,Na] :
      ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))
     => ( Xa != zero_zero(real)
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,Na)),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))) ) ) ).

fof(fact_2721_psubset__imp__ex__mem,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
     => ? [B_4] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_4),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2))) ) ).

fof(fact_2722_Union__least,axiom,
    ! [X_a,C_2,A_2] :
      ( ! [X_3] :
          ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_3),A_2))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),X_3),C_2)) )
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)),C_2)) ) ).

fof(fact_2723_DERIV__isconst__all,axiom,
    ! [Xa,Ya,F] :
      ( ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real)))
     => hAPP(real,real,F,Xa) = hAPP(real,real,F,Ya) ) ).

fof(fact_2724_reals__Archimedean2,axiom,
    ! [X] :
    ? [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),N_1))) ).

fof(fact_2725_Union__SetS__finite,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
     => hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)))) ) ).

fof(fact_2726_complete__lattice__class_OSup__least,axiom,
    ! [X_a] :
      ( comple44059174attice(X_a)
     => ! [Z,A_2] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z)) )
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),Z)) ) ) ).

fof(fact_2727_less__mono__imp__le__mono,axiom,
    ! [I,J,F] :
      ( ! [I_2,J_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,I_2)),hAPP(nat,nat,F,J_1))) )
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,F,I)),hAPP(nat,nat,F,J))) ) ) ).

fof(fact_2728_Suc__le__D,axiom,
    ! [N,M_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),M_4))
     => ? [M_2] : M_4 = hAPP(nat,nat,suc,M_2) ) ).

fof(fact_2729_wset__fin,axiom,
    ! [A_3,P_4] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) ).

fof(fact_2730_d22set__fin,axiom,
    ! [A_3] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),d22set,A_3))) ).

fof(fact_2731_SetS__elems__finite,axiom,
    ! [A_3,P_4,X_1] :
      ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)))
     => hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_1)) ) ).

fof(fact_2732_neg__one__special,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))) = one_one(int) ) ).

fof(fact_2733_SRStar__finite,axiom,
    ! [P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
     => hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),sRStar,P_4))) ) ).

fof(fact_2734_card__psubset,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) ) ) ) ).

fof(fact_2735_diff__card__le__card__Diff,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1))),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))) ) ).

fof(fact_2736_card__Diff__subset,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
       => hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)) ) ) ).

fof(fact_2737_finite__Diff,axiom,
    ! [X_a,B_1,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) ) ).

fof(fact_2738_finite,axiom,
    ! [X_a] :
      ( finite_finite(X_a)
     => ! [A_2] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).

fof(fact_2739_finite__code,axiom,
    ! [X_a] :
      ( finite_finite(X_a)
     => ! [A_2] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).

fof(fact_2740_SetS__finite,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
     => hBOOL(hAPP(fun(fun(int,bool),bool),bool,finite_finite_1(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))) ) ).

fof(fact_2741_rev__finite__subset,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
       => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).

fof(fact_2742_finite__subset,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
       => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).

fof(fact_2743_finite__Diff2,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
      <=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).

fof(fact_2744_finite__UnionD,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)))
     => hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),A_2)) ) ).

fof(fact_2745_card__infinite,axiom,
    ! [X_a,A_2] :
      ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2) = zero_zero(nat) ) ).

fof(fact_2746_card__ge__0__finite,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)))
     => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).

fof(fact_2747_card__seteq,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)))
         => ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ) ).

fof(fact_2748_card__mono,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1))) ) ) ).

fof(fact_2749_psubset__card__mono,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1))) ) ) ).

fof(fact_2750_finite__Union,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),A_2))
     => ( ! [M_3] :
            ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),M_3),A_2))
           => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),M_3)) )
       => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2))) ) ) ).

fof(fact_2751_card_Oneutral,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => ( ( ? [X_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
         => one_one(nat) = zero_zero(nat) )
       => hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2) = zero_zero(nat) ) ) ).

fof(fact_2752_Suc__lessE,axiom,
    ! [I_1,K_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I_1)),K_4))
     => ~ ! [J_1] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1))
           => K_4 != hAPP(nat,nat,suc,J_1) ) ) ).

fof(fact_2753_lessE,axiom,
    ! [I_1,K_4] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_4))
     => ( K_4 != hAPP(nat,nat,suc,I_1)
       => ~ ! [J_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1))
             => K_4 != hAPP(nat,nat,suc,J_1) ) ) ) ).

fof(fact_2754_real__lbound__gt__zero,axiom,
    ! [D2,D1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D1))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ? [E] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),E),D1))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),E),D2)) ) ) ) ).

fof(fact_2755_complex__norm,axiom,
    ! [X,Y] : hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_2756_exists__least__lemma,axiom,
    ! [P_1] :
      ( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
     => ( ? [X1] : hBOOL(hAPP(nat,bool,P_1,X1))
       => ? [N_1] :
            ( ~ hBOOL(hAPP(nat,bool,P_1,N_1))
            & hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_1))) ) ) ) ).

fof(fact_2757_Complex__eq__0,axiom,
    ! [A_3,B_2] :
      ( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2) = zero_zero(complex)
    <=> ( A_3 = zero_zero(real)
        & B_2 = zero_zero(real) ) ) ).

fof(fact_2758_complex__zero__def,axiom,
    zero_zero(complex) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),zero_zero(real)) ).

fof(fact_2759_complex_Oinject,axiom,
    ! [Real1,Real2,Real1_2,Real2_2] :
      ( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_2),Real2_2)
    <=> ( Real1 = Real1_2
        & Real2 = Real2_2 ) ) ).

fof(fact_2760_Complex__eq__number__of,axiom,
    ! [A_3,B_2,Wa] :
      ( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2) = hAPP(int,complex,number_number_of(complex),Wa)
    <=> ( A_3 = hAPP(int,real,number_number_of(real),Wa)
        & B_2 = zero_zero(real) ) ) ).

fof(fact_2761_Complex__eq__1,axiom,
    ! [A_3,B_2] :
      ( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2) = one_one(complex)
    <=> ( A_3 = one_one(real)
        & B_2 = zero_zero(real) ) ) ).

fof(fact_2762_complex__one__def,axiom,
    one_one(complex) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,one_one(real)),zero_zero(real)) ).

fof(fact_2763_complex__add,axiom,
    ! [A_1,B,C_1,D_3] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C_1),D_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),C_1)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D_3)) ).

fof(fact_2764_complex__minus,axiom,
    ! [A_1,B] : hAPP(complex,complex,uminus_uminus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),A_1)),hAPP(real,real,uminus_uminus(real),B)) ).

fof(fact_2765_complex__diff,axiom,
    ! [A_1,B,C_1,D_3] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C_1),D_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),C_1)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D_3)) ).

fof(fact_2766_complex__mult,axiom,
    ! [A_1,B,C_1,D_3] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C_1),D_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),C_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D_3))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),D_3)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C_1))) ).

fof(fact_2767_cmod__unit__one,axiom,
    ! [A_1] : hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1))) = one_one(real) ).

fof(fact_2768_complex__inverse,axiom,
    ! [A_1,B] : hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A_1),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).

fof(fact_2769_complex_Osize_I1_J,axiom,
    ! [Real1_1,Real2_1] : hAPP(complex,nat,complex_size,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_1),Real2_1)) = zero_zero(nat) ).

fof(fact_2770_cos__arg__i__mult__zero__pos,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))) = zero_zero(real) ) ).

fof(fact_2771_cos__arg__i__mult__zero__neg,axiom,
    ! [Y] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),zero_zero(real)))
     => hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))) = zero_zero(real) ) ).

fof(fact_2772_complex__divide__def,axiom,
    ! [X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),X),Y) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),hAPP(complex,complex,inverse_inverse(complex),Y)) ).

fof(fact_2773_cos__arg__i__mult__zero,axiom,
    ! [Y] :
      ( Y != zero_zero(real)
     => hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))) = zero_zero(real) ) ).

fof(fact_2774_complex_Osize_I2_J,axiom,
    ! [Real1_1,Real2_1] : hAPP(complex,nat,size_size(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_1),Real2_1)) = zero_zero(nat) ).

fof(fact_2775_power2__i,axiom,
    hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),ii),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,complex,number_number_of(complex),min) ).

fof(fact_2776_complex_Orecs,axiom,
    ! [X_a,F1,Real1,Real2] : hAPP(complex,X_a,hAPP(fun(real,fun(real,X_a)),fun(complex,X_a),complex_rec(X_a),F1),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2)) = hAPP(real,X_a,hAPP(real,fun(real,X_a),F1,Real1),Real2) ).

fof(fact_2777_complex__i__not__zero,axiom,
    ii != zero_zero(complex) ).

fof(fact_2778_complex__i__not__one,axiom,
    ii != one_one(complex) ).

fof(fact_2779_complex__i__not__number__of,axiom,
    ! [W] : ii != hAPP(int,complex,number_number_of(complex),W) ).

fof(fact_2780_complex__i__mult__minus,axiom,
    ! [X] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),X)) = hAPP(complex,complex,uminus_uminus(complex),X) ).

fof(fact_2781_inverse__i,axiom,
    hAPP(complex,complex,inverse_inverse(complex),ii) = hAPP(complex,complex,uminus_uminus(complex),ii) ).

fof(fact_2782_i__def,axiom,
    ii = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),one_one(real)) ).

fof(fact_2783_Complex__eq__i,axiom,
    ! [Xa,Ya] :
      ( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Xa),Ya) = ii
    <=> ( Xa = zero_zero(real)
        & Ya = one_one(real) ) ) ).

fof(fact_2784_Complex__mult__i,axiom,
    ! [A_1,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),ii) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),B)),A_1) ).

fof(fact_2785_i__mult__Complex,axiom,
    ! [A_1,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),B)),A_1) ).

fof(fact_2786_i__squared,axiom,
    hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),ii) = hAPP(int,complex,number_number_of(complex),min) ).

fof(fact_2787_i__mult__eq2,axiom,
    hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),ii) = hAPP(complex,complex,uminus_uminus(complex),one_one(complex)) ).

fof(fact_2788_complex__inverse__complex__split,axiom,
    ! [X,Y] : hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,of_real(complex),X)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),Y)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Y),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ).

fof(fact_2789_complex_Osimps_I2_J,axiom,
    ! [X_a,F1,Real1,Real2] : hAPP(complex,X_a,hAPP(fun(real,fun(real,X_a)),fun(complex,X_a),complex_case(X_a),F1),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2)) = hAPP(real,X_a,hAPP(real,fun(real,X_a),F1,Real1),Real2) ).

fof(fact_2790_not0__implies__Suc,axiom,
    ! [N] :
      ( N != zero_zero(nat)
     => ? [M_2] : N = hAPP(nat,nat,suc,M_2) ) ).

fof(fact_2791_of__real__eq__iff,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [Xa,Ya] :
          ( hAPP(real,X_a,of_real(X_a),Xa) = hAPP(real,X_a,of_real(X_a),Ya)
        <=> Xa = Ya ) ) ).

fof(fact_2792_of__real__0,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => hAPP(real,X_a,of_real(X_a),zero_zero(real)) = zero_zero(X_a) ) ).

fof(fact_2793_of__real_Ozero,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => hAPP(real,X_a,of_real(X_a),zero_zero(real)) = zero_zero(X_a) ) ).

fof(fact_2794_of__real__eq__0__iff,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [Xa] :
          ( hAPP(real,X_a,of_real(X_a),Xa) = zero_zero(X_a)
        <=> Xa = zero_zero(real) ) ) ).

fof(fact_2795_of__real__mult,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).

fof(fact_2796_of__real__number__of__eq,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & real_algebra_1(X_a) )
     => ! [W] : hAPP(real,X_a,of_real(X_a),hAPP(int,real,number_number_of(real),W)) = hAPP(int,X_a,number_number_of(X_a),W) ) ).

fof(fact_2797_of__real__add,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).

fof(fact_2798_of__real_Oadd,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).

fof(fact_2799_of__real__1,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => hAPP(real,X_a,of_real(X_a),one_one(real)) = one_one(X_a) ) ).

fof(fact_2800_of__real__power,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [X,N] : hAPP(real,X_a,of_real(X_a),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(real,X_a,of_real(X_a),X)),N) ) ).

fof(fact_2801_complex__eq__cancel__iff2,axiom,
    ! [Xa,Ya,Xaa] :
      ( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Xa),Ya) = hAPP(real,complex,of_real(complex),Xaa)
    <=> ( Xa = Xaa
        & Ya = zero_zero(real) ) ) ).

fof(fact_2802_complex__of__real__def,axiom,
    ! [R_3] : hAPP(real,complex,of_real(complex),R_3) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,R_3),zero_zero(real)) ).

fof(fact_2803_of__real_Odiff,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).

fof(fact_2804_of__real__diff,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).

fof(fact_2805_of__real__divide,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & real_field(X_a) )
     => ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).

fof(fact_2806_of__real__minus,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [X] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,uminus_uminus(real),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(real,X_a,of_real(X_a),X)) ) ).

fof(fact_2807_of__real_Ominus,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [X] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,uminus_uminus(real),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(real,X_a,of_real(X_a),X)) ) ).

fof(fact_2808_norm__of__real,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => ! [R_3] : hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),R_3)) = hAPP(real,real,abs_abs(real),R_3) ) ).

fof(fact_2809_of__real__inverse,axiom,
    ! [X_a] :
      ( ( divisi14063676e_zero(X_a)
        & real_div_algebra(X_a) )
     => ! [X] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,inverse_inverse(real),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(real,X_a,of_real(X_a),X)) ) ).

fof(fact_2810_exp__of__real,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [X] : hAPP(X_a,X_a,exp(X_a),hAPP(real,X_a,of_real(X_a),X)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,exp(real),X)) ) ).

fof(fact_2811_of__real__of__nat__eq,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [N] : hAPP(real,X_a,of_real(X_a),hAPP(nat,real,semiring_1_of_nat(real),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N) ) ).

fof(fact_2812_sgn__of__real,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => ! [R_3] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(real,X_a,of_real(X_a),R_3)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,sgn_sgn(real),R_3)) ) ).

fof(fact_2813_sgn__eq,axiom,
    ! [Z_1] : hAPP(complex,complex,sgn_sgn(complex),Z_1) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),Z_1),hAPP(real,complex,of_real(complex),hAPP(complex,real,norm_norm(complex),Z_1))) ).

fof(fact_2814_nonzero__of__real__divide,axiom,
    ! [X_a] :
      ( real_field(X_a)
     => ! [X,Y] :
          ( Y != zero_zero(real)
         => hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ) ).

fof(fact_2815_nonzero__of__real__inverse,axiom,
    ! [X_a] :
      ( real_div_algebra(X_a)
     => ! [X] :
          ( X != zero_zero(real)
         => hAPP(real,X_a,of_real(X_a),hAPP(real,real,inverse_inverse(real),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(real,X_a,of_real(X_a),X)) ) ) ).

fof(fact_2816_complex__of__real__mult__Complex,axiom,
    ! [R_3,X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_3)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),Y)) ).

fof(fact_2817_Complex__mult__complex__of__real,axiom,
    ! [X,Y,R_3] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)),hAPP(real,complex,of_real(complex),R_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),R_3)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),R_3)) ).

fof(fact_2818_Complex__add__complex__of__real,axiom,
    ! [X,Y,R_3] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)),hAPP(real,complex,of_real(complex),R_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),R_3)),Y) ).

fof(fact_2819_complex__of__real__add__Complex,axiom,
    ! [R_3,X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,of_real(complex),R_3)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_3),X)),Y) ).

fof(fact_2820_i__mult__eq,axiom,
    hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),ii) = hAPP(real,complex,of_real(complex),hAPP(int,real,number_number_of(real),min)) ).

fof(fact_2821_i__complex__of__real,axiom,
    ! [R_3] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),R_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),R_3) ).

fof(fact_2822_complex__of__real__i,axiom,
    ! [R_3] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_3)),ii) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),R_3) ).

fof(fact_2823_complex__of__real__minus__one,axiom,
    hAPP(real,complex,of_real(complex),hAPP(real,real,uminus_uminus(real),one_one(real))) = hAPP(complex,complex,uminus_uminus(complex),one_one(complex)) ).

fof(fact_2824_cmod__complex__polar,axiom,
    ! [R_3,A_1] : hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_3)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1)))) = hAPP(real,real,abs_abs(real),R_3) ).

fof(fact_2825_of__real_Opos__bounded,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2))) ) ) ).

fof(fact_2826_of__real_Ononneg__bounded,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2))) ) ) ).

fof(fact_2827_expi__two__pi__i,axiom,
    hAPP(complex,complex,expi,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,complex,of_real(complex),pi))),ii)) = one_one(complex) ).

fof(fact_2828_expi__add,axiom,
    ! [A_1,B] : hAPP(complex,complex,expi,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A_1),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,expi,A_1)),hAPP(complex,complex,expi,B)) ).

fof(fact_2829_expi__zero,axiom,
    hAPP(complex,complex,expi,zero_zero(complex)) = one_one(complex) ).

fof(fact_2830_of__real_Obounded,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ? [K_2] :
        ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2))) ) ).

fof(fact_2831_complex__split__polar,axiom,
    ! [Z_1] :
    ? [R_1,A_4] : Z_1 = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_1)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_4)),hAPP(real,real,sin,A_4))) ).

fof(fact_2832_nat__induct,axiom,
    ! [Na,P_1] :
      ( hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
     => ( ! [N_1] :
            ( hBOOL(hAPP(nat,bool,P_1,N_1))
           => hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_1))) )
       => hBOOL(hAPP(nat,bool,P_1,Na)) ) ) ).

fof(fact_2833_nat_Oexhaust,axiom,
    ! [Y] :
      ( Y != zero_zero(nat)
     => ~ ! [Nat_2] : Y != hAPP(nat,nat,suc,Nat_2) ) ).

fof(fact_2834_zero__induct,axiom,
    ! [P_1,K] :
      ( hBOOL(hAPP(nat,bool,P_1,K))
     => ( ! [N_1] :
            ( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_1)))
           => hBOOL(hAPP(nat,bool,P_1,N_1)) )
       => hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) ) ) ).

fof(fact_2835_ex__less__of__nat,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
        ? [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))) ) ).

fof(fact_2836_ex__le__of__nat,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
        ? [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))) ) ).

fof(fact_2837_le__funI,axiom,
    ! [X_a,X_b] :
      ( ord(X_b)
     => ! [F,G] :
          ( ! [X_1] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1)))
         => hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)) ) ) ).

fof(fact_2838_dvdE,axiom,
    ! [X_a] :
      ( dvd(X_a)
     => ! [B,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),A_1))
         => ~ ! [K_1] : ti(X_a,A_1) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),K_1) ) ) ).

fof(fact_2839_card__Pow,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => hAPP(fun(fun(X_a,bool),bool),nat,finite_card(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)) ) ).

fof(fact_2840_xt6,axiom,
    ! [X_b,X_a] :
      ( ( order(X_a)
        & ord(X_b) )
     => ! [C,F,B_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,B_2)),A_3))
         => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),B_2))
           => ( ! [X_1,Y_1] :
                  ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_1))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))) )
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,C)),A_3)) ) ) ) ) ).

fof(fact_2841_xt4,axiom,
    ! [X_b,X_a] :
      ( ( order(X_a)
        & order(X_b) )
     => ! [C,F,B_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,B_2)),A_3))
         => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C),B_2))
           => ( ! [X_1,Y_1] :
                  ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_1))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))) )
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,C)),A_3)) ) ) ) ) ).

fof(fact_2842_PowI,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) ) ).

fof(fact_2843_PowD,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1)))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).

fof(fact_2844_Pow__iff,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1)))
    <=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).

fof(fact_2845_Pow__def,axiom,
    ! [X_a,A_2] : hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2) = hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),combc(fun(X_a,bool),fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool))),A_2)) ).

fof(fact_2846_Pow__top,axiom,
    ! [X_a,A_2] : hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2))) ).

fof(fact_2847_Union__Pow__eq,axiom,
    ! [X_a,A_2] : hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)) = ti(fun(X_a,bool),A_2) ).

fof(fact_2848_Pow__mono,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) ) ).

fof(fact_2849_finite__Pow__iff,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)))
    <=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).

fof(fact_2850_subset__Pow__Union,axiom,
    ! [X_a,A_2] : hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)))) ).

fof(fact_2851_minf_I8_J,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [T_5] :
        ? [Z_2] :
        ! [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_2))
         => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_5),X_1)) ) ) ).

fof(fact_2852_pinf_I8_J,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [T_5] :
        ? [Z_2] :
        ! [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_5),X_1)) ) ) ).

fof(fact_2853_minf_I6_J,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [T_5] :
        ? [Z_2] :
        ! [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_2))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),T_5)) ) ) ).

fof(fact_2854_order__le__less__subst1,axiom,
    ! [X_a,X_b] :
      ( ( order(X_b)
        & order(X_a) )
     => ! [C,A_3,F,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_b,X_a,F,B_2)))
         => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),B_2),C))
           => ( ! [X_1,Y_1] :
                  ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_1))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,X_1)),hAPP(X_b,X_a,F,Y_1))) )
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_b,X_a,F,C))) ) ) ) ) ).

fof(fact_2855_order__less__le__subst1,axiom,
    ! [X_a,X_b] :
      ( ( order(X_b)
        & order(X_a) )
     => ! [C,A_3,F,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_b,X_a,F,B_2)))
         => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B_2),C))
           => ( ! [X_1,Y_1] :
                  ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_1)),hAPP(X_b,X_a,F,Y_1))) )
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_b,X_a,F,C))) ) ) ) ) ).

fof(fact_2856_order__le__less__subst2,axiom,
    ! [X_a,X_b] :
      ( ( order(X_b)
        & order(X_a) )
     => ! [F,C,A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))
         => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,B_2)),C))
           => ( ! [X_1,Y_1] :
                  ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
                 => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,F,Y_1))) )
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,A_3)),C)) ) ) ) ) ).

fof(fact_2857_pinf_I6_J,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [T_5] :
        ? [Z_2] :
        ! [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_1))
         => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),T_5)) ) ) ).

fof(fact_2858_dense__le,axiom,
    ! [X_a] :
      ( dense_linorder(X_a)
     => ! [Z_1,Y] :
          ( ! [X_1] :
              ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z_1)) )
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1)) ) ) ).

fof(fact_2859_dense__le__bounded,axiom,
    ! [X_a] :
      ( dense_linorder(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
         => ( ! [W_1] :
                ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),W_1))
               => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W_1),Y))
                 => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W_1),Z_1)) ) )
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1)) ) ) ) ).

fof(fact_2860_order__less__le__subst2,axiom,
    ! [X_a,X_b] :
      ( ( order(X_b)
        & order(X_a) )
     => ! [F,C,A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
         => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,B_2)),C))
           => ( ! [X_1,Y_1] :
                  ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
                 => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,F,Y_1))) )
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,A_3)),C)) ) ) ) ) ).

fof(fact_2861_xt7,axiom,
    ! [X_b,X_a] :
      ( ( order(X_a)
        & order(X_b) )
     => ! [C,F,B_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3))
         => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,B_2)))
           => ( ! [X_1,Y_1] :
                  ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1))
                 => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Y_1)),hAPP(X_a,X_b,F,X_1))) )
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,A_3))) ) ) ) ) ).

fof(fact_2862_xt5,axiom,
    ! [X_b,X_a] :
      ( ( order(X_a)
        & order(X_b) )
     => ! [C,F,B_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3))
         => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C),hAPP(X_a,X_b,F,B_2)))
           => ( ! [X_1,Y_1] :
                  ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
                 => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Y_1)),hAPP(X_a,X_b,F,X_1))) )
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,A_3))) ) ) ) ) ).

fof(fact_2863_power__power__power,axiom,
    ! [X_a] :
      ( power(X_a)
     => power_power(X_a) = hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),one_one(X_a)),times_times(X_a)) ) ).

fof(fact_2864_DERIV__neg__imp__decreasing,axiom,
    ! [F,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2))
     => ( ! [X_1] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
           => ? [Y_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),Y_1))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),zero_zero(real))) ) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))) ) ) ).

fof(fact_2865_DERIV__pos__imp__increasing,axiom,
    ! [F,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2))
     => ( ! [X_1] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
           => ? [Y_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),Y_1))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1)) ) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,A_3)),hAPP(real,real,F,B_2))) ) ) ).

fof(fact_2866_power_Opower_Opower__0,axiom,
    ! [X_a,One,Times,A_3] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),A_3),zero_zero(nat)) = ti(X_a,One) ).

fof(fact_2867_power_Opower_Opower__Suc,axiom,
    ! [X_a,One,Times,A_3,Na] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),A_3),hAPP(nat,nat,suc,Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),Times,A_3),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),A_3),Na)) ).

fof(fact_2868_int__induct,axiom,
    ! [I,P_1,K] :
      ( hBOOL(hAPP(int,bool,P_1,K))
     => ( ! [I_2] :
            ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I_2))
           => ( hBOOL(hAPP(int,bool,P_1,I_2))
             => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int)))) ) )
       => ( ! [I_2] :
              ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K))
             => ( hBOOL(hAPP(int,bool,P_1,I_2))
               => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int)))) ) )
         => hBOOL(hAPP(int,bool,P_1,I)) ) ) ) ).

fof(fact_2869_minusinfinity,axiom,
    ! [P_1,P1,D_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
     => ( ! [X_1,K_1] :
            ( hBOOL(hAPP(int,bool,P1,X_1))
          <=> hBOOL(hAPP(int,bool,P1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))) )
       => ( ? [Z_2] :
            ! [X_1] :
              ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Z_2))
             => ( hBOOL(hAPP(int,bool,P_1,X_1))
              <=> hBOOL(hAPP(int,bool,P1,X_1)) ) )
         => ( ? [X1] : hBOOL(hAPP(int,bool,P1,X1))
           => ? [X1] : hBOOL(hAPP(int,bool,P_1,X1)) ) ) ) ) ).

fof(fact_2870_plusinfinity,axiom,
    ! [P_1,P_2,D_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
     => ( ! [X_1,K_1] :
            ( hBOOL(hAPP(int,bool,P_2,X_1))
          <=> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))) )
       => ( ? [Z_2] :
            ! [X_1] :
              ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),X_1))
             => ( hBOOL(hAPP(int,bool,P_1,X_1))
              <=> hBOOL(hAPP(int,bool,P_2,X_1)) ) )
         => ( ? [X1] : hBOOL(hAPP(int,bool,P_2,X1))
           => ? [X1] : hBOOL(hAPP(int,bool,P_1,X1)) ) ) ) ) ).

fof(fact_2871_coprime__sos,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),Y))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).

fof(fact_2872_floor__le__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),one_one(int)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ).

fof(fact_2873_one__less__floor,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Xa)) ) ) ).

fof(fact_2874_coprime__exp,axiom,
    ! [N,D_3,A_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),A_1))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N))) ) ).

fof(fact_2875_coprime__exp__imp,axiom,
    ! [N,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N))) ) ).

fof(fact_2876_coprime__commute,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,B_2),A_3)) ) ).

fof(fact_2877_coprime__divisors,axiom,
    ! [E_1,B,D_3,A_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),A_1))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E_1),B))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),E_1)) ) ) ) ).

fof(fact_2878_coprime__lmul2,axiom,
    ! [D_3,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),B)) ) ).

fof(fact_2879_coprime__rmul2,axiom,
    ! [D_3,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),A_1)) ) ).

fof(fact_2880_coprime__mul,axiom,
    ! [B,D_3,A_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),A_1))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),B))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))) ) ) ).

fof(fact_2881_coprime__mul__eq,axiom,
    ! [D_1,A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),B_2)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),A_3))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),B_2)) ) ) ).

fof(fact_2882_coprime__1_H,axiom,
    ! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,one_one(nat)),A_1)) ).

fof(fact_2883_coprime__1,axiom,
    ! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),one_one(nat))) ).

fof(fact_2884_coprime__refl,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,Na),Na))
    <=> Na = one_one(nat) ) ).

fof(fact_2885_floor__real__of__nat,axiom,
    ! [N] : hAPP(real,int,archim2085082626_floor(real),hAPP(nat,real,real_1(nat),N)) = hAPP(nat,int,semiring_1_of_nat(int),N) ).

fof(fact_2886_natfloor__def,axiom,
    ! [X] : hAPP(real,nat,natfloor,X) = hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X)) ).

fof(fact_2887_coprime__Suc0,axiom,
    ! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),hAPP(nat,nat,suc,zero_zero(nat)))) ).

fof(fact_2888_coprime__Suc0_H,axiom,
    ! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,suc,zero_zero(nat))),A_1)) ).

fof(fact_2889_floor__zero,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => hAPP(X_a,int,archim2085082626_floor(X_a),zero_zero(X_a)) = zero_zero(int) ) ).

fof(fact_2890_floor__less__cancel,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(X_a,int,archim2085082626_floor(X_a),Y)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).

fof(fact_2891_floor__mono,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(X_a,int,archim2085082626_floor(X_a),Y))) ) ) ).

fof(fact_2892_coprime__0,axiom,
    ! [D_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),zero_zero(nat)))
    <=> D_1 = one_one(nat) ) ).

fof(fact_2893_coprime__0_H,axiom,
    ! [D_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,zero_zero(nat)),D_1))
    <=> D_1 = one_one(nat) ) ).

fof(fact_2894_divides__mul,axiom,
    ! [N,M,R_3] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),R_3))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),R_3))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,M),N))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),R_3)) ) ) ) ).

fof(fact_2895_coprime__divprod,axiom,
    ! [D_3,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),A_1))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),B)) ) ) ).

fof(fact_2896_floor__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => hAPP(X_a,int,archim2085082626_floor(X_a),one_one(X_a)) = one_one(int) ) ).

fof(fact_2897_coprime,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2))
    <=> ! [D_2] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),A_3))
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),B_2)) )
        <=> D_2 = one_one(nat) ) ) ).

fof(fact_2898_floor__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [V] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(int,X_a,number_number_of(X_a),V)) = hAPP(int,int,number_number_of(int),V) ) ).

fof(fact_2899_coprime__plus1,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),N)) ).

fof(fact_2900_coprime__exp2,axiom,
    ! [A_3,Na,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_3),hAPP(nat,nat,suc,Na))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B_2),hAPP(nat,nat,suc,Na))))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2)) ) ).

fof(fact_2901_floor__of__nat,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [N] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) = hAPP(nat,int,semiring_1_of_nat(int),N) ) ).

fof(fact_2902_coprime__minus1,axiom,
    ! [N] :
      ( N != zero_zero(nat)
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),N)) ) ).

fof(fact_2903_floor__eq3,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N))))
       => hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X)) = N ) ) ).

fof(fact_2904_floor__minus__real__of__nat,axiom,
    ! [N] : hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,real_1(nat),N))) = hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N)) ).

fof(fact_2905_floor__less__zero,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),zero_zero(int)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),zero_zero(X_a))) ) ) ).

fof(fact_2906_zero__le__floor,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa)) ) ) ).

fof(fact_2907_zero__less__floor,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),Xa)) ) ) ).

fof(fact_2908_floor__le__zero,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),zero_zero(int)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),one_one(X_a))) ) ) ).

fof(fact_2909_floor__less__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),one_one(int)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),one_one(X_a))) ) ) ).

fof(fact_2910_one__le__floor,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),Xa)) ) ) ).

fof(fact_2911_floor__less__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa,Va] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),hAPP(int,int,number_number_of(int),Va)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(int,X_a,number_number_of(X_a),Va))) ) ) ).

fof(fact_2912_number__of__le__floor,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Va,Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),Va)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),Xa)) ) ) ).

fof(fact_2913_floor__add__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int)) ) ).

fof(fact_2914_floor__add__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,V] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(int,int,number_number_of(int),V)) ) ).

fof(fact_2915_floor__diff__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int)) ) ).

fof(fact_2916_floor__diff__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,V] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(int,int,number_number_of(int),V)) ) ).

fof(fact_2917_le__mult__floor,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(real,int,archim2085082626_floor(real),A_1)),hAPP(real,int,archim2085082626_floor(real),B))),hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)))) ) ) ).

fof(fact_2918_floor__eq4,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N))))
       => hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X)) = N ) ) ).

fof(fact_2919_number__of__less__floor,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Va,Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),Va)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),one_one(X_a))),Xa)) ) ) ).

fof(fact_2920_floor__le__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa,Va] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),hAPP(int,int,number_number_of(int),Va)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),one_one(X_a)))) ) ) ).

fof(fact_2921_coprime__bezout,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2))
    <=> ? [X_1,Y_1] :
          ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_2),Y_1)) = one_one(nat)
          | hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_2),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),Y_1)) = one_one(nat) ) ) ).

fof(fact_2922_coprime__bezout__strong,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
     => ( B != one_one(nat)
       => ? [X_1,Y_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),Y_1)),one_one(nat)) ) ) ).

fof(fact_2923_nat__number__of__diff__1,axiom,
    ! [V] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = zero_zero(nat) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,pred,V)) ) ) ).

fof(fact_2924_succ__pred,axiom,
    ! [X] : hAPP(int,int,succ,hAPP(int,int,pred,X)) = ti(int,X) ).

fof(fact_2925_pred__Bit0,axiom,
    ! [K_4] : hAPP(int,int,pred,hAPP(int,int,bit0,K_4)) = hAPP(int,int,bit1,hAPP(int,int,pred,K_4)) ).

fof(fact_2926_pred__Bit1,axiom,
    ! [K_4] : hAPP(int,int,pred,hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit0,K_4) ).

fof(fact_2927_le__iff__pred__less,axiom,
    ! [K,L] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,pred,K)),L)) ) ).

fof(fact_2928_pred__Pls,axiom,
    hAPP(int,int,pred,pls) = min ).

fof(fact_2929_pred__Min,axiom,
    hAPP(int,int,pred,min) = hAPP(int,int,bit0,min) ).

fof(fact_2930_pred__def,axiom,
    ! [K_4] : hAPP(int,int,pred,K_4) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),one_one(int)) ).

fof(fact_2931_minus__Bit1,axiom,
    ! [K_4] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit1,hAPP(int,int,pred,hAPP(int,int,uminus_uminus(int),K_4))) ).

fof(fact_2932_add__Min__right,axiom,
    ! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),min) = hAPP(int,int,pred,K_4) ).

fof(fact_2933_add__Min,axiom,
    ! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),min),K_4) = hAPP(int,int,pred,K_4) ).

fof(fact_2934_diff__bin__simps_I8_J,axiom,
    ! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,pred,K_4)),L_4)) ).

fof(fact_2935_number__of__pred,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,pred,W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),hAPP(int,X_a,number_number_of(X_a),W)) ) ).

fof(fact_2936_neg__number__of__pred__iff__0,axiom,
    ! [Va] :
      ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
    <=> hAPP(int,nat,number_number_of(nat),Va) = zero_zero(nat) ) ).

fof(fact_2937_Suc__diff__number__of,axiom,
    ! [M,V] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),hAPP(int,nat,number_number_of(nat),V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,pred,V))) ) ).

fof(fact_2938_accp__subset,axiom,
    ! [X_a,R1_1,R2_1] :
      ( hBOOL(hAPP(fun(X_a,fun(X_a,bool)),bool,hAPP(fun(X_a,fun(X_a,bool)),fun(fun(X_a,fun(X_a,bool)),bool),ord_less_eq(fun(X_a,fun(X_a,bool))),R1_1),R2_1))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R2_1)),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R1_1))) ) ).

fof(fact_2939_complex__expi__Ex,axiom,
    ! [Z_1] :
    ? [A_4,R_1] : Z_1 = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_1)),hAPP(complex,complex,expi,A_4)) ).

fof(fact_2940_ResSet__finite,axiom,
    ! [X_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Ma),X_2))
       => hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_2)) ) ) ).

fof(fact_2941_ResSet__def,axiom,
    ! [Ma,X_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Ma),X_2))
    <=> ! [Y1,Y2] :
          ( ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y1),X_2))
            & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y2),X_2))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y1),Y2),Ma)) )
         => ti(int,Y1) = ti(int,Y2) ) ) ).

fof(fact_2942_ResSet__SRStar__prop,axiom,
    ! [P_4] : hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,P_4),hAPP(int,fun(int,bool),sRStar,P_4))) ).

fof(fact_2943_accp__downward,axiom,
    ! [X_a,A_3,Ra,B_2] :
      ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),Ra),B_2))
     => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),Ra,A_3),B_2))
       => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),Ra),A_3)) ) ) ).

fof(fact_2944_accp_Osimps,axiom,
    ! [X_a,Ra,A_3] :
      ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),Ra),A_3))
    <=> ? [X_1] :
          ( ti(X_a,A_3) = ti(X_a,X_1)
          & ! [Xa_1] :
              ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),Ra,Xa_1),X_1))
             => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),Ra),Xa_1)) ) ) ) ).

fof(fact_2945_nat_Osize_I2_J,axiom,
    ! [Nat_1] : hAPP(nat,nat,nat_size,hAPP(nat,nat,suc,Nat_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,nat_size,Nat_1)),hAPP(nat,nat,suc,zero_zero(nat))) ).

fof(fact_2946_nat_Osize_I4_J,axiom,
    ! [Nat_1] : hAPP(nat,nat,size_size(nat),hAPP(nat,nat,suc,Nat_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,size_size(nat),Nat_1)),hAPP(nat,nat,suc,zero_zero(nat))) ).

fof(fact_2947_code__numeral_Osize_I1_J,axiom,
    hAPP(code_code_numeral,nat,code_c271388182l_size,zero_zero(code_code_numeral)) = zero_zero(nat) ).

fof(fact_2948_nat__size,axiom,
    ! [N] : hAPP(nat,nat,size_size(nat),N) = N ).

fof(fact_2949_nat_Osize_I3_J,axiom,
    hAPP(nat,nat,size_size(nat),zero_zero(nat)) = zero_zero(nat) ).

fof(fact_2950_nat_Osize_I1_J,axiom,
    hAPP(nat,nat,nat_size,zero_zero(nat)) = zero_zero(nat) ).

fof(fact_2951_code__numeral_Osize_I2_J,axiom,
    ! [Code_numeral_2] : hAPP(code_code_numeral,nat,code_c271388182l_size,hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(code_code_numeral,nat,code_c271388182l_size,Code_numeral_2)),hAPP(nat,nat,suc,zero_zero(nat))) ).

fof(fact_2952_complex__mod__mult__cnj,axiom,
    ! [Z_1] : hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z_1),hAPP(complex,complex,cnj,Z_1))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,norm_norm(complex),Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).

fof(fact_2953_fun__Compl__def,axiom,
    ! [X_a,X_b] :
      ( uminus(X_b)
     => ! [A_2,X_1] : hAPP(X_a,X_b,hAPP(fun(X_a,X_b),fun(X_a,X_b),uminus_uminus(fun(X_a,X_b)),A_2),X_1) = hAPP(X_b,X_b,uminus_uminus(X_b),hAPP(X_a,X_b,A_2,X_1)) ) ).

fof(fact_2954_complex__cnj__complex__of__real,axiom,
    ! [X] : hAPP(complex,complex,cnj,hAPP(real,complex,of_real(complex),X)) = hAPP(real,complex,of_real(complex),X) ).

fof(fact_2955_complex__cnj__divide,axiom,
    ! [X,Y] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),X),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) ).

fof(fact_2956_complex__cnj__power,axiom,
    ! [X,N] : hAPP(complex,complex,cnj,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),X),N)) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,cnj,X)),N) ).

fof(fact_2957_cnj_Oadd,axiom,
    ! [X,Y] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) ).

fof(fact_2958_cnj_Ominus,axiom,
    ! [X] : hAPP(complex,complex,cnj,hAPP(complex,complex,uminus_uminus(complex),X)) = hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,cnj,X)) ).

fof(fact_2959_complex__cnj__cnj,axiom,
    ! [Z_1] : hAPP(complex,complex,cnj,hAPP(complex,complex,cnj,Z_1)) = Z_1 ).

fof(fact_2960_complex__cnj__cancel__iff,axiom,
    ! [Xa,Ya] :
      ( hAPP(complex,complex,cnj,Xa) = hAPP(complex,complex,cnj,Ya)
    <=> Xa = Ya ) ).

fof(fact_2961_code__numeral_Oinject,axiom,
    ! [Code_numeral,Code_numeral_5] :
      ( hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral) = hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_5)
    <=> Code_numeral = Code_numeral_5 ) ).

fof(fact_2962_complex__cnj__of__nat,axiom,
    ! [N] : hAPP(complex,complex,cnj,hAPP(nat,complex,semiring_1_of_nat(complex),N)) = hAPP(nat,complex,semiring_1_of_nat(complex),N) ).

fof(fact_2963_complex__mod__cnj,axiom,
    ! [Z_1] : hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,Z_1)) = hAPP(complex,real,norm_norm(complex),Z_1) ).

fof(fact_2964_cnj_Odiff,axiom,
    ! [X,Y] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) ).

fof(fact_2965_complex__cnj__inverse,axiom,
    ! [X] : hAPP(complex,complex,cnj,hAPP(complex,complex,inverse_inverse(complex),X)) = hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,cnj,X)) ).

fof(fact_2966_cnj_Ozero,axiom,
    hAPP(complex,complex,cnj,zero_zero(complex)) = zero_zero(complex) ).

fof(fact_2967_complex__cnj__zero__iff,axiom,
    ! [Z] :
      ( hAPP(complex,complex,cnj,Z) = zero_zero(complex)
    <=> Z = zero_zero(complex) ) ).

fof(fact_2968_complex__cnj__number__of,axiom,
    ! [W] : hAPP(complex,complex,cnj,hAPP(int,complex,number_number_of(complex),W)) = hAPP(int,complex,number_number_of(complex),W) ).

fof(fact_2969_complex__cnj__mult,axiom,
    ! [X,Y] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) ).

fof(fact_2970_complex__cnj__one,axiom,
    hAPP(complex,complex,cnj,one_one(complex)) = one_one(complex) ).

fof(fact_2971_code__numeral_Osimps_I3_J,axiom,
    ! [Code_numeral_4] : hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_4) != zero_zero(code_code_numeral) ).

fof(fact_2972_code__numeral_Osimps_I2_J,axiom,
    ! [Code_numeral_3] : zero_zero(code_code_numeral) != hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_3) ).

fof(fact_2973_complex__cnj,axiom,
    ! [A_1,B] : hAPP(complex,complex,cnj,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),hAPP(real,real,uminus_uminus(real),B)) ).

fof(fact_2974_complex__cnj__i,axiom,
    hAPP(complex,complex,cnj,ii) = hAPP(complex,complex,uminus_uminus(complex),ii) ).

fof(fact_2975_mult__left__idem,axiom,
    ! [X_a] :
      ( ab_sem1668676832m_mult(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) ) ).

fof(fact_2976_mult__idem,axiom,
    ! [X_a] :
      ( ab_sem1668676832m_mult(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X) = ti(X_a,X) ) ).

fof(fact_2977_times_Oidem,axiom,
    ! [X_a] :
      ( ab_sem1668676832m_mult(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1) = ti(X_a,A_1) ) ).

fof(fact_2978_compl__eq__compl__iff,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [Xa,Ya] :
          ( hAPP(X_a,X_a,uminus_uminus(X_a),Xa) = hAPP(X_a,X_a,uminus_uminus(X_a),Ya)
        <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ).

fof(fact_2979_double__compl,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [X] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = ti(X_a,X) ) ).

fof(fact_2980_Suc__code__numeral__minus__one,axiom,
    ! [N] : hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,N)),one_one(code_code_numeral)) = N ).

fof(fact_2981_compl__le__swap2,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),X))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)) ) ) ).

fof(fact_2982_compl__le__swap1,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),hAPP(X_a,X_a,uminus_uminus(X_a),X)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y))) ) ) ).

fof(fact_2983_compl__mono,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),hAPP(X_a,X_a,uminus_uminus(X_a),X))) ) ) ).

fof(fact_2984_compl__le__compl__iff,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Xa)),hAPP(X_a,X_a,uminus_uminus(X_a),Ya)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) ) ) ).

fof(fact_2985_compl__less__swap2,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),X))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)) ) ) ).

fof(fact_2986_compl__less__swap1,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),hAPP(X_a,X_a,uminus_uminus(X_a),X)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y))) ) ) ).

fof(fact_2987_compl__less__compl__iff,axiom,
    ! [X_a] :
      ( boolean_algebra(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Xa)),hAPP(X_a,X_a,uminus_uminus(X_a),Ya)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) ) ) ).

fof(fact_2988_minus__apply,axiom,
    ! [X_b,X_a] :
      ( minus(X_a)
     => ! [A_2,B_1,Xa] : hAPP(X_b,X_a,hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),minus_minus(fun(X_b,X_a)),A_2),B_1),Xa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_b,X_a,A_2,Xa)),hAPP(X_b,X_a,B_1,Xa)) ) ).

fof(fact_2989_fun__diff__def,axiom,
    ! [X_a,X_b] :
      ( minus(X_b)
     => ! [A_2,B_1,X_1] : hAPP(X_a,X_b,hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),minus_minus(fun(X_a,X_b)),A_2),B_1),X_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,A_2,X_1)),hAPP(X_a,X_b,B_1,X_1)) ) ).

fof(fact_2990_uminus__apply,axiom,
    ! [X_b,X_a] :
      ( uminus(X_a)
     => ! [A_2,Xa] : hAPP(X_b,X_a,hAPP(fun(X_b,X_a),fun(X_b,X_a),uminus_uminus(fun(X_b,X_a)),A_2),Xa) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_b,X_a,A_2,Xa)) ) ).

fof(fact_2991_cnj_Opos__bounded,axiom,
    ? [K_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
      & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ) ).

fof(fact_2992_cnj_Ononneg__bounded,axiom,
    ? [K_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
      & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ) ).

fof(fact_2993_cnj_Obounded,axiom,
    ? [K_2] :
    ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ).

fof(fact_2994_code__numeral_Osize_I4_J,axiom,
    ! [Code_numeral_2] : hAPP(code_code_numeral,nat,size_size(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(code_code_numeral,nat,size_size(code_code_numeral),Code_numeral_2)),hAPP(nat,nat,suc,zero_zero(nat))) ).

fof(fact_2995_complex__diff__cnj,axiom,
    ! [Z_1] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),Z_1),hAPP(complex,complex,cnj,Z_1)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(complex,real,im,Z_1)))),ii) ).

fof(fact_2996_code__numeral_Oexhaust,axiom,
    ! [Y] :
      ( Y != zero_zero(code_code_numeral)
     => ~ ! [Code_numeral_1] : Y != hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_1) ) ).

fof(fact_2997_Im,axiom,
    ! [X,Y] : hAPP(complex,real,im,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = Y ).

fof(fact_2998_complex__Im__cnj,axiom,
    ! [X] : hAPP(complex,real,im,hAPP(complex,complex,cnj,X)) = hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X)) ).

fof(fact_2999_Im__complex__of__real,axiom,
    ! [Z_1] : hAPP(complex,real,im,hAPP(real,complex,of_real(complex),Z_1)) = zero_zero(real) ).

fof(fact_3000_complex__Im__i,axiom,
    hAPP(complex,real,im,ii) = one_one(real) ).

fof(fact_3001_Im_Ozero,axiom,
    hAPP(complex,real,im,zero_zero(complex)) = zero_zero(real) ).

fof(fact_3002_complex__Im__one,axiom,
    hAPP(complex,real,im,one_one(complex)) = zero_zero(real) ).

fof(fact_3003_Im_Oadd,axiom,
    ! [X,Y] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)) ).

fof(fact_3004_complex__Im__number__of,axiom,
    ! [V] : hAPP(complex,real,im,hAPP(int,complex,number_number_of(complex),V)) = zero_zero(real) ).

fof(fact_3005_Im_Ominus,axiom,
    ! [X] : hAPP(complex,real,im,hAPP(complex,complex,uminus_uminus(complex),X)) = hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X)) ).

fof(fact_3006_Im_Odiff,axiom,
    ! [X,Y] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)) ).

fof(fact_3007_complex__Im__of__nat,axiom,
    ! [N] : hAPP(complex,real,im,hAPP(nat,complex,semiring_1_of_nat(complex),N)) = zero_zero(real) ).

fof(fact_3008_abs__Im__le__cmod,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(complex,real,im,X))),hAPP(complex,real,norm_norm(complex),X))) ).

fof(fact_3009_complex__In__mult__cnj__zero,axiom,
    ! [Z_1] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z_1),hAPP(complex,complex,cnj,Z_1))) = zero_zero(real) ).

fof(fact_3010_Im__sgn,axiom,
    ! [Z_1] : hAPP(complex,real,im,hAPP(complex,complex,sgn_sgn(complex),Z_1)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,im,Z_1)),hAPP(complex,real,norm_norm(complex),Z_1)) ).

fof(fact_3011_code__numeral_Osize_I3_J,axiom,
    hAPP(code_code_numeral,nat,size_size(code_code_numeral),zero_zero(code_code_numeral)) = zero_zero(nat) ).

fof(fact_3012_Im_Opos__bounded,axiom,
    ? [K_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
      & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ) ).

fof(fact_3013_Im_Ononneg__bounded,axiom,
    ? [K_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
      & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ) ).

fof(fact_3014_Im_Obounded,axiom,
    ? [K_2] :
    ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ).

fof(fact_3015_complex__inverse__def,axiom,
    ! [X] : hAPP(complex,complex,inverse_inverse(complex),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).

fof(fact_3016_complex__mult__cnj,axiom,
    ! [Z_1] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z_1),hAPP(complex,complex,cnj,Z_1)) = hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_3017_complex__Im__inverse,axiom,
    ! [X] : hAPP(complex,real,im,hAPP(complex,complex,inverse_inverse(complex),X)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_3018_Re__complex__of__real,axiom,
    ! [Z_1] : hAPP(complex,real,re,hAPP(real,complex,of_real(complex),Z_1)) = Z_1 ).

fof(fact_3019_Re,axiom,
    ! [X,Y] : hAPP(complex,real,re,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = X ).

fof(fact_3020_complex__Re__cnj,axiom,
    ! [X] : hAPP(complex,real,re,hAPP(complex,complex,cnj,X)) = hAPP(complex,real,re,X) ).

fof(fact_3021_complex__eqI,axiom,
    ! [X,Y] :
      ( hAPP(complex,real,re,X) = hAPP(complex,real,re,Y)
     => ( hAPP(complex,real,im,X) = hAPP(complex,real,im,Y)
       => X = Y ) ) ).

fof(fact_3022_complex__eq__iff,axiom,
    ! [Xa,Ya] :
      ( Xa = Ya
    <=> ( hAPP(complex,real,re,Xa) = hAPP(complex,real,re,Ya)
        & hAPP(complex,real,im,Xa) = hAPP(complex,real,im,Ya) ) ) ).

fof(fact_3023_complex__Re__le__cmod,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,re,X)),hAPP(complex,real,norm_norm(complex),X))) ).

fof(fact_3024_complex__Re__i,axiom,
    hAPP(complex,real,re,ii) = zero_zero(real) ).

fof(fact_3025_complex__surj,axiom,
    ! [Z_1] : hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(complex,real,re,Z_1)),hAPP(complex,real,im,Z_1)) = Z_1 ).

fof(fact_3026_Re_Ozero,axiom,
    hAPP(complex,real,re,zero_zero(complex)) = zero_zero(real) ).

fof(fact_3027_Re_Oadd,axiom,
    ! [X,Y] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y)) ).

fof(fact_3028_complex__Re__one,axiom,
    hAPP(complex,real,re,one_one(complex)) = one_one(real) ).

fof(fact_3029_Re_Ominus,axiom,
    ! [X] : hAPP(complex,real,re,hAPP(complex,complex,uminus_uminus(complex),X)) = hAPP(real,real,uminus_uminus(real),hAPP(complex,real,re,X)) ).

fof(fact_3030_complex__Re__number__of,axiom,
    ! [V] : hAPP(complex,real,re,hAPP(int,complex,number_number_of(complex),V)) = hAPP(int,real,number_number_of(real),V) ).

fof(fact_3031_Re_Odiff,axiom,
    ! [X,Y] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y)) ).

fof(fact_3032_complex__Re__of__nat,axiom,
    ! [N] : hAPP(complex,real,re,hAPP(nat,complex,semiring_1_of_nat(complex),N)) = hAPP(nat,real,semiring_1_of_nat(real),N) ).

fof(fact_3033_abs__Re__le__cmod,axiom,
    ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(complex,real,re,X))),hAPP(complex,real,norm_norm(complex),X))) ).

fof(fact_3034_Re__sgn,axiom,
    ! [Z_1] : hAPP(complex,real,re,hAPP(complex,complex,sgn_sgn(complex),Z_1)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,Z_1)),hAPP(complex,real,norm_norm(complex),Z_1)) ).

fof(fact_3035_complex__Im__mult,axiom,
    ! [X,Y] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,im,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,re,Y))) ).

fof(fact_3036_complex__Re__mult,axiom,
    ! [X,Y] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y))) ).

fof(fact_3037_cnj__def,axiom,
    ! [Z_1] : hAPP(complex,complex,cnj,Z_1) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(complex,real,re,Z_1)),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,Z_1))) ).

fof(fact_3038_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z_1] : hAPP(complex,real,norm_norm(complex),Z_1) = hAPP(real,real,sqrt,hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z_1),hAPP(complex,complex,cnj,Z_1)))) ).

fof(fact_3039_complex__add__def,axiom,
    ! [X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y))) ).

fof(fact_3040_complex__minus__def,axiom,
    ! [X] : hAPP(complex,complex,uminus_uminus(complex),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),hAPP(complex,real,re,X))),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))) ).

fof(fact_3041_complex__mult__def,axiom,
    ! [X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,im,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,re,Y)))) ).

fof(fact_3042_complex__add__cnj,axiom,
    ! [Z_1] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),Z_1),hAPP(complex,complex,cnj,Z_1)) = hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(complex,real,re,Z_1))) ).

fof(fact_3043_cmod__def,axiom,
    ! [Z_1] : hAPP(complex,real,norm_norm(complex),Z_1) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_3044_complex__Re__inverse,axiom,
    ! [X] : hAPP(complex,real,re,hAPP(complex,complex,inverse_inverse(complex),X)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_3045_Re_Opos__bounded,axiom,
    ? [K_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
      & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ) ).

fof(fact_3046_Re_Ononneg__bounded,axiom,
    ? [K_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
      & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ) ).

fof(fact_3047_Re_Obounded,axiom,
    ? [K_2] :
    ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2))) ).

fof(fact_3048_expi__def,axiom,
    ! [Z_1] : hAPP(complex,complex,expi,Z_1) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,exp(real),hAPP(complex,real,re,Z_1)))),hAPP(real,complex,cis,hAPP(complex,real,im,Z_1))) ).

fof(fact_3049_sin__n__Im__cis__pow__n,axiom,
    ! [N,A_1] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1)) = hAPP(complex,real,im,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N)) ).

fof(fact_3050_cos__n__Re__cis__pow__n,axiom,
    ! [N,A_1] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1)) = hAPP(complex,real,re,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N)) ).

fof(fact_3051_Re__cis,axiom,
    ! [A_1] : hAPP(complex,real,re,hAPP(real,complex,cis,A_1)) = hAPP(real,real,cos,A_1) ).

fof(fact_3052_Im__cis,axiom,
    ! [A_1] : hAPP(complex,real,im,hAPP(real,complex,cis,A_1)) = hAPP(real,real,sin,A_1) ).

fof(fact_3053_cis__mult,axiom,
    ! [A_1,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,B)) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)) ).

fof(fact_3054_cis__zero,axiom,
    hAPP(real,complex,cis,zero_zero(real)) = one_one(complex) ).

fof(fact_3055_cis__inverse,axiom,
    ! [A_1] : hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,cis,A_1)) = hAPP(real,complex,cis,hAPP(real,real,uminus_uminus(real),A_1)) ).

fof(fact_3056_cis__divide,axiom,
    ! [A_1,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,B)) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)) ).

fof(fact_3057_cis__def,axiom,
    ! [A_1] : hAPP(real,complex,cis,A_1) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1)) ).

fof(fact_3058_DeMoivre,axiom,
    ! [A_1,N] : hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1)) ).

fof(fact_3059_cis__real__of__nat__Suc__mult,axiom,
    ! [N,A_1] : hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N))),A_1)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1))) ).

fof(fact_3060_bezw__0,axiom,
    ! [X] : hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),zero_zero(nat)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

fof(fact_3061_Diff__infinite__finite,axiom,
    ! [X_a,S,T_3] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3))
     => ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
       => ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),S),T_3))) ) ) ).

fof(fact_3062_termination__basic__simps_I3_J,axiom,
    ! [Z_1,X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))) ) ).

fof(fact_3063_infinite__super,axiom,
    ! [X_a,S,T_3] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3))
     => ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
       => ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3)) ) ) ).

fof(fact_3064_termination__basic__simps_I2_J,axiom,
    ! [Y,X,Z_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Z_1))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))) ) ).

fof(fact_3065_termination__basic__simps_I1_J,axiom,
    ! [Z_1,X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))) ) ).

fof(fact_3066_termination__basic__simps_I5_J,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ).

fof(fact_3067_termination__basic__simps_I4_J,axiom,
    ! [Y,X,Z_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Z_1))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))) ) ).

fof(fact_3068_in__rel__def,axiom,
    ! [X_b,X_a,R_2,Xa,Ya] :
      ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),in_rel(X_a,X_b),R_2),Xa),Ya))
    <=> hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),Ya)),R_2)) ) ).

fof(fact_3069_prime__dvd__power__two,axiom,
    ! [M,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),M)) ) ) ).

fof(fact_3070_divmod__int__def,axiom,
    ! [B,A_1] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
       => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
           => ( ( ti(int,A_1) = zero_zero(int)
               => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) )
              & ( ti(int,A_1) != zero_zero(int)
               => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B))) ) ) ) ) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
       => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B))) ) ) ) ) ).

fof(fact_3071_primes__eq,axiom,
    ! [Q_3,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,prime,Q_3))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),Q_3))
         => P_5 = Q_3 ) ) ) ).

fof(fact_3072_prime__1,axiom,
    ~ hBOOL(hAPP(nat,bool,prime,one_one(nat))) ).

fof(fact_3073_prime__0,axiom,
    ~ hBOOL(hAPP(nat,bool,prime,zero_zero(nat))) ).

fof(fact_3074_distinct__prime__coprime,axiom,
    ! [Q_3,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,prime,Q_3))
       => ( P_5 != Q_3
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_5),Q_3)) ) ) ) ).

fof(fact_3075_prime__Suc0,axiom,
    ~ hBOOL(hAPP(nat,bool,prime,hAPP(nat,nat,suc,zero_zero(nat)))) ).

fof(fact_3076_prime__g__zero,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),P_5)) ) ).

fof(fact_3077_prime__divprod__eq,axiom,
    ! [A_3,B_2,P_4] :
      ( hBOOL(hAPP(nat,bool,prime,P_4))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),B_2)))
      <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),A_3))
          | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),B_2)) ) ) ) ).

fof(fact_3078_prime__divprod,axiom,
    ! [A_1,B,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),A_1))
          | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),B)) ) ) ) ).

fof(fact_3079_prime__dvd__mult,axiom,
    ! [M,N,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),M))
          | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),N)) ) ) ) ).

fof(fact_3080_prime__product,axiom,
    ! [P_5,Q_3] :
      ( hBOOL(hAPP(nat,bool,prime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),Q_3)))
     => ( P_5 = one_one(nat)
        | Q_3 = one_one(nat) ) ) ).

fof(fact_3081_prime__dvd__power,axiom,
    ! [A_1,N,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),A_1)) ) ) ).

fof(fact_3082_prime__divexp__n,axiom,
    ! [X,N,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N)))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N))) ) ) ).

fof(fact_3083_prime__divexp,axiom,
    ! [X,N,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N)))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),X)) ) ) ).

fof(fact_3084_prime__exp,axiom,
    ! [P_4,Na] :
      ( hBOOL(hAPP(nat,bool,prime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),Na)))
    <=> ( hBOOL(hAPP(nat,bool,prime,P_4))
        & Na = one_one(nat) ) ) ).

fof(fact_3085_prime__impl__zprime__int,axiom,
    ! [A_1] :
      ( hBOOL(hAPP(nat,bool,prime,A_1))
     => hBOOL(hAPP(int,bool,zprime,hAPP(nat,int,semiring_1_of_nat(int),A_1))) ) ).

fof(fact_3086_coprime__prime,axiom,
    ! [P_5,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
     => ~ ( hBOOL(hAPP(nat,bool,prime,P_5))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),A_1))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),B)) ) ) ).

fof(fact_3087_prime__coprime__strong,axiom,
    ! [N,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),N))
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_5),N)) ) ) ).

fof(fact_3088_coprime__prime__eq,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2))
    <=> ! [P_6] :
          ~ ( hBOOL(hAPP(nat,bool,prime,P_6))
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),A_3))
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),B_2)) ) ) ).

fof(fact_3089_prime__g__one,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),P_5)) ) ).

fof(fact_3090_prime__nd__one,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,suc,zero_zero(nat)))) ) ).

fof(fact_3091_prime__factor__lt,axiom,
    ! [M,N,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( N != zero_zero(nat)
       => ( N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),M)
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

fof(fact_3092_prime__def,axiom,
    ! [P_4] :
      ( hBOOL(hAPP(nat,bool,prime,P_4))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),P_4))
        & ! [M_2] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_2),P_4))
           => ( M_2 = one_one(nat)
              | M_2 = P_4 ) ) ) ) ).

fof(fact_3093_divides__primepow,axiom,
    ! [D_1,K,P_4] :
      ( hBOOL(hAPP(nat,bool,prime,P_4))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),K)))
      <=> ? [I_2] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),K))
            & D_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),I_2) ) ) ) ).

fof(fact_3094_prime__power__dvd__cancel__right,axiom,
    ! [N,A_1,B,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),B))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),A_1)) ) ) ) ).

fof(fact_3095_prime__coprime__lt,axiom,
    ! [X,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),P_5))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),P_5)) ) ) ) ).

fof(fact_3096_prime__coprime,axiom,
    ! [N,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( N = one_one(nat)
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),N))
        | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_5),N)) ) ) ).

fof(fact_3097_negateSnd__eq,axiom,
    ! [Q_3,R_3] : hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),hAPP(int,int,uminus_uminus(int),R_3)) ).

fof(fact_3098_divmod__int__rel__neg,axiom,
    ! [A_1,B,Qr_1] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)),Qr_1))
     => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,Qr_1))) ) ).

fof(fact_3099_two__is__prime,axiom,
    hBOOL(hAPP(nat,bool,prime,hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_3100_divmod__int__correct,axiom,
    ! [A_1,B] :
      ( ti(int,B) != zero_zero(int)
     => hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B))) ) ).

fof(fact_3101_prime__dvd__square,axiom,
    ! [M,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),M)) ) ) ).

fof(fact_3102_prime__divprod__pow,axiom,
    ! [N,A_1,B,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),A_1))
            | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),B)) ) ) ) ) ).

fof(fact_3103_prime__ge__2,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) ) ).

fof(fact_3104_prime__odd,axiom,
    ! [P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( P_5 = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))
        | ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),P_5)) ) ) ).

fof(fact_3105_divmod__int__mod__div,axiom,
    ! [P_5,Q_3] : hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,P_5),Q_3) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),P_5),Q_3)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),P_5),Q_3)) ).

fof(fact_3106_div__mod__code__numeral__def,axiom,
    ! [N,M] : hAPP(code_code_numeral,product_prod(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,fun(code_code_numeral,product_prod(code_code_numeral,code_code_numeral)),code_d418564891umeral,N),M) = hAPP(code_code_numeral,product_prod(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,fun(code_code_numeral,product_prod(code_code_numeral,code_code_numeral)),product_Pair(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),N),M)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),N),M)) ).

fof(fact_3107_xzgcd__linear,axiom,
    ! [M,R_3,S_3,T_5,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
     => ( hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,M),N) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),R_3),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_3),T_5))
       => ti(int,R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_3),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_5),N)) ) ) ).

fof(fact_3108_xzgcda_Osimps,axiom,
    ! [M,N,R_4,S_4,S_3,T_6,T_5,R_3] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int)))
       => hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),R_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_4),T_6)) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int)))
       => hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_3),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),R_4),R_3)),S_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),S_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_4),R_3)),S_3))),T_5),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),T_6),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_4),R_3)),T_5))) ) ) ).

fof(fact_3109_xzgcd__def,axiom,
    ! [M,N] : hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,M),N) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),M),N),one_one(int)),zero_zero(int)),zero_zero(int)),one_one(int)) ).

fof(fact_3110_xzgcda__linear,axiom,
    ! [M,N,R_4,S_4,S_3,T_6,T_5,Rn,Sn_1,Tn_1,R_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_3))
     => ( hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),Rn),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn_1),Tn_1))
       => ( ti(int,R_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_6),N))
         => ( ti(int,R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_3),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_5),N))
           => ti(int,Rn) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Sn_1),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Tn_1),N)) ) ) ) ) ).

fof(fact_3111_coprime__prime__dvd__ex,axiom,
    ! [X,Y] :
      ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),Y))
     => ? [P_6] :
          ( hBOOL(hAPP(nat,bool,prime,P_6))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),X))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),Y)) ) ) ).

fof(fact_3112_prime__power__exp,axiom,
    ! [X,K_4,N,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( N != zero_zero(nat)
       => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),K_4)
         => ? [I_2] : X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),I_2) ) ) ) ).

fof(fact_3113_prime__factor,axiom,
    ! [N] :
      ( N != one_one(nat)
     => ? [P_6] :
          ( hBOOL(hAPP(nat,bool,prime,P_6))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),N)) ) ) ).

fof(fact_3114_divmod__nat__step,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ).

fof(fact_3115_bezout__prime,axiom,
    ! [A_1,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),A_1))
       => ? [X_1,Y_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),Y_1)),one_one(nat)) ) ) ).

fof(fact_3116_not__prime__ex__mk,axiom,
    ! [N] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N))
        & ~ hBOOL(hAPP(nat,bool,prime,N)) )
     => ? [M_2,K_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_2))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_1))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),N))
          & N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_2),K_1) ) ) ).

fof(fact_3117_divmod__nat__zero,axiom,
    ! [M] : hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),zero_zero(nat)) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) ).

fof(fact_3118_divmod__nat__base,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
     => hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) ) ).

fof(fact_3119_divmod__nat__div__mod,axiom,
    ! [M,N] : hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) ).

fof(fact_3120_divmod__nat__rel__mult2__eq,axiom,
    ! [C_1,A_1,B,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),B),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_3)))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C_1))
         => hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Q_3),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Q_3),C_1))),R_3)))) ) ) ) ).

fof(fact_3121_divmod__nat__rel__mult1__eq,axiom,
    ! [A_1,B,C_1,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,B),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_3)))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C_1))
       => hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),Q_3)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),R_3)),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),R_3)),C_1)))) ) ) ).

fof(fact_3122_divmod__nat__rel__add1__eq,axiom,
    ! [B,Bq,Br,A_1,C_1,Aq,Ar] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Aq),Ar)))
     => ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,B),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Bq),Br)))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C_1))
         => hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B)),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Aq),Bq)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ar),Br)),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ar),Br)),C_1)))) ) ) ) ).

fof(fact_3123_divmod__nat__rel__unique,axiom,
    ! [Qr_2,M,N,Qr_1] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),Qr_1))
     => ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),Qr_2))
       => Qr_1 = Qr_2 ) ) ).

fof(fact_3124_divmod__nat__rel__divmod__nat,axiom,
    ! [M,N] : hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N))) ).

fof(fact_3125_divmod__nat__eq,axiom,
    ! [M,N,Qr_1] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),Qr_1))
     => hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = Qr_1 ) ).

fof(fact_3126_div__eq,axiom,
    ! [M,N,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_3)))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = Q_3 ) ).

fof(fact_3127_mod__eq,axiom,
    ! [M,N,Q_3,R_3] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_3)))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) = R_3 ) ).

fof(fact_3128_divmod__nat__rel,axiom,
    ! [M,N] : hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)))) ).

fof(fact_3129_divmod__nat__rel__ex,axiom,
    ! [M,N] :
      ~ ! [Q_4,R_1] : ~ hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_4),R_1))) ).

fof(fact_3130_euclid,axiom,
    ! [N] :
    ? [P_6] :
      ( hBOOL(hAPP(nat,bool,prime,P_6))
      & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),P_6)) ) ).

fof(fact_3131_div__pos__neg__1__number__of,axiom,
    ! [W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),zero_zero(int)))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),one_one(int))),hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),W))))) ) ).

fof(fact_3132_div__int__def,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B)) ).

fof(fact_3133_div__neg__pos,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B)) ) ) ).

fof(fact_3134_div__pos__pos,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B)) ) ) ).

fof(fact_3135_div__pos__pos__1__number__of,axiom,
    ! [W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
     => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W))) ) ).

fof(fact_3136_div__pos__neg,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) ) ) ).

fof(fact_3137_div__neg__neg,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) ) ) ).

fof(fact_3138_bezw_Osimps,axiom,
    ! [X,Y] :
      ( ( Y = zero_zero(nat)
       => hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) )
      & ( Y != zero_zero(nat)
       => hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y))))) ) ) ).

fof(fact_3139_bezw__non__0,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
     => hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y))))) ) ).

fof(fact_3140_mod__pos__neg__1__number__of,axiom,
    ! [W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),zero_zero(int)))
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),one_one(int))),hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),W))))) ) ).

fof(fact_3141_mod__int__def,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B)) ).

fof(fact_3142_mod__neg__pos,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B)) ) ) ).

fof(fact_3143_mod__pos__pos,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B)) ) ) ).

fof(fact_3144_mod__pos__pos__1__number__of,axiom,
    ! [W] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
     => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W))) ) ).

fof(fact_3145_mod__pos__neg,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) ) ) ).

fof(fact_3146_mod__neg__neg,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int)))
       => hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) ) ) ).

fof(fact_3147_prod__size__simp,axiom,
    ! [X_a,X_b,F,G,P_4] : hAPP(product_prod(X_a,X_b),nat,hAPP(fun(X_b,nat),fun(product_prod(X_a,X_b),nat),hAPP(fun(X_a,nat),fun(fun(X_b,nat),fun(product_prod(X_a,X_b),nat)),product_prod_size(X_a,X_b),F),G),P_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(X_a,nat,F,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_4))),hAPP(X_b,nat,G,hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),P_4)))),hAPP(nat,nat,suc,zero_zero(nat))) ).

fof(fact_3148_surjective__pairing,axiom,
    ! [X_a,X_b,T_5] : ti(product_prod(X_a,X_b),T_5) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),T_5)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),T_5)) ).

fof(fact_3149_pair__collapse,axiom,
    ! [X_b,X_a,P_5] : hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_5)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),P_5)) = ti(product_prod(X_a,X_b),P_5) ).

fof(fact_3150_div__nat__def,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N)) ).

fof(fact_3151_mod__nat__def,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N) = hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N)) ).

fof(fact_3152_divmod__nat__rel__def,axiom,
    ! [Ma,Na,Qr] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,Ma),Na),Qr))
    <=> ( Ma = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),Qr)),Na)),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr))
        & ( Na = zero_zero(nat)
         => hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),Qr) = zero_zero(nat) )
        & ( Na != zero_zero(nat)
         => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
             => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)))
                & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)),Na)) ) )
            & ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
             => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)))
                & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)),zero_zero(nat))) ) ) ) ) ) ) ).

fof(fact_3153_split__paired__All,axiom,
    ! [X_a,X_b,P_1] :
      ( ! [X1] : hBOOL(hAPP(product_prod(X_a,X_b),bool,P_1,X1))
    <=> ! [A_4,B_4] : hBOOL(hAPP(product_prod(X_a,X_b),bool,P_1,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_4),B_4))) ) ).

fof(fact_3154_Pair__eq,axiom,
    ! [X_b,X_a,A_3,B_2,A_8,B_7] :
      ( hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_8),B_7)
    <=> ( ti(X_a,A_3) = ti(X_a,A_8)
        & ti(X_b,B_2) = ti(X_b,B_7) ) ) ).

fof(fact_3155_Pair__inject,axiom,
    ! [X_b,X_a,A_1,B,A_6,B_5] :
      ( hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_1),B) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_6),B_5)
     => ~ ( ti(X_a,A_1) = ti(X_a,A_6)
         => ti(X_b,B) != ti(X_b,B_5) ) ) ).

fof(fact_3156_snd__conv,axiom,
    ! [X_b,X_a,A_1,B] : hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),A_1),B)) = ti(X_a,B) ).

fof(fact_3157_snd__eqD,axiom,
    ! [X_b,X_a,X,Y,A_1] :
      ( hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),X),Y)) = ti(X_a,A_1)
     => ti(X_a,Y) = ti(X_a,A_1) ) ).

fof(fact_3158_fst__conv,axiom,
    ! [X_a,X_b,A_1,B] : hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_1),B)) = ti(X_a,A_1) ).

fof(fact_3159_fst__eqD,axiom,
    ! [X_a,X_b,X,Y,A_1] :
      ( hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X),Y)) = ti(X_a,A_1)
     => ti(X_a,X) = ti(X_a,A_1) ) ).

fof(fact_3160_prod__eqI,axiom,
    ! [X_a,X_b,P_5,Q_3] :
      ( hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_5) = hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Q_3)
     => ( hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),P_5) = hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Q_3)
       => ti(product_prod(X_a,X_b),P_5) = ti(product_prod(X_a,X_b),Q_3) ) ) ).

fof(fact_3161_prod__eq__iff,axiom,
    ! [X_a,X_b,Sb,Ta] :
      ( ti(product_prod(X_a,X_b),Sb) = ti(product_prod(X_a,X_b),Ta)
    <=> ( hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Sb) = hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Ta)
        & hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Sb) = hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Ta) ) ) ).

fof(fact_3162_Bolzano__bisect__diff,axiom,
    ! [P_1,Na,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
     => hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na))),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)) ) ).

fof(fact_3163_exI__realizer,axiom,
    ! [X_b,X_a,P_1,Ya,Xa] :
      ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Ya),Xa))
     => hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),Xa),Ya))),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),Xa),Ya)))) ) ).

fof(fact_3164_conjI__realizer,axiom,
    ! [X_b,X_a,Q_1,Q_2,P_1,P_4] :
      ( hBOOL(hAPP(X_a,bool,P_1,P_4))
     => ( hBOOL(hAPP(X_b,bool,Q_1,Q_2))
       => ( hBOOL(hAPP(X_a,bool,P_1,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),P_4),Q_2))))
          & hBOOL(hAPP(X_b,bool,Q_1,hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),P_4),Q_2)))) ) ) ) ).

fof(fact_3165_Bolzano__bisect__le,axiom,
    ! [P_1,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
     => ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1)))) ) ).

fof(fact_3166_Bolzano__bisect_Osimps_I1_J,axiom,
    ! [P_1,A_3,B_2] : hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),zero_zero(nat)) = hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_2) ).

fof(fact_3167_Bolzano__bisect__fst__le__Suc,axiom,
    ! [P_1,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
     => ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),hAPP(nat,nat,suc,N_1))))) ) ).

fof(fact_3168_Bolzano__bisect__Suc__le__snd,axiom,
    ! [P_1,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
     => ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),hAPP(nat,nat,suc,N_1)))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1)))) ) ).

fof(fact_3169_not__P__Bolzano__bisect,axiom,
    ! [Na,A_3,B_2,P_1] :
      ( ! [A_4,B_4,C_4] :
          ( hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4)))
         => ( hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_4),C_4)))
           => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4))
             => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_4),C_4))
               => hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),C_4))) ) ) ) )
     => ( ~ hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_2)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
         => ~ hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na))))) ) ) ) ).

fof(fact_3170_not__P__Bolzano__bisect_H,axiom,
    ! [A_3,B_2,P_1] :
      ( ! [A_4,B_4,C_4] :
          ( ( hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4)))
            & hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_4),C_4)))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_4),C_4)) )
         => hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),C_4))) )
     => ( ~ hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_2)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
         => ! [N_1] : ~ hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))))) ) ) ) ).

fof(fact_3171_exE__realizer_H,axiom,
    ! [X_b,X_a,P_1,P_4] :
      ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),P_4)),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),P_4)))
     => ~ ! [X_1,Y_1] : ~ hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Y_1),X_1)) ) ).

fof(fact_3172_prod__induct6,axiom,
    ! [X_a,X_b,X_c,X_d,X_e,X_f,Xa,P_1] :
      ( ! [A_4,B_4,C_4,D_2,E,F_3] : hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),A_4),hAPP(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),B_4),hAPP(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),hAPP(X_c,fun(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_Pair(X_c,product_prod(X_d,product_prod(X_e,X_f))),C_4),hAPP(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f)),hAPP(X_d,fun(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f))),product_Pair(X_d,product_prod(X_e,X_f)),D_2),hAPP(X_f,product_prod(X_e,X_f),hAPP(X_e,fun(X_f,product_prod(X_e,X_f)),product_Pair(X_e,X_f),E),F_3)))))))
     => hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),bool,P_1,Xa)) ) ).

fof(fact_3173_prod__cases6,axiom,
    ! [X_a,X_b,X_c,X_d,X_e,X_f,Y] :
      ~ ! [A_4,B_4,C_4,D_2,E,F_3] : ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),Y) != hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),A_4),hAPP(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),B_4),hAPP(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),hAPP(X_c,fun(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_Pair(X_c,product_prod(X_d,product_prod(X_e,X_f))),C_4),hAPP(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f)),hAPP(X_d,fun(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f))),product_Pair(X_d,product_prod(X_e,X_f)),D_2),hAPP(X_f,product_prod(X_e,X_f),hAPP(X_e,fun(X_f,product_prod(X_e,X_f)),product_Pair(X_e,X_f),E),F_3))))) ).

fof(fact_3174_prod__induct5,axiom,
    ! [X_a,X_b,X_c,X_d,X_e,Xa,P_1] :
      ( ! [A_4,B_4,C_4,D_2,E] : hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),A_4),hAPP(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,X_e))),B_4),hAPP(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e)),hAPP(X_c,fun(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e))),product_Pair(X_c,product_prod(X_d,X_e)),C_4),hAPP(X_e,product_prod(X_d,X_e),hAPP(X_d,fun(X_e,product_prod(X_d,X_e)),product_Pair(X_d,X_e),D_2),E))))))
     => hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),bool,P_1,Xa)) ) ).

fof(fact_3175_prod__cases5,axiom,
    ! [X_a,X_b,X_c,X_d,X_e,Y] :
      ~ ! [A_4,B_4,C_4,D_2,E] : ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),Y) != hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),A_4),hAPP(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,X_e))),B_4),hAPP(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e)),hAPP(X_c,fun(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e))),product_Pair(X_c,product_prod(X_d,X_e)),C_4),hAPP(X_e,product_prod(X_d,X_e),hAPP(X_d,fun(X_e,product_prod(X_d,X_e)),product_Pair(X_d,X_e),D_2),E)))) ).

fof(fact_3176_prod__cases4,axiom,
    ! [X_a,X_b,X_c,X_d,Y] :
      ~ ! [A_4,B_4,C_4,D_2] : ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),Y) != hAPP(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d)))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,X_d))),A_4),hAPP(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d)),hAPP(X_b,fun(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d))),product_Pair(X_b,product_prod(X_c,X_d)),B_4),hAPP(X_d,product_prod(X_c,X_d),hAPP(X_c,fun(X_d,product_prod(X_c,X_d)),product_Pair(X_c,X_d),C_4),D_2))) ).

fof(fact_3177_prod__induct4,axiom,
    ! [X_a,X_b,X_c,X_d,Xa,P_1] :
      ( ! [A_4,B_4,C_4,D_2] : hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d)))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,X_d))),A_4),hAPP(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d)),hAPP(X_b,fun(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d))),product_Pair(X_b,product_prod(X_c,X_d)),B_4),hAPP(X_d,product_prod(X_c,X_d),hAPP(X_c,fun(X_d,product_prod(X_c,X_d)),product_Pair(X_c,X_d),C_4),D_2)))))
     => hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),bool,P_1,Xa)) ) ).

fof(fact_3178_prod__induct3,axiom,
    ! [X_a,X_b,X_c,Xa,P_1] :
      ( ! [A_4,B_4,C_4] : hBOOL(hAPP(product_prod(X_a,product_prod(X_b,X_c)),bool,P_1,hAPP(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c)),hAPP(X_a,fun(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c))),product_Pair(X_a,product_prod(X_b,X_c)),A_4),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),B_4),C_4))))
     => hBOOL(hAPP(product_prod(X_a,product_prod(X_b,X_c)),bool,P_1,Xa)) ) ).

fof(fact_3179_prod__cases3,axiom,
    ! [X_a,X_b,X_c,Y] :
      ~ ! [A_4,B_4,C_4] : ti(product_prod(X_a,product_prod(X_b,X_c)),Y) != hAPP(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c)),hAPP(X_a,fun(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c))),product_Pair(X_a,product_prod(X_b,X_c)),A_4),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),B_4),C_4)) ).

fof(fact_3180_prod_Orecs,axiom,
    ! [X_b,X_c,X_a,F1,A_3,B_2] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_rec(X_b,X_c,X_a),F1),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),A_3),B_2)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F1,A_3),B_2) ).

fof(fact_3181_one__code__int__code,axiom,
    one_one(quickcheck_code_int) = hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit1,pls)) ).

fof(fact_3182_Quickcheck__Narrowing_Oint__of__code,axiom,
    ! [K_4] :
      ( ( K_4 = zero_zero(quickcheck_code_int)
       => hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4) = zero_zero(int) )
      & ( K_4 != zero_zero(quickcheck_code_int)
       => ( ( hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(quickcheck_code_int)
           => hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) )
          & ( hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(quickcheck_code_int)
           => hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int)) ) ) ) ) ).

fof(fact_3183_DeMoivre2,axiom,
    ! [R_3,A_1,N] : hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)),N) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_3),N)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1)) ).

fof(fact_3184_less__eq__code__int__def,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),Na),Ma))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,Na)),hAPP(quickcheck_code_int,int,quickcheck_int_of,Ma))) ) ).

fof(fact_3185_int__of__number,axiom,
    ! [K_4] : hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),K_4)) = hAPP(int,int,number_number_of(int),K_4) ).

fof(fact_3186_less__code__int__def,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),Na),Ma))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,Na)),hAPP(quickcheck_code_int,int,quickcheck_int_of,Ma))) ) ).

fof(fact_3187_Quickcheck__Narrowing_Oint__of__inject,axiom,
    ! [K,L] :
      ( hAPP(quickcheck_code_int,int,quickcheck_int_of,K) = hAPP(quickcheck_code_int,int,quickcheck_int_of,L)
    <=> K = L ) ).

fof(fact_3188_code__int_Oint__of__inject,axiom,
    ! [Xa,Ya] :
      ( hAPP(quickcheck_code_int,int,quickcheck_int_of,Xa) = hAPP(quickcheck_code_int,int,quickcheck_int_of,Ya)
    <=> Xa = Ya ) ).

fof(fact_3189_zero__code__int__code,axiom,
    zero_zero(quickcheck_code_int) = hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),pls) ).

fof(fact_3190_rcis__zero__arg,axiom,
    ! [R_3] : hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),zero_zero(real)) = hAPP(real,complex,of_real(complex),R_3) ).

fof(fact_3191_complex__mod__rcis,axiom,
    ! [R_3,A_1] : hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)) = hAPP(real,real,abs_abs(real),R_3) ).

fof(fact_3192_cis__rcis__eq,axiom,
    ! [A_1] : hAPP(real,complex,cis,A_1) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,one_one(real)),A_1) ).

fof(fact_3193_rcis__zero__mod,axiom,
    ! [A_1] : hAPP(real,complex,hAPP(real,fun(real,complex),rcis,zero_zero(real)),A_1) = zero_zero(complex) ).

fof(fact_3194_Re__rcis,axiom,
    ! [R_3,A_1] : hAPP(complex,real,re,hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(real,real,cos,A_1)) ).

fof(fact_3195_Im__rcis,axiom,
    ! [R_3,A_1] : hAPP(complex,real,im,hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(real,real,sin,A_1)) ).

fof(fact_3196_rcis__mult,axiom,
    ! [R1,A_1,R2,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R1),A_1)),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R2),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R1),R2)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)) ).

fof(fact_3197_rcis__divide,axiom,
    ! [R1,A_1,R2,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R1),A_1)),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R2),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),R1),R2)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)) ).

fof(fact_3198_rcis__def,axiom,
    ! [R_3,A_1] : hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_3)),hAPP(real,complex,cis,A_1)) ).

fof(fact_3199_rcis__inverse,axiom,
    ! [R_3,A_1] : hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),R_3)),hAPP(real,real,uminus_uminus(real),A_1)) ).

fof(fact_3200_div__mod__code__int__def,axiom,
    ! [N,M] : hAPP(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,fun(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int)),quickc495462417de_int,N),M) = hAPP(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,fun(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int)),product_Pair(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),N),M)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),N),M)) ).

fof(fact_3201_nat__of__def,axiom,
    ! [I_1] : hAPP(quickcheck_code_int,nat,quickcheck_nat_of,I_1) = hAPP(int,nat,nat_1,hAPP(quickcheck_code_int,int,quickcheck_int_of,I_1)) ).

fof(fact_3202_around__zero_Opinduct,axiom,
    ! [P_1,A0] :
      ( hBOOL(hAPP(int,bool,hAPP(fun(int,fun(int,bool)),fun(int,bool),accp(int),quickc1265749348ro_rel),A0))
     => ( ! [I_2] :
            ( hBOOL(hAPP(int,bool,hAPP(fun(int,fun(int,bool)),fun(int,bool),accp(int),quickc1265749348ro_rel),I_2))
           => ( ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),zero_zero(int)))
               => ( ti(int,I_2) != zero_zero(int)
                 => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int)))) ) )
             => hBOOL(hAPP(int,bool,P_1,I_2)) ) )
       => hBOOL(hAPP(int,bool,P_1,A0)) ) ) ).

fof(fact_3203_fact__int__def,axiom,
    ! [X] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
       => hAPP(int,int,fact_fact(int),X) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),hAPP(int,nat,nat_1,X))) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
       => hAPP(int,int,fact_fact(int),X) = zero_zero(int) ) ) ).

fof(fact_3204_in__finite__psubset,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_a,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(X_a,bool),product_prod(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),product_prod(fun(X_a,bool),fun(X_a,bool))),product_Pair(fun(X_a,bool),fun(X_a,bool)),A_2),B_1)),finite_psubset(X_a)))
    <=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
        & hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) ) ) ).

fof(fact_3205_card__Plus__conv__if,axiom,
    ! [X_b,X_a,B_1,A_2] :
      ( ( ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
          & hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) )
       => hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_1)) )
      & ( ~ ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
            & hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) )
       => hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)) = zero_zero(nat) ) ) ).

fof(fact_3206_fact__mono__int,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),N))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,fact_fact(int),M)),hAPP(int,int,fact_fact(int),N))) ) ).

fof(fact_3207_finite__Plus__iff,axiom,
    ! [X_a,X_b,A_2,B_1] :
      ( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)))
    <=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
        & hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) ) ) ).

fof(fact_3208_finite__Plus,axiom,
    ! [X_b,X_a,B_1,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1))
       => hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1))) ) ) ).

fof(fact_3209_finite__PlusD_I1_J,axiom,
    ! [X_a,X_b,A_2,B_1] :
      ( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)))
     => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).

fof(fact_3210_finite__PlusD_I2_J,axiom,
    ! [X_a,X_b,A_2,B_1] :
      ( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)))
     => hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) ) ).

fof(fact_3211_fact__1__int,axiom,
    hAPP(int,int,fact_fact(int),one_one(int)) = one_one(int) ).

fof(fact_3212_fact__0__int,axiom,
    hAPP(int,int,fact_fact(int),zero_zero(int)) = one_one(int) ).

fof(fact_3213_fact__neg__int,axiom,
    ! [M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M),zero_zero(int)))
     => hAPP(int,int,fact_fact(int),M) = zero_zero(int) ) ).

fof(fact_3214_fact__less__mono__int,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M),N))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,fact_fact(int),M)),hAPP(int,int,fact_fact(int),N))) ) ) ).

fof(fact_3215_fact__ge__zero__int,axiom,
    ! [M] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,fact_fact(int),M))) ).

fof(fact_3216_fact__nonzero__int,axiom,
    ! [N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
     => hAPP(int,int,fact_fact(int),N) != zero_zero(int) ) ).

fof(fact_3217_transfer__nat__int__factorial__closure,axiom,
    ! [X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,fact_fact(int),X))) ) ).

fof(fact_3218_transfer__int__nat__factorial,axiom,
    ! [X] : hAPP(int,int,fact_fact(int),hAPP(nat,int,semiring_1_of_nat(int),X)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),X)) ).

fof(fact_3219_fact__ge__one__int,axiom,
    ! [N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(int,int,fact_fact(int),N))) ) ).

fof(fact_3220_fact__gt__zero__int,axiom,
    ! [N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,fact_fact(int),N))) ) ).

fof(fact_3221_fact__mono__int__aux,axiom,
    ! [M,K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,fact_fact(int),M)),hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),K_4)))) ) ).

fof(fact_3222_dvd__fact__int,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),N))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),hAPP(int,int,fact_fact(int),N))) ) ) ).

fof(fact_3223_card__Plus,axiom,
    ! [X_b,X_a,B_1,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1))
       => hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_1)) ) ) ).

fof(fact_3224_transfer__nat__int__factorial,axiom,
    ! [X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => hAPP(nat,nat,fact_fact(nat),hAPP(int,nat,nat_1,X)) = hAPP(int,nat,nat_1,hAPP(int,int,fact_fact(int),X)) ) ).

fof(fact_3225_fact__less__mono__int__aux,axiom,
    ! [M,K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,fact_fact(int),M)),hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),one_one(int))),K_4)))) ) ) ).

fof(fact_3226_fact__plus__one__int,axiom,
    ! [N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
     => hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int))),hAPP(int,int,fact_fact(int),N)) ) ).

fof(fact_3227_fact__reduce__int,axiom,
    ! [N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
     => hAPP(int,int,fact_fact(int),N) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N),one_one(int)))) ) ).

fof(fact_3228_length__around__zero,axiom,
    ! [I_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I_1))
     => hAPP(list(int),nat,size_size(list(int)),hAPP(int,list(int),quickc666637781d_zero,I_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,nat,nat_1,I_1))),one_one(nat)) ) ).

fof(fact_3229_Bnor__prime,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),P_4))
       => hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),P_4)) = hAPP(int,nat,nat_1,A_3) ) ) ).

fof(fact_3230_lemma__BOLZANO2,axiom,
    ! [P_1] :
      ( ( ! [A_4,B_4,C_4] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_4),C_4))
              & hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4)))
              & hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_4),C_4))) )
           => hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),C_4))) )
        & ! [X_1] :
          ? [D_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
            & ! [A_4,B_4] :
                ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),X_1))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_4))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_4),A_4)),D_2)) )
               => hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))) ) ) )
     => ! [A_4,B_4] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4))
         => hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))) ) ) ).

fof(fact_3231_Bnor__mem__zle,axiom,
    ! [B_2,A_3,Ma] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) ) ).

fof(fact_3232_Bnor__mem__zle__swap,axiom,
    ! [Ma,A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2))
     => ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))) ) ).

fof(fact_3233_Bnor__fin,axiom,
    ! [A_3,Ma] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))) ).

fof(fact_3234_Bnor__mem__zg,axiom,
    ! [B_2,A_3,Ma] :
      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) ) ).

fof(fact_3235_lemma__BOLZANO,axiom,
    ! [A_3,B_2,P_1] :
      ( ! [A_4,B_4,C_4] :
          ( ( hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4)))
            & hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_4),C_4)))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_4),C_4)) )
         => hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),C_4))) )
     => ( ! [X_1] :
          ? [D_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
            & ! [A_4,B_4] :
                ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),X_1))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_4))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_4),A_4)),D_2)) )
               => hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))) ) )
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
         => hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_2))) ) ) ) ).

fof(fact_3236_number__of__le__ceiling,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Va,Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),Va)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),one_one(X_a))),Xa)) ) ) ).

fof(fact_3237_bezw__aux,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y))),hAPP(nat,int,semiring_1_of_nat(int),X))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y))),hAPP(nat,int,semiring_1_of_nat(int),Y))) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) ).

fof(fact_3238_gcd__dvd1__nat,axiom,
    ! [M,N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)),M)) ).

fof(fact_3239_gcd__dvd2__nat,axiom,
    ! [M,N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)),N)) ).

fof(fact_3240_gcd__dvd__prod__nat,axiom,
    ! [M,N,K_4] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N))) ).

fof(fact_3241_coprime__exp2__nat,axiom,
    ! [N,M,A_1,B] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) = one_one(nat)
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),M)) = one_one(nat) ) ).

fof(fact_3242_gcd__mult__distrib__nat,axiom,
    ! [K_4,M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) ).

fof(fact_3243_gcd__semilattice__nat_Oinf__assoc,axiom,
    ! [X,Y,Z_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)),Z_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),Z_1)) ).

fof(fact_3244_gcd__assoc__nat,axiom,
    ! [A_1,B,C_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),C_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),C_1)) ).

fof(fact_3245_gcd__semilattice__nat_Oinf__left__commute,axiom,
    ! [X,Y,Z_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),Z_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Z_1)) ).

fof(fact_3246_gcd__left__commute__nat,axiom,
    ! [B,A_1,C_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),C_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),C_1)) ).

fof(fact_3247_gcd__semilattice__nat_Oinf__left__idem,axiom,
    ! [X,Y] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) ).

fof(fact_3248_gcd__semilattice__nat_Oinf_Oleft__idem,axiom,
    ! [A_1,B] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) ).

fof(fact_3249_gcd__semilattice__nat_Oinf__commute,axiom,
    ! [X,Y] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),X) ).

fof(fact_3250_gcd__commute__nat,axiom,
    ! [A_1,B] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),A_1) ).

fof(fact_3251_gcd__idem__nat,axiom,
    ! [X] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),X) = X ).

fof(fact_3252_gcd__semilattice__nat_Oinf_Oidem,axiom,
    ! [A_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),A_1) = A_1 ).

fof(fact_3253_gcd__semilattice__nat_Oinf__le1,axiom,
    ! [X,Y] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)),X)) ).

fof(fact_3254_gcd__semilattice__nat_Oinf__le2,axiom,
    ! [X,Y] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)),Y)) ).

fof(fact_3255_gcd__semilattice__nat_Ole__iff__inf,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
    <=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Xa),Ya) = Xa ) ).

fof(fact_3256_gcd__greatest__iff__nat,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ma),Na)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Ma))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Na)) ) ) ).

fof(fact_3257_gcd__semilattice__nat_Ole__inf__iff,axiom,
    ! [Xa,Ya,Z] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ya),Z)))
    <=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Z)) ) ) ).

fof(fact_3258_gcd__unique__nat,axiom,
    ! [B_2,D_1,A_3] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),A_3))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),B_2))
        & ! [E] :
            ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),A_3))
              & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),B_2)) )
           => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),D_1)) ) )
    <=> D_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_3),B_2) ) ).

fof(fact_3259_gcd__semilattice__nat_Ole__infI1,axiom,
    ! [B,A_1,X] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),X))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X)) ) ).

fof(fact_3260_gcd__semilattice__nat_Ole__infI2,axiom,
    ! [A_1,B,X] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),X))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X)) ) ).

fof(fact_3261_gcd__proj1__if__dvd__nat,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X ) ).

fof(fact_3262_gcd__proj2__if__dvd__nat,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = Y ) ).

fof(fact_3263_gcd__greatest__nat,axiom,
    ! [N,K_4,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N))) ) ) ).

fof(fact_3264_gcd__semilattice__nat_Ole__infI,axiom,
    ! [B,X,A_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) ) ) ).

fof(fact_3265_gcd__semilattice__nat_Oinf__greatest,axiom,
    ! [Z_1,X,Y] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),Z_1))) ) ) ).

fof(fact_3266_gcd__semilattice__nat_Oinf__mono,axiom,
    ! [B,D_3,A_1,C_1] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),D_3))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),C_1),D_3))) ) ) ).

fof(fact_3267_dvd__gcd__D1__nat,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M)) ) ).

fof(fact_3268_dvd__gcd__D2__nat,axiom,
    ! [K_4,M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N)) ) ).

fof(fact_3269_gcd__semilattice__nat_Ole__infE,axiom,
    ! [X,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)))
     => ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1))
         => ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B)) ) ) ).

fof(fact_3270_gcd__semilattice__nat_Oless__infI1,axiom,
    ! [B,A_1,X] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),X))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1)) )
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) ) ) ).

fof(fact_3271_gcd__semilattice__nat_Oless__infI2,axiom,
    ! [A_1,B,X] :
      ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),X))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B)) )
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X))
        & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) ) ) ).

fof(fact_3272_gcd__add2__nat,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) ).

fof(fact_3273_gcd__add1__nat,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) ).

fof(fact_3274_gcd__exp__nat,axiom,
    ! [A_1,N,B] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),N) ).

fof(fact_3275_gcd__red__nat,axiom,
    ! [X,Y] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ).

fof(fact_3276_gcd__lcm__complete__lattice__nat_Oinf__bot__right,axiom,
    ! [X] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),one_one(nat)) = one_one(nat) ).

fof(fact_3277_gcd__1__nat,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),one_one(nat)) = one_one(nat) ).

fof(fact_3278_gcd__lcm__complete__lattice__nat_Oinf__bot__left,axiom,
    ! [X] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),one_one(nat)),X) = one_one(nat) ).

fof(fact_3279_gcd__0__left__nat,axiom,
    ! [X] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X ).

fof(fact_3280_gcd__0__nat,axiom,
    ! [X] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat)) = X ).

fof(fact_3281_gcd__zero__nat,axiom,
    ! [Ma,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ma),Na) = zero_zero(nat)
    <=> ( Ma = zero_zero(nat)
        & Na = zero_zero(nat) ) ) ).

fof(fact_3282_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff,axiom,
    ! [Xa,Ya] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Xa),Ya) = zero_zero(nat)
    <=> ( Xa = zero_zero(nat)
        & Ya = zero_zero(nat) ) ) ).

fof(fact_3283_gcd__Suc__0,axiom,
    ! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),hAPP(nat,nat,suc,zero_zero(nat))) = hAPP(nat,nat,suc,zero_zero(nat)) ).

fof(fact_3284_gcd__pos__nat,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ma),Na)))
    <=> ( Ma != zero_zero(nat)
        | Na != zero_zero(nat) ) ) ).

fof(fact_3285_gcd__le2__nat,axiom,
    ! [A_1,B] :
      ( B != zero_zero(nat)
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),B)) ) ).

fof(fact_3286_gcd__le1__nat,axiom,
    ! [B,A_1] :
      ( A_1 != zero_zero(nat)
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),A_1)) ) ).

fof(fact_3287_coprime__Suc__nat,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,suc,N)),N) = one_one(nat) ).

fof(fact_3288_gcd__add__mult__nat,axiom,
    ! [M,K_4,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) ).

fof(fact_3289_coprime__common__divisor__nat,axiom,
    ! [X,A_1,B] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) = one_one(nat)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1))
       => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B))
         => X = one_one(nat) ) ) ) ).

fof(fact_3290_coprime__nat,axiom,
    ! [A_3,B_2] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_3),B_2) = one_one(nat)
    <=> ! [D_2] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),A_3))
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),B_2)) )
        <=> D_2 = one_one(nat) ) ) ).

fof(fact_3291_coprime__plus__one__nat,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),N) = one_one(nat) ).

fof(fact_3292_coprime__mul__eq__nat,axiom,
    ! [D_1,A_3,B_2] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),B_2)) = one_one(nat)
    <=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),A_3) = one_one(nat)
        & hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),B_2) = one_one(nat) ) ) ).

fof(fact_3293_gcd__mult__cancel__nat,axiom,
    ! [M,K_4,N] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K_4),N) = one_one(nat)
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ).

fof(fact_3294_coprime__mult__nat,axiom,
    ! [B,D_3,A_1] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),A_1) = one_one(nat)
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),B) = one_one(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)) = one_one(nat) ) ) ).

fof(fact_3295_coprime__crossproduct__nat,axiom,
    ! [B_2,C,A_3,D_1] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_3),D_1) = one_one(nat)
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B_2),C) = one_one(nat)
       => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_2),D_1)
        <=> ( A_3 = B_2
            & C = D_1 ) ) ) ) ).

fof(fact_3296_coprime__lmult__nat,axiom,
    ! [D_3,A_1,B] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)) = one_one(nat)
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),A_1) = one_one(nat) ) ).

fof(fact_3297_coprime__rmult__nat,axiom,
    ! [D_3,A_1,B] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)) = one_one(nat)
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),B) = one_one(nat) ) ).

fof(fact_3298_gcd__diff2__nat,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ).

fof(fact_3299_gcd__diff1__nat,axiom,
    ! [N,M] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ).

fof(fact_3300_gcd__nat_Osimps,axiom,
    ! [X,Y] :
      ( ( Y = zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X )
      & ( Y != zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ) ) ).

fof(fact_3301_gcd__non__0__nat,axiom,
    ! [X,Y] :
      ( Y != zero_zero(nat)
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ) ).

fof(fact_3302_coprime__exp__nat,axiom,
    ! [N,D_3,A_1] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),A_1) = one_one(nat)
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)) = one_one(nat) ) ).

fof(fact_3303_ceiling__real__of__nat,axiom,
    ! [N] : hAPP(real,int,archimedean_ceiling(real),hAPP(nat,real,real_1(nat),N)) = hAPP(nat,int,semiring_1_of_nat(int),N) ).

fof(fact_3304_ceiling__zero,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => hAPP(X_a,int,archimedean_ceiling(X_a),zero_zero(X_a)) = zero_zero(int) ) ).

fof(fact_3305_ceiling__less__cancel,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(X_a,int,archimedean_ceiling(X_a),Y)))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).

fof(fact_3306_ceiling__mono,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Y)),hAPP(X_a,int,archimedean_ceiling(X_a),X))) ) ) ).

fof(fact_3307_ceiling__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => hAPP(X_a,int,archimedean_ceiling(X_a),one_one(X_a)) = one_one(int) ) ).

fof(fact_3308_ceiling__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [V] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(int,X_a,number_number_of(X_a),V)) = hAPP(int,int,number_number_of(int),V) ) ).

fof(fact_3309_natceiling__def,axiom,
    ! [X] : hAPP(real,nat,natceiling,X) = hAPP(int,nat,nat_1,hAPP(real,int,archimedean_ceiling(real),X)) ).

fof(fact_3310_ceiling__of__nat,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [N] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) = hAPP(nat,int,semiring_1_of_nat(int),N) ) ).

fof(fact_3311_gcd__coprime__nat,axiom,
    ! [B_5,A_6,A_1,B] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) != zero_zero(nat)
     => ( A_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_6),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))
       => ( B = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))
         => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_6),B_5) = one_one(nat) ) ) ) ).

fof(fact_3312_coprime__minus__one__nat,axiom,
    ! [N] :
      ( N != zero_zero(nat)
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),N) = one_one(nat) ) ).

fof(fact_3313_divides__mult__nat,axiom,
    ! [N,M,R_3] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),R_3))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),R_3))
       => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) = one_one(nat)
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),R_3)) ) ) ) ).

fof(fact_3314_coprime__dvd__mult__iff__nat,axiom,
    ! [Ma,K,Na] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K),Na) = one_one(nat)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na)))
      <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Ma)) ) ) ).

fof(fact_3315_coprime__dvd__mult__nat,axiom,
    ! [M,K_4,N] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K_4),N) = one_one(nat)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M)) ) ) ).

fof(fact_3316_div__gcd__coprime__nat,axiom,
    ! [B,A_1] :
      ( ( A_1 != zero_zero(nat)
        | B != zero_zero(nat) )
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) = one_one(nat) ) ).

fof(fact_3317_invertible__coprime__nat,axiom,
    ! [X,Y,M] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)),M) = one_one(nat)
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),M) = one_one(nat) ) ).

fof(fact_3318_floor__minus,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)) ) ).

fof(fact_3319_ceiling__minus,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)) ) ).

fof(fact_3320_ceiling__def,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hAPP(X_a,int,archimedean_ceiling(X_a),X) = hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X))) ) ).

fof(fact_3321_coprime__Suc__0__nat,axiom,
    ! [A_3,B_2] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_3),B_2) = one_one(nat)
    <=> ! [D_2] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),A_3))
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),B_2)) )
        <=> D_2 = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).

fof(fact_3322_zero__less__ceiling,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)) ) ) ).

fof(fact_3323_ceiling__le__zero,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),zero_zero(int)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a))) ) ) ).

fof(fact_3324_ceiling__less__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),one_one(int)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a))) ) ) ).

fof(fact_3325_one__le__ceiling,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)) ) ) ).

fof(fact_3326_one__less__ceiling,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),Xa)) ) ) ).

fof(fact_3327_ceiling__le__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),one_one(int)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),one_one(X_a))) ) ) ).

fof(fact_3328_number__of__less__ceiling,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Va,Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),Va)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),Xa)) ) ) ).

fof(fact_3329_ceiling__le__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa,Va] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),hAPP(int,int,number_number_of(int),Va)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(int,X_a,number_number_of(X_a),Va))) ) ) ).

fof(fact_3330_ceiling__add__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),one_one(int)) ) ).

fof(fact_3331_ceiling__add__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,V] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(int,int,number_number_of(int),V)) ) ).

fof(fact_3332_ceiling__diff__one,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),one_one(int)) ) ).

fof(fact_3333_ceiling__diff__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,V] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(int,int,number_number_of(int),V)) ) ).

fof(fact_3334_ceiling__less__zero,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),zero_zero(int)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(int,X_a,number_number_of(X_a),min))) ) ) ).

fof(fact_3335_zero__le__ceiling,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),min)),Xa)) ) ) ).

fof(fact_3336_ceiling__less__number__of,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa,Va] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),hAPP(int,int,number_number_of(int),Va)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),one_one(X_a)))) ) ) ).

fof(fact_3337_gcd__coprime__exists__nat,axiom,
    ! [A_1,B] :
      ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) != zero_zero(nat)
     => ? [A_5,B_3] :
          ( A_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))
          & B = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))
          & hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_5),B_3) = one_one(nat) ) ) ).

fof(fact_3338_bezout__nat,axiom,
    ! [B,A_1] :
      ( A_1 != zero_zero(nat)
     => ? [X_1,Y_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),Y_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) ) ).

fof(fact_3339_Cauchy__iff2,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_2))
    <=> ! [J_1] :
        ? [M_3] :
        ! [M_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2))
         => ! [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,X_2,M_2)),hAPP(nat,real,X_2,N_1)))),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,J_1))))) ) ) ) ).

fof(fact_3340_Cauchy__iff,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
        <=> ! [E] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M_3] :
                ! [M_2] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2))
                 => ! [N_1] :
                      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1))
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1)))),E)) ) ) ) ) ) ).

fof(fact_3341_CauchyD,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [E_2,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E_2))
           => ? [M_3] :
              ! [M_2] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2))
               => ! [N_1] :
                    ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1))
                   => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1)))),E_2)) ) ) ) ) ) ).

fof(fact_3342_CauchyI,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( ! [E] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M_3] :
                ! [M_2] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2))
                 => ! [N_1] :
                      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1))
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1)))),E)) ) ) )
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) ) ) ).

fof(fact_3343_ceiling__eq2,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real))))
       => hAPP(real,int,archimedean_ceiling(real),X) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int)) ) ) ).

fof(fact_3344_in__measure,axiom,
    ! [X_a,Xa,Ya,F] :
      ( hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Xa),Ya)),hAPP(fun(X_a,nat),fun(product_prod(X_a,X_a),bool),measure(X_a),F)))
    <=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(X_a,nat,F,Xa)),hAPP(X_a,nat,F,Ya))) ) ).

fof(fact_3345_ln__powr__bound,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),A_1))) ) ) ).

fof(fact_3346_real__number__of,axiom,
    ! [V] : hAPP(int,real,real_1(int),hAPP(int,int,number_number_of(int),V)) = hAPP(int,real,number_number_of(real),V) ).

fof(fact_3347_real__of__int__zero__cancel,axiom,
    ! [Xa] :
      ( hAPP(int,real,real_1(int),Xa) = zero_zero(real)
    <=> ti(int,Xa) = zero_zero(int) ) ).

fof(fact_3348_real__of__int__zero,axiom,
    hAPP(int,real,real_1(int),zero_zero(int)) = zero_zero(real) ).

fof(fact_3349_powr__not__zero,axiom,
    ! [X,A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1) != zero_zero(real) ).

fof(fact_3350_powr__one__eq__one,axiom,
    ! [A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,one_one(real)),A_1) = one_one(real) ).

fof(fact_3351_real__of__int__floor__cancel,axiom,
    ! [Xa] :
      ( hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),Xa)) = Xa
    <=> ? [N_1] : Xa = hAPP(int,real,real_1(int),N_1) ) ).

fof(fact_3352_floor__real__of__int,axiom,
    ! [N] : hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),N)) = ti(int,N) ).

fof(fact_3353_real__of__int__inject,axiom,
    ! [Xa,Ya] :
      ( hAPP(int,real,real_1(int),Xa) = hAPP(int,real,real_1(int),Ya)
    <=> ti(int,Xa) = ti(int,Ya) ) ).

fof(fact_3354_powr__powr__swap,axiom,
    ! [X,A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),B) = hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)),A_1) ).

fof(fact_3355_powr__powr,axiom,
    ! [X,A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),B) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)) ).

fof(fact_3356_real__of__int__less__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Xa)),hAPP(int,real,real_1(int),Ya)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) ) ).

fof(fact_3357_real__of__int__le__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),Xa)),hAPP(int,real,real_1(int),Ya)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)) ) ).

fof(fact_3358_real__of__one,axiom,
    hAPP(int,real,real_1(int),one_one(int)) = one_one(real) ).

fof(fact_3359_real__of__int__mult,axiom,
    ! [X,Y] : hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) ).

fof(fact_3360_real__of__int__diff,axiom,
    ! [X,Y] : hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) ).

fof(fact_3361_real__of__int__power,axiom,
    ! [X,N] : hAPP(int,real,real_1(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,real_1(int),X)),N) ).

fof(fact_3362_power__real__of__int,axiom,
    ! [X,N] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,real_1(int),X)),N) = hAPP(int,real,real_1(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N)) ).

fof(fact_3363_real__of__int__of__nat__eq,axiom,
    ! [N] : hAPP(int,real,real_1(int),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,real,real_1(nat),N) ).

fof(fact_3364_real__of__int__add,axiom,
    ! [X,Y] : hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) ).

fof(fact_3365_real__of__int__ceiling__cancel,axiom,
    ! [Xa] :
      ( hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),Xa)) = Xa
    <=> ? [N_1] : Xa = hAPP(int,real,real_1(int),N_1) ) ).

fof(fact_3366_ceiling__real__of__int,axiom,
    ! [N] : hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),N)) = ti(int,N) ).

fof(fact_3367_real__of__int__minus,axiom,
    ! [X] : hAPP(int,real,real_1(int),hAPP(int,int,uminus_uminus(int),X)) = hAPP(real,real,uminus_uminus(real),hAPP(int,real,real_1(int),X)) ).

fof(fact_3368_real__of__int__abs,axiom,
    ! [X] : hAPP(int,real,real_1(int),hAPP(int,int,abs_abs(int),X)) = hAPP(real,real,abs_abs(real),hAPP(int,real,real_1(int),X)) ).

fof(fact_3369_real__of__int__floor__le,axiom,
    ! [R_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3))),R_3)) ).

fof(fact_3370_real__of__int__ceiling__ge,axiom,
    ! [R_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_3),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_3)))) ).

fof(fact_3371_powr__ge__pzero,axiom,
    ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y))) ).

fof(fact_3372_powr__gt__zero,axiom,
    ! [X,A_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1))) ).

fof(fact_3373_powr__less__mono2,axiom,
    ! [Y,X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1))) ) ) ) ).

fof(fact_3374_powr__less__mono2__neg,axiom,
    ! [Y,X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1))) ) ) ) ).

fof(fact_3375_powr__mono,axiom,
    ! [X,A_1,B] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_1),B))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B))) ) ) ).

fof(fact_3376_powr__zero__eq__one,axiom,
    ! [X] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),zero_zero(real)) = one_one(real) ).

fof(fact_3377_powr__less__cancel,axiom,
    ! [X,A_1,B] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),B)) ) ) ).

fof(fact_3378_powr__less__mono,axiom,
    ! [X,A_1,B] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),B))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B))) ) ) ).

fof(fact_3379_powr__less__cancel__iff,axiom,
    ! [A_3,B_2,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),A_3)),hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),B_2)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) ) ) ).

fof(fact_3380_floor__ceiling,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [R_3] : hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),hAPP(X_a,int,archimedean_ceiling(X_a),R_3))) = hAPP(X_a,int,archimedean_ceiling(X_a),R_3) ) ).

fof(fact_3381_powr__add,axiom,
    ! [X,A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)) ).

fof(fact_3382_ceiling__floor,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [R_3] : hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),hAPP(X_a,int,archim2085082626_floor(X_a),R_3))) = hAPP(X_a,int,archim2085082626_floor(X_a),R_3) ) ).

fof(fact_3383_powr__divide2,axiom,
    ! [X,A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)) ).

fof(fact_3384_powr__minus,axiom,
    ! [X,A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,uminus_uminus(real),A_1)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)) ).

fof(fact_3385_real__of__int__div4,axiom,
    ! [N,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),X))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N)),hAPP(int,real,real_1(int),X)))) ).

fof(fact_3386_floor__less__eq,axiom,
    ! [Xa,A_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),Xa)),A_3))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),hAPP(int,real,real_1(int),A_3))) ) ).

fof(fact_3387_le__floor,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),A_1)),X))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(real,int,archim2085082626_floor(real),X))) ) ).

fof(fact_3388_real__le__floor,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(real,int,archim2085082626_floor(real),X)))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),A_1)),X)) ) ).

fof(fact_3389_le__floor__eq,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_3),hAPP(real,int,archim2085082626_floor(real),Xa)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),A_3)),Xa)) ) ).

fof(fact_3390_floor__add,axiom,
    ! [X,A_1] : hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archim2085082626_floor(real),X)),A_1) ).

fof(fact_3391_less__ceiling__eq,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(real,int,archimedean_ceiling(real),Xa)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),A_3)),Xa)) ) ).

fof(fact_3392_ceiling__le__eq,axiom,
    ! [Xa,A_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),Xa)),A_3))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(int,real,real_1(int),A_3))) ) ).

fof(fact_3393_ceiling__le,axiom,
    ! [X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),A_1)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1)) ) ).

fof(fact_3394_ceiling__le__real,axiom,
    ! [X,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),A_1))) ) ).

fof(fact_3395_floor__power,axiom,
    ! [N,X] :
      ( X = hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),X))
     => hAPP(real,int,archim2085082626_floor(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(real,int,archim2085082626_floor(real),X)),N) ) ).

fof(fact_3396_floor__subtract,axiom,
    ! [X,A_1] : hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archim2085082626_floor(real),X)),A_1) ).

fof(fact_3397_ceiling__add,axiom,
    ! [X,A_1] : hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1) ).

fof(fact_3398_floor__minus__real__of__int,axiom,
    ! [N] : hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,uminus_uminus(real),hAPP(int,real,real_1(int),N))) = hAPP(int,int,uminus_uminus(int),N) ).

fof(fact_3399_ceiling__subtract,axiom,
    ! [X,A_1] : hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1) ).

fof(fact_3400_powr__mono2,axiom,
    ! [Y,X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1))) ) ) ) ).

fof(fact_3401_powr__mult,axiom,
    ! [A_1,Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)),A_1) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1)) ) ) ).

fof(fact_3402_ge__one__powr__ge__zero,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1))) ) ) ).

fof(fact_3403_powr__le__cancel__iff,axiom,
    ! [A_3,B_2,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),A_3)),hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),B_2)))
      <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) ) ) ).

fof(fact_3404_powr__one__gt__zero__iff,axiom,
    ! [Xa] :
      ( hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),one_one(real)) = Xa
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) ) ).

fof(fact_3405_powr__divide,axiom,
    ! [A_1,Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)),A_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1)) ) ) ).

fof(fact_3406_powr__minus__divide,axiom,
    ! [X,A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,uminus_uminus(real),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)) ).

fof(fact_3407_powr__def,axiom,
    ! [X,A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1) = hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),hAPP(real,real,ln,X))) ).

fof(fact_3408_real__of__int__lt__zero__cancel__iff,axiom,
    ! [Na] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Na)),zero_zero(real)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Na),zero_zero(int))) ) ).

fof(fact_3409_real__of__int__gt__zero__cancel__iff,axiom,
    ! [Na] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(int,real,real_1(int),Na)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Na)) ) ).

fof(fact_3410_real__of__int__ge__zero__cancel__iff,axiom,
    ! [Na] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(int,real,real_1(int),Na)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na)) ) ).

fof(fact_3411_real__of__int__le__zero__cancel__iff,axiom,
    ! [Na] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),Na)),zero_zero(real)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Na),zero_zero(int))) ) ).

fof(fact_3412_number__of__less__real__of__int__iff2,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Ma)),hAPP(int,real,number_number_of(real),Na)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ma),hAPP(int,int,number_number_of(int),Na))) ) ).

fof(fact_3413_number__of__less__real__of__int__iff,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),Na)),hAPP(int,real,real_1(int),Ma)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),Na)),Ma)) ) ).

fof(fact_3414_number__of__le__real__of__int__iff,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),Na)),hAPP(int,real,real_1(int),Ma)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),Na)),Ma)) ) ).

fof(fact_3415_number__of__le__real__of__int__iff2,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),Ma)),hAPP(int,real,number_number_of(real),Na)))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ma),hAPP(int,int,number_number_of(int),Na))) ) ).

fof(fact_3416_int__less__real__le,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Na),Ma))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),Na)),one_one(real))),hAPP(int,real,real_1(int),Ma))) ) ).

fof(fact_3417_int__le__real__less,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Na),Ma))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Na)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),Ma)),one_one(real)))) ) ).

fof(fact_3418_lemma__floor2,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),X)),one_one(real))))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),X)) ) ).

fof(fact_3419_real__nat__eq__real,axiom,
    ! [X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => hAPP(nat,real,real_1(nat),hAPP(int,nat,nat_1,X)) = hAPP(int,real,real_1(int),X) ) ).

fof(fact_3420_real__of__int__div,axiom,
    ! [N,D_3] :
      ( ti(int,D_3) != zero_zero(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),N))
       => hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),D_3)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N)),hAPP(int,real,real_1(int),D_3)) ) ) ).

fof(fact_3421_real__of__int__floor__add__one__ge,axiom,
    ! [R_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_3),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3))),one_one(real)))) ).

fof(fact_3422_real__of__int__floor__add__one__gt,axiom,
    ! [R_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3))),one_one(real)))) ).

fof(fact_3423_floor__eq,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real))))
       => hAPP(real,int,archim2085082626_floor(real),X) = ti(int,N) ) ) ).

fof(fact_3424_tan__periodic__int,axiom,
    ! [X,I_1] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,real_1(int),I_1)),pi))) = hAPP(real,real,tan,X) ).

fof(fact_3425_real__of__int__floor__ge__diff__one,axiom,
    ! [R_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_3),one_one(real))),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3)))) ).

fof(fact_3426_real__of__int__floor__gt__diff__one,axiom,
    ! [R_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_3),one_one(real))),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3)))) ).

fof(fact_3427_real__of__int__ceiling__le__add__one,axiom,
    ! [R_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_3))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_3),one_one(real)))) ).

fof(fact_3428_real__of__int__ceiling__diff__one__le,axiom,
    ! [R_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_3))),one_one(real))),R_3)) ).

fof(fact_3429_powr__realpow,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(nat,real,real_1(nat),N)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N) ) ).

fof(fact_3430_ln__powr,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,ln,X)) ) ) ).

fof(fact_3431_powr__log__cancel,axiom,
    ! [X,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ( A_1 != one_one(real)
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)) = X ) ) ) ).

fof(fact_3432_log__powr__cancel,axiom,
    ! [Y,A_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
     => ( A_1 != one_one(real)
       => hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),Y)) = Y ) ) ).

fof(fact_3433_lemma__floor,axiom,
    ! [N,M,R_3] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),M)),R_3))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real))))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),N)) ) ) ).

fof(fact_3434_real__of__int__div2,axiom,
    ! [N,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N)),hAPP(int,real,real_1(int),X))),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),X))))) ).

fof(fact_3435_real__of__int__div3,axiom,
    ! [N,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N)),hAPP(int,real,real_1(int),X))),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),X)))),one_one(real))) ).

fof(fact_3436_real__of__int__div__aux,axiom,
    ! [X,D_3] :
      ( ti(int,D_3) != zero_zero(int)
     => hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),D_3)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),D_3))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),D_3))),hAPP(int,real,real_1(int),D_3))) ) ).

fof(fact_3437_floor__eq2,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real))))
       => hAPP(real,int,archim2085082626_floor(real),X) = ti(int,N) ) ) ).

fof(fact_3438_less__floor__eq,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(real,int,archim2085082626_floor(real),Xa)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),A_3)),one_one(real))),Xa)) ) ).

fof(fact_3439_floor__le__eq,axiom,
    ! [Xa,A_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),Xa)),A_3))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),A_3)),one_one(real)))) ) ).

fof(fact_3440_ceiling__eq3,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),N)),one_one(real))),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),N)))
       => hAPP(real,int,archimedean_ceiling(real),X) = ti(int,N) ) ) ).

fof(fact_3441_ceiling__less__eq,axiom,
    ! [Xa,A_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),Xa)),A_3))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),A_3)),one_one(real)))) ) ).

fof(fact_3442_le__ceiling__eq,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_3),hAPP(real,int,archimedean_ceiling(real),Xa)))
    <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),A_3)),one_one(real))),Xa)) ) ).

fof(fact_3443_log__powr,axiom,
    ! [B,Y,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => hAPP(real,real,hAPP(real,fun(real,real),log,B),hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,hAPP(real,fun(real,real),log,B),X)) ) ) ).

fof(fact_3444_ceiling__eq,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real))))
       => hAPP(real,int,archimedean_ceiling(real),X) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int)) ) ) ).

fof(fact_3445_powr__realpow2,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( X = zero_zero(real)
           => hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N) = zero_zero(real) )
          & ( X != zero_zero(real)
           => hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(nat,real,real_1(nat),N)) ) ) ) ) ).

fof(fact_3446_ln__powr__bound2,axiom,
    ! [A_1,X] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,ln,X)),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),A_1)),X))) ) ) ).

fof(fact_3447_reals__Archimedean__6b__int,axiom,
    ! [R_3] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_3))
     => ? [N_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_1)),R_3))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))))) ) ) ).

fof(fact_3448_reals__Archimedean__6c__int,axiom,
    ! [R_3] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),zero_zero(real)))
     => ? [N_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_1)),R_3))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))))) ) ) ).

fof(fact_3449_real__lb__ub__int,axiom,
    ! [R_3] :
    ? [N_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_1)),R_3))
      & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))))) ) ).

fof(fact_3450_coprime__pow,axiom,
    ! [C_1,N,A_1,B] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C_1),N)
       => ? [R_1,S_1] :
            ( A_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),R_1),N)
            & B = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),S_1),N) ) ) ) ).

fof(fact_3451_prime__power__mult,axiom,
    ! [X,Y,K_4,P_5] :
      ( hBOOL(hAPP(nat,bool,prime,P_5))
     => ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),K_4)
       => ? [I_2,J_1] :
            ( X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),I_2)
            & Y = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),J_1) ) ) ) ).

fof(fact_3452_Euler__Fermat,axiom,
    ! [X,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X),M) = one_one(int)
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,phi,M))),one_one(int)),M)) ) ) ).

fof(fact_3453_zgcd__zdvd2,axiom,
    ! [I_1,J_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)),J_2)) ).

fof(fact_3454_zgcd__zdvd1,axiom,
    ! [I_1,J_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)),I_1)) ).

fof(fact_3455_zgcd__zdvd__zgcd__zmult,axiom,
    ! [M,N,K_4] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N))) ).

fof(fact_3456_zgcd__zmult__cancel,axiom,
    ! [M,K_4,N] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) ) ).

fof(fact_3457_zgcd__zgcd__zmult,axiom,
    ! [N,K_4,M] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),M) = one_one(int)
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),M) = one_one(int)
       => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)),M) = one_one(int) ) ) ).

fof(fact_3458_zgcd__1,axiom,
    ! [M] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),one_one(int)) = one_one(int) ).

fof(fact_3459_zgcd__1__left,axiom,
    ! [M] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,one_one(int)),M) = one_one(int) ).

fof(fact_3460_zgcd0,axiom,
    ! [I,J] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I),J) = zero_zero(int)
    <=> ( ti(int,I) = zero_zero(int)
        & ti(int,J) = zero_zero(int) ) ) ).

fof(fact_3461_zgcd__eq,axiom,
    ! [M,N] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N)) ).

fof(fact_3462_zgcd__power__distrib,axiom,
    ! [A_1,B,N] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N)) ).

fof(fact_3463_zgcd__greatest,axiom,
    ! [N,K_4,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N))) ) ) ).

fof(fact_3464_zgcd__greatest__iff,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Ma),Na)))
    <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Ma))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Na)) ) ) ).

fof(fact_3465_zgcd__commute,axiom,
    ! [I_1,J_2] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,J_2),I_1) ).

fof(fact_3466_zgcd__left__commute,axiom,
    ! [K_4,M,N] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N)) ).

fof(fact_3467_zgcd__assoc,axiom,
    ! [K_4,M,N] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),M)),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)) ).

fof(fact_3468_zgcd__zminus,axiom,
    ! [I_1,J_2] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,uminus_uminus(int),I_1)),J_2) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) ).

fof(fact_3469_zgcd__zminus2,axiom,
    ! [I_1,J_2] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),hAPP(int,int,uminus_uminus(int),J_2)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) ).

fof(fact_3470_zgcd__zadd__zmult,axiom,
    ! [M,N,K_4] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),K_4))),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) ).

fof(fact_3471_zgcd__pos,axiom,
    ! [I_1,J_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2))) ).

fof(fact_3472_zgcd__self,axiom,
    ! [M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),M) = ti(int,M) ) ).

fof(fact_3473_zgcd__geq__zero,axiom,
    ! [X,Y] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X),Y))) ).

fof(fact_3474_zgcd__1__power__distrib,axiom,
    ! [N,A_1,B] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N)) = one_one(int) ) ).

fof(fact_3475_zgcd__1__power__left__distrib,axiom,
    ! [N,A_1,B] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),B) = one_one(int) ) ).

fof(fact_3476_zgcd__zmult__distrib2__abs,axiom,
    ! [K_4,M,N] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)) ).

fof(fact_3477_zgcd__0__left,axiom,
    ! [M] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,zero_zero(int)),M) = hAPP(int,int,abs_abs(int),M) ).

fof(fact_3478_zgcd__0,axiom,
    ! [M] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),zero_zero(int)) = hAPP(int,int,abs_abs(int),M) ).

fof(fact_3479_zgcd__zmult__eq__self2,axiom,
    ! [N,K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)),K_4) = ti(int,K_4) ) ).

fof(fact_3480_zgcd__zmult__distrib2,axiom,
    ! [M,N,K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
     => hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)) ) ).

fof(fact_3481_zgcd__zmult__eq__self,axiom,
    ! [N,K_4] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)) = ti(int,K_4) ) ).

fof(fact_3482_zdvd__iff__zgcd,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Na))
      <=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Na),Ma) = ti(int,Ma) ) ) ).

fof(fact_3483_div__zgcd__relprime,axiom,
    ! [B,A_1] :
      ( ( ti(int,A_1) != zero_zero(int)
        | ti(int,B) != zero_zero(int) )
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B))) = one_one(int) ) ).

fof(fact_3484_zcong__zgcd__zmult__zmod,axiom,
    ! [N,A_1,B,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),N))
       => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = one_one(int)
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N))) ) ) ) ).

fof(fact_3485_zgcd__non__0,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N)) ) ).

fof(fact_3486_zgcd__zmult__zdvd__zgcd,axiom,
    ! [M,K_4,N] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N) = one_one(int)
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N))) ) ).

fof(fact_3487_zrelprime__zdvd__zmult,axiom,
    ! [M,N,K_4] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),K_4) = one_one(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M)) ) ) ).

fof(fact_3488_zrelprime__dvd__mult,axiom,
    ! [K_4,I_1,J_2] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) = one_one(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),J_2)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),K_4)) ) ) ).

fof(fact_3489_zgcd__0__1__iff,axiom,
    ! [Ma] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,zero_zero(int)),Ma) = one_one(int)
    <=> hAPP(int,int,abs_abs(int),Ma) = one_one(int) ) ).

fof(fact_3490_zgcd__code,axiom,
    ! [K,L] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),L) = hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),L),zero_zero(int))),K),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,L),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K)),hAPP(int,int,abs_abs(int),L))))) ).

fof(fact_3491_zprime__imp__zrelprime,axiom,
    ! [N,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),N))
       => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),P_5) = one_one(int) ) ) ).

fof(fact_3492_zgcd1__iff__no__common__primedivisor,axiom,
    ! [A_3,B_2] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),B_2) = one_one(int)
    <=> ~ ? [P_6] :
            ( hBOOL(hAPP(int,bool,zprime,P_6))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_6),A_3))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_6),B_2)) ) ) ).

fof(fact_3493_zgcd__zcong__zgcd,axiom,
    ! [B,A_1,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),M) = one_one(int)
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))
         => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B),M) = one_one(int) ) ) ) ).

fof(fact_3494_zless__zprime__imp__zrelprime,axiom,
    ! [N,P_5] :
      ( hBOOL(hAPP(int,bool,zprime,P_5))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),N),P_5))
         => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),P_5) = one_one(int) ) ) ) ).

fof(fact_3495_zcong__cancel2,axiom,
    ! [A_3,B_2,K,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ma))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),Ma) = one_one(int)
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),A_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),B_2)),Ma))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) ) ) ) ).

fof(fact_3496_zcong__cancel,axiom,
    ! [A_3,B_2,K,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ma))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),Ma) = one_one(int)
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),K)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),K)),Ma))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) ) ) ) ).

fof(fact_3497_zrelprime__zdvd__zmult__aux,axiom,
    ! [M,N,K_4] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),K_4) = one_one(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M)) ) ) ) ).

fof(fact_3498_Bnor__mem__if,axiom,
    ! [A_3,B_2,Ma] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B_2),Ma) = one_one(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3))
         => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))) ) ) ) ).

fof(fact_3499_int__relprime__odd__power__divisors,axiom,
    ! [A_3,B_2,C,Xa] :
      ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),B_2) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),hAPP(int,nat,nat_1,Xa))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(int,nat,nat_1,Xa)))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
         => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),B_2) = one_one(int)
           => ? [K_1] : ti(int,A_3) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),hAPP(int,nat,nat_1,Xa)) ) ) ) ) ).

fof(fact_3500_xzgcd__correct,axiom,
    ! [Ma,K,Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Na))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Ma),Na) = ti(int,K)
      <=> ? [S_1,T_4] : hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,Ma),Na) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_1),T_4)) ) ) ).

fof(fact_3501_int__relprime__power__divisors,axiom,
    ! [A_1,B,C_1,N] :
      ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C_1),N)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
       => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int)
         => ? [K_1] : hAPP(int,int,abs_abs(int),A_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),N) ) ) ) ).

fof(fact_3502_xzgcd__correct__aux2,axiom,
    ! [M,N,R_4,R_3,S_4,S_3,T_6,T_5,K_4] :
      ( ? [Sn,Tn] : hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn),Tn))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_3))
       => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,R_4),R_3) = ti(int,K_4) ) ) ).

fof(fact_3503_zcong__lineq__unique,axiom,
    ! [B,A_1,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),N) = one_one(int)
       => ? [X_1] :
            ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),N))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),X_1)),B),N))
            & ! [Y_1] :
                ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
                  & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),N))
                  & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Y_1)),B),N)) )
               => ti(int,Y_1) = ti(int,X_1) ) ) ) ) ).

fof(fact_3504_zcong__lineq__ex,axiom,
    ! [A_1,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),N) = one_one(int)
       => ? [X_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),X_1)),one_one(int)),N)) ) ) ).

fof(fact_3505_xzgcd__correct__aux1,axiom,
    ! [M,N,S_4,S_3,T_6,T_5,R_4,R_3,K_4] :
      ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,R_4),R_3) = ti(int,K_4)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_3))
       => ? [Sn,Tn] : hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn),Tn)) ) ) ).

fof(fact_3506_int__triple__relprime__odd__power__divisors,axiom,
    ! [A_3,B_2,C,D_1,Xa] :
      ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),B_2)),C) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),D_1),hAPP(int,nat,nat_1,Xa))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(int,nat,nat_1,Xa)))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))
         => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),B_2) = one_one(int)
           => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B_2),C) = one_one(int)
             => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C),A_3) = one_one(int)
               => ? [K_1,L_3,M_2] :
                    ( ti(int,A_3) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),hAPP(int,nat,nat_1,Xa))
                    & ti(int,B_2) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),L_3),hAPP(int,nat,nat_1,Xa))
                    & ti(int,C) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),M_2),hAPP(int,nat,nat_1,Xa)) ) ) ) ) ) ) ) ).

fof(fact_3507_zgcd__ex__linear,axiom,
    ! [M,K_4,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = ti(int,K_4)
       => ? [S_1,T_4] : ti(int,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_1),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_4),N)) ) ) ).

fof(fact_3508_int__triple__relprime__power__divisors,axiom,
    ! [A_1,B,C_1,D_3,N] :
      ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),D_3),N)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
       => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int)
         => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B),C_1) = one_one(int)
           => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C_1),A_1) = one_one(int)
             => ? [K_1,L_3,M_2] :
                  ( hAPP(int,int,abs_abs(int),A_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),N)
                  & hAPP(int,int,abs_abs(int),B) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),L_3),N)
                  & hAPP(int,int,abs_abs(int),C_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),M_2),N) ) ) ) ) ) ) ).

fof(fact_3509_make__zrelprime,axiom,
    ! [B,A_1] :
      ( ( ti(int,A_1) != zero_zero(int)
        | ti(int,B) != zero_zero(int) )
     => ? [C_4,D_2] :
          ( ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),C_4)
          & ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),D_2)
          & hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C_4),D_2) = one_one(int) ) ) ).

fof(fact_3510_pair__lessI2,axiom,
    ! [Sb,Ta,A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_3),B_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Sb),Ta))
       => hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A_3),Sb)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B_2),Ta))),pair_less)) ) ) ).

fof(fact_3511_pair__lessI1,axiom,
    ! [Sb,Ta,A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2))
     => hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A_3),Sb)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B_2),Ta))),pair_less)) ) ).

fof(fact_3512_in__lex__prod,axiom,
    ! [X_a,X_b,A_3,B_2,A_8,B_7,Ra,Sb] :
      ( hBOOL(hAPP(fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),bool,hAPP(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),fun(fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),bool),member(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b))),hAPP(product_prod(X_a,X_b),product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),hAPP(product_prod(X_a,X_b),fun(product_prod(X_a,X_b),product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b))),product_Pair(product_prod(X_a,X_b),product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_8),B_7))),hAPP(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),hAPP(fun(product_prod(X_a,X_a),bool),fun(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool)),lex_prod(X_a,X_b),Ra),Sb)))
    <=> ( hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),A_3),A_8)),Ra))
        | ( ti(X_a,A_3) = ti(X_a,A_8)
          & hBOOL(hAPP(fun(product_prod(X_b,X_b),bool),bool,hAPP(product_prod(X_b,X_b),fun(fun(product_prod(X_b,X_b),bool),bool),member(product_prod(X_b,X_b)),hAPP(X_b,product_prod(X_b,X_b),hAPP(X_b,fun(X_b,product_prod(X_b,X_b)),product_Pair(X_b,X_b),B_2),B_7)),Sb)) ) ) ) ).

fof(fact_3513_pair__leqI2,axiom,
    ! [Sb,Ta,A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_3),B_2))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Sb),Ta))
       => hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A_3),Sb)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B_2),Ta))),pair_leq)) ) ) ).

fof(fact_3514_pair__leqI1,axiom,
    ! [Sb,Ta,A_3,B_2] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2))
     => hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A_3),Sb)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B_2),Ta))),pair_leq)) ) ).

fof(fact_3515_Bseq__iff2,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))
        <=> ? [K_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_1))
              & ? [X_1] :
                ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(X_a,X_a,uminus_uminus(X_a),X_1)))),K_1)) ) ) ) ).

fof(fact_3516_Bseq__iff3,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))
        <=> ? [K_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_1))
              & ? [N_2] :
                ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,X_2,N_2))))),K_1)) ) ) ) ).

fof(fact_3517_polar__Ex,axiom,
    ! [Y,X] :
    ? [R_1,A_4] :
      ( X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,cos,A_4))
      & Y = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,sin,A_4)) ) ).

fof(fact_3518_Cauchy__Bseq,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) ) ) ).

fof(fact_3519_Bseq__def,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))
        <=> ? [K_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
              & ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_2)) ) ) ) ).

fof(fact_3520_Bseq__iff,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))
        <=> ? [N_2] :
            ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2)))) ) ) ).

fof(fact_3521_Bseq__iff1a,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))
        <=> ? [N_2] :
            ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2)))) ) ) ).

fof(fact_3522_BseqD,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))
         => ? [K_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
              & ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_2)) ) ) ) ).

fof(fact_3523_BseqI,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,K_3] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_3))
         => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_3))
           => hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) ) ) ) ).

fof(fact_3524_BseqI2_H,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,K_3,N_3] :
          ( ! [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_3)) )
         => hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) ) ) ).

fof(fact_3525_BseqI_H,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,K_3] :
          ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_3))
         => hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) ) ) ).

fof(fact_3526_BseqE,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))
         => ~ ! [K_2] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
               => ~ ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_2)) ) ) ) ).

fof(fact_3527_dist__triangle__half__l,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X2_1,X1_1,Y,E_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X2_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),X2_1)),E_1)) ) ) ) ).

fof(fact_3528_dist__triangle__le,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Z_1,Y,E_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1))),E_1))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),E_1)) ) ) ).

fof(fact_3529_dist__triangle__alt,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [Y,Z_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)))) ) ).

fof(fact_3530_dist__triangle,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Z_1,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1)))) ) ).

fof(fact_3531_dist__triangle2,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Y,Z_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1)))) ) ).

fof(fact_3532_dist__triangle3,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Y,A_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),A_1),X)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),A_1),Y)))) ) ).

fof(fact_3533_dist__triangle__lt,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Z_1,Y,E_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1))),E_1))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),E_1)) ) ) ).

fof(fact_3534_dist__le__zero__iff,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Xa),Ya)),zero_zero(real)))
        <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ).

fof(fact_3535_zero__le__dist,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y))) ) ).

fof(fact_3536_dist__pos__lt,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Y] :
          ( ti(X_a,X) != ti(X_a,Y)
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y))) ) ) ).

fof(fact_3537_zero__less__dist__iff,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Xa),Ya)))
        <=> ti(X_a,Xa) != ti(X_a,Ya) ) ) ).

fof(fact_3538_dist__nz,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [Xa,Ya] :
          ( ti(X_a,Xa) != ti(X_a,Ya)
        <=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Xa),Ya))) ) ) ).

fof(fact_3539_dist__not__less__zero,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Y] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),zero_zero(real))) ) ).

fof(fact_3540_dist__norm,axiom,
    ! [X_a] :
      ( dist_norm(X_a)
     => ! [X,Y] : hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) ) ).

fof(fact_3541_dist__self,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X] : hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),X) = zero_zero(real) ) ).

fof(fact_3542_dist__eq__0__iff,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [Xa,Ya] :
          ( hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Xa),Ya) = zero_zero(real)
        <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ).

fof(fact_3543_dist__commute,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X,Y] : hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y) = hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X) ) ).

fof(fact_3544_dist__real__def,axiom,
    ! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),dist_dist(real),X),Y) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) ).

fof(fact_3545_dist__complex__def,axiom,
    ! [X,Y] : hAPP(complex,real,hAPP(complex,fun(complex,real),dist_dist(complex),X),Y) = hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) ).

fof(fact_3546_dist__triangle__half__r,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X2_1,Y,X1_1,E_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X1_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X2_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),X2_1)),E_1)) ) ) ) ).

fof(fact_3547_metric__CauchyD,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [E_2,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E_2))
           => ? [M_3] :
              ! [M_2] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2))
               => ! [N_1] :
                    ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1))
                   => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))),E_2)) ) ) ) ) ) ).

fof(fact_3548_Cauchy__def,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
        <=> ! [E] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M_3] :
                ! [M_2] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2))
                 => ! [N_1] :
                      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1))
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))),E)) ) ) ) ) ) ).

fof(fact_3549_norm__conv__dist,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] : hAPP(X_a,real,norm_norm(X_a),X) = hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),zero_zero(X_a)) ) ).

fof(fact_3550_metric__CauchyI,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2] :
          ( ! [E] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M_3] :
                ! [M_2] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2))
                 => ! [N_1] :
                      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1))
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))),E)) ) ) )
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) ) ) ).

fof(fact_3551_DERIV__nonneg__imp__nonincreasing,axiom,
    ! [F,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
     => ( ! [X_1] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
           => ? [Y_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),Y_1))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1)) ) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,A_3)),hAPP(real,real,F,B_2))) ) ) ).

fof(fact_3552_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [F,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
     => ( ! [X_1] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
           => ? [Y_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),Y_1))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),zero_zero(real))) ) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))) ) ) ).

fof(fact_3553_int__cases,axiom,
    ! [Z_1] :
      ( ! [N_1] : ti(int,Z_1) != hAPP(nat,int,semiring_1_of_nat(int),N_1)
     => ~ ! [N_1] : ti(int,Z_1) != hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1))) ) ).

fof(fact_3554_int__of__nat__induct,axiom,
    ! [Z,P_1] :
      ( ! [N_1] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),N_1)))
     => ( ! [N_1] : hBOOL(hAPP(int,bool,P_1,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1)))))
       => hBOOL(hAPP(int,bool,P_1,Z)) ) ) ).

fof(fact_3555_gcd__nat__induct,axiom,
    ! [Ma,Na,P_1] :
      ( ! [M_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,M_2),zero_zero(nat)))
     => ( ! [M_2,N_1] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
           => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_2),N_1)))
             => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,M_2),N_1)) ) )
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Ma),Na)) ) ) ).

fof(fact_3556_code__numeral_Osimps_I5_J,axiom,
    ! [X_a,F1,F2,Code_numeral] : hAPP(code_code_numeral,X_a,hAPP(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a),hAPP(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a)),code_c600343653l_case(X_a),F1),F2),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral)) = hAPP(code_code_numeral,X_a,F2,Code_numeral) ).

fof(fact_3557_divides__fact,axiom,
    ! [N,P_5] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P_5))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_5),N))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,fact,N))) ) ) ).

fof(fact_3558_mod__code__int__def,axiom,
    ! [N,M] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),N),M) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) ).

fof(fact_3559_number__of__code__int__def,axiom,
    number_number_of(quickcheck_code_int) = quickcheck_of_int ).

fof(fact_3560_int__of__of__int,axiom,
    ! [N] : hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(int,quickcheck_code_int,quickcheck_of_int,N)) = ti(int,N) ).

fof(fact_3561_of__int__int__of,axiom,
    ! [K_4] : hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4)) = K_4 ).

fof(fact_3562_int__of__inverse,axiom,
    ! [X] : hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(quickcheck_code_int,int,quickcheck_int_of,X)) = X ).

fof(fact_3563_Quickcheck__Narrowing_Oof__int__inject,axiom,
    ! [Na,Ma] :
      ( hAPP(int,quickcheck_code_int,quickcheck_of_int,Na) = hAPP(int,quickcheck_code_int,quickcheck_of_int,Ma)
    <=> ti(int,Na) = ti(int,Ma) ) ).

fof(fact_3564_fact__mono,axiom,
    ! [M,N] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,fact,M)),hAPP(nat,nat,fact,N))) ) ).

fof(fact_3565_times__code__int__def,axiom,
    ! [N,M] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),N),M) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) ).

fof(fact_3566_plus__code__int__def,axiom,
    ! [N,M] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),N),M) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) ).

fof(fact_3567_minus__code__int__def,axiom,
    ! [N,M] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),minus_minus(quickcheck_code_int),N),M) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) ).

fof(fact_3568_zero__code__int__def,axiom,
    zero_zero(quickcheck_code_int) = hAPP(int,quickcheck_code_int,quickcheck_of_int,zero_zero(int)) ).

fof(fact_3569_fact__lt,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,fact,N))) ).

fof(fact_3570_fact_Osimps_I1_J,axiom,
    hAPP(nat,nat,fact,zero_zero(nat)) = one_one(nat) ).

fof(fact_3571_fact_Osimps_I2_J,axiom,
    ! [N] : hAPP(nat,nat,fact,hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,N)),hAPP(nat,nat,fact,N)) ).

fof(fact_3572_fact__le,axiom,
    ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(nat,nat,fact,N))) ).

fof(fact_3573_one__code__int__def,axiom,
    one_one(quickcheck_code_int) = hAPP(int,quickcheck_code_int,quickcheck_of_int,one_one(int)) ).

fof(fact_3574_code__numeral_Osimps_I4_J,axiom,
    ! [X_a,F1,F2] : hAPP(code_code_numeral,X_a,hAPP(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a),hAPP(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a)),code_c600343653l_case(X_a),F1),F2),zero_zero(code_code_numeral)) = ti(X_a,F1) ).

fof(fact_3575_div__code__int__def,axiom,
    ! [N,M] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),N),M) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) ).

fof(fact_3576_euclid__bound,axiom,
    ! [N] :
    ? [P_6] :
      ( hBOOL(hAPP(nat,bool,prime,P_6))
      & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),P_6))
      & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_6),hAPP(nat,nat,suc,hAPP(nat,nat,fact,N)))) ) ).

fof(fact_3577_weak__decr__stable,axiom,
    ! [F] :
      ( ! [I_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,F,hAPP(nat,nat,suc,I_2))),hAPP(nat,nat,F,I_2)))
     => ? [I_2] :
        ! [K_1] : hAPP(nat,nat,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_2),K_1)) = hAPP(nat,nat,F,I_2) ) ).

fof(fact_3578_norm__frac_Opinduct,axiom,
    ! [P_1,A0,A1] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
     => ( ! [A_4,B_4] :
            ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_4),B_4)))
           => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_4),zero_zero(int)))
               => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,uminus_uminus(int),A_4)),hAPP(int,int,uminus_uminus(int),B_4))) )
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),B_4)) ) )
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).

fof(fact_3579_Nitpick_Orefl_H__def,axiom,
    ! [X_a,Ra] :
      ( hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,refl(X_a),Ra))
    <=> ! [X_1] : hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),X_1),X_1)),Ra)) ) ).

fof(fact_3580_upto_Opinduct,axiom,
    ! [P_1,A0,A1] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),upto_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
     => ( ! [I_2,J_1] :
            ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),upto_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),I_2),J_1)))
           => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),J_1))
               => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int))),J_1)) )
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,I_2),J_1)) ) )
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).

fof(fact_3581_neq__if__length__neq,axiom,
    ! [X_a,Xs,Ys] :
      ( hAPP(list(X_a),nat,size_size(list(X_a)),Xs) != hAPP(list(X_a),nat,size_size(list(X_a)),Ys)
     => Xs != Ys ) ).

fof(fact_3582_finite__maxlen,axiom,
    ! [X_a,M_1] :
      ( hBOOL(hAPP(fun(list(X_a),bool),bool,finite_finite_1(list(X_a)),M_1))
     => ? [N_1] :
        ! [X_1] :
          ( hBOOL(hAPP(fun(list(X_a),bool),bool,hAPP(list(X_a),fun(fun(list(X_a),bool),bool),member(list(X_a)),X_1),M_1))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(list(X_a),nat,size_size(list(X_a)),X_1)),N_1)) ) ) ).

fof(fact_3583_small__lazy_H_Opinduct,axiom,
    ! [P_1,A0,A1] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),lazy_small_lazy_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
     => ( ! [D_2,I_2] :
            ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),lazy_small_lazy_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),D_2),I_2)))
           => ( ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),D_2),I_2))
               => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,D_2),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int)))) )
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,D_2),I_2)) ) )
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).

fof(fact_3584_of__int__of__nat,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [K_4] :
          ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int)))
           => hAPP(int,X_a,ring_1_of_int(X_a),K_4) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),K_4)))) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int)))
           => hAPP(int,X_a,ring_1_of_int(X_a),K_4) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,K_4)) ) ) ) ).

fof(fact_3585_norRRset__def,axiom,
    ! [Ma] : hAPP(int,fun(int,bool),norRRset,Ma) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ma),one_one(int))),Ma) ).

fof(fact_3586_of__int__diff,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [W,Z_1] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) ) ).

fof(fact_3587_of__int__mult,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [W,Z_1] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) ) ).

fof(fact_3588_of__int__add,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [W,Z_1] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) ) ).

fof(fact_3589_of__int__0,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => hAPP(int,X_a,ring_1_of_int(X_a),zero_zero(int)) = zero_zero(X_a) ) ).

fof(fact_3590_of__int__0__eq__iff,axiom,
    ! [X_a] :
      ( ring_char_0(X_a)
     => ! [Z] :
          ( zero_zero(X_a) = hAPP(int,X_a,ring_1_of_int(X_a),Z)
        <=> ti(int,Z) = zero_zero(int) ) ) ).

fof(fact_3591_of__int__eq__0__iff,axiom,
    ! [X_a] :
      ( ring_char_0(X_a)
     => ! [Z] :
          ( hAPP(int,X_a,ring_1_of_int(X_a),Z) = zero_zero(X_a)
        <=> ti(int,Z) = zero_zero(int) ) ) ).

fof(fact_3592_number__of__real__def,axiom,
    ! [X] : hAPP(int,real,number_number_of(real),X) = hAPP(int,real,ring_1_of_int(real),X) ).

fof(fact_3593_int__number__of__def,axiom,
    ! [W] : hAPP(int,int,number_number_of(int),W) = hAPP(int,int,ring_1_of_int(int),W) ).

fof(fact_3594_number__of__eq,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [K_4] : hAPP(int,X_a,number_number_of(X_a),K_4) = hAPP(int,X_a,ring_1_of_int(X_a),K_4) ) ).

fof(fact_3595_complex__number__of__def,axiom,
    ! [W] : hAPP(int,complex,number_number_of(complex),W) = hAPP(int,complex,ring_1_of_int(complex),W) ).

fof(fact_3596_of__int__int__eq,axiom,
    ! [N] : hAPP(int,int,ring_1_of_int(int),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),N) ).

fof(fact_3597_floor__of__int,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Z_1] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) = ti(int,Z_1) ) ).

fof(fact_3598_of__real__of__int__eq,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [Z_1] : hAPP(real,X_a,of_real(X_a),hAPP(int,real,ring_1_of_int(real),Z_1)) = hAPP(int,X_a,ring_1_of_int(X_a),Z_1) ) ).

fof(fact_3599_norm__of__int,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => ! [Z_1] : hAPP(X_a,real,norm_norm(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) = hAPP(real,real,abs_abs(real),hAPP(int,real,ring_1_of_int(real),Z_1)) ) ).

fof(fact_3600_of__int__eq__iff,axiom,
    ! [X_a] :
      ( ring_char_0(X_a)
     => ! [Wa,Z] :
          ( hAPP(int,X_a,ring_1_of_int(X_a),Wa) = hAPP(int,X_a,ring_1_of_int(X_a),Z)
        <=> ti(int,Wa) = ti(int,Z) ) ) ).

fof(fact_3601_complex__cnj__of__int,axiom,
    ! [Z_1] : hAPP(complex,complex,cnj,hAPP(int,complex,ring_1_of_int(complex),Z_1)) = hAPP(int,complex,ring_1_of_int(complex),Z_1) ).

fof(fact_3602_complex__Re__of__int,axiom,
    ! [Z_1] : hAPP(complex,real,re,hAPP(int,complex,ring_1_of_int(complex),Z_1)) = hAPP(int,real,ring_1_of_int(real),Z_1) ).

fof(fact_3603_of__int__1,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => hAPP(int,X_a,ring_1_of_int(X_a),one_one(int)) = one_one(X_a) ) ).

fof(fact_3604_of__int__le__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Wa,Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Wa)),hAPP(int,X_a,ring_1_of_int(X_a),Z)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z)) ) ) ).

fof(fact_3605_of__int__less__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Wa,Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Wa)),hAPP(int,X_a,ring_1_of_int(X_a),Z)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) ) ) ).

fof(fact_3606_of__int__number__of__eq,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [V] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,number_number_of(int),V)) = hAPP(int,X_a,number_number_of(X_a),V) ) ).

fof(fact_3607_le__of__int__ceiling,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X)))) ) ).

fof(fact_3608_complex__Im__of__int,axiom,
    ! [Z_1] : hAPP(complex,real,im,hAPP(int,complex,ring_1_of_int(complex),Z_1)) = zero_zero(real) ).

fof(fact_3609_of__int__minus,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [Z_1] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,uminus_uminus(int),Z_1)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) ) ).

fof(fact_3610_of__int__floor__le,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archim2085082626_floor(X_a),X))),X)) ) ).

fof(fact_3611_of__int__of__nat__eq,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [N] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N) ) ).

fof(fact_3612_of__int__power,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [Z_1,N] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),N) ) ).

fof(fact_3613_ceiling__of__int,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Z_1] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) = ti(int,Z_1) ) ).

fof(fact_3614_real__eq__of__int,axiom,
    real_1(int) = ring_1_of_int(real) ).

fof(fact_3615_real__of__int__def,axiom,
    real_1(int) = ring_1_of_int(real) ).

fof(fact_3616_of__int__m1,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,number_number_of(int),min)) = hAPP(int,X_a,number_number_of(X_a),min) ) ).

fof(fact_3617_floor__less__iff,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa,Z] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),Z))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(int,X_a,ring_1_of_int(X_a),Z))) ) ) ).

fof(fact_3618_le__floor__iff,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Z,Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),Xa)) ) ) ).

fof(fact_3619_floor__add__of__int,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,Z_1] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),Z_1) ) ).

fof(fact_3620_less__ceiling__iff,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Z,Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),Xa)) ) ) ).

fof(fact_3621_ceiling__le__iff,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa,Z] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),Z))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(int,X_a,ring_1_of_int(X_a),Z))) ) ) ).

fof(fact_3622_ceiling__add__of__int,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,Z_1] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),Z_1) ) ).

fof(fact_3623_floor__diff__of__int,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,Z_1] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),Z_1) ) ).

fof(fact_3624_ceiling__diff__of__int,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X,Z_1] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),Z_1) ) ).

fof(fact_3625_of__int__0__less__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,ring_1_of_int(X_a),Z)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ) ).

fof(fact_3626_of__int__less__0__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),zero_zero(X_a)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ) ).

fof(fact_3627_of__int__0__le__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,ring_1_of_int(X_a),Z)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

fof(fact_3628_of__int__le__0__iff,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),zero_zero(X_a)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int))) ) ) ).

fof(fact_3629_of__nat__nat,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [Z_1] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
         => hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,Z_1)) = hAPP(int,X_a,ring_1_of_int(X_a),Z_1) ) ) ).

fof(fact_3630_floor__unique,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Z_1,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),one_one(X_a))))
           => hAPP(X_a,int,archim2085082626_floor(X_a),X) = ti(int,Z_1) ) ) ) ).

fof(fact_3631_ceiling__unique,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Z_1,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),one_one(X_a))),X))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)))
           => hAPP(X_a,int,archimedean_ceiling(X_a),X) = ti(int,Z_1) ) ) ) ).

fof(fact_3632_ceiling__correct,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X))),one_one(X_a))),X))
          & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X)))) ) ) ).

fof(fact_3633_less__floor__iff,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Z,Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a))),Xa)) ) ) ).

fof(fact_3634_floor__le__iff,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa,Z] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),Z))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a)))) ) ) ).

fof(fact_3635_floor__correct,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archim2085082626_floor(X_a),X))),X))
          & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int))))) ) ) ).

fof(fact_3636_ceiling__less__iff,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Xa,Z] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),Z))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a)))) ) ) ).

fof(fact_3637_le__ceiling__iff,axiom,
    ! [X_a] :
      ( archim1727834104eiling(X_a)
     => ! [Z,Xa] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a))),Xa)) ) ) ).

fof(fact_3638_phi__def,axiom,
    ! [Ma] : hAPP(int,nat,phi,Ma) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),norRRset,Ma)) ).

fof(fact_3639_norR__mem__unique,axiom,
    ! [A_3,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),Ma) = one_one(int)
       => ? [X_1] :
            ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),X_1),Ma))
            & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),norRRset,Ma)))
            & ! [Y_1] :
                ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),Y_1),Ma))
                  & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),hAPP(int,fun(int,bool),norRRset,Ma))) )
               => ti(int,Y_1) = ti(int,X_1) ) ) ) ) ).

fof(fact_3640_card__nor__eq__noX,axiom,
    ! [Xa,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Xa),Ma) = one_one(int)
       => hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,Ma),Xa)) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),norRRset,Ma)) ) ) ).

fof(fact_3641_floor__exists1,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
        ? [X_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),X_1)),X))
          & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),one_one(int)))))
          & ! [Y_1] :
              ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Y_1)),X))
                & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Y_1),one_one(int))))) )
             => ti(int,Y_1) = ti(int,X_1) ) ) ) ).

fof(fact_3642_floor__exists,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
        ? [Z_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_2)),X))
          & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_2),one_one(int))))) ) ) ).

fof(fact_3643_noX__is__RRset,axiom,
    ! [Xa,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Xa),Ma) = one_one(int)
       => hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,Ma),Xa)),Ma)) ) ) ).

fof(fact_3644_ex__of__int__less,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
        ? [Z_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_2)),X)) ) ).

fof(fact_3645_RRset__gcd,axiom,
    ! [A_3,A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2))
       => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),Ma) = one_one(int) ) ) ).

fof(fact_3646_RRset__zcong__eq,axiom,
    ! [A_3,B_2,A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))
         => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2))
           => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),A_2))
             => ti(int,A_3) = ti(int,B_2) ) ) ) ) ) ).

fof(fact_3647_RRset2norRR__correct,axiom,
    ! [A_3,A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)),Ma))
            & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)),hAPP(int,fun(int,bool),norRRset,Ma))) ) ) ) ) ).

fof(fact_3648_RRset2norRR__correct2,axiom,
    ! [A_3,A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2))
         => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)),hAPP(int,fun(int,bool),norRRset,Ma))) ) ) ) ).

fof(fact_3649_RRset2norRR__correct1,axiom,
    ! [A_3,A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)),Ma)) ) ) ) ).

fof(fact_3650_ex__less__of__int,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
        ? [Z_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_2))) ) ).

fof(fact_3651_ex__le__of__int,axiom,
    ! [X_a] :
      ( archim1804426504_field(X_a)
     => ! [X] :
        ? [Z_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_2))) ) ).

fof(fact_3652_is__RRset__def,axiom,
    ! [A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
    <=> ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma)))
        & hAPP(fun(int,bool),nat,finite_card(int),A_2) = hAPP(int,nat,phi,Ma) ) ) ).

fof(fact_3653_RRset2norRR__eq__norR,axiom,
    ! [A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
       => hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma)),A_2) = hAPP(int,fun(int,bool),norRRset,Ma) ) ) ).

fof(fact_3654_image__eqI,axiom,
    ! [X_a,X_b,A_2,B_2,F,Xa] :
      ( ti(X_a,B_2) = hAPP(X_b,X_a,F,Xa)
     => ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Xa),A_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2))) ) ) ).

fof(fact_3655_finite__imageI,axiom,
    ! [X_b,X_a,H,F_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
     => hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),H),F_1))) ) ).

fof(fact_3656_finite__surj,axiom,
    ! [X_b,X_a,B_1,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_1),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)))
       => hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) ) ) ).

fof(fact_3657_image__diff__subset,axiom,
    ! [X_a,X_b,F,A_2,B_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),B_1))),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_2),B_1)))) ).

fof(fact_3658_image__Pow__surj,axiom,
    ! [X_b,X_a,F,A_2,B_1] :
      ( hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2) = ti(fun(X_a,bool),B_1)
     => hAPP(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_b,bool),fun(X_a,bool)),fun(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool)),image(fun(X_b,bool),fun(X_a,bool)),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),pow(X_b),A_2)) = hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1) ) ).

fof(fact_3659_image__iff,axiom,
    ! [X_a,X_b,Z,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
    <=> ? [X_1] :
          ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_2))
          & ti(X_a,Z) = hAPP(X_b,X_a,F,X_1) ) ) ).

fof(fact_3660_imageI,axiom,
    ! [X_b,X_a,F,Xa,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
     => hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_a,X_b,F,Xa)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))) ) ).

fof(fact_3661_rev__image__eqI,axiom,
    ! [X_b,X_a,B_2,F,Xa,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
     => ( ti(X_b,B_2) = hAPP(X_a,X_b,F,Xa)
       => hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_2),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))) ) ) ).

fof(fact_3662_image__Pow__mono,axiom,
    ! [X_a,X_b,F,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)),B_1))
     => hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_b,bool),fun(X_a,bool)),fun(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool)),image(fun(X_b,bool),fun(X_a,bool)),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),pow(X_b),A_2))),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) ) ).

fof(fact_3663_subset__image__iff,axiom,
    ! [X_a,X_b,B_1,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
    <=> ? [AA] :
          ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),AA),A_2))
          & ti(fun(X_a,bool),B_1) = hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),AA) ) ) ).

fof(fact_3664_image__mono,axiom,
    ! [X_b,X_a,F,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),B_1))) ) ).

fof(fact_3665_RsetR__fin,axiom,
    ! [A_2,Ma] :
      ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma)))
     => hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),A_2)) ) ).

fof(fact_3666_card__image__le,axiom,
    ! [X_b,X_a,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))) ) ).

fof(fact_3667_Bnor__in__RsetR,axiom,
    ! [A_3,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),Ma))
     => hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma)),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma))) ) ).

fof(fact_3668_ResSet__image,axiom,
    ! [F,A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Ma),A_2))
       => ( ! [X_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),A_2))
             => ! [Xa_1] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),A_2))
                 => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,F,X_1)),hAPP(int,int,F,Xa_1)),Ma))
                   => ti(int,X_1) = ti(int,Xa_1) ) ) )
         => hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Ma),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),F),A_2))) ) ) ) ).

fof(fact_3669_finite__subset__image,axiom,
    ! [X_b,X_a,F,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
       => ? [C_3] :
            ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C_3),A_2))
            & hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),C_3))
            & ti(fun(X_a,bool),B_1) = hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),C_3) ) ) ) ).

fof(fact_3670_transfer__nat__int__set__relations_I1_J,axiom,
    ! [A_2] :
      ( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A_2))
    <=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2))) ) ).

fof(fact_3671_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [A_2] : hAPP(fun(nat,bool),nat,finite_card(nat),A_2) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) ).

fof(fact_3672_transfer__nat__int__set__relations_I3_J,axiom,
    ! [A_2,B_1] :
      ( A_2 = B_1
    <=> hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1) ) ).

fof(fact_3673_transfer__nat__int__set__relations_I2_J,axiom,
    ! [Xa,A_2] :
      ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),Xa),A_2))
    <=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2))) ) ).

fof(fact_3674_transfer__int__nat__set__return__embed,axiom,
    ! [A_2] : hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) = A_2 ).

fof(fact_3675_transfer__nat__int__set__relations_I4_J,axiom,
    ! [A_2,B_1] :
      ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),A_2),B_1))
    <=> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1))) ) ).

fof(fact_3676_transfer__nat__int__set__relations_I5_J,axiom,
    ! [A_2,B_1] :
      ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),A_2),B_1))
    <=> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1))) ) ).

fof(fact_3677_SetS__def,axiom,
    ! [A_3,P_4] : hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4) = hAPP(fun(int,bool),fun(fun(int,bool),bool),hAPP(fun(int,fun(int,bool)),fun(fun(int,bool),fun(fun(int,bool),bool)),image(int,fun(int,bool)),hAPP(int,fun(int,fun(int,bool)),hAPP(int,fun(int,fun(int,fun(int,bool))),multInvPair,A_3),P_4)),hAPP(int,fun(int,bool),sRStar,P_4)) ).

fof(fact_3678_finite__surjI,axiom,
    ! [X_a,X_b,B_1,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
     => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
       => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) ) ) ).

fof(fact_3679_image__subsetI,axiom,
    ! [X_b,X_a,F,B_1,A_2] :
      ( ! [X_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
         => hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_a,X_b,F,X_1)),B_1)) )
     => hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)),B_1)) ) ).

fof(fact_3680_infinite__nat__iff__unbounded__le,axiom,
    ! [S] :
      ( ~ hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
    <=> ! [M_2] :
        ? [N_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1))
          & hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),N_1),S)) ) ) ).

fof(fact_3681_infinite__nat__iff__unbounded,axiom,
    ! [S] :
      ( ~ hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
    <=> ! [M_2] :
        ? [N_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N_1))
          & hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),N_1),S)) ) ) ).

fof(fact_3682_transfer__int__nat__set__relations_I4_J,axiom,
    ! [B_1,A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),A_2),B_1))
        <=> hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1))) ) ) ) ).

fof(fact_3683_finite__nat__set__iff__bounded__le,axiom,
    ! [N_3] :
      ( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3))
    <=> ? [M_2] :
        ! [X_1] :
          ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),N_3))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),M_2)) ) ) ).

fof(fact_3684_nat__set__def,axiom,
    ! [S] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,S))
    <=> ! [X_1] :
          ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),S))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) ) ) ).

fof(fact_3685_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
    ! [Xa,A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),A_2))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) ) ) ).

fof(fact_3686_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J,axiom,
    ! [C_2] : hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),C_2))) ).

fof(fact_3687_transfer__int__nat__set__relations_I3_J,axiom,
    ! [B_1,A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
       => ( A_2 = B_1
        <=> hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1) ) ) ) ).

fof(fact_3688_transfer__nat__int__set__return__embed,axiom,
    ! [A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) = A_2 ) ).

fof(fact_3689_transfer__int__nat__set__relations_I1_J,axiom,
    ! [A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => ( hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),A_2))
      <=> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2))) ) ) ).

fof(fact_3690_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
    ! [A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => hAPP(fun(int,bool),nat,finite_card(int),A_2) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) ) ).

fof(fact_3691_transfer__int__nat__set__relations_I5_J,axiom,
    ! [B_1,A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),A_2),B_1))
        <=> hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1))) ) ) ) ).

fof(fact_3692_finite__nat__set__iff__bounded,axiom,
    ! [N_3] :
      ( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3))
    <=> ? [M_2] :
        ! [X_1] :
          ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),N_3))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),M_2)) ) ) ).

fof(fact_3693_bounded__nat__set__is__finite,axiom,
    ! [Na,N_3] :
      ( ! [X_1] :
          ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),N_3))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Na)) )
     => hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3)) ) ).

fof(fact_3694_card__greaterThanLessThan__int,axiom,
    ! [L,U] : hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

fof(fact_3695_finite__greaterThanLessThan__int,axiom,
    ! [L,U] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U))) ).

fof(fact_3696_image__uminus__greaterThanLessThan,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [Xa,Ya] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),uminus_uminus(X_a)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),Xa),Ya)) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Ya)),hAPP(X_a,X_a,uminus_uminus(X_a),Xa)) ) ).

fof(fact_3697_Rats__abs__nat__div__natE,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),field_char_0_Rats(real)))
     => ~ ! [M_2,N_1] :
            ( N_1 != zero_zero(nat)
           => ( hAPP(real,real,abs_abs(real),Xa) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),M_2)),hAPP(nat,real,real_1(nat),N_1))
             => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_2),N_1) != one_one(nat) ) ) ) ).

fof(fact_3698_Nitpick_Oint__lcm__def,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),int_lcm,X),Y) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_lcm,hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y)))) ).

fof(fact_3699_finite__greaterThanLessThan,axiom,
    ! [L,U] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U))) ).

fof(fact_3700_Rats__real__nat,axiom,
    ! [Na] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(nat,real,real_1(nat),Na)),field_char_0_Rats(real))) ).

fof(fact_3701_card__greaterThanLessThan,axiom,
    ! [L,U] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),U),hAPP(nat,nat,suc,L)) ).

fof(fact_3702_Rats__dense__in__nn__real,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))
       => ? [X_1] :
            ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),field_char_0_Rats(real)))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),X_1))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Ya)) ) ) ) ).

fof(fact_3703_DERIV__isconst3,axiom,
    ! [F,Ya,Xa,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2))
     => ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),A_3),B_2)))
       => ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Ya),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),A_3),B_2)))
         => ( ! [X_1] :
                ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),A_3),B_2)))
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real))) )
           => hAPP(real,real,F,Xa) = hAPP(real,real,F,Ya) ) ) ) ) ).

fof(fact_3704_nonzero__Rats__inverse,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => ( ti(X_a,A_3) != zero_zero(X_a)
           => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),field_char_0_Rats(X_a))) ) ) ) ).

fof(fact_3705_nonzero__Rats__divide,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a)))
           => ( ti(X_a,B_2) != zero_zero(X_a)
             => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),field_char_0_Rats(X_a))) ) ) ) ) ).

fof(fact_3706_iszero__rat,axiom,
    ! [K] :
      ( hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),K)))
    <=> hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),K))) ) ).

fof(fact_3707_Rats__0,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),zero_zero(X_a)),field_char_0_Rats(X_a))) ) ).

fof(fact_3708_Rats__mult,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a)))
           => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),field_char_0_Rats(X_a))) ) ) ) ).

fof(fact_3709_Rats__add,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a)))
           => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)),field_char_0_Rats(X_a))) ) ) ) ).

fof(fact_3710_Rats__number__of,axiom,
    ! [X_a] :
      ( ( number_ring(X_a)
        & field_char_0(X_a) )
     => ! [Wa] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),field_char_0_Rats(X_a))) ) ).

fof(fact_3711_Rats__1,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),one_one(X_a)),field_char_0_Rats(X_a))) ) ).

fof(fact_3712_Rats__diff,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a)))
           => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),field_char_0_Rats(X_a))) ) ) ) ).

fof(fact_3713_Rats__power,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [Na,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na)),field_char_0_Rats(X_a))) ) ) ).

fof(fact_3714_Rats__minus,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),field_char_0_Rats(X_a))) ) ) ).

fof(fact_3715_Rats__divide,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & field_char_0(X_a) )
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a)))
           => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),field_char_0_Rats(X_a))) ) ) ) ).

fof(fact_3716_Rats__of__nat,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [Na] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),field_char_0_Rats(X_a))) ) ).

fof(fact_3717_Rats__inverse,axiom,
    ! [X_a] :
      ( ( field_inverse_zero(X_a)
        & field_char_0(X_a) )
     => ! [A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a)))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),field_char_0_Rats(X_a))) ) ) ).

fof(fact_3718_Rats__of__int,axiom,
    ! [X_a] :
      ( field_char_0(X_a)
     => ! [Z] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),field_char_0_Rats(X_a))) ) ).

fof(fact_3719_Rats__dense__in__real,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))
     => ? [X_1] :
          ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),field_char_0_Rats(real)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),X_1))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Ya)) ) ) ).

fof(fact_3720_Sup__greaterThanLessThan,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),Xa))
     => hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),Ya),Xa)) = Xa ) ).

fof(fact_3721_Sup__upper__EX,axiom,
    ! [Xa,X_2] :
      ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),X_2))
     => ( ? [Z_2] :
          ! [X_1] :
            ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_2))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Z_2)) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_2))) ) ) ).

fof(fact_3722_SupInf_OSup__upper,axiom,
    ! [Z,Xa,X_2] :
      ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),X_2))
     => ( ! [X_1] :
            ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_2))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Z)) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_2))) ) ) ).

fof(fact_3723_Frct__code__post_I5_J,axiom,
    ! [K_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),hAPP(int,int,number_number_of(int),K_4))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),K_4)) ).

fof(fact_3724_sgn__rat__def,axiom,
    ! [Q_3] :
      ( ( Q_3 = zero_zero(rat)
       => hAPP(rat,rat,sgn_sgn(rat),Q_3) = zero_zero(rat) )
      & ( Q_3 != zero_zero(rat)
       => ( ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),Q_3))
           => hAPP(rat,rat,sgn_sgn(rat),Q_3) = one_one(rat) )
          & ( ~ hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),Q_3))
           => hAPP(rat,rat,sgn_sgn(rat),Q_3) = hAPP(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ).

fof(fact_3725_divide__rat__def,axiom,
    ! [Q_3,R_3] : hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),Q_3),R_3) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),Q_3),hAPP(rat,rat,inverse_inverse(rat),R_3)) ).

fof(fact_3726_Frct__code__post_I2_J,axiom,
    ! [K_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),K_4),zero_zero(int))) = zero_zero(rat) ).

fof(fact_3727_Frct__code__post_I1_J,axiom,
    ! [K_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K_4)) = zero_zero(rat) ).

fof(fact_3728_Frct__code__post_I3_J,axiom,
    hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).

fof(fact_3729_Frct__code__post_I6_J,axiom,
    ! [K_4,L_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),hAPP(int,int,number_number_of(int),L_4))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),K_4)),hAPP(int,rat,number_number_of(rat),L_4)) ).

fof(fact_3730_Frct__code__post_I4_J,axiom,
    ! [K_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),one_one(int))) = hAPP(int,rat,number_number_of(rat),K_4) ).

fof(fact_3731_Sup__eq__maximum,axiom,
    ! [Z,X_2] :
      ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Z),X_2))
     => ( ! [X_1] :
            ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_2))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Z)) )
       => hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_2) = Z ) ) ).

fof(fact_3732_SupInf_OSup__upper2,axiom,
    ! [Z,Ya,Xa,X_2] :
      ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),X_2))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),Xa))
       => ( ! [X_1] :
              ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_2))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Z)) )
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_2))) ) ) ) ).

fof(fact_3733_diff__rat__def,axiom,
    ! [Q_3,R_3] : hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),Q_3),R_3) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),Q_3),hAPP(rat,rat,uminus_uminus(rat),R_3)) ).

fof(fact_3734_abs__rat__def,axiom,
    ! [Q_3] :
      ( ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Q_3),zero_zero(rat)))
       => hAPP(rat,rat,abs_abs(rat),Q_3) = hAPP(rat,rat,uminus_uminus(rat),Q_3) )
      & ( ~ hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Q_3),zero_zero(rat)))
       => hAPP(rat,rat,abs_abs(rat),Q_3) = Q_3 ) ) ).

fof(fact_3735_less__rat__def,axiom,
    ! [Z,Wa] :
      ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Z),Wa))
    <=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),Z),Wa))
        & Z != Wa ) ) ).

fof(fact_3736_obtain__pos__sum,axiom,
    ! [R_3] :
      ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_3))
     => ~ ! [S_1] :
            ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),S_1))
           => ! [T_4] :
                ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),T_4))
               => R_3 != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),S_1),T_4) ) ) ) ).

fof(fact_3737_rat__sgn__code,axiom,
    ! [P_5] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,sgn_sgn(rat),P_5)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,sgn_sgn(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(rat,product_prod(int,int),quotient_of,P_5)))),one_one(int)) ).

fof(fact_3738_quotient__of__inject__eq,axiom,
    ! [A_3,B_2] :
      ( hAPP(rat,product_prod(int,int),quotient_of,A_3) = hAPP(rat,product_prod(int,int),quotient_of,B_2)
    <=> A_3 = B_2 ) ).

fof(fact_3739_quotient__of__inject,axiom,
    ! [A_1,B] :
      ( hAPP(rat,product_prod(int,int),quotient_of,A_1) = hAPP(rat,product_prod(int,int),quotient_of,B)
     => A_1 = B ) ).

fof(fact_3740_quotient__of__denom__pos,axiom,
    ! [R_3,P_5,Q_3] :
      ( hAPP(rat,product_prod(int,int),quotient_of,R_3) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_3)) ) ).

fof(fact_3741_rat__one__code,axiom,
    hAPP(rat,product_prod(int,int),quotient_of,one_one(rat)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).

fof(fact_3742_rat__zero__code,axiom,
    hAPP(rat,product_prod(int,int),quotient_of,zero_zero(rat)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

fof(fact_3743_quotient__of__number_I3_J,axiom,
    ! [K_4] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,number_number_of(rat),K_4)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),one_one(int)) ).

fof(fact_3744_Nitpick_Onat__lcm__def,axiom,
    ! [X,Y] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_lcm,X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y)) ).

fof(fact_3745_normalize__negative,axiom,
    ! [P_5,Q_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Q_3),zero_zero(int)))
     => hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,uminus_uminus(int),P_5)),hAPP(int,int,uminus_uminus(int),Q_3))) ) ).

fof(fact_3746_gcd__eq__nitpick__gcd,axiom,
    ! [X,Y] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y) ).

fof(fact_3747_nat__gcd_Osimps,axiom,
    ! [X,Y] :
      ( ( Y = zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y) = X )
      & ( Y != zero_zero(nat)
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ) ) ).

fof(fact_3748_normalize__denom__zero,axiom,
    ! [P_5] : hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),zero_zero(int))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

fof(fact_3749_normalize__denom__pos,axiom,
    ! [R_3,P_5,Q_3] :
      ( hAPP(product_prod(int,int),product_prod(int,int),normalize,R_3) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_3)) ) ).

fof(fact_3750_normalize__crossproduct,axiom,
    ! [P_5,R_3,S_3,Q_3] :
      ( ti(int,Q_3) != zero_zero(int)
     => ( ti(int,S_3) != zero_zero(int)
       => ( hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),R_3),S_3))
         => hAPP(int,int,hAPP(int,fun(int,int),times_times(int),P_5),S_3) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_3),Q_3) ) ) ) ).

fof(fact_3751_nat__gcd_Opsimps,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Ya)))
     => ( ( Ya = zero_zero(nat)
         => hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Xa),Ya) = Xa )
        & ( Ya != zero_zero(nat)
         => hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Xa),Ya) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Ya),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Xa),Ya)) ) ) ) ).

fof(fact_3752_Nitpick_Oint__gcd__def,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),int_gcd,X),Y) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y)))) ).

fof(fact_3753_nat__gcd_Opinduct,axiom,
    ! [P_1,A0,A1] :
      ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A0),A1)))
     => ( ! [X_1,Y_1] :
            ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X_1),Y_1)))
           => ( ( Y_1 != zero_zero(nat)
               => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Y_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X_1),Y_1))) )
             => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,X_1),Y_1)) ) )
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,A0),A1)) ) ) ).

fof(fact_3754_ratrel__iff,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),bool,hAPP(product_prod(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),bool),member(product_prod(product_prod(int,int),product_prod(int,int))),hAPP(product_prod(int,int),product_prod(product_prod(int,int),product_prod(int,int)),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(product_prod(int,int),product_prod(int,int))),product_Pair(product_prod(int,int),product_prod(int,int)),Xa),Ya)),ratrel))
    <=> ( hAPP(product_prod(int,int),int,product_snd(int,int),Xa) != zero_zero(int)
        & hAPP(product_prod(int,int),int,product_snd(int,int),Ya) != zero_zero(int)
        & hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_fst(int,int),Xa)),hAPP(product_prod(int,int),int,product_snd(int,int),Ya)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_fst(int,int),Ya)),hAPP(product_prod(int,int),int,product_snd(int,int),Xa)) ) ) ).

fof(fact_3755_Fract__1__number__of,axiom,
    ! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,one_one(int)),hAPP(int,int,number_number_of(int),K_4)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),K_4)) ).

fof(fact_3756_Ratreal__number__of__quotient2,axiom,
    ! [R_3,S_3] : hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),R_3)),hAPP(int,rat,number_number_of(rat),S_3))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),R_3)),hAPP(int,real,number_number_of(real),S_3)) ).

fof(fact_3757_quotient__of__eq,axiom,
    ! [A_1,B,P_5,Q_3] :
      ( hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)
     => hAPP(int,rat,hAPP(int,fun(int,rat),fract,P_5),Q_3) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ) ).

fof(fact_3758_minus__rat,axiom,
    ! [A_1,B] : hAPP(rat,rat,uminus_uminus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,uminus_uminus(int),A_1)),B) ).

fof(fact_3759_eq__rat_I2_J,axiom,
    ! [A_1] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),zero_zero(int)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

fof(fact_3760_eq__rat_I1_J,axiom,
    ! [A_3,C,D_1,B_2] :
      ( ti(int,B_2) != zero_zero(int)
     => ( ti(int,D_1) != zero_zero(int)
       => ( hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D_1)
        <=> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),D_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B_2) ) ) ) ).

fof(fact_3761_mult__rat__cancel,axiom,
    ! [A_1,B,C_1] :
      ( ti(int,C_1) != zero_zero(int)
     => hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),A_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ) ).

fof(fact_3762_eq__rat_I3_J,axiom,
    ! [A_1,C_1] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),A_1) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),C_1) ).

fof(fact_3763_rat__number__collapse_I4_J,axiom,
    ! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),zero_zero(int)) = zero_zero(rat) ).

fof(fact_3764_rat__number__collapse_I1_J,axiom,
    ! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),K_4) = zero_zero(rat) ).

fof(fact_3765_inverse__rat,axiom,
    ! [A_1,B] : hAPP(rat,rat,inverse_inverse(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,B),A_1) ).

fof(fact_3766_minus__rat__cancel,axiom,
    ! [A_1,B] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ).

fof(fact_3767_divide__rat,axiom,
    ! [A_1,B,C_1,D_3] : hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_3)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),D_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)) ).

fof(fact_3768_real__floor__code,axiom,
    ! [X] : hAPP(real,int,archim2085082626_floor(real),hAPP(rat,real,ratreal,X)) = hAPP(rat,int,archim2085082626_floor(rat),X) ).

fof(fact_3769_floor__Fract,axiom,
    ! [A_1,B] : hAPP(rat,int,archim2085082626_floor(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) ).

fof(fact_3770_Fract__of__int__eq,axiom,
    ! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),one_one(int)) = hAPP(int,rat,ring_1_of_int(rat),K_4) ).

fof(fact_3771_of__int__rat,axiom,
    ! [K_4] : hAPP(int,rat,ring_1_of_int(rat),K_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),one_one(int)) ).

fof(fact_3772_Fract__of__int__quotient,axiom,
    ! [K_4,L_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),L_4) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,ring_1_of_int(rat),K_4)),hAPP(int,rat,ring_1_of_int(rat),L_4)) ).

fof(fact_3773_rat__number__collapse_I2_J,axiom,
    hAPP(int,rat,hAPP(int,fun(int,rat),fract,one_one(int)),one_one(int)) = one_one(rat) ).

fof(fact_3774_one__rat,axiom,
    one_one(rat) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,one_one(int)),one_one(int)) ).

fof(fact_3775_of__nat__rat,axiom,
    ! [K_4] : hAPP(nat,rat,semiring_1_of_nat(rat),K_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(nat,int,semiring_1_of_nat(int),K_4)),one_one(int)) ).

fof(fact_3776_Fract__of__nat__eq,axiom,
    ! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(nat,int,semiring_1_of_nat(int),K_4)),one_one(int)) = hAPP(nat,rat,semiring_1_of_nat(rat),K_4) ).

fof(fact_3777_rat__number__of__def,axiom,
    ! [W] : hAPP(int,rat,number_number_of(rat),W) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,W),one_one(int)) ).

fof(fact_3778_mult__rat,axiom,
    ! [A_1,B,C_1,D_3] : hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_3)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D_3)) ).

fof(fact_3779_abs__rat,axiom,
    ! [A_1,B] : hAPP(rat,rat,abs_abs(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,abs_abs(int),A_1)),hAPP(int,int,abs_abs(int),B)) ).

fof(fact_3780_normalize__eq,axiom,
    ! [A_1,B,P_5,Q_3] :
      ( hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_1),B)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)
     => hAPP(int,rat,hAPP(int,fun(int,rat),fract,P_5),Q_3) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ) ).

fof(fact_3781_real__less__code,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(rat,real,ratreal,Xa)),hAPP(rat,real,ratreal,Ya)))
    <=> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Xa),Ya)) ) ).

fof(fact_3782_Ratreal__number__collapse_I1_J,axiom,
    hAPP(rat,real,ratreal,zero_zero(rat)) = zero_zero(real) ).

fof(fact_3783_zero__real__code,axiom,
    zero_zero(real) = hAPP(rat,real,ratreal,zero_zero(rat)) ).

fof(fact_3784_number__of__real__code,axiom,
    ! [K_4] : hAPP(int,real,number_number_of(real),K_4) = hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),K_4)) ).

fof(fact_3785_Ratreal__number__collapse_I3_J,axiom,
    ! [K_4] : hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),K_4)) = hAPP(int,real,number_number_of(real),K_4) ).

fof(fact_3786_one__real__code,axiom,
    one_one(real) = hAPP(rat,real,ratreal,one_one(rat)) ).

fof(fact_3787_Ratreal__number__collapse_I2_J,axiom,
    hAPP(rat,real,ratreal,one_one(rat)) = one_one(real) ).

fof(fact_3788_real__less__eq__code,axiom,
    ! [Xa,Ya] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(rat,real,ratreal,Xa)),hAPP(rat,real,ratreal,Ya)))
    <=> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),Xa),Ya)) ) ).

fof(fact_3789_real__plus__code,axiom,
    ! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),X),Y)) ).

fof(fact_3790_real__minus__code,axiom,
    ! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),X),Y)) ).

fof(fact_3791_real__uminus__code,axiom,
    ! [X] : hAPP(real,real,uminus_uminus(real),hAPP(rat,real,ratreal,X)) = hAPP(rat,real,ratreal,hAPP(rat,rat,uminus_uminus(rat),X)) ).

fof(fact_3792_real__divide__code,axiom,
    ! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),X),Y)) ).

fof(fact_3793_real__times__code,axiom,
    ! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),X),Y)) ).

fof(fact_3794_real__inverse__code,axiom,
    ! [X] : hAPP(real,real,inverse_inverse(real),hAPP(rat,real,ratreal,X)) = hAPP(rat,real,ratreal,hAPP(rat,rat,inverse_inverse(rat),X)) ).

fof(fact_3795_zero__rat,axiom,
    zero_zero(rat) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

fof(fact_3796_rat__number__collapse_I3_J,axiom,
    ! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),one_one(int)) = hAPP(int,rat,number_number_of(rat),K_4) ).

fof(fact_3797_rat__number__expand_I3_J,axiom,
    ! [K_4] : hAPP(int,rat,number_number_of(rat),K_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),one_one(int)) ).

fof(fact_3798_Fract__number__of__quotient,axiom,
    ! [K_4,L_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),hAPP(int,int,number_number_of(int),L_4)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),K_4)),hAPP(int,rat,number_number_of(rat),L_4)) ).

fof(fact_3799_quotient__of__Fract,axiom,
    ! [A_1,B] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_1),B)) ).

fof(fact_3800_Frct__def,axiom,
    ! [P_5] : hAPP(product_prod(int,int),rat,frct,P_5) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)) ).

fof(fact_3801_less__rat,axiom,
    ! [A_3,C,D_1,B_2] :
      ( ti(int,B_2) != zero_zero(int)
     => ( ti(int,D_1) != zero_zero(int)
       => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D_1)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),D_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B_2)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1)))) ) ) ) ).

fof(fact_3802_Ratreal__number__of__quotient,axiom,
    ! [R_3,S_3] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),R_3))),hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),S_3))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),R_3)),hAPP(int,real,number_number_of(real),S_3)) ).

fof(fact_3803_sgn__rat,axiom,
    ! [A_1,B] : hAPP(rat,rat,sgn_sgn(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,sgn_sgn(int),A_1)),hAPP(int,int,sgn_sgn(int),B))) ).

fof(fact_3804_zero__less__Fract__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_3)) ) ) ).

fof(fact_3805_Fract__less__zero__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),zero_zero(rat)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),zero_zero(int))) ) ) ).

fof(fact_3806_le__rat,axiom,
    ! [A_3,C,D_1,B_2] :
      ( ti(int,B_2) != zero_zero(int)
     => ( ti(int,D_1) != zero_zero(int)
       => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D_1)))
        <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),D_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B_2)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1)))) ) ) ) ).

fof(fact_3807_add__rat,axiom,
    ! [A_1,C_1,D_3,B] :
      ( ti(int,B) != zero_zero(int)
     => ( ti(int,D_3) != zero_zero(int)
       => hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_3)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),D_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D_3)) ) ) ).

fof(fact_3808_diff__rat,axiom,
    ! [A_1,C_1,D_3,B] :
      ( ti(int,B) != zero_zero(int)
     => ( ti(int,D_3) != zero_zero(int)
       => hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_3)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),D_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D_3)) ) ) ).

fof(fact_3809_one__less__Fract__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),A_3)) ) ) ).

fof(fact_3810_Fract__less__one__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),one_one(rat)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2)) ) ) ).

fof(fact_3811_Fract__add__one,axiom,
    ! [M,N] :
      ( ti(int,N) != zero_zero(int)
     => hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),N)),N) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,M),N)),one_one(rat)) ) ).

fof(fact_3812_zero__le__Fract__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_3)) ) ) ).

fof(fact_3813_Fract__le__zero__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),zero_zero(rat)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_3),zero_zero(int))) ) ) ).

fof(fact_3814_rat__floor__lemma,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)))
      & hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),one_one(int))))) ) ).

fof(fact_3815_one__le__Fract__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) ) ) ).

fof(fact_3816_Fract__le__one__iff,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),one_one(rat)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_3),B_2)) ) ) ).

fof(fact_3817_Rat__induct__pos,axiom,
    ! [Q_2,P_1] :
      ( ! [A_4,B_4] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
         => hBOOL(hAPP(rat,bool,P_1,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_4),B_4))) )
     => hBOOL(hAPP(rat,bool,P_1,Q_2)) ) ).

fof(fact_3818_Nitpick_Ozero__frac__def,axiom,
    ! [X_a] : zero_frac(X_a) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).

fof(fact_3819_Nitpick_Oone__frac__def,axiom,
    ! [X_a] : one_frac(X_a) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).

fof(fact_3820_Nitpick_Onumber__of__frac__def,axiom,
    ! [X_a,N] : hAPP(int,X_a,number_of_frac(X_a),N) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),N),one_one(int))) ).

fof(fact_3821_mlex__leq,axiom,
    ! [X_a,R_2,F,Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(X_a,nat,F,Xa)),hAPP(X_a,nat,F,Ya)))
     => ( hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Xa),Ya)),R_2))
       => hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Xa),Ya)),hAPP(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool),hAPP(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool)),mlex_prod(X_a),F),R_2))) ) ) ).

fof(fact_3822_Ints__odd__less__0,axiom,
    ! [X_a] :
      ( linordered_idom(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),A_3)),A_3)),zero_zero(X_a)))
          <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) ) ) ) ).

fof(fact_3823_Ints__add,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a)))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a)))
           => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)),ring_1_Ints(X_a))) ) ) ) ).

fof(fact_3824_Ints__diff,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a)))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a)))
           => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),ring_1_Ints(X_a))) ) ) ) ).

fof(fact_3825_Ints__1,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),one_one(X_a)),ring_1_Ints(X_a))) ) ).

fof(fact_3826_Ints__real__of__nat,axiom,
    ! [Na] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(nat,real,real_1(nat),Na)),ring_1_Ints(real))) ).

fof(fact_3827_Ints__real__of__int,axiom,
    ! [Xa] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(int,real,real_1(int),Xa)),ring_1_Ints(real))) ).

fof(fact_3828_Ints__power,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [Na,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a)))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na)),ring_1_Ints(X_a))) ) ) ).

fof(fact_3829_Ints__number__of,axiom,
    ! [X_a] :
      ( number_ring(X_a)
     => ! [Wa] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),ring_1_Ints(X_a))) ) ).

fof(fact_3830_Ints__mult,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a)))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a)))
           => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),ring_1_Ints(X_a))) ) ) ) ).

fof(fact_3831_Ints__0,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),zero_zero(X_a)),ring_1_Ints(X_a))) ) ).

fof(fact_3832_Ints__minus,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a)))
         => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),ring_1_Ints(X_a))) ) ) ).

fof(fact_3833_Ints__of__int,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [Z] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),ring_1_Ints(X_a))) ) ).

fof(fact_3834_Ints__of__nat,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [Na] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),ring_1_Ints(X_a))) ) ).

fof(fact_3835_Ints__double__eq__0__iff,axiom,
    ! [X_a] :
      ( ring_char_0(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a)))
         => ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3) = zero_zero(X_a)
          <=> ti(X_a,A_3) = zero_zero(X_a) ) ) ) ).

fof(fact_3836_Ints__odd__nonzero,axiom,
    ! [X_a] :
      ( ring_char_0(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a)))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),A_3)),A_3) != zero_zero(X_a) ) ) ).

fof(fact_3837_mlex__less,axiom,
    ! [X_a,R_2,F,Xa,Ya] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(X_a,nat,F,Xa)),hAPP(X_a,nat,F,Ya)))
     => hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Xa),Ya)),hAPP(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool),hAPP(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool)),mlex_prod(X_a),F),R_2))) ) ).

fof(fact_3838_Ints__induct,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [P_1,Q_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Q_2),ring_1_Ints(X_a)))
         => ( ! [Z_2] : hBOOL(hAPP(X_a,bool,P_1,hAPP(int,X_a,ring_1_of_int(X_a),Z_2)))
           => hBOOL(hAPP(X_a,bool,P_1,Q_2)) ) ) ) ).

fof(fact_3839_Ints__cases,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [Q_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Q_2),ring_1_Ints(X_a)))
         => ~ ! [Z_2] : ti(X_a,Q_2) != hAPP(int,X_a,ring_1_of_int(X_a),Z_2) ) ) ).

fof(fact_3840_adjust__eq,axiom,
    ! [B_2,Q_2,Ra] : hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),Ra)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ra),B_2))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Q_2)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ra),B_2))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Q_2)),Ra)) ).

fof(fact_3841_lemma__DERIV__sin__cos__minus,axiom,
    ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),uminus_uminus(real)))),sin))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),uminus_uminus(real)))),cos))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1),zero_zero(real))) ).

fof(fact_3842_finite__Collect__conjI,axiom,
    ! [X_a,Q_1,P_1] :
      ( ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)))
        | hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Q_1))) )
     => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),P_1)),Q_1)))) ) ).

fof(fact_3843_finite__Collect__less__nat,axiom,
    ! [K] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),K)))) ).

fof(fact_3844_finite__Collect__le__nat,axiom,
    ! [K] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),K)))) ).

fof(fact_3845_finite__Collect__subsets,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),combc(fun(X_a,bool),fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool))),A_2)))) ) ).

fof(fact_3846_DERIV__divide,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [G,E_3,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3))
           => ( hAPP(X_a,X_a,G,Xa) != zero_zero(X_a)
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D),hAPP(X_a,X_a,G,Xa))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,F,Xa)),E_3))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,G,Xa)),hAPP(X_a,X_a,G,Xa))))) ) ) ) ) ).

fof(fact_3847_Bseq__realpow,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real)))
       => hBOOL(hAPP(fun(nat,real),bool,bseq(real),hAPP(real,fun(nat,real),power_power(real),Xa))) ) ) ).

fof(fact_3848_DERIV__cos__add,axiom,
    ! [K,Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),K))),Xa),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),K))))) ).

fof(fact_3849_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J,axiom,
    ! [P_1] : hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1)))) ).

fof(fact_3850_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J,axiom,
    ! [P_1] : hAPP(fun(nat,bool),fun(nat,bool),collect(nat),P_1) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(fun(int,nat),fun(int,bool),hAPP(fun(nat,bool),fun(fun(int,nat),fun(int,bool)),combb(nat,bool,int),P_1),nat_1)))) ).

fof(fact_3851_bdd__int__set__l__le__finite,axiom,
    ! [Na] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na))))) ).

fof(fact_3852_bdd__int__set__l__finite,axiom,
    ! [Na] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na))))) ).

fof(fact_3853_bdd__int__set__le__finite,axiom,
    ! [Na] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na))))) ).

fof(fact_3854_card__bdd__int__set__l,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na))
     => hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na)))) = hAPP(int,nat,nat_1,Na) ) ).

fof(fact_3855_card__bdd__int__set__l__le,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na))
     => hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na)))) = hAPP(int,nat,nat_1,Na) ) ).

fof(fact_3856_less__eq__Suc__le__raw,axiom,
    ! [X_1] : hAPP(nat,fun(nat,bool),ord_less(nat),X_1) = hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,X_1)) ).

fof(fact_3857_DERIV__mult_H,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [G,E_3,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),times_times(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,F,Xa)),E_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D),hAPP(X_a,X_a,G,Xa))))) ) ) ) ).

fof(fact_3858_DERIV__mult,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [G,Db,F,Xa,Da] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),Da))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),Db))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),times_times(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Da),hAPP(X_a,X_a,G,Xa))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Db),hAPP(X_a,X_a,F,Xa))))) ) ) ) ).

fof(fact_3859_DERIV__add__minus,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [G,E_3,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),plus_plus(X_a)),F)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),uminus_uminus(X_a)),G))),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),D),hAPP(X_a,X_a,uminus_uminus(X_a),E_3)))) ) ) ) ).

fof(fact_3860_diff__nat__eq__if,axiom,
    ! [Z,Z_3] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,Z_3))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Z)),hAPP(int,nat,nat_1,Z_3)) = hAPP(int,nat,nat_1,Z) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,Z_3))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Z)),hAPP(int,nat,nat_1,Z_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),Z_3))),zero_zero(nat)),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),Z_3))) ) ) ).

fof(fact_3861_DERIV__sin__sin__mult,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),sin)),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Xa)),hAPP(real,real,sin,Xa))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Xa)),hAPP(real,real,sin,Xa))))) ).

fof(fact_3862_nat__number__of__Bit0,axiom,
    ! [W] : hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),W)),hAPP(int,nat,number_number_of(nat),W)) ).

fof(fact_3863_Bseq__offset,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,K] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_2),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))))
         => hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) ) ) ).

fof(fact_3864_Bseq__ignore__initial__segment,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [K,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_2),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))) ) ) ).

fof(fact_3865_DERIV__add,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [G,E_3,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),plus_plus(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),D),E_3))) ) ) ) ).

fof(fact_3866_DERIV__diff,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [G,E_3,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),D),E_3))) ) ) ) ).

fof(fact_3867_DERIV__sin__add,axiom,
    ! [K,Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),K))),Xa),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),K)))) ).

fof(fact_3868_card__Collect__le__nat,axiom,
    ! [Na] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),Na))) = hAPP(nat,nat,suc,Na) ).

fof(fact_3869_card__bdd__nat__set__le,axiom,
    ! [Xa] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),Xa))) = hAPP(nat,nat,suc,Xa) ).

fof(fact_3870_bdd__nat__set__le__finite,axiom,
    ! [Xa] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),Xa)))) ).

fof(fact_3871_DERIV__ident,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Xa] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),combi(X_a)),Xa),one_one(X_a))) ) ).

fof(fact_3872_DERIV__cdivide,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [C,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),C)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),D),C))) ) ) ).

fof(fact_3873_image__image,axiom,
    ! [X_b,X_a,X_c,F,G,A_2] : hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),G),A_2)) = hAPP(fun(X_c,bool),fun(X_a,bool),hAPP(fun(X_c,X_a),fun(fun(X_c,bool),fun(X_a,bool)),image(X_c,X_a),hAPP(fun(X_c,X_b),fun(X_c,X_a),hAPP(fun(X_b,X_a),fun(fun(X_c,X_b),fun(X_c,X_a)),combb(X_b,X_a,X_c),F),G)),A_2) ).

fof(fact_3874_image__ident,axiom,
    ! [X_a,Y_2] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),combi(X_a)),Y_2) = ti(fun(X_a,bool),Y_2) ).

fof(fact_3875_finite__Collect__disjI,axiom,
    ! [X_a,P_1,Q_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fdisj),P_1)),Q_1))))
    <=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)))
        & hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Q_1))) ) ) ).

fof(fact_3876_primes__infinite,axiom,
    ~ hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),prime))) ).

fof(fact_3877_Re_OCauchy,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_2))
     => hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2))) ) ).

fof(fact_3878_Im_OCauchy,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_2))
     => hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2))) ) ).

fof(fact_3879_cnj_OCauchy,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_2))
     => hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2))) ) ).

fof(fact_3880_zpower__number__of__even,axiom,
    ! [Z_1,W] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),W))) ).

fof(fact_3881_power__number__of__even,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [Z_1,W] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),W))) ) ).

fof(fact_3882_power__number__of__even__number__of,axiom,
    ! [X_b] :
      ( ( monoid_mult(X_b)
        & number(X_b) )
     => ! [V,W] : hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W))) ) ).

fof(fact_3883_DERIV__fun__sin,axiom,
    ! [G,Xa,Ma] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),Xa),Ma))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),G)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,hAPP(real,real,G,Xa))),Ma))) ) ).

fof(fact_3884_DERIV__fun__exp,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [G,Xa,Ma] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),Ma))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),exp(X_a)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,G,Xa))),Ma))) ) ) ).

fof(fact_3885_DERIV__chain2,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Db,F,G,Xa,Da] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),hAPP(X_a,X_a,G,Xa)),Da))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),Db))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Da),Db))) ) ) ) ).

fof(fact_3886_DERIV__chain_H,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [G,E_3,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),hAPP(X_a,X_a,F,Xa)),E_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),G),F)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),E_3),D))) ) ) ) ).

fof(fact_3887_DERIV__cmult,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [C,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),F)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),D))) ) ) ).

fof(fact_3888_DERIV__inverse_H,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => ( hAPP(X_a,X_a,F,Xa) != zero_zero(X_a)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),inverse_inverse(X_a)),F)),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,F,Xa))),D)),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,F,Xa)))))) ) ) ) ).

fof(fact_3889_DERIV__const,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [K,Xa] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),combk(X_a,X_a),K)),Xa),zero_zero(X_a))) ) ).

fof(fact_3890_Collect__neg__eq,axiom,
    ! [X_a,P_1] : hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),P_1)) = hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)) ).

fof(fact_3891_DERIV__minus,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),uminus_uminus(X_a)),F)),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),D))) ) ) ).

fof(fact_3892_Bseq__minus__iff,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)))
        <=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) ) ) ).

fof(fact_3893_card__bdd__nat__set__l,axiom,
    ! [Xa] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),Xa))) = Xa ).

fof(fact_3894_card__Collect__less__nat,axiom,
    ! [Na] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),Na))) = Na ).

fof(fact_3895_bdd__int__set__l__l__finite,axiom,
    ! [Na] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na))))) ).

fof(fact_3896_bdd__nat__set__l__finite,axiom,
    ! [Xa] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),Xa)))) ).

fof(fact_3897_finite__divisors__int,axiom,
    ! [I] :
      ( ti(int,I) != zero_zero(int)
     => hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),dvd_dvd(int)),I)))) ) ).

fof(fact_3898_finite__divisors__nat,axiom,
    ! [Ma] :
      ( Ma != zero_zero(nat)
     => hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat)),Ma)))) ) ).

fof(fact_3899_nat__seg__image__imp__finite,axiom,
    ! [X_a,A_2,F,Na] :
      ( ti(fun(X_a,bool),A_2) = hAPP(fun(nat,bool),fun(X_a,bool),hAPP(fun(nat,X_a),fun(fun(nat,bool),fun(X_a,bool)),image(nat,X_a),F),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),Na)))
     => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).

fof(fact_3900_DERIV__fun__cos,axiom,
    ! [G,Xa,Ma] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),Xa),Ma))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),G)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,G,Xa)))),Ma))) ) ).

fof(fact_3901_finite__M__bounded__by__nat,axiom,
    ! [P_1,I] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),P_1)),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I))))) ).

fof(fact_3902_card__less__Suc2,axiom,
    ! [I,M_1] :
      ( ~ hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_1))
     => hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),hAPP(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool)),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool))),combb(nat,fun(fun(nat,bool),bool),nat),member(nat)),suc)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I)))) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))) ) ).

fof(fact_3903_card__less,axiom,
    ! [I,M_1] :
      ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_1))
     => hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))) != zero_zero(nat) ) ).

fof(fact_3904_card__less__Suc,axiom,
    ! [I,M_1] :
      ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_1))
     => hAPP(nat,nat,suc,hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),hAPP(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool)),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool))),combb(nat,fun(fun(nat,bool),bool),nat),member(nat)),suc)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I))))) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))) ) ).

fof(fact_3905_transfer__morphism__int__nat,axiom,
    hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,int),fun(fun(nat,bool),bool),nat_tr1645093318rphism(nat,int),semiring_1_of_nat(int)),hAPP(bool,fun(nat,bool),combk(bool,nat),fTrue))) ).

fof(fact_3906_int__card__bdd__int__set__l__le,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na))
     => hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na))))) = ti(int,Na) ) ).

fof(fact_3907_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
    ! [P_1] : hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1)) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,int),fun(nat,bool),hAPP(fun(int,bool),fun(fun(nat,int),fun(nat,bool)),combb(int,bool,nat),P_1),semiring_1_of_nat(int)))) ).

fof(fact_3908_noXRRset__def,axiom,
    ! [Ma,Xa] : hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,Ma),Xa) = hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),Xa)),hAPP(int,fun(int,bool),norRRset,Ma)) ).

fof(fact_3909_DERIV__fun__pow,axiom,
    ! [Na,G,Xa,Ma] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),Xa),Ma))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),G)),Na)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,G,Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))))),Ma))) ) ).

fof(fact_3910_int__card__bdd__int__set__l__l,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Na))
     => hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na))))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Na),one_one(int)) ) ).

fof(fact_3911_card__bdd__int__set__le,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na))
     => hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,Na)),one_one(nat)) ) ).

fof(fact_3912_card__bdd__int__set__l__l,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Na))
     => hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Na)),one_one(nat)) ) ).

fof(fact_3913_DERIV__cos__cos__mult,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))),hAPP(real,real,cos,Xa))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))),hAPP(real,real,cos,Xa))))) ).

fof(fact_3914_eq__number__of__Suc,axiom,
    ! [Va,Na] :
      ( hAPP(int,nat,number_number_of(nat),Va) = hAPP(nat,nat,suc,Na)
    <=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
        & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
         => hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))) = Na ) ) ) ).

fof(fact_3915_Suc__eq__number__of,axiom,
    ! [Na,Va] :
      ( hAPP(nat,nat,suc,Na) = hAPP(int,nat,number_number_of(nat),Va)
    <=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
        & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
         => hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))) = Na ) ) ) ).

fof(fact_3916_DERIV__power__Suc,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Na,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(nat,fun(X_a,X_a)),combc(X_a,nat,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),F)),hAPP(nat,nat,suc,Na))),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,Xa)),Na))))) ) ) ).

fof(fact_3917_DERIV__inverse,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Xa] :
          ( ti(X_a,Xa) != zero_zero(X_a)
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),inverse_inverse(X_a)),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Xa)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))) ) ) ).

fof(fact_3918_DERIV__power,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Na,F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(nat,fun(X_a,X_a)),combc(X_a,nat,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),F)),Na)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))) ) ) ).

fof(fact_3919_nat__number__of__Bit1,axiom,
    ! [W] :
      ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),W)))
       => hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W)) = zero_zero(nat) )
      & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),W)))
       => hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),W)),hAPP(int,nat,number_number_of(nat),W))) ) ) ).

fof(fact_3920_DERIV__pow,axiom,
    ! [Na,Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),Na)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))))) ).

fof(fact_3921_DERIV__cos__cos__mult2,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,min))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa)))) ).

fof(fact_3922_less__Suc__number__of,axiom,
    ! [Na,Va] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),hAPP(int,nat,number_number_of(nat),Va)))
    <=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
        & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))) ) ) ) ).

fof(fact_3923_less__number__of__Suc,axiom,
    ! [Va,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(nat,nat,suc,Na)))
    <=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)) ) ) ).

fof(fact_3924_le__Suc__number__of,axiom,
    ! [Na,Va] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Na)),hAPP(int,nat,number_number_of(nat),Va)))
    <=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
        & ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
         => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))) ) ) ) ).

fof(fact_3925_le__number__of__Suc,axiom,
    ! [Va,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(nat,nat,suc,Na)))
    <=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)) ) ) ).

fof(fact_3926_SR__pos,axiom,
    ! [X_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
     => hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),standardRes,Ma)),X_2)),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Ma))))) ) ).

fof(fact_3927_power__number__of__odd,axiom,
    ! [X_a] :
      ( monoid_mult(X_a)
     => ! [Z_1,W] :
          ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),W))) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
           => hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = one_one(X_a) ) ) ) ).

fof(fact_3928_DERIV__quotient,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [G,E_2,F,Xa,D_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_2))
           => ( hAPP(X_a,X_a,G,Xa) != zero_zero(X_a)
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_1),hAPP(X_a,X_a,G,Xa))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),E_2),hAPP(X_a,X_a,F,Xa)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,G,Xa)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))) ) ) ) ) ).

fof(fact_3929_RsetR__zmult__mono,axiom,
    ! [Xa,A_2,Ma] :
      ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma)))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma))
       => ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Xa),Ma) = one_one(int)
         => hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),Xa)),A_2)),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma))) ) ) ) ).

fof(fact_3930_DERIV__inverse__fun,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F,Xa,D_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D_1))
         => ( hAPP(X_a,X_a,F,Xa) != zero_zero(X_a)
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),inverse_inverse(X_a)),F)),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_1),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,Xa)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))))) ) ) ) ).

fof(fact_3931_zpower__number__of__odd,axiom,
    ! [Z_1,W] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
       => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),W))) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
       => hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = one_one(int) ) ) ).

fof(fact_3932_DERIV__sin__sin__mult2,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),sin)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa)))) ).

fof(fact_3933_DERIV__log,axiom,
    ! [B_2,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(real,fun(real,real),log,B_2)),Xa),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,ln,B_2)),Xa)))) ) ).

fof(fact_3934_power__number__of__odd__number__of,axiom,
    ! [X_b] :
      ( ( monoid_mult(X_b)
        & number(X_b) )
     => ! [V,W] :
          ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
           => hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W))) )
          & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
           => hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = one_one(X_b) ) ) ) ).

fof(fact_3935_DERIV__cos__cos__mult3,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),Xa),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa))))) ).

fof(fact_3936_DERIV__sin__circle__all__zero,axiom,
    ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1),zero_zero(real))) ).

fof(fact_3937_DERIV__sin__realpow2a,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa)))) ).

fof(fact_3938_diff__nat__number__of,axiom,
    ! [Va,V_1] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(int,nat,number_number_of(nat),V_1)) = hAPP(int,nat,number_number_of(nat),Va) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(int,nat,number_number_of(nat),V_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Va),hAPP(int,int,uminus_uminus(int),V_1))))),zero_zero(nat)),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Va),hAPP(int,int,uminus_uminus(int),V_1))))) ) ) ).

fof(fact_3939_DERIV__sin__realpow2,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Xa)),hAPP(real,real,sin,Xa))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Xa)),hAPP(real,real,sin,Xa))))) ).

fof(fact_3940_DERIV__cos__realpow2a,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,min))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa)))) ).

fof(fact_3941_DERIV__cos__realpow2b,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa))))) ).

fof(fact_3942_DERIV__cos__realpow2,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))),hAPP(real,real,cos,Xa))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))),hAPP(real,real,cos,Xa))))) ).

fof(fact_3943_lemma__DERIV__tan,axiom,
    ! [Xa] :
      ( hAPP(real,real,cos,Xa) != zero_zero(real)
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),sin)),cos)),Xa),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,Xa)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).

fof(fact_3944_lemma__DERIV__sin__cos__add,axiom,
    ! [Ya,X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),Ya)))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),hAPP(real,real,cos,Ya)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),hAPP(real,real,sin,Ya)))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),Ya)))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),hAPP(real,real,cos,Ya)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),hAPP(real,real,sin,Ya)))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1),zero_zero(real))) ).

fof(fact_3945_DERIV__sin__circle__all,axiom,
    ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1))))) ).

fof(fact_3946_norm__frac_Opsimps,axiom,
    ! [A_3,B_2] :
      ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_2)))
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),zero_zero(int)))
         => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_3),B_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,hAPP(int,int,uminus_uminus(int),A_3)),hAPP(int,int,uminus_uminus(int),B_2)) )
        & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),zero_zero(int)))
         => ( ( ( ti(int,A_3) = zero_zero(int)
                | ti(int,B_2) = zero_zero(int) )
             => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_3),B_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) )
            & ( ~ ( ti(int,A_3) = zero_zero(int)
                  | ti(int,B_2) = zero_zero(int) )
             => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_3),B_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_3),B_2))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B_2),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_3),B_2))) ) ) ) ) ) ).

fof(fact_3947_monoseq__arctan__series,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real)))
     => hBOOL(hAPP(fun(nat,real),bool,monoseq(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))) ) ).

fof(fact_3948_monoseq__minus,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [A_3] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),A_3))
         => hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),A_3))) ) ) ).

fof(fact_3949_monoseq__Suc,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2))
        <=> ( ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(nat,X_a,X_2,hAPP(nat,nat,suc,N_1))))
            | ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,hAPP(nat,nat,suc,N_1))),hAPP(nat,X_a,X_2,N_1))) ) ) ) ).

fof(fact_3950_monoseq__def,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2))
        <=> ( ! [M_2,N_1] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))) )
            | ! [M_2,N_1] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(nat,X_a,X_2,M_2))) ) ) ) ) ).

fof(fact_3951_monoseq__realpow,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real)))
       => hBOOL(hAPP(fun(nat,real),bool,monoseq(real),hAPP(real,fun(nat,real),power_power(real),Xa))) ) ) ).

fof(fact_3952_norm__frac_Osimps,axiom,
    ! [A_1,B] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
       => ( ( ( ti(int,A_1) = zero_zero(int)
              | ti(int,B) = zero_zero(int) )
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) )
          & ( ~ ( ti(int,A_1) = zero_zero(int)
                | ti(int,B) = zero_zero(int) )
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_1),B))) ) ) ) ) ).

fof(fact_3953_arg__def,axiom,
    ! [Z] : hAPP(complex,real,arg,Z) = hAPP(fun(real,bool),real,hilbert_Eps(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(fun(real,real),fun(real,bool),hAPP(fun(real,bool),fun(fun(real,real),fun(real,bool)),combb(real,bool,real),hAPP(real,fun(real,bool),fequal(real),hAPP(complex,real,re,hAPP(complex,complex,sgn_sgn(complex),Z)))),cos))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(fun(real,real),fun(real,bool),hAPP(fun(real,bool),fun(fun(real,real),fun(real,bool)),combb(real,bool,real),hAPP(real,fun(real,bool),fequal(real),hAPP(complex,real,im,hAPP(complex,complex,sgn_sgn(complex),Z)))),sin))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),pi)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),pi))))) ).

fof(fact_3954_Union__SetS__setprod__prop1,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_4)) ) ) ) ) ).

fof(fact_3955_of__nat__setprod,axiom,
    ! [X_b,X_a] :
      ( ( comm_monoid_mult(X_a)
        & semiring_1(X_a) )
     => ! [F,A_2] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co302546470etprod(X_b,nat),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,nat),fun(X_b,X_a),hAPP(fun(nat,X_a),fun(fun(X_b,nat),fun(X_b,X_a)),combb(nat,X_a,X_b),semiring_1_of_nat(X_a)),F)),A_2) ) ).

fof(fact_3956_of__int__setprod,axiom,
    ! [X_b,X_a] :
      ( ( comm_monoid_mult(X_a)
        & ring_1(X_a) )
     => ! [F,A_2] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(fun(X_b,bool),int,hAPP(fun(X_b,int),fun(fun(X_b,bool),int),big_co302546470etprod(X_b,int),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,int),fun(X_b,X_a),hAPP(fun(int,X_a),fun(fun(X_b,int),fun(X_b,X_a)),combb(int,X_a,X_b),ring_1_of_int(X_a)),F)),A_2) ) ).

fof(fact_3957_d22set__prod__zfact,axiom,
    ! [A_3] : hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),d22set,A_3)) = hAPP(int,int,zfact,A_3) ).

fof(fact_3958_Bnor__prod__zgcd,axiom,
    ! [A_3,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),Ma))
     => hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))),Ma) = one_one(int) ) ).

fof(fact_3959_Bnor__prod__power,axiom,
    ! [A_3,Ma,Xa] :
      ( ti(int,Xa) != zero_zero(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),Ma))
       => hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),Xa)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Xa),hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma)))) ) ) ).

fof(fact_3960_SetS__setprod__prop,axiom,
    ! [Xa,A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4))
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3))
           => ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),Xa),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)))
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),Xa)),A_3),P_4)) ) ) ) ) ) ).

fof(fact_3961_wset__zcong__prod__1,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_4))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))),one_one(int)),P_4)) ) ) ) ).

fof(fact_3962_Union__SetS__setprod__prop2,axiom,
    ! [A_3,P_4] :
      ( hBOOL(hAPP(int,bool,zprime,P_4))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
       => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4))
         => hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))) = hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))) ) ) ) ).

fof(fact_3963_setprod__gen__delta,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [B_2,C,A_3,S] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
         => ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),C)),S) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_a,X_b,B_2,A_3)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),S)),one_one(nat)))) )
            & ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),C)),S) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),C),hAPP(fun(X_a,bool),nat,finite_card(X_a),S)) ) ) ) ) ).

fof(fact_3964_transfer__nat__int__sum__prod__closure_I2_J,axiom,
    ! [F,A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => ( ! [X_1] :
            ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,F,X_1))) )
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),F),A_2))) ) ) ).

fof(fact_3965_setprod__eq__1__iff,axiom,
    ! [X_a,F,F_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
     => ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),F_1) = one_one(nat)
      <=> ! [X_1] :
            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),F_1))
           => hAPP(X_a,nat,F,X_1) = one_one(nat) ) ) ) ).

fof(fact_3966_setprod__pos__nat__iff,axiom,
    ! [X_a,F,S] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),S)))
      <=> ! [X_1] :
            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S))
           => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_1))) ) ) ) ).

fof(fact_3967_int__setprod,axiom,
    ! [X_a,F,A_2] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2) ).

fof(fact_3968_real__of__int__setprod,axiom,
    ! [X_a,F,A_2] : hAPP(int,real,real_1(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co302546470etprod(X_a,real),hAPP(fun(X_a,int),fun(X_a,real),hAPP(fun(int,real),fun(fun(X_a,int),fun(X_a,real)),combb(int,real,X_a),real_1(int)),F)),A_2) ).

fof(fact_3969_real__of__nat__setprod,axiom,
    ! [X_a,F,A_2] : hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co302546470etprod(X_a,real),hAPP(fun(X_a,nat),fun(X_a,real),hAPP(fun(nat,real),fun(fun(X_a,nat),fun(X_a,real)),combb(nat,real,X_a),real_1(nat)),F)),A_2) ).

fof(fact_3970_transfer__nat__int__sum__prod2_I2_J,axiom,
    ! [X_a,F,A_2] : hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2) = hAPP(int,nat,nat_1,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2)) ).

fof(fact_3971_transfer__nat__int__sum__prod_I2_J,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & comm_monoid_mult(X_a) )
     => ! [F,A_2] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co302546470etprod(nat,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),hAPP(fun(int,nat),fun(int,X_a),hAPP(fun(nat,X_a),fun(fun(int,nat),fun(int,X_a)),combb(nat,X_a,int),F),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) ) ).

fof(fact_3972_transfer__int__nat__sum__prod_I2_J,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & comm_monoid_mult(X_a) )
     => ! [F,A_2] :
          ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
         => hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),F),A_2) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co302546470etprod(nat,X_a),hAPP(fun(nat,int),fun(nat,X_a),hAPP(fun(int,X_a),fun(fun(nat,int),fun(nat,X_a)),combb(int,X_a,nat),F),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) ) ) ).

fof(fact_3973_setprod__timesf,axiom,
    ! [X_b,X_a] :
      ( comm_monoid_mult(X_a)
     => ! [F,G,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F)),G)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),G),A_2)) ) ).

fof(fact_3974_setprod__1,axiom,
    ! [X_b,X_a] :
      ( comm_monoid_mult(X_a)
     => ! [A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),one_one(X_a))),A_2) = one_one(X_a) ) ).

fof(fact_3975_abs__setprod,axiom,
    ! [X_b,X_a] :
      ( linordered_field(X_a)
     => ! [F,A_2] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2) ) ).

fof(fact_3976_setprod__zero,axiom,
    ! [X_a,X_b] :
      ( comm_semiring_1(X_b)
     => ! [F,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( ? [X_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
                & hAPP(X_a,X_b,F,X_1) = zero_zero(X_b) )
           => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) = zero_zero(X_b) ) ) ) ).

fof(fact_3977_setprod__zero__iff,axiom,
    ! [X_a,X_b] :
      ( ( comm_semiring_1(X_b)
        & no_zero_divisors(X_b) )
     => ! [F,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) = zero_zero(X_b)
          <=> ? [X_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
                & hAPP(X_a,X_b,F,X_1) = zero_zero(X_b) ) ) ) ) ).

fof(fact_3978_setprod__infinite,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [F,A_2] :
          ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) = one_one(X_b) ) ) ).

fof(fact_3979_setprod_Oinfinite,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [G,A_2] :
          ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = one_one(X_b) ) ) ).

fof(fact_3980_dvd__setprod,axiom,
    ! [X_a,X_b] :
      ( comm_semiring_1(X_b)
     => ! [F,I,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),A_2))
           => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,I)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2))) ) ) ) ).

fof(fact_3981_setprod_Odistrib,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [G,H,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),H)),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),H),A_2)) ) ) ).

fof(fact_3982_setprod__delta,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [B_2,A_3,S] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
         => ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),one_one(X_b))),S) = hAPP(X_a,X_b,B_2,A_3) )
            & ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),one_one(X_b))),S) = one_one(X_b) ) ) ) ) ).

fof(fact_3983_setprod__delta_H,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [B_2,A_3,S] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
         => ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),A_3))),B_2)),one_one(X_b))),S) = hAPP(X_a,X_b,B_2,A_3) )
            & ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),A_3))),B_2)),one_one(X_b))),S) = one_one(X_b) ) ) ) ) ).

fof(fact_3984_setprod__dividef,axiom,
    ! [X_a,X_b] :
      ( field_inverse_zero(X_b)
     => ! [F,G,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),F)),G)),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2)) ) ) ).

fof(fact_3985_setprod__dvd__setprod__subset,axiom,
    ! [X_a,X_b] :
      ( ( comm_monoid_mult(X_b)
        & dvd(X_b) )
     => ! [F,A_2,B_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
           => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),B_1))) ) ) ) ).

fof(fact_3986_setprod__constant,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [Ya,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),Ya)),A_2) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),Ya),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)) ) ) ).

fof(fact_3987_setprod__mono__one__right,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [F,S,T_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S)))
                 => hAPP(X_a,X_b,F,X_1) = one_one(X_b) )
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),T_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),S) ) ) ) ) ).

fof(fact_3988_setprod__mono__one__left,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [F,S,T_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S)))
                 => hAPP(X_a,X_b,F,X_1) = one_one(X_b) )
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),S) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),T_3) ) ) ) ) ).

fof(fact_3989_transfer__nat__int__sum__prod__cong_I2_J,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & comm_monoid_mult(X_a) )
     => ! [F,G,A_2,B_1] :
          ( A_2 = B_1
         => ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
                 => hAPP(int,X_a,F,X_1) = hAPP(int,X_a,G,X_1) )
             => hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),G),B_1) ) ) ) ) ).

fof(fact_3990_setprod__pos__nat,axiom,
    ! [X_a,F,S] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
     => ( ! [X_1] :
            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S))
           => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_1))) )
       => hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),S))) ) ) ).

fof(fact_3991_imageE,axiom,
    ! [X_a,X_b,B_2,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
     => ~ ! [X_1] :
            ( ti(X_a,B_2) = hAPP(X_b,X_a,F,X_1)
           => ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_2)) ) ) ).

fof(fact_3992_setprod__dvd__setprod__subset2,axiom,
    ! [X_a,X_b] :
      ( comm_semiring_1(X_b)
     => ! [F,G,A_2,B_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
                 => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) )
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),B_1))) ) ) ) ) ).

fof(fact_3993_arctan__series,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real)))
     => hAPP(real,real,arctan,Xa) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))) ) ).

fof(fact_3994_DERIV__arctan__series,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real)))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,fun(nat,real)),fun(real,real),hAPP(fun(fun(nat,real),real),fun(fun(real,fun(nat,real)),fun(real,real)),combb(fun(nat,real),real,real),suminf(real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))))),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))),hAPP(fun(nat,nat),fun(real,fun(nat,real)),hAPP(fun(real,fun(fun(nat,nat),fun(nat,real))),fun(fun(nat,nat),fun(real,fun(nat,real))),combc(real,fun(nat,nat),fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real))),hAPP(fun(fun(nat,real),fun(fun(nat,nat),fun(nat,real))),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real)))),combb(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),real),combb(nat,real,nat)),power_power(real))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))),Xa),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ) ).

fof(fact_3995_suminf__zero,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & t2_space(X_a) )
     => hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a))) = zero_zero(X_a) ) ).

fof(fact_3996_sin__def,axiom,
    ! [Xa] : hAPP(real,real,sin,Xa) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))) ).

fof(fact_3997_lemma__sin__ext,axiom,
    ! [X_1] : hAPP(real,real,sin,X_1) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1))) ).

fof(fact_3998_cos__def,axiom,
    ! [Xa] : hAPP(real,real,cos,Xa) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))) ).

fof(fact_3999_lemma__cos__ext,axiom,
    ! [X_1] : hAPP(real,real,cos,X_1) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1))) ).

fof(fact_4000_powser__zero,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => ! [F] : hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)))) = hAPP(nat,X_a,F,zero_zero(nat)) ) ).

fof(fact_4001_lemma__sin__minus,axiom,
    ! [Xa] : hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa)) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa)))) ).

fof(fact_4002_exp__first__two__terms,axiom,
    ! [Xa] : hAPP(real,real,exp(real),Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).

fof(fact_4003_pi__series,axiom,
    hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),one_one(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))) ).

fof(fact_4004_ln__series,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
       => hAPP(real,real,ln,Xa) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),one_one(real)))),suc))) ) ) ).

fof(fact_4005_of__int__num,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [K] :
          ( ( ti(int,K) = zero_zero(int)
           => hAPP(int,X_a,ring_1_of_int(X_a),K) = zero_zero(X_a) )
          & ( ti(int,K) != zero_zero(int)
           => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
               => hAPP(int,X_a,ring_1_of_int(X_a),K) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,uminus_uminus(int),K))) )
              & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
               => hAPP(int,X_a,ring_1_of_int(X_a),K) = hAPP(product_prod(int,int),X_a,hAPP(fun(int,fun(int,X_a)),fun(product_prod(int,int),X_a),product_prod_case(int,int,X_a),hAPP(fun(int,X_a),fun(int,fun(int,X_a)),hAPP(fun(int,fun(X_a,fun(int,X_a))),fun(fun(int,X_a),fun(int,fun(int,X_a))),combs(int,X_a,fun(int,X_a)),hAPP(fun(int,fun(int,fun(X_a,X_a))),fun(int,fun(X_a,fun(int,X_a))),hAPP(fun(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a))),fun(fun(int,fun(int,fun(X_a,X_a))),fun(int,fun(X_a,fun(int,X_a)))),combb(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a)),int),combc(int,X_a,X_a)),hAPP(fun(int,X_a),fun(int,fun(int,fun(X_a,X_a))),hAPP(fun(X_a,fun(int,fun(X_a,X_a))),fun(fun(int,X_a),fun(int,fun(int,fun(X_a,X_a)))),combb(X_a,fun(int,fun(X_a,X_a)),int),hAPP(fun(int,fun(X_a,fun(X_a,X_a))),fun(X_a,fun(int,fun(X_a,X_a))),combc(int,X_a,fun(X_a,X_a)),hAPP(fun(int,bool),fun(int,fun(X_a,fun(X_a,X_a))),hAPP(fun(bool,fun(X_a,fun(X_a,X_a))),fun(fun(int,bool),fun(int,fun(X_a,fun(X_a,X_a)))),combb(bool,fun(X_a,fun(X_a,X_a)),int),if(X_a)),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(fun(int,X_a),fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,X_a)),combs(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),ring_1_of_int(X_a))),ring_1_of_int(X_a))))),hAPP(X_a,fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a)),combc(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),hAPP(fun(int,X_a),fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,X_a)),combs(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),ring_1_of_int(X_a))),ring_1_of_int(X_a)))),one_one(X_a)))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ) ) ) ).

fof(fact_4006_summable__arctan__series,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real)))
     => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))) ) ).

fof(fact_4007_summable__zero,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a)))) ) ).

fof(fact_4008_pair__imageI,axiom,
    ! [X_c,X_a,X_b,F,A_3,B_2,A_2] :
      ( hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2)),A_2))
     => hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,A_3),B_2)),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_c,bool),hAPP(fun(product_prod(X_a,X_b),X_c),fun(fun(product_prod(X_a,X_b),bool),fun(X_c,bool)),image(product_prod(X_a,X_b),X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F)),A_2))) ) ).

fof(fact_4009_suminf__add,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y_2,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_2))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)),hAPP(fun(nat,X_a),X_a,suminf(X_a),Y_2)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),Y_2)) ) ) ) ).

fof(fact_4010_suminf__diff,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y_2,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_2))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)),hAPP(fun(nat,X_a),X_a,suminf(X_a),Y_2)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_2)),Y_2)) ) ) ) ).

fof(fact_4011_divide_Osuminf,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Ya,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)),Ya) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya)) ) ) ).

fof(fact_4012_suminf__divide,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [C,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),C) ) ) ).

fof(fact_4013_mult__left_Osuminf,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Ya,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)),Ya) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya)) ) ) ).

fof(fact_4014_mult__right_Osuminf,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2)) ) ) ).

fof(fact_4015_suminf__mult2,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [C,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),C) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C)) ) ) ).

fof(fact_4016_suminf__mult,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [C,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),F)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)) ) ) ).

fof(fact_4017_suminf__minus,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)) ) ) ).

fof(fact_4018_prod__case__beta,axiom,
    ! [X_b,X_c,X_a,F,P_4] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),P_4) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F,hAPP(product_prod(X_b,X_c),X_b,product_fst(X_b,X_c),P_4)),hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),P_4)) ).

fof(fact_4019_summable__comparison__test,axiom,
    ! [X_a] :
      ( banach(X_a)
     => ! [F,G] :
          ( ? [N_2] :
            ! [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_1))),hAPP(nat,real,G,N_1))) )
         => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
           => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ) ).

fof(fact_4020_split__weak__cong,axiom,
    ! [X_c,X_a,X_b,C,P_4,Q_2] :
      ( ti(product_prod(X_a,X_b),P_4) = ti(product_prod(X_a,X_b),Q_2)
     => hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),C),P_4) = hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),C),Q_2) ) ).

fof(fact_4021_split__conv,axiom,
    ! [X_b,X_c,X_a,F,A_3,B_2] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),A_3),B_2)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F,A_3),B_2) ).

fof(fact_4022_Product__Type_Oprod_Osimps_I2_J,axiom,
    ! [X_b,X_c,X_a,F1,A_3,B_2] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F1),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),A_3),B_2)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F1,A_3),B_2) ).

fof(fact_4023_split__eta,axiom,
    ! [X_a,X_b,X_c,F] : hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,X_c)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,X_c))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c),X_a),hAPP(fun(product_prod(X_a,X_b),X_c),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c)),combb(product_prod(X_a,X_b),X_c,X_b),F)),product_Pair(X_a,X_b))) = ti(fun(product_prod(X_a,X_b),X_c),F) ).

fof(fact_4024_powser__insidea,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [Z,F,Xa] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Z)),hAPP(X_a,real,norm_norm(X_a),Xa)))
           => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z))))) ) ) ) ).

fof(fact_4025_summable__minus,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2))) ) ) ).

fof(fact_4026_summable__mult,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [C,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),F))) ) ) ).

fof(fact_4027_summable__mult2,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [C,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C))) ) ) ).

fof(fact_4028_mult__right_Osummable,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2))) ) ) ).

fof(fact_4029_mult__left_Osummable,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Ya,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya))) ) ) ).

fof(fact_4030_split__twice,axiom,
    ! [X_b,X_c,X_a,X_d,X_e,F,G,P_4] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),hAPP(product_prod(X_d,X_e),product_prod(X_b,X_c),hAPP(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(product_prod(X_d,X_e),product_prod(X_b,X_c)),product_prod_case(X_d,X_e,product_prod(X_b,X_c)),G),P_4)) = hAPP(product_prod(X_d,X_e),X_a,hAPP(fun(X_d,fun(X_e,X_a)),fun(product_prod(X_d,X_e),X_a),product_prod_case(X_d,X_e,X_a),hAPP(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(X_d,fun(X_e,X_a)),hAPP(fun(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a)),fun(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(X_d,fun(X_e,X_a))),combb(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a),X_d),hAPP(fun(product_prod(X_b,X_c),X_a),fun(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a)),combb(product_prod(X_b,X_c),X_a,X_e),hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F))),G)),P_4) ).

fof(fact_4031_of__real_Osummable,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,real),bool,summable(real),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2))) ) ) ).

fof(fact_4032_summable__norm__cancel,axiom,
    ! [X_a] :
      ( banach(X_a)
     => ! [F] :
          ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F)))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ).

fof(fact_4033_fst__def,axiom,
    ! [X_a,X_b,P_4] : hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_4) = hAPP(product_prod(X_a,X_b),X_a,hAPP(fun(X_a,fun(X_b,X_a)),fun(product_prod(X_a,X_b),X_a),product_prod_case(X_a,X_b,X_a),combk(X_a,X_b)),P_4) ).

fof(fact_4034_snd__def,axiom,
    ! [X_b,X_a,P_4] : hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),P_4) = hAPP(product_prod(X_b,X_a),X_a,hAPP(fun(X_b,fun(X_a,X_a)),fun(product_prod(X_b,X_a),X_a),product_prod_case(X_b,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_b,fun(X_a,X_a)),combk(fun(X_a,X_a),X_b),combi(X_a))),P_4) ).

fof(fact_4035_summable__divide,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [C,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C))) ) ) ).

fof(fact_4036_divide_Osummable,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Ya,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya))) ) ) ).

fof(fact_4037_summable__rabs__cancel,axiom,
    ! [F] :
      ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)))
     => hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) ) ).

fof(fact_4038_summable__diff,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y_2,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_2))
           => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_2)),Y_2))) ) ) ) ).

fof(fact_4039_summable__add,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y_2,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_2))
           => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),Y_2))) ) ) ) ).

fof(fact_4040_summable__ignore__initial__segment,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [K,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))) ) ) ).

fof(fact_4041_split__def,axiom,
    ! [X_a,X_b,X_c,X_1,Xa_1] : hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),X_1),Xa_1) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),X_1,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Xa_1)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Xa_1)) ).

fof(fact_4042_split__comp__eq,axiom,
    ! [X_a,X_b,X_c,X_d,F,G] : hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),X_c),hAPP(fun(product_prod(X_a,X_b),fun(X_b,X_c)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),X_c)),combs(product_prod(X_a,X_b),X_b,X_c),hAPP(fun(product_prod(X_a,X_b),X_d),fun(product_prod(X_a,X_b),fun(X_b,X_c)),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(product_prod(X_a,X_b),X_d),fun(product_prod(X_a,X_b),fun(X_b,X_c))),combb(X_d,fun(X_b,X_c),product_prod(X_a,X_b)),F),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),X_d),hAPP(fun(X_a,X_d),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),X_d)),combb(X_a,X_d,product_prod(X_a,X_b)),G),product_fst(X_a,X_b)))),product_snd(X_a,X_b)) = hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),hAPP(fun(X_a,X_d),fun(X_a,fun(X_b,X_c)),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(X_a,X_d),fun(X_a,fun(X_b,X_c))),combb(X_d,fun(X_b,X_c),X_a),F),G)) ).

fof(fact_4043_split__beta,axiom,
    ! [X_b,X_c,X_a,P_1,Z] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),P_1),Z) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),P_1,hAPP(product_prod(X_b,X_c),X_b,product_fst(X_b,X_c),Z)),hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),Z)) ).

fof(fact_4044_summable__norm__comparison__test,axiom,
    ! [X_a] :
      ( banach(X_a)
     => ! [F,G] :
          ( ? [N_2] :
            ! [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_1))),hAPP(nat,real,G,N_1))) )
         => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
           => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F))) ) ) ) ).

fof(fact_4045_summable__rabs__comparison__test,axiom,
    ! [F,G] :
      ( ? [N_2] :
        ! [N_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(nat,real,F,N_1))),hAPP(nat,real,G,N_1))) )
     => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
       => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F))) ) ) ).

fof(fact_4046_summable__rabs,axiom,
    ! [F] :
      ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)))
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(fun(nat,real),real,suminf(real),F))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)))) ) ).

fof(fact_4047_of__real_Osuminf,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,real),bool,summable(real),X_2))
         => hAPP(real,X_a,of_real(X_a),hAPP(fun(nat,real),real,suminf(real),X_2)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2)) ) ) ).

fof(fact_4048_summable__sin,axiom,
    ! [Xa] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa)))) ).

fof(fact_4049_summable__cos,axiom,
    ! [Xa] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa)))) ).

fof(fact_4050_summable__norm,axiom,
    ! [X_a] :
      ( banach(X_a)
     => ! [F] :
          ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F)))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F)))) ) ) ).

fof(fact_4051_powser__inside,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [Z,F,Xa] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Z)),hAPP(X_a,real,norm_norm(X_a),Xa)))
           => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z)))) ) ) ) ).

fof(fact_4052_summable__geometric,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),one_one(real)))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))) ) ) ).

fof(fact_4053_complete__algebra__summable__geometric,axiom,
    ! [X_a] :
      ( ( real_n2089651433ebra_1(X_a)
        & banach(X_a) )
     => ! [Xa] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),one_one(real)))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))) ) ) ).

fof(fact_4054_rat__floor__code,axiom,
    ! [P_4] : hAPP(rat,int,archim2085082626_floor(rat),P_4) = hAPP(product_prod(int,int),int,hAPP(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_prod_case(int,int,int),div_div(int)),hAPP(rat,product_prod(int,int),quotient_of,P_4)) ).

fof(fact_4055_rat__uminus__code,axiom,
    ! [P_4] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,uminus_uminus(rat),P_4)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),hAPP(rat,product_prod(int,int),quotient_of,P_4)) ).

fof(fact_4056_summable__exp,axiom,
    ! [Xa] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa)))) ).

fof(fact_4057_rat__abs__code,axiom,
    ! [P_4] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,abs_abs(rat),P_4)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),abs_abs(int))),hAPP(rat,product_prod(int,int),quotient_of,P_4)) ).

fof(fact_4058_divmod__int__rel__def,axiom,
    ! [A_3,B_2] : hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_3),B_2) = hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,bool),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,bool),fun(int,fun(int,bool))),combc(int,fun(int,bool),fun(int,bool)),hAPP(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool))),hAPP(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),int),combs(int,bool,bool)),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool))),hAPP(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),int),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj)),hAPP(fun(int,fun(int,int)),fun(int,fun(int,bool)),hAPP(fun(fun(int,int),fun(int,bool)),fun(fun(int,fun(int,int)),fun(int,fun(int,bool))),combb(fun(int,int),fun(int,bool),int),hAPP(fun(int,bool),fun(fun(int,int),fun(int,bool)),combb(int,bool,int),hAPP(int,fun(int,bool),fequal(int),A_3))),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),plus_plus(int)),hAPP(int,fun(int,int),times_times(int),B_2)))))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),combb(bool,bool,int),hAPP(bool,fun(bool,bool),fimplies,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),B_2))))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),combb(bool,bool,int),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),B_2))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),zero_zero(int))))))) ).

fof(fact_4059_sum2sq__def,axiom,
    twoSqu1094008024sum2sq = hAPP(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_prod_case(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(fun(int,int),fun(int,int))),fun(fun(int,int),fun(int,fun(int,int))),combc(int,fun(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(int,fun(fun(int,int),fun(int,int))),hAPP(fun(fun(int,int),fun(fun(int,int),fun(int,int))),fun(fun(int,fun(int,int)),fun(int,fun(fun(int,int),fun(int,int)))),combb(fun(int,int),fun(fun(int,int),fun(int,int)),int),combb(int,int,int)),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),plus_plus(int)),hAPP(nat,fun(int,int),hAPP(fun(int,fun(nat,int)),fun(nat,fun(int,int)),combc(int,nat,int),power_power(int)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,fun(int,int),hAPP(fun(int,fun(nat,int)),fun(nat,fun(int,int)),combc(int,nat,int),power_power(int)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_4060_rat__inverse__code,axiom,
    ! [P_4] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,inverse_inverse(rat),P_4)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(product_prod(int,int),product_prod(int,int)),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(product_prod(int,int),product_prod(int,int)),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combb(product_prod(int,int),product_prod(int,int),int)),hAPP(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int)))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combs(int,int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(int,fun(int,fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(int,fun(int,fun(int,product_prod(int,int))))),combb(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),int),combc(int,int,product_prod(int,int))),hAPP(fun(int,fun(int,int)),fun(int,fun(int,fun(int,product_prod(int,int)))),hAPP(fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),fun(fun(int,fun(int,int)),fun(int,fun(int,fun(int,product_prod(int,int))))),combb(fun(int,int),fun(int,fun(int,product_prod(int,int))),int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int))),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),times_times(int)),sgn_sgn(int))))),abs_abs(int)))),hAPP(rat,product_prod(int,int),quotient_of,P_4)) ).

fof(fact_4061_exp__tail__after__first__two__terms__summable,axiom,
    ! [Xa] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).

fof(fact_4062_adjust__def,axiom,
    ! [B_2] : hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B_2) = hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combs(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(fun(int,int),fun(int,bool),hAPP(fun(int,bool),fun(fun(int,int),fun(int,bool)),combb(int,bool,int),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int))),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int)),B_2))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),plus_plus(int)),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(int))))),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int)),B_2))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).

fof(fact_4063_DERIV__power__series_H,axiom,
    ! [X0,F,R_2] :
      ( ! [X_1] :
          ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),hAPP(real,real,uminus_uminus(real),R_2)),R_2)))
         => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),hAPP(real,fun(nat,real),power_power(real),X_1)))) )
     => ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X0),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),hAPP(real,real,uminus_uminus(real),R_2)),R_2)))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,fun(nat,real)),fun(real,real),hAPP(fun(fun(nat,real),real),fun(fun(real,fun(nat,real)),fun(real,real)),combb(fun(nat,real),real,real),suminf(real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F))),hAPP(fun(nat,nat),fun(real,fun(nat,real)),hAPP(fun(real,fun(fun(nat,nat),fun(nat,real))),fun(fun(nat,nat),fun(real,fun(nat,real))),combc(real,fun(nat,nat),fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real))),hAPP(fun(fun(nat,real),fun(fun(nat,nat),fun(nat,real))),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real)))),combb(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),real),combb(nat,real,nat)),power_power(real))),suc)))),X0),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),hAPP(real,fun(nat,real),power_power(real),X0))))) ) ) ) ).

fof(fact_4064_Nitpick_OFrac__def,axiom,
    frac_1 = hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),hAPP(fun(fun(int,bool),fun(int,bool)),fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),combb(fun(int,bool),fun(int,bool),int),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int))))),hAPP(int,fun(int,fun(int,bool)),hAPP(fun(int,fun(int,fun(int,bool))),fun(int,fun(int,fun(int,bool))),combc(int,int,fun(int,bool)),hAPP(fun(int,fun(int,fun(int,bool))),fun(int,fun(int,fun(int,bool))),hAPP(fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),fun(fun(int,fun(int,fun(int,bool))),fun(int,fun(int,fun(int,bool)))),combb(fun(int,fun(int,bool)),fun(int,fun(int,bool)),int),combc(int,int,bool)),hAPP(fun(int,fun(int,int)),fun(int,fun(int,fun(int,bool))),hAPP(fun(fun(int,int),fun(int,fun(int,bool))),fun(fun(int,fun(int,int)),fun(int,fun(int,fun(int,bool)))),combb(fun(int,int),fun(int,fun(int,bool)),int),hAPP(fun(int,fun(int,bool)),fun(fun(int,int),fun(int,fun(int,bool))),combb(int,fun(int,bool),int),fequal(int))),int_gcd))),one_one(int)))) ).

fof(fact_4065_mem__splitI,axiom,
    ! [X_a,X_c,X_b,Z,C,A_3,B_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(X_c,fun(X_a,bool),hAPP(X_b,fun(X_c,fun(X_a,bool)),C,A_3),B_2)))
     => hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(product_prod(X_b,X_c),fun(X_a,bool),hAPP(fun(X_b,fun(X_c,fun(X_a,bool))),fun(product_prod(X_b,X_c),fun(X_a,bool)),product_prod_case(X_b,X_c,fun(X_a,bool)),C),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),A_3),B_2)))) ) ).

fof(fact_4066_splitI,axiom,
    ! [X_b,X_a,F,A_3,B_2] :
      ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F,A_3),B_2))
     => hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2))) ) ).

fof(fact_4067_prod__caseI,axiom,
    ! [X_b,X_a,F1,A_3,B_2] :
      ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F1,A_3),B_2))
     => hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F1),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2))) ) ).

fof(fact_4068_cnj_Osuminf,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2))
     => hAPP(complex,complex,cnj,hAPP(fun(nat,complex),complex,suminf(complex),X_2)) = hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2)) ) ).

fof(fact_4069_splitD,axiom,
    ! [X_a,X_b,F,A_3,B_2] :
      ( hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2)))
     => hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F,A_3),B_2)) ) ).

fof(fact_4070_splitD_H,axiom,
    ! [X_c,X_a,X_b,R_2,A_3,B_2,C] :
      ( hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),R_2),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2)),C))
     => hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),R_2,A_3),B_2),C)) ) ).

fof(fact_4071_split__part,axiom,
    ! [X_a,X_b,P_1,Q_1,X_1] :
      ( hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fconj,P_1))),Q_1)),X_1))
    <=> ( hBOOL(P_1)
        & hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),Q_1),X_1)) ) ) ).

fof(fact_4072_cnj_Osummable,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2))
     => hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2))) ) ).

fof(fact_4073_Re_Osummable,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2))
     => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2))) ) ).

fof(fact_4074_Im_Osummable,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2))
     => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2))) ) ).

fof(fact_4075_Re_Osuminf,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2))
     => hAPP(complex,real,re,hAPP(fun(nat,complex),complex,suminf(complex),X_2)) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2)) ) ).

fof(fact_4076_Im_Osuminf,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2))
     => hAPP(complex,real,im,hAPP(fun(nat,complex),complex,suminf(complex),X_2)) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2)) ) ).

fof(fact_4077_finite__psubset__def,axiom,
    ! [X_a] : finite_psubset(X_a) = hAPP(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),collect(product_prod(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),product_prod_case(fun(X_a,bool),fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combs(fun(X_a,bool),bool,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool)),fun(X_a,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),combb(bool,fun(bool,bool),fun(X_a,bool)),fconj)),ord_less(fun(X_a,bool))))),finite_finite_1(X_a)))) ).

fof(fact_4078_ratrel__def,axiom,
    ratrel = hAPP(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),collect(product_prod(product_prod(int,int),product_prod(int,int))),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),product_prod_case(product_prod(int,int),product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),fun(product_prod(int,int),bool)),hAPP(fun(product_prod(int,int),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool))),fun(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),fun(product_prod(int,int),bool))),combs(product_prod(int,int),fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(product_prod(int,int),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool))),hAPP(fun(fun(bool,bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool))),fun(fun(product_prod(int,int),fun(bool,bool)),fun(product_prod(int,int),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)))),combb(fun(bool,bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),product_prod(int,int)),combb(bool,bool,product_prod(int,int))),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(bool,bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combb(bool,bool,product_prod(int,int)),fNot),hAPP(int,fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(int,bool)),fun(int,fun(product_prod(int,int),bool)),combc(product_prod(int,int),int,bool),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int)),product_snd(int,int))),zero_zero(int)))))),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),fun(product_prod(int,int),bool)),hAPP(fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),fun(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),fun(product_prod(int,int),bool))),combb(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),product_prod(int,int)),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combs(product_prod(int,int),bool,bool),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(bool,bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combb(bool,bool,product_prod(int,int)),fNot),hAPP(int,fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(int,bool)),fun(int,fun(product_prod(int,int),bool)),combc(product_prod(int,int),int,bool),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int)),product_snd(int,int))),zero_zero(int)))))),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),int)),fun(product_prod(int,int),fun(product_prod(int,int),bool)),hAPP(fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool))),fun(fun(product_prod(int,int),fun(product_prod(int,int),int)),fun(product_prod(int,int),fun(product_prod(int,int),bool))),combs(product_prod(int,int),fun(product_prod(int,int),int),fun(product_prod(int,int),bool)),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),fun(int,bool))),fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool))),hAPP(fun(fun(product_prod(int,int),fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool))),fun(fun(product_prod(int,int),fun(product_prod(int,int),fun(int,bool))),fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool)))),combb(fun(product_prod(int,int),fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool)),product_prod(int,int)),combs(product_prod(int,int),int,bool)),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),int)),fun(product_prod(int,int),fun(product_prod(int,int),fun(int,bool))),hAPP(fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),fun(fun(product_prod(int,int),fun(product_prod(int,int),int)),fun(product_prod(int,int),fun(product_prod(int,int),fun(int,bool)))),combb(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),product_prod(int,int)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int))),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(product_prod(int,int),int)),hAPP(fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int))),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(product_prod(int,int),int))),combc(product_prod(int,int),fun(product_prod(int,int),int),fun(product_prod(int,int),int)),hAPP(fun(product_prod(int,int),fun(int,int)),fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int))),hAPP(fun(fun(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int))),fun(fun(product_prod(int,int),fun(int,int)),fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int)))),combb(fun(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int)),product_prod(int,int)),combb(int,int,product_prod(int,int))),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int))),combb(int,fun(int,int),product_prod(int,int)),times_times(int)),product_fst(int,int)))),product_snd(int,int))))),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(product_prod(int,int),int)),hAPP(fun(int,fun(product_prod(int,int),int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(product_prod(int,int),int))),combb(int,fun(product_prod(int,int),int),product_prod(int,int)),hAPP(fun(product_prod(int,int),fun(int,int)),fun(int,fun(product_prod(int,int),int)),combc(product_prod(int,int),int,int),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int))),combb(int,fun(int,int),product_prod(int,int)),times_times(int)),product_fst(int,int)))),product_snd(int,int))))))) ).

fof(fact_4079_divmod__nat__if,axiom,
    ! [Ma,Na] :
      ( ( ( Na = zero_zero(nat)
          | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) )
       => hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,Ma),Na) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),Ma) )
      & ( ~ ( Na = zero_zero(nat)
            | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) )
       => hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,Ma),Na) = hAPP(product_prod(nat,nat),product_prod(nat,nat),hAPP(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_prod_case(nat,nat,product_prod(nat,nat)),hAPP(fun(nat,nat),fun(nat,fun(nat,product_prod(nat,nat))),hAPP(fun(nat,fun(nat,product_prod(nat,nat))),fun(fun(nat,nat),fun(nat,fun(nat,product_prod(nat,nat)))),combb(nat,fun(nat,product_prod(nat,nat)),nat),product_Pair(nat,nat)),suc)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),Na)),Na)) ) ) ).

fof(fact_4080_Bolzano__bisect_Osimps_I2_J,axiom,
    ! [P_1,A_3,B_2,Na] : hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),hAPP(nat,nat,suc,Na)) = hAPP(product_prod(real,real),product_prod(real,real),hAPP(fun(real,fun(real,product_prod(real,real))),fun(product_prod(real,real),product_prod(real,real)),product_prod_case(real,real,product_prod(real,real)),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,product_prod(real,real)),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,fun(product_prod(real,real),product_prod(real,real))),fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real))))),combb(fun(real,fun(product_prod(real,real),product_prod(real,real))),fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real))),real),combs(real,product_prod(real,real),product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real))))),combs(real,fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),hAPP(fun(fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))))),combb(fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))),real),combs(real,product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),hAPP(fun(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,bool)),fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))))),combb(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),real),hAPP(fun(bool,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),combb(bool,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))),real),if(product_prod(real,real)))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,bool)),hAPP(fun(fun(real,product_prod(real,real)),fun(real,bool)),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,bool))),combb(fun(real,product_prod(real,real)),fun(real,bool),real),hAPP(fun(product_prod(real,real),bool),fun(fun(real,product_prod(real,real)),fun(real,bool)),combb(product_prod(real,real),bool,real),P_1)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real))),real),combb(real,product_prod(real,real),real)),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(real,real),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),combc(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,fun(real,product_prod(real,real)))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,fun(real,product_prod(real,real)))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,product_prod(real,real))),real),combs(real,real,product_prod(real,real))),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,product_prod(real,real)))),hAPP(fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,product_prod(real,real))))),combb(fun(real,real),fun(real,fun(real,product_prod(real,real))),real),hAPP(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),combb(real,fun(real,product_prod(real,real)),real),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),combi(real))))),hAPP(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real))),real),combb(real,product_prod(real,real),real)),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na)) ).

fof(fact_4081_ratio__test,axiom,
    ! [X_a] :
      ( banach(X_a)
     => ! [F,N_3,C] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),one_one(real)))
         => ( ! [N_1] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_1))
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_1))))) )
           => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ) ).

fof(fact_4082_ratio__test__lemma2,axiom,
    ! [X_a] :
      ( banach(X_a)
     => ! [F,C,N_3] :
          ( ! [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_1))))) )
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C))
            | hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ) ).

fof(fact_4083_suminf__exist__split,axiom,
    ! [A_3,Ra] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra))
     => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),A_3))
       => ? [N_2] :
          ! [N_1] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),A_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),N_1))))),Ra)) ) ) ) ).

fof(fact_4084_splitE_H,axiom,
    ! [X_c,X_a,X_b,C,P_4,Z] :
      ( hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C),P_4),Z))
     => ~ ! [X_1,Y_1] :
            ( ti(product_prod(X_a,X_b),P_4) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1)
           => ~ hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C,X_1),Y_1),Z)) ) ) ).

fof(fact_4085_mem__splitI2,axiom,
    ! [X_c,X_a,X_b,Z,C,P_4] :
      ( ! [A_4,B_4] :
          ( ti(product_prod(X_a,X_b),P_4) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_4),B_4)
         => hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Z),hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C,A_4),B_4))) )
     => hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Z),hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C),P_4))) ) ).

fof(fact_4086_splitI2_H,axiom,
    ! [X_c,X_b,X_a,C,Xa,P_4] :
      ( ! [A_4,B_4] :
          ( hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_4),B_4) = ti(product_prod(X_a,X_b),P_4)
         => hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C,A_4),B_4),Xa)) )
     => hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C),P_4),Xa)) ) ).

fof(fact_4087_mem__splitE,axiom,
    ! [X_a,X_b,X_c,Z,C,P_4] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(product_prod(X_b,X_c),fun(X_a,bool),hAPP(fun(X_b,fun(X_c,fun(X_a,bool))),fun(product_prod(X_b,X_c),fun(X_a,bool)),product_prod_case(X_b,X_c,fun(X_a,bool)),C),P_4)))
     => ~ ! [X_1,Y_1] :
            ( ti(product_prod(X_b,X_c),P_4) = hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),X_1),Y_1)
           => ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(X_c,fun(X_a,bool),hAPP(X_b,fun(X_c,fun(X_a,bool)),C,X_1),Y_1))) ) ) ).

fof(fact_4088_splitE,axiom,
    ! [X_a,X_b,C,P_4] :
      ( hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),C),P_4))
     => ~ ! [X_1,Y_1] :
            ( ti(product_prod(X_a,X_b),P_4) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1)
           => ~ hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),C,X_1),Y_1)) ) ) ).

fof(fact_4089_splitI2,axiom,
    ! [X_a,X_b,C,P_4] :
      ( ! [A_4,B_4] :
          ( ti(product_prod(X_a,X_b),P_4) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_4),B_4)
         => hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),C,A_4),B_4)) )
     => hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),C),P_4)) ) ).

fof(fact_4090_divmod__int__pdivmod,axiom,
    ! [L,K] :
      ( ( ti(int,K) = zero_zero(int)
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) )
      & ( ti(int,K) != zero_zero(int)
       => ( ( ti(int,L) = zero_zero(int)
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K) )
          & ( ti(int,L) != zero_zero(int)
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,int),fun(product_prod(int,int),product_prod(int,int)),product_apsnd(int,int,int),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,sgn_sgn(int),L))),hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),L)),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))),hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),L),zero_zero(int))),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)),hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(int,fun(int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),combc(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),zero_zero(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),minus_minus(int)),uminus_uminus(int))),one_one(int))))),hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,abs_abs(int),L))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)))) ) ) ) ) ).

fof(fact_4091_apsnd__eq__conv,axiom,
    ! [X_a,X_c,X_b,F,Xa,G] :
      ( hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),Xa) = hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),G),Xa)
    <=> hAPP(X_c,X_b,F,hAPP(product_prod(X_a,X_c),X_c,product_snd(X_a,X_c),Xa)) = hAPP(X_c,X_b,G,hAPP(product_prod(X_a,X_c),X_c,product_snd(X_a,X_c),Xa)) ) ).

fof(fact_4092_snd__apsnd,axiom,
    ! [X_b,X_a,X_c,F,Xa] : hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(product_prod(X_b,X_c),product_prod(X_b,X_a),hAPP(fun(X_c,X_a),fun(product_prod(X_b,X_c),product_prod(X_b,X_a)),product_apsnd(X_c,X_a,X_b),F),Xa)) = hAPP(X_c,X_a,F,hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),Xa)) ).

fof(fact_4093_fst__apsnd,axiom,
    ! [X_a,X_b,X_c,F,Xa] : hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),Xa)) = hAPP(product_prod(X_a,X_c),X_a,product_fst(X_a,X_c),Xa) ).

fof(fact_4094_apsnd__conv,axiom,
    ! [X_a,X_c,X_b,F,Xa,Ya] : hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),hAPP(X_c,product_prod(X_a,X_c),hAPP(X_a,fun(X_c,product_prod(X_a,X_c)),product_Pair(X_a,X_c),Xa),Ya)) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),hAPP(X_c,X_b,F,Ya)) ).

fof(fact_4095_negateSnd__def,axiom,
    negateSnd = hAPP(fun(int,int),fun(product_prod(int,int),product_prod(int,int)),product_apsnd(int,int,int),uminus_uminus(int)) ).

fof(fact_4096_divmod__int__code,axiom,
    ! [L,K] :
      ( ( ti(int,K) = zero_zero(int)
       => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) )
      & ( ti(int,K) != zero_zero(int)
       => ( ( ti(int,L) = zero_zero(int)
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K) )
          & ( ti(int,L) != zero_zero(int)
           => hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,int),fun(product_prod(int,int),product_prod(int,int)),product_apsnd(int,int,int),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,sgn_sgn(int),L))),hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),hAPP(int,int,sgn_sgn(int),K)),hAPP(int,int,sgn_sgn(int),L))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)),hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(int,fun(int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),combc(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),zero_zero(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),minus_minus(int)),uminus_uminus(int))),one_one(int))))),hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,abs_abs(int),L))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)))) ) ) ) ) ).

fof(fact_4097_summable__le2,axiom,
    ! [F,G] :
      ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(nat,real,F,N_1))),hAPP(nat,real,G,N_1)))
     => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
       => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),hAPP(fun(nat,real),real,suminf(real),G))) ) ) ) ).

fof(fact_4098_suminf__gt__zero,axiom,
    ! [F] :
      ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,F,N_1)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).

fof(fact_4099_suminf__0__le,axiom,
    ! [F] :
      ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_1)))
     => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).

fof(fact_4100_suminf__ge__zero,axiom,
    ! [F] :
      ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_1)))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).

fof(fact_4101_int__ge__less__than2__def,axiom,
    ! [D_1] : hAPP(int,fun(product_prod(int,int),bool),int_ge_less_than2,D_1) = hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),collect(product_prod(int,int)),hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),hAPP(fun(fun(int,bool),fun(int,bool)),fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),combb(fun(int,bool),fun(int,bool),int),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),D_1)))),ord_less(int)))) ).

fof(fact_4102_int__ge__less__than__def,axiom,
    ! [D_1] : hAPP(int,fun(product_prod(int,int),bool),int_ge_less_than,D_1) = hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),collect(product_prod(int,int)),hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),combs(int,fun(int,bool),fun(int,bool)),hAPP(fun(int,fun(bool,bool)),fun(int,fun(fun(int,bool),fun(int,bool))),hAPP(fun(fun(bool,bool),fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(bool,bool)),fun(int,fun(fun(int,bool),fun(int,bool)))),combb(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),int),combb(bool,bool,int)),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),D_1)))),ord_less(int)))) ).

fof(fact_4103_sin__paired,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),hAPP(real,real,sin,Xa))) ).

fof(fact_4104_of__real_OCauchy,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [X_2] :
          ( hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2))) ) ) ).

fof(fact_4105_sums__zero,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a))),zero_zero(X_a))) ) ).

fof(fact_4106_sums__minus,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) ) ) ).

fof(fact_4107_sums__mult,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [C,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),F)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3))) ) ) ).

fof(fact_4108_sums__mult2,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [C,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) ) ) ).

fof(fact_4109_mult__right_Osums,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa,X_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),A_3))) ) ) ).

fof(fact_4110_mult__left_Osums,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Ya,X_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),Ya))) ) ) ).

fof(fact_4111_of__real_Osums,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),X_2),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2)),hAPP(real,X_a,of_real(X_a),A_3))) ) ) ).

fof(fact_4112_sums__divide,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [C,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),C))) ) ) ).

fof(fact_4113_divide_Osums,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Ya,X_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),Ya))) ) ) ).

fof(fact_4114_sums__diff,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y_2,B_2,X_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3))
         => ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),Y_2),B_2))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2))) ) ) ) ).

fof(fact_4115_sums__add,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y_2,B_2,X_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3))
         => ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),Y_2),B_2))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2))) ) ) ) ).

fof(fact_4116_sums__unique,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & t2_space(X_a) )
     => ! [F,Sb] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Sb))
         => ti(X_a,Sb) = hAPP(fun(nat,X_a),X_a,suminf(X_a),F) ) ) ).

fof(fact_4117_summable__def,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ! [F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
        <=> ? [X1] : hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),X1)) ) ) ).

fof(fact_4118_sums__summable,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ! [F,L] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),L))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ).

fof(fact_4119_summable__sums,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & t2_space(X_a) )
     => ! [F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(fun(nat,X_a),X_a,suminf(X_a),F))) ) ) ).

fof(fact_4120_sums__iff,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & t2_space(X_a) )
     => ! [F,Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Xa))
        <=> ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
            & hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = ti(X_a,Xa) ) ) ) ).

fof(fact_4121_sums__Suc__imp,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ! [Sb,F] :
          ( hAPP(nat,X_a,F,zero_zero(nat)) = zero_zero(X_a)
         => ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),Sb))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Sb)) ) ) ) ).

fof(fact_4122_sums__Suc,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [F,L] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),L),hAPP(nat,X_a,F,zero_zero(nat))))) ) ) ).

fof(fact_4123_sin__converges,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(real,real,sin,Xa))) ).

fof(fact_4124_cos__converges,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(real,real,cos,Xa))) ).

fof(fact_4125_geometric__sums,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),one_one(real)))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),one_one(X_a)),Xa)))) ) ) ).

fof(fact_4126_power__half__series,axiom,
    hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),suc)),one_one(real))) ).

fof(fact_4127_mult__right_OCauchy,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2))) ) ) ).

fof(fact_4128_mult__left_OCauchy,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Ya,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya))) ) ) ).

fof(fact_4129_divide_OCauchy,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Ya,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya))) ) ) ).

fof(fact_4130_Ln_Oaux2,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).

fof(fact_4131_sums__if_H,axiom,
    ! [G,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),G),Xa))
     => hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),one_one(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),Xa)) ) ).

fof(fact_4132_sums__if,axiom,
    ! [F,Ya,G,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),G),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),F),Ya))
       => hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),div_div(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),one_one(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya))) ) ) ).

fof(fact_4133_cos__paired,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,cos,Xa))) ).

fof(fact_4134_diffs__equiv,axiom,
    ! [X_a] :
      ( ( real_normed_vector(X_a)
        & ring_1(X_a) )
     => ! [C,Xa] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),C))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))) ) ) ).

fof(fact_4135_nat__case__add__eq__if,axiom,
    ! [X_a,A_3,F,Va,Na] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),A_3),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),Va)),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),A_3),F),Na)),hAPP(nat,X_a,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na))) ).

fof(fact_4136_cnj_Osums,axiom,
    ! [X_2,A_3] :
      ( hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_2),A_3))
     => hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2)),hAPP(complex,complex,cnj,A_3))) ) ).

fof(fact_4137_diffs__of__real,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [F,X_1] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),F)),X_1) = hAPP(real,X_a,of_real(X_a),hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),F),X_1)) ) ).

fof(fact_4138_nat__case__Suc,axiom,
    ! [X_a,F1,F2,Nat] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),F1),F2),hAPP(nat,nat,suc,Nat)) = hAPP(nat,X_a,F2,Nat) ).

fof(fact_4139_nat__case__0,axiom,
    ! [X_a,F1,F2] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),F1),F2),zero_zero(nat)) = ti(X_a,F1) ).

fof(fact_4140_diffs__minus,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [C,X_1] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),C)),X_1) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C),X_1)) ) ).

fof(fact_4141_Re_Osums,axiom,
    ! [X_2,A_3] :
      ( hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_2),A_3))
     => hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2)),hAPP(complex,real,re,A_3))) ) ).

fof(fact_4142_Im_Osums,axiom,
    ! [X_2,A_3] :
      ( hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_2),A_3))
     => hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2)),hAPP(complex,real,im,A_3))) ) ).

fof(fact_4143_sin__fdiffs2,axiom,
    ! [Na] : hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),sin_coeff),Na) = hAPP(nat,real,cos_coeff,Na) ).

fof(fact_4144_sin__fdiffs,axiom,
    hAPP(fun(nat,real),fun(nat,real),diffs(real),sin_coeff) = cos_coeff ).

fof(fact_4145_diffs__def,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [C,X_1] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C),X_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,X_1))),hAPP(nat,X_a,C,hAPP(nat,nat,suc,X_1))) ) ).

fof(fact_4146_exp__fdiffs,axiom,
    ! [X_1] : hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),X_1) = hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),X_1))) ).

fof(fact_4147_less__eq__nat_Osimps_I2_J,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Ma)),Na))
    <=> hBOOL(hAPP(nat,bool,hAPP(fun(nat,bool),fun(nat,bool),hAPP(bool,fun(fun(nat,bool),fun(nat,bool)),nat_case(bool),fFalse),hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma)),Na)) ) ).

fof(fact_4148_cos__fdiffs,axiom,
    ! [X_1] : hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),cos_coeff),X_1) = hAPP(real,real,uminus_uminus(real),hAPP(nat,real,sin_coeff,X_1)) ).

fof(fact_4149_cos__fdiffs2,axiom,
    ! [Na] : hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),cos_coeff),Na) = hAPP(real,real,uminus_uminus(real),hAPP(nat,real,sin_coeff,Na)) ).

fof(fact_4150_diff__Suc,axiom,
    ! [Ma,Na] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),combi(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),Na)) ).

fof(fact_4151_termdiffs,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [Xa,C,K_3] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C)),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3))))
         => ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3))))
           => ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3))))
             => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),hAPP(X_a,real,norm_norm(X_a),K_3)))
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,X_a),hAPP(fun(fun(nat,X_a),X_a),fun(fun(X_a,fun(nat,X_a)),fun(X_a,X_a)),combb(fun(nat,X_a),X_a,X_a),suminf(X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C))),power_power(X_a)))),Xa),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))) ) ) ) ) ) ).

fof(fact_4152_nat__case__number__of,axiom,
    ! [X_a,A_3,F,Va] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),A_3),F),hAPP(int,nat,number_number_of(nat),Va)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),A_3),hAPP(nat,X_a,F,hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))) ).

fof(fact_4153_exE__realizer,axiom,
    ! [X_c,X_b,X_a,Q_1,F,P_1,P_4] :
      ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),P_4)),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),P_4)))
     => ( ! [X_1,Y_1] :
            ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Y_1),X_1))
           => hBOOL(hAPP(X_c,bool,Q_1,hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),F,X_1),Y_1))) )
       => hBOOL(hAPP(X_c,bool,Q_1,hAPP(product_prod(X_b,X_a),X_c,hAPP(fun(X_b,fun(X_a,X_c)),fun(product_prod(X_b,X_a),X_c),product_prod_case(X_b,X_a,X_c),F),P_4))) ) ) ).

fof(fact_4154_finite__conv__nat__seg__image,axiom,
    ! [X_a,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
    <=> ? [N_1,F_3] : ti(fun(X_a,bool),A_2) = hAPP(fun(nat,bool),fun(X_a,bool),hAPP(fun(nat,X_a),fun(fun(nat,bool),fun(X_a,bool)),image(nat,X_a),F_3),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N_1))) ) ).

fof(fact_4155_setprod__coprime__nat,axiom,
    ! [X_a,F,Xa,A_2] :
      ( ! [I_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2))
         => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(X_a,nat,F,I_2)),Xa) = one_one(nat) )
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2)),Xa) = one_one(nat) ) ).

fof(fact_4156_summable__le,axiom,
    ! [F,G] :
      ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),hAPP(nat,real,G,N_1)))
     => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
       => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),hAPP(fun(nat,real),real,suminf(real),G))) ) ) ) ).

fof(fact_4157_monoI2,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X_2] :
          ( ! [M_2,N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(nat,X_a,X_2,M_2))) )
         => hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)) ) ) ).

fof(fact_4158_monoI1,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X_2] :
          ( ! [M_2,N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))) )
         => hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)) ) ) ).

fof(fact_4159_arctan__def,axiom,
    ! [Ya] : hAPP(real,real,arctan,Ya) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),tan)),Ya)))) ).

fof(fact_4160_arcsin__def,axiom,
    ! [Ya] : hAPP(real,real,arcsin,Ya) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),sin)),Ya)))) ).

fof(fact_4161_ln__def,axiom,
    ! [Xa] : hAPP(real,real,ln,Xa) = hAPP(fun(real,bool),real,the(real),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),exp(real))),Xa)) ).

fof(fact_4162_suminf__def,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ! [F] : hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(fun(X_a,bool),X_a,the(X_a),hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F)) ) ).

fof(fact_4163_root__def,axiom,
    ! [Na,Xa] :
      ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
       => hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),Na))),Xa))) )
      & ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
       => ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real)))
           => hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = hAPP(real,real,uminus_uminus(real),hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),Na))),hAPP(real,real,uminus_uminus(real),Xa))))) )
          & ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real)))
           => hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = zero_zero(real) ) ) ) ) ).

fof(fact_4164_arccos__def,axiom,
    ! [Ya] : hAPP(real,real,arccos,Ya) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),pi))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),cos)),Ya)))) ).

fof(fact_4165_pi__def,axiom,
    pi = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),cos)),zero_zero(real)))))) ).

fof(fact_4166_pi__half,axiom,
    hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),cos)),zero_zero(real))))) ).

fof(fact_4167_mono__SucI1,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X_2] :
          ( ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(nat,X_a,X_2,hAPP(nat,nat,suc,N_1))))
         => hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)) ) ) ).

fof(fact_4168_mono__SucI2,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X_2] :
          ( ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,hAPP(nat,nat,suc,N_1))),hAPP(nat,X_a,X_2,N_1)))
         => hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)) ) ) ).

fof(fact_4169_split__paired__The,axiom,
    ! [X_a,X_b,P_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),P_1) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool),X_a),hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),combb(product_prod(X_a,X_b),bool,X_b),P_1)),product_Pair(X_a,X_b)))) ).

fof(fact_4170_The__split__eq,axiom,
    ! [X_a,X_b,Xa,Ya] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(X_a,fun(X_a,bool),fequal(X_a),Xa)))),hAPP(X_b,fun(X_b,bool),fequal(X_b),Ya)))) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),Ya) ).

fof(fact_4171_divmod__nat__def,axiom,
    ! [Ma,Na] : hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,Ma),Na) = hAPP(fun(product_prod(nat,nat),bool),product_prod(nat,nat),the(product_prod(nat,nat)),hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,Ma),Na)) ).

fof(fact_4172_The__split,axiom,
    ! [X_a,X_b,P_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),P_1)) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool),hAPP(fun(product_prod(X_a,X_b),fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool)),combs(product_prod(X_a,X_b),X_b,bool),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool))),combb(X_a,fun(X_b,bool),product_prod(X_a,X_b)),P_1),product_fst(X_a,X_b))),product_snd(X_a,X_b))) ).

fof(fact_4173_floor__real__def,axiom,
    ! [Xa] : hAPP(real,int,archim2085082626_floor(real),Xa) = hAPP(fun(int,bool),int,the(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(real,fun(int,bool),hAPP(fun(int,fun(real,bool)),fun(real,fun(int,bool)),combc(int,real,bool),hAPP(fun(int,real),fun(int,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(int,real),fun(int,fun(real,bool))),combb(real,fun(real,bool),int),ord_less_eq(real)),ring_1_of_int(real))),Xa))),hAPP(fun(int,real),fun(int,bool),hAPP(fun(real,bool),fun(fun(int,real),fun(int,bool)),combb(real,bool,int),hAPP(real,fun(real,bool),ord_less(real),Xa)),hAPP(fun(int,int),fun(int,real),hAPP(fun(int,real),fun(fun(int,int),fun(int,real)),combb(int,real,int),ring_1_of_int(real)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),one_one(int)))))) ).

fof(fact_4174_floor__rat__def,axiom,
    ! [Xa] : hAPP(rat,int,archim2085082626_floor(rat),Xa) = hAPP(fun(int,bool),int,the(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(rat,fun(int,bool),hAPP(fun(int,fun(rat,bool)),fun(rat,fun(int,bool)),combc(int,rat,bool),hAPP(fun(int,rat),fun(int,fun(rat,bool)),hAPP(fun(rat,fun(rat,bool)),fun(fun(int,rat),fun(int,fun(rat,bool))),combb(rat,fun(rat,bool),int),ord_less_eq(rat)),ring_1_of_int(rat))),Xa))),hAPP(fun(int,rat),fun(int,bool),hAPP(fun(rat,bool),fun(fun(int,rat),fun(int,bool)),combb(rat,bool,int),hAPP(rat,fun(rat,bool),ord_less(rat),Xa)),hAPP(fun(int,int),fun(int,rat),hAPP(fun(int,rat),fun(fun(int,int),fun(int,rat)),combb(int,rat,int),ring_1_of_int(rat)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),one_one(int)))))) ).

fof(fact_4175_setprod_Oneutral,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [G,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( ! [X_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
               => hAPP(X_a,X_b,G,X_1) = one_one(X_b) )
           => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = one_one(X_b) ) ) ) ).

fof(fact_4176_setprod__pos,axiom,
    ! [X_a,X_b] :
      ( linordered_semidom(X_b)
     => ! [F,A_2] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2))) ) ) ).

fof(fact_4177_setprod__mono,axiom,
    ! [X_a,X_b] :
      ( linordered_semidom(X_b)
     => ! [G,F,A_2] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1)))
                & hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) ) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2))) ) ) ).

fof(fact_4178_setprod__nonneg,axiom,
    ! [X_a,X_b] :
      ( linordered_semidom(X_b)
     => ! [F,A_2] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2))) ) ) ).

fof(fact_4179_nat__rec__add__eq__if,axiom,
    ! [X_a,A_3,F,Va,Na] : hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),Va)),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),Na)),hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)))) ).

fof(fact_4180_RRset2norRR__def,axiom,
    ! [A_3,A_2,Ma] :
      ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
          & hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
          & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) )
       => hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3) = hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3)),Ma))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,Ma)))) )
      & ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
            & hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
            & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) )
       => hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3) = zero_zero(int) ) ) ).

fof(fact_4181_nat__rec__0,axiom,
    ! [X_a,F1,F2] : hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),zero_zero(nat)) = ti(X_a,F1) ).

fof(fact_4182_nat__rec__Suc,axiom,
    ! [X_a,F1,F2,Nat] : hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),hAPP(nat,nat,suc,Nat)) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F2,Nat),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),Nat)) ).

fof(fact_4183_aux__some,axiom,
    ! [A_3,A_2,Ma] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma))
     => ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3)),Ma))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,Ma))))),Ma))
            & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3)),Ma))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,Ma))))),hAPP(int,fun(int,bool),norRRset,Ma))) ) ) ) ) ).

fof(fact_4184_nat__rec__number__of,axiom,
    ! [X_a,A_3,F,Va] : hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),hAPP(int,nat,number_number_of(nat),Va)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),A_3),hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F,hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))))) ).

fof(fact_4185_semiring__1__class_Oof__nat__code,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [Na] : hAPP(nat,X_a,semiring_1_of_nat(X_a),Na) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),plus_plus(X_a)),one_one(X_a))),Na),zero_zero(X_a)) ) ).

fof(fact_4186_Nitpick_Oprod__def,axiom,
    ! [X_b,X_a,A_2,B_1] : hAPP(fun(X_b,bool),fun(product_prod(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(product_prod(X_a,X_b),bool)),prod(X_a,X_b),A_2),B_1) = hAPP(fun(product_prod(X_a,X_b),bool),fun(product_prod(X_a,X_b),bool),collect(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_1)))) ).

fof(fact_4187_of__nat__aux_Osimps_I1_J,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [Inc,I] : hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),zero_zero(nat)),I) = ti(X_a,I) ) ).

fof(fact_4188_of__nat__aux_Osimps_I2_J,axiom,
    ! [X_a] :
      ( semiring_1(X_a)
     => ! [Inc,Na,I] : hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),hAPP(nat,nat,suc,Na)),I) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),Na),hAPP(X_a,X_a,Inc,I)) ) ).

fof(fact_4189_Maclaurin__sin__bound,axiom,
    ! [Xa,Na] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,sin,Xa)),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,abs_abs(real),Xa)),Na)))) ).

fof(fact_4190_finite__less__ub,axiom,
    ! [U,F] :
      ( ! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,F,N_1)))
     => hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less_eq(nat)),F)),U)))) ) ).

fof(fact_4191_finite__atLeastLessThan,axiom,
    ! [L,U] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),U))) ).

fof(fact_4192_setsum__abs,axiom,
    ! [X_b,X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [F,A_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2))),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2))) ) ).

fof(fact_4193_setsum__abs__ge__zero,axiom,
    ! [X_b,X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [F,A_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2))) ) ).

fof(fact_4194_sums__split__initial__segment,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [K,F,Sb] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Sb))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Sb),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))))) ) ) ).

fof(fact_4195_suminf__split__initial__segment,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [K,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))) ) ) ).

fof(fact_4196_suminf__minus__initial__segment,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [K,Sb,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => ( hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = ti(X_a,Sb)
           => hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Sb),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ) ) ) ).

fof(fact_4197_geometric__sum,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [Na,Xa] :
          ( ti(X_a,Xa) != one_one(X_a)
         => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)),one_one(X_a))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Xa),one_one(X_a))) ) ) ).

fof(fact_4198_sumr__group,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,K,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),K)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K))) ) ).

fof(fact_4199_sumr__offset3,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Na,K] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ) ).

fof(fact_4200_sumr__offset4,axiom,
    ! [K,N_1,F_3] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_1))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ).

fof(fact_4201_lemma__realpow__rev__sumr,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & power(X_a) )
     => ! [Xa,Ya,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na))) ) ).

fof(fact_4202_lemma__realpow__diff__sumr,axiom,
    ! [X_a] :
      ( ( monoid_mult(X_a)
        & comm_semiring_0(X_a) )
     => ! [Xa,Ya,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,Na))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na)))) ) ).

fof(fact_4203_sumr__offset,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [F,K,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K)))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ) ).

fof(fact_4204_sumr__offset2,axiom,
    ! [K,Na,F_3] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K)))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ).

fof(fact_4205_sumr__diff__mult__const,axiom,
    ! [F,Na,Ra] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),Ra)) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),F)),Ra)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) ).

fof(fact_4206_norm__setsum,axiom,
    ! [X_b,X_a] :
      ( real_normed_vector(X_a)
     => ! [F,A_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2))),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),hAPP(fun(X_b,X_a),fun(X_b,real),hAPP(fun(X_a,real),fun(fun(X_b,X_a),fun(X_b,real)),combb(X_a,real,X_b),norm_norm(X_a)),F)),A_2))) ) ).

fof(fact_4207_setsum__shift__bounds__nat__ivl,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Ma,K,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na)) ) ).

fof(fact_4208_divide_Osetsum,axiom,
    ! [X_b,X_a] :
      ( real_normed_field(X_a)
     => ! [G,A_2,Ya] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)),Ya) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),inverse_divide(X_a)),G)),Ya)),A_2) ) ).

fof(fact_4209_setsum__shift__bounds__Suc__ivl,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Ma,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,Ma)),hAPP(nat,nat,suc,Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na)) ) ).

fof(fact_4210_setsum__setsum__restrict,axiom,
    ! [X_b,X_a,X_c] :
      ( comm_monoid_add(X_c)
     => ! [F,R_2,T_3,S] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
         => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_3))
           => hAPP(fun(X_a,bool),X_c,hAPP(fun(X_a,X_c),fun(fun(X_a,bool),X_c),big_co1399186613setsum(X_a,X_c),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,X_c),hAPP(fun(X_a,fun(fun(X_b,bool),X_c)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,X_c)),combs(X_a,fun(X_b,bool),X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_a,fun(fun(X_b,bool),X_c)),hAPP(fun(fun(X_b,X_c),fun(fun(X_b,bool),X_c)),fun(fun(X_a,fun(X_b,X_c)),fun(X_a,fun(fun(X_b,bool),X_c))),combb(fun(X_b,X_c),fun(fun(X_b,bool),X_c),X_a),big_co1399186613setsum(X_b,X_c)),F)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),T_3)))),R_2)))),S) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,X_c)),combs(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),collect(X_a)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S)))),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_2))))),T_3) ) ) ) ).

fof(fact_4211_of__real_Osetsum,axiom,
    ! [X_b,X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [G,A_2] : hAPP(real,X_a,of_real(X_a),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),G),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,real),fun(X_b,X_a),hAPP(fun(real,X_a),fun(fun(X_b,real),fun(X_b,X_a)),combb(real,X_a,X_b),of_real(X_a)),G)),A_2) ) ).

fof(fact_4212_mult__left_Osetsum,axiom,
    ! [X_b,X_a] :
      ( real_normed_algebra(X_a)
     => ! [G,A_2,Ya] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)),Ya) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),G)),Ya)),A_2) ) ).

fof(fact_4213_mult__right_Osetsum,axiom,
    ! [X_b,X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa,G,A_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),G)),A_2) ) ).

fof(fact_4214_sumr__one__lb__realpow__zero,axiom,
    ! [F,Na] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(real,fun(nat,real),power_power(real),zero_zero(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,zero_zero(nat))),Na)) = zero_zero(real) ).

fof(fact_4215_setsum__restrict__set_H,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [F,B_1,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1)))) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1))),F)),zero_zero(X_b))),A_2) ) ) ).

fof(fact_4216_setsum__restrict__set_H_H,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [F,P_1,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),P_1))) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),P_1)),F)),zero_zero(X_b))),A_2) ) ) ).

fof(fact_4217_of__int__setsum,axiom,
    ! [X_b,X_a] :
      ( ring_1(X_a)
     => ! [F,A_2] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(fun(X_b,bool),int,hAPP(fun(X_b,int),fun(fun(X_b,bool),int),big_co1399186613setsum(X_b,int),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,int),fun(X_b,X_a),hAPP(fun(int,X_a),fun(fun(X_b,int),fun(X_b,X_a)),combb(int,X_a,X_b),ring_1_of_int(X_a)),F)),A_2) ) ).

fof(fact_4218_of__nat__setsum,axiom,
    ! [X_b,X_a] :
      ( semiring_1(X_a)
     => ! [F,A_2] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,nat),fun(X_b,X_a),hAPP(fun(nat,X_a),fun(fun(X_b,nat),fun(X_b,X_a)),combb(nat,X_a,X_b),semiring_1_of_nat(X_a)),F)),A_2) ) ).

fof(fact_4219_sumr__diff__mult__const2,axiom,
    ! [X_a] :
      ( ring_1(X_a)
     => ! [F,Na,Ra] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),Ra)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),Ra)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) ) ).

fof(fact_4220_ivl__diff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Ma,I,Na] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),I),Na))
         => hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),I),Ma)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),I),Na)) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),Na),Ma) ) ) ).

fof(fact_4221_setsum__shift__lb__Suc0__0__upt,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [K,F] :
          ( hAPP(nat,X_a,F,zero_zero(nat)) = zero_zero(X_a)
         => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,zero_zero(nat))),K)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K)) ) ) ).

fof(fact_4222_atLeastLessThan__inj_I2_J,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
             => ti(X_a,B_2) = ti(X_a,D_1) ) ) ) ) ).

fof(fact_4223_atLeastLessThan__inj_I1_J,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1)
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
             => ti(X_a,A_3) = ti(X_a,C) ) ) ) ) ).

fof(fact_4224_atLeastLessThan__eq__iff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [C,D_1,A_3,B_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
           => ( hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1)
            <=> ( ti(X_a,A_3) = ti(X_a,C)
                & ti(X_a,B_2) = ti(X_a,D_1) ) ) ) ) ) ).

fof(fact_4225_ex__nat__less__eq,axiom,
    ! [P_1,Na] :
      ( ? [M_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
          & hBOOL(hAPP(nat,bool,P_1,M_2)) )
    <=> ? [X_1] :
          ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)))
          & hBOOL(hAPP(nat,bool,P_1,X_1)) ) ) ).

fof(fact_4226_all__nat__less__eq,axiom,
    ! [P_1,Na] :
      ( ! [M_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
         => hBOOL(hAPP(nat,bool,P_1,M_2)) )
    <=> ! [X_1] :
          ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)))
         => hBOOL(hAPP(nat,bool,P_1,X_1)) ) ) ).

fof(fact_4227_card__atLeastLessThan,axiom,
    ! [L,U] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),U)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),U),L) ).

fof(fact_4228_setsum__add__nat__ivl,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,P_4,Ma,Na] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),P_4))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Na),P_4))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),P_4)) ) ) ) ).

fof(fact_4229_setsum__diff__nat__ivl,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [F,P_4,Ma,Na] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),P_4))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),P_4))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Na),P_4)) ) ) ) ).

fof(fact_4230_atLeastLessThan__subset__iff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1)))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3))
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) ) ) ) ) ).

fof(fact_4231_setsum__op__ivl__Suc,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Na,Ma] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma))
           => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),hAPP(nat,nat,suc,Na))) = zero_zero(X_a) )
          & ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma))
           => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na))),hAPP(nat,X_a,F,Na)) ) ) ) ).

fof(fact_4232_setsum__head__upt__Suc,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Ma,Na] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))
         => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,Ma)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,Ma)),Na))) ) ) ).

fof(fact_4233_sumr__geometric,axiom,
    ! [Na,Xa] :
      ( Xa != one_one(real)
     => hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na)),one_one(real))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),one_one(real))) ) ).

fof(fact_4234_setsum__infinite,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [F,A_2] :
          ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = zero_zero(X_b) ) ) ).

fof(fact_4235_setsum_Oinfinite,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [G,A_2] :
          ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = zero_zero(X_b) ) ) ).

fof(fact_4236_setsum__delta,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [B_2,A_3,S] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
         => ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),zero_zero(X_b))),S) = hAPP(X_a,X_b,B_2,A_3) )
            & ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),zero_zero(X_b))),S) = zero_zero(X_b) ) ) ) ) ).

fof(fact_4237_setsum__delta_H,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [B_2,A_3,S] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
         => ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),A_3))),B_2)),zero_zero(X_b))),S) = hAPP(X_a,X_b,B_2,A_3) )
            & ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S))
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),A_3))),B_2)),zero_zero(X_b))),S) = zero_zero(X_b) ) ) ) ) ).

fof(fact_4238_setsum__negf,axiom,
    ! [X_b,X_a] :
      ( ab_group_add(X_a)
     => ! [F,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),uminus_uminus(X_a)),F)),A_2) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)) ) ).

fof(fact_4239_setsum__0,axiom,
    ! [X_b,X_a] :
      ( comm_monoid_add(X_a)
     => ! [A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),zero_zero(X_a))),A_2) = zero_zero(X_a) ) ).

fof(fact_4240_setsum__product,axiom,
    ! [X_b,X_c,X_a] :
      ( semiring_0(X_a)
     => ! [F,A_2,G,B_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_c,bool),X_a,hAPP(fun(X_c,X_a),fun(fun(X_c,bool),X_a),big_co1399186613setsum(X_c,X_a),G),B_1)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_c,bool),fun(X_b,X_a),hAPP(fun(X_b,fun(fun(X_c,bool),X_a)),fun(fun(X_c,bool),fun(X_b,X_a)),combc(X_b,fun(X_c,bool),X_a),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a)),hAPP(fun(fun(X_c,X_a),fun(fun(X_c,bool),X_a)),fun(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a))),combb(fun(X_c,X_a),fun(fun(X_c,bool),X_a),X_b),big_co1399186613setsum(X_c,X_a)),hAPP(fun(X_c,X_a),fun(X_b,fun(X_c,X_a)),hAPP(fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a))),fun(fun(X_c,X_a),fun(X_b,fun(X_c,X_a))),combc(X_b,fun(X_c,X_a),fun(X_c,X_a)),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a))),hAPP(fun(fun(X_a,X_a),fun(fun(X_c,X_a),fun(X_c,X_a))),fun(fun(X_b,fun(X_a,X_a)),fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a)))),combb(fun(X_a,X_a),fun(fun(X_c,X_a),fun(X_c,X_a)),X_b),combb(X_a,X_a,X_c)),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F))),G))),B_1)),A_2) ) ).

fof(fact_4241_setsum__left__distrib,axiom,
    ! [X_b,X_a] :
      ( semiring_0(X_a)
     => ! [F,A_2,Ra] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),Ra) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F)),Ra)),A_2) ) ).

fof(fact_4242_setsum__right__distrib,axiom,
    ! [X_b,X_a] :
      ( semiring_0(X_a)
     => ! [Ra,F,A_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ra),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ra)),F)),A_2) ) ).

fof(fact_4243_setsum__commute,axiom,
    ! [X_b,X_c,X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,B_1,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_c,bool),fun(X_b,X_a),hAPP(fun(X_b,fun(fun(X_c,bool),X_a)),fun(fun(X_c,bool),fun(X_b,X_a)),combc(X_b,fun(X_c,bool),X_a),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a)),hAPP(fun(fun(X_c,X_a),fun(fun(X_c,bool),X_a)),fun(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a))),combb(fun(X_c,X_a),fun(fun(X_c,bool),X_a),X_b),big_co1399186613setsum(X_c,X_a)),F)),B_1)),A_2) = hAPP(fun(X_c,bool),X_a,hAPP(fun(X_c,X_a),fun(fun(X_c,bool),X_a),big_co1399186613setsum(X_c,X_a),hAPP(fun(X_b,bool),fun(X_c,X_a),hAPP(fun(X_c,fun(fun(X_b,bool),X_a)),fun(fun(X_b,bool),fun(X_c,X_a)),combc(X_c,fun(X_b,bool),X_a),hAPP(fun(X_c,fun(X_b,X_a)),fun(X_c,fun(fun(X_b,bool),X_a)),hAPP(fun(fun(X_b,X_a),fun(fun(X_b,bool),X_a)),fun(fun(X_c,fun(X_b,X_a)),fun(X_c,fun(fun(X_b,bool),X_a))),combb(fun(X_b,X_a),fun(fun(X_b,bool),X_a),X_c),big_co1399186613setsum(X_b,X_a)),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_c,fun(X_b,X_a)),combc(X_b,X_c,X_a),F))),A_2)),B_1) ) ).

fof(fact_4244_setsum__divide__distrib,axiom,
    ! [X_b,X_a] :
      ( field(X_a)
     => ! [F,A_2,Ra] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),Ra) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),inverse_divide(X_a)),F)),Ra)),A_2) ) ).

fof(fact_4245_setsum_Odistrib,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [G,H,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),G)),H)),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),H),A_2)) ) ) ).

fof(fact_4246_abs__setsum__abs,axiom,
    ! [X_b,X_a] :
      ( ordere142940540dd_abs(X_a)
     => ! [F,A_2] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2) ) ).

fof(fact_4247_setsum__subtractf,axiom,
    ! [X_b,X_a] :
      ( ab_group_add(X_a)
     => ! [F,G,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),minus_minus(X_a)),F)),G)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) ) ).

fof(fact_4248_setsum__addf,axiom,
    ! [X_b,X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,G,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),plus_plus(X_a)),F)),G)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) ) ).

fof(fact_4249_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L,U] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,L)),U) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U) ).

fof(fact_4250_image__add__atLeastLessThan,axiom,
    ! [K,I,J] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

fof(fact_4251_setsum__image__gen,axiom,
    ! [X_c,X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [G,F,S] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),S) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,X_b),hAPP(fun(fun(X_a,bool),X_b),fun(fun(X_c,fun(X_a,bool)),fun(X_c,X_b)),combb(fun(X_a,bool),X_b,X_c),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G)),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_c),collect(X_a)),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_c),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S)))),hAPP(fun(X_a,fun(X_c,bool)),fun(X_c,fun(X_a,bool)),combc(X_a,X_c,bool),hAPP(fun(X_a,X_c),fun(X_a,fun(X_c,bool)),hAPP(fun(X_c,fun(X_c,bool)),fun(fun(X_a,X_c),fun(X_a,fun(X_c,bool))),combb(X_c,fun(X_c,bool),X_a),fequal(X_c)),F)))))),hAPP(fun(X_a,bool),fun(X_c,bool),hAPP(fun(X_a,X_c),fun(fun(X_a,bool),fun(X_c,bool)),image(X_a,X_c),F),S)) ) ) ).

fof(fact_4252_image__Suc__atLeastLessThan,axiom,
    ! [I,J] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),suc),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,I)),hAPP(nat,nat,suc,J)) ).

fof(fact_4253_lemma__termdiff1,axiom,
    ! [X_a] :
      ( ( monoid_mult(X_a)
        & comm_ring(X_a) )
     => ! [Z,H,Ma] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H))),hAPP(nat,fun(nat,nat),minus_minus(nat),Ma)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),Ma))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Ma)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H))),hAPP(nat,fun(nat,nat),minus_minus(nat),Ma)))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z)),hAPP(nat,fun(nat,nat),minus_minus(nat),Ma))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Ma)) ) ).

fof(fact_4254_lemma__realpow__diff__sumr2,axiom,
    ! [X_a] :
      ( ( monoid_mult(X_a)
        & comm_ring(X_a) )
     => ! [Xa,Na,Ya] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(nat,nat,suc,Na))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Xa),Ya)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na)))) ) ).

fof(fact_4255_setsum__constant,axiom,
    ! [X_b,X_a] :
      ( semiring_1(X_a)
     => ! [Ya,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),Ya)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_2))),Ya) ) ).

fof(fact_4256_real__of__card,axiom,
    ! [X_a,A_2] : hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(real,fun(X_a,real),combk(real,X_a),one_one(real))),A_2) ).

fof(fact_4257_Maclaurin__zero,axiom,
    ! [Diff,Na,Xa] :
      ( Xa = zero_zero(real)
     => ( Na != zero_zero(nat)
       => hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(real,real,hAPP(nat,fun(real,real),Diff,zero_zero(nat)),zero_zero(real)) ) ) ).

fof(fact_4258_sums__group,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [K,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),K)))),hAPP(fun(nat,X_a),X_a,suminf(X_a),F))) ) ) ) ).

fof(fact_4259_subset__card__intvl__is__intvl,axiom,
    ! [A_2,K] :
      ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),A_2),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(fun(nat,bool),nat,finite_card(nat),A_2)))))
     => A_2 = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(fun(nat,bool),nat,finite_card(nat),A_2))) ) ).

fof(fact_4260_setsum__diff,axiom,
    ! [X_a,X_b] :
      ( ab_group_add(X_b)
     => ! [F,B_1,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
           => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1)) ) ) ) ).

fof(fact_4261_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [X_a] :
      ( dense_linorder(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) ) ) ) ) ).

fof(fact_4262_sum__split__even__odd,axiom,
    ! [F,G,Na] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),F)),G)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))) ).

fof(fact_4263_Maclaurin__bi__le__lemma,axiom,
    ! [X_a] :
      ( zero(X_a)
     => ! [Diff,Na] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
         => hAPP(X_a,real,hAPP(nat,fun(X_a,real),Diff,zero_zero(nat)),zero_zero(X_a)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(X_a,fun(nat,real),hAPP(fun(nat,fun(X_a,real)),fun(X_a,fun(nat,real)),combc(nat,X_a,real),Diff),zero_zero(X_a)))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,hAPP(nat,fun(X_a,real),Diff,Na),zero_zero(X_a))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),Na))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))) ) ) ).

fof(fact_4264_sumr__minus__one__realpow__zero,axiom,
    ! [Na] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na))) = zero_zero(real) ).

fof(fact_4265_lemma__termdiff2,axiom,
    ! [X_a] :
      ( field(X_a)
     => ! [Z,Na,H] :
          ( ti(X_a,H) != zero_zero(X_a)
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H)),Na)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),Na))),H)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),H),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H)))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))) ) ) ).

fof(fact_4266_lemma__STAR__cos2,axiom,
    ! [Na] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),div_div(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real))))),zero_zero(real))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),one_one(nat)),Na)) = zero_zero(real) ).

fof(fact_4267_sumr__cos__zero__one,axiom,
    ! [Na] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),div_div(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),zero_zero(real)))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na))) = one_one(real) ).

fof(fact_4268_Maclaurin__sin__expansion2,axiom,
    ! [Na,Xa] :
    ? [T_4] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
      & hAPP(real,real,sin,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_4),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),Na))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ).

fof(fact_4269_Maclaurin__cos__expansion,axiom,
    ! [Na,Xa] :
    ? [T_4] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
      & hAPP(real,real,cos,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),div_div(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),zero_zero(real)))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_4),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),Na))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ).

fof(fact_4270_finite__atLeastLessThan__int,axiom,
    ! [L,U] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U))) ).

fof(fact_4271_real__of__nat__setsum,axiom,
    ! [X_a,F,A_2] : hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,nat),fun(X_a,real),hAPP(fun(nat,real),fun(fun(X_a,nat),fun(X_a,real)),combb(nat,real,X_a),real_1(nat)),F)),A_2) ).

fof(fact_4272_real__of__int__setsum,axiom,
    ! [X_a,F,A_2] : hAPP(int,real,real_1(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,int),fun(X_a,real),hAPP(fun(int,real),fun(fun(X_a,int),fun(X_a,real)),combb(int,real,X_a),real_1(int)),F)),A_2) ).

fof(fact_4273_Re_Osetsum,axiom,
    ! [X_a,G,A_2] : hAPP(complex,real,re,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),A_2) ).

fof(fact_4274_Im_Osetsum,axiom,
    ! [X_a,G,A_2] : hAPP(complex,real,im,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),A_2) ).

fof(fact_4275_cnj_Osetsum,axiom,
    ! [X_a,G,A_2] : hAPP(complex,complex,cnj,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2)) = hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),A_2) ).

fof(fact_4276_setsum__const__mult,axiom,
    ! [X_a,C,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,int),fun(X_a,int),hAPP(fun(int,int),fun(fun(X_a,int),fun(X_a,int)),combb(int,int,X_a),hAPP(int,fun(int,int),times_times(int),C)),F)),A_2) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2)) ) ).

fof(fact_4277_transfer__nat__int__sum__prod2_I1_J,axiom,
    ! [X_a,F,A_2] : hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = hAPP(int,nat,nat_1,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2)) ).

fof(fact_4278_int__setsum,axiom,
    ! [X_a,F,A_2] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2) ).

fof(fact_4279_finite__atLeastZeroLessThan__int,axiom,
    ! [U] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U))) ).

fof(fact_4280_Sup__atLeastLessThan,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),Xa))
     => hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),Ya),Xa)) = Xa ) ).

fof(fact_4281_setsum__eq__0__iff,axiom,
    ! [X_a,F,F_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
     => ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),F_1) = zero_zero(nat)
      <=> ! [X_1] :
            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),F_1))
           => hAPP(X_a,nat,F,X_1) = zero_zero(nat) ) ) ) ).

fof(fact_4282_card__eq__setsum,axiom,
    ! [X_a,A_2] : hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2) = hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(nat,fun(X_a,nat),combk(nat,X_a),one_one(nat))),A_2) ).

fof(fact_4283_setsum__eq__Suc0__iff,axiom,
    ! [X_a,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = hAPP(nat,nat,suc,zero_zero(nat))
      <=> ? [X_1] :
            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
            & hAPP(X_a,nat,F,X_1) = hAPP(nat,nat,suc,zero_zero(nat))
            & ! [Xa_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa_1),A_2))
               => ( ti(X_a,X_1) != ti(X_a,Xa_1)
                 => hAPP(X_a,nat,F,Xa_1) = zero_zero(nat) ) ) ) ) ) ).

fof(fact_4284_setsum__eq__1__iff,axiom,
    ! [X_a,F,A_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = one_one(nat)
      <=> ? [X_1] :
            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
            & hAPP(X_a,nat,F,X_1) = one_one(nat)
            & ! [Xa_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa_1),A_2))
               => ( ti(X_a,X_1) != ti(X_a,Xa_1)
                 => hAPP(X_a,nat,F,Xa_1) = zero_zero(nat) ) ) ) ) ) ).

fof(fact_4285_card__atLeastZeroLessThan__int,axiom,
    ! [U] : hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U)) = hAPP(int,nat,nat_1,U) ).

fof(fact_4286_card__atLeastLessThan__int,axiom,
    ! [L,U] : hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U),L)) ).

fof(fact_4287_setsum__diff__nat,axiom,
    ! [X_a,F,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
       => hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),B_1)) ) ) ).

fof(fact_4288_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L,U] : hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),L),one_one(int))),U) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U) ).

fof(fact_4289_setsum__const,axiom,
    ! [X_a,C,X_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_2))
     => hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(int,fun(X_a,int),combk(int,X_a),C)),X_2) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,finite_card(X_a),X_2))) ) ).

fof(fact_4290_setsum__const2,axiom,
    ! [X_a,C,X_2] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_2))
     => hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(nat,fun(X_a,nat),combk(nat,X_a),C)),X_2)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),C)),hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,finite_card(X_a),X_2))) ) ).

fof(fact_4291_transfer__nat__int__sum__prod_I1_J,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & comm_monoid_mult(X_a) )
     => ! [F,A_2] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),hAPP(fun(int,nat),fun(int,X_a),hAPP(fun(nat,X_a),fun(fun(int,nat),fun(int,X_a)),combb(nat,X_a,int),F),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) ) ).

fof(fact_4292_image__add__int__atLeastLessThan,axiom,
    ! [L,U] : hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),L)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U),L))) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U) ).

fof(fact_4293_transfer__int__nat__sum__prod_I1_J,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & comm_monoid_mult(X_a) )
     => ! [F,A_2] :
          ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
         => hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),F),A_2) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,int),fun(nat,X_a),hAPP(fun(int,X_a),fun(fun(nat,int),fun(nat,X_a)),combb(int,X_a,nat),F),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) ) ) ).

fof(fact_4294_Maclaurin__sin__expansion,axiom,
    ! [Xa,Na] :
    ? [T_4] : hAPP(real,real,sin,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_4),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),Na))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ).

fof(fact_4295_Maclaurin__lemma2,axiom,
    ! [B_1,K,Diff,H,Na] :
      ( ! [M_2,T_4] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) )
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))) )
     => ( Na = hAPP(nat,nat,suc,K)
       => ! [M_2,T_4] :
            ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) )
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(nat,fun(real,real),Diff,M_2))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(fun(nat,bool),fun(real,real),hAPP(fun(real,fun(fun(nat,bool),real)),fun(fun(nat,bool),fun(real,real)),combc(real,fun(nat,bool),real),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real)),hAPP(fun(fun(nat,real),fun(fun(nat,bool),real)),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real))),combb(fun(nat,real),fun(fun(nat,bool),real),real),big_co1399186613setsum(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,nat),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,real)),fun(fun(nat,nat),fun(nat,fun(real,real))),combb(nat,fun(real,real),nat),Diff),hAPP(nat,fun(nat,nat),plus_plus(nat),M_2))),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),power_power(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),M_2))))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),hAPP(real,fun(real,real),times_times(real),B_1)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),M_2)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),M_2)))))))),T_4),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,nat),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,real)),fun(fun(nat,nat),fun(nat,fun(real,real))),combb(nat,fun(real,real),nat),Diff),hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M_2)))),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),T_4))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,M_2))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),T_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,M_2)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,M_2)))))))))) ) ) ) ).

fof(fact_4296_Maclaurin__exp__lt,axiom,
    ! [Na,Xa] :
      ( Xa != zero_zero(real)
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
       => ? [T_4] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_4)))
            & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
            & hAPP(real,real,exp(real),Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,exp(real),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ).

fof(fact_4297_Maclaurin__exp__le,axiom,
    ! [Na,Xa] :
    ? [T_4] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
      & hAPP(real,real,exp(real),Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,exp(real),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ).

fof(fact_4298_sumr__pos__lt__pair,axiom,
    ! [K,F] :
      ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
     => ( ! [D_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),D_2)))),hAPP(nat,real,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),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,suc,hAPP(nat,nat,suc,zero_zero(nat)))),D_2)),one_one(nat)))))))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).

fof(fact_4299_Maclaurin__lemma,axiom,
    ! [F,J,Na,H] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ? [B_6] : hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),J)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B_6),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na))))) ) ).

fof(fact_4300_real__setsum__nat__ivl__bounded2,axiom,
    ! [X_a] :
      ( linordered_semidom(X_a)
     => ! [K,F,K_3,Na] :
          ( ! [P_6] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),P_6),Na))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,P_6)),K_3)) )
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),K_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),K)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),K_3))) ) ) ) ).

fof(fact_4301_series__pos__less,axiom,
    ! [Na,F] :
      ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
     => ( ! [M_2] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),M_2))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,F,M_2))) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).

fof(fact_4302_series__pos__le,axiom,
    ! [Na,F] :
      ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
     => ( ! [M_2] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),M_2))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,M_2))) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).

fof(fact_4303_Maclaurin__all__lt__objl,axiom,
    ! [Na,Xa,Diff,F] :
      ( ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
        & ! [M_2,X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),X_1)))
        & Xa != zero_zero(real)
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) )
     => ? [T_4] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_4)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
          & hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ).

fof(fact_4304_Maclaurin__all__lt,axiom,
    ! [Xa,Na,Diff,F] :
      ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
       => ( Xa != zero_zero(real)
         => ( ! [M_2,X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),X_1)))
           => ? [T_4] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_4)))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
                & hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ) ).

fof(fact_4305_Maclaurin__bi__le,axiom,
    ! [Xa,Na,Diff,F] :
      ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
     => ( ! [M_2,T_4] :
            ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))) )
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))) )
       => ? [T_4] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
            & hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ).

fof(fact_4306_Maclaurin__objl,axiom,
    ! [Diff,F,Na,H] :
      ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
        & hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
        & ! [M_2,T_4] :
            ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) )
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))) ) )
     => ? [T_4] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_4))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_4),H))
          & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) ) ) ).

fof(fact_4307_Maclaurin2__objl,axiom,
    ! [Na,Diff,F,H] :
      ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
        & hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
        & ! [M_2,T_4] :
            ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) )
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))) ) )
     => ? [T_4] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_4))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H))
          & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) ) ) ).

fof(fact_4308_Maclaurin__minus,axiom,
    ! [Diff,F,Na,H] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),zero_zero(real)))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
       => ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
         => ( ! [M_2,T_4] :
                ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),H),T_4))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),zero_zero(real))) )
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))) )
           => ? [T_4] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),T_4))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_4),zero_zero(real)))
                & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) ) ) ) ) ) ).

fof(fact_4309_Maclaurin2,axiom,
    ! [Na,Diff,F,H] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
       => ( ! [M_2,T_4] :
              ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) )
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))) )
         => ? [T_4] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_4))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H))
              & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) ) ) ) ) ).

fof(fact_4310_Maclaurin__minus__objl,axiom,
    ! [Diff,F,Na,H] :
      ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),zero_zero(real)))
        & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
        & hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
        & ! [M_2,T_4] :
            ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),H),T_4))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),zero_zero(real))) )
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))) ) )
     => ? [T_4] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),T_4))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_4),zero_zero(real)))
          & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) ) ) ).

fof(fact_4311_Maclaurin,axiom,
    ! [Diff,F,Na,H] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
       => ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
         => ( ! [M_2,T_4] :
                ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4))
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) )
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))) )
           => ? [T_4] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_4))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_4),H))
                & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) ) ) ) ) ) ).

fof(fact_4312_Maclaurin__all__le,axiom,
    ! [Na,Xa,Diff,F] :
      ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
     => ( ! [M_2,X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),X_1)))
       => ? [T_4] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
            & hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ).

fof(fact_4313_Maclaurin__all__le__objl,axiom,
    ! [Na,Xa,Diff,F] :
      ( ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
        & ! [M_2,X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),X_1))) )
     => ? [T_4] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))
          & hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ).

fof(fact_4314_summable__Cauchy,axiom,
    ! [X_a] :
      ( banach(X_a)
     => ! [F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
        <=> ! [E] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [N_2] :
                ! [M_2] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M_2))
                 => ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M_2),N_1)))),E)) ) ) ) ) ).

fof(fact_4315_real__setsum__nat__ivl__bounded,axiom,
    ! [F,K_3,Na] :
      ( ! [P_6] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),P_6),Na))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,P_6)),K_3)) )
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),K_3))) ) ).

fof(fact_4316_DERIV__sumr,axiom,
    ! [F,Xa,F_2,Na,Ma] :
      ( ! [R_1] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),R_1))
            & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na))) )
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),F,R_1)),Xa),hAPP(real,real,hAPP(nat,fun(real,real),F_2,R_1),Xa))) )
     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(nat,bool),fun(real,real),hAPP(fun(real,fun(fun(nat,bool),real)),fun(fun(nat,bool),fun(real,real)),combc(real,fun(nat,bool),real),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real)),hAPP(fun(fun(nat,real),fun(fun(nat,bool),real)),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real))),combb(fun(nat,real),fun(fun(nat,bool),real),real),big_co1399186613setsum(nat,real)),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),F))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na))),Xa),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),F_2),Xa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na)))) ) ).

fof(fact_4317_suminf__le,axiom,
    ! [Xa,F] :
      ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_1))),Xa))
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),Xa)) ) ) ).

fof(fact_4318_series__zero,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & t2_space(X_a) )
     => ! [F,Na] :
          ( ! [M_2] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),M_2))
             => hAPP(nat,X_a,F,M_2) = zero_zero(X_a) )
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)))) ) ) ).

fof(fact_4319_setsum__mono3,axiom,
    ! [X_a,X_b] :
      ( ( comm_monoid_add(X_b)
        & ordere779506340up_add(X_b) )
     => ! [F,A_2,B_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2)))
                 => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))) )
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1))) ) ) ) ) ).

fof(fact_4320_setsum__mono2,axiom,
    ! [X_a,X_b] :
      ( ordere216010020id_add(X_b)
     => ! [F,A_2,B_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
           => ( ! [B_4] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_4),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2)))
                 => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,B_4))) )
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1))) ) ) ) ) ).

fof(fact_4321_setsum__mono__zero__left,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [F,S,T_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S)))
                 => hAPP(X_a,X_b,F,X_1) = zero_zero(X_b) )
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_3) ) ) ) ) ).

fof(fact_4322_setsum__mono__zero__right,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [F,S,T_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S)))
                 => hAPP(X_a,X_b,F,X_1) = zero_zero(X_b) )
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S) ) ) ) ) ).

fof(fact_4323_setsum__bounded,axiom,
    ! [X_a,X_b] :
      ( ( ordere779506340up_add(X_b)
        & semiring_1(X_b) )
     => ! [F,K_3,A_2] :
          ( ! [I_2] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2))
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I_2)),K_3)) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))),K_3))) ) ) ).

fof(fact_4324_transfer__nat__int__sum__prod__cong_I1_J,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & comm_monoid_mult(X_a) )
     => ! [F,G,A_2,B_1] :
          ( A_2 = B_1
         => ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
                 => hAPP(int,X_a,F,X_1) = hAPP(int,X_a,G,X_1) )
             => hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),G),B_1) ) ) ) ) ).

fof(fact_4325_setsum__multicount,axiom,
    ! [X_b,X_a,R_2,K,T_3,S] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
     => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_3))
       => ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),T_3))
             => hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_2),X_1)))) = K )
         => hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,nat),hAPP(fun(fun(X_b,bool),nat),fun(fun(X_a,fun(X_b,bool)),fun(X_a,nat)),combb(fun(X_b,bool),nat,X_a),finite_card(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),T_3)))),R_2)))),S) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),hAPP(fun(X_b,bool),nat,finite_card(X_b),T_3)) ) ) ) ).

fof(fact_4326_setsum__nonneg__leq__bound,axiom,
    ! [X_a,X_b] :
      ( ordered_ab_group_add(X_b)
     => ! [I,B_1,F,Sb] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),Sb))
         => ( ! [I_2] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),Sb))
               => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,I_2))) )
           => ( hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),Sb) = B_1
             => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),Sb))
               => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I)),B_1)) ) ) ) ) ) ).

fof(fact_4327_setsum__nonneg__0,axiom,
    ! [X_a,X_b] :
      ( ordered_ab_group_add(X_b)
     => ! [I,F,Sb] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),Sb))
         => ( ! [I_2] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),Sb))
               => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,I_2))) )
           => ( hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),Sb) = zero_zero(X_b)
             => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),Sb))
               => hAPP(X_a,X_b,F,I) = zero_zero(X_b) ) ) ) ) ) ).

fof(fact_4328_setsum__SucD,axiom,
    ! [X_a,F,A_2,Na] :
      ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = hAPP(nat,nat,suc,Na)
     => ? [X_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
          & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_1))) ) ) ).

fof(fact_4329_setsum__ivl__cong,axiom,
    ! [X_b,X_a] :
      ( ( ord(X_a)
        & comm_monoid_add(X_b) )
     => ! [F,G,B_2,D_1,A_3,C] :
          ( ti(X_a,A_3) = ti(X_a,C)
         => ( ti(X_a,B_2) = ti(X_a,D_1)
           => ( ! [X_1] :
                  ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),X_1))
                 => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),D_1))
                   => hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) ) )
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2)) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1)) ) ) ) ) ).

fof(fact_4330_setsum__multicount__gen,axiom,
    ! [X_b,X_a,R_2,K,Ta,Sb] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),Sb))
     => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),Ta))
       => ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),Ta))
             => hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),Sb))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_2),X_1)))) = hAPP(X_b,nat,K,X_1) )
         => hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,nat),hAPP(fun(fun(X_b,bool),nat),fun(fun(X_a,fun(X_b,bool)),fun(X_a,nat)),combb(fun(X_b,bool),nat,X_a),finite_card(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),Ta)))),R_2)))),Sb) = hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),K),Ta) ) ) ) ).

fof(fact_4331_DERIV__setsum,axiom,
    ! [X_a,X_b] :
      ( real_normed_field(X_b)
     => ! [F,Xa,F_2,S] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
         => ( ! [N_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_1),S))
               => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),deriv(X_b),hAPP(X_a,fun(X_b,X_b),hAPP(fun(X_b,fun(X_a,X_b)),fun(X_a,fun(X_b,X_b)),combc(X_b,X_a,X_b),F),N_1)),Xa),hAPP(X_a,X_b,hAPP(X_b,fun(X_a,X_b),F_2,Xa),N_1))) )
           => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),deriv(X_b),hAPP(fun(X_a,bool),fun(X_b,X_b),hAPP(fun(X_b,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(X_b,X_b)),combc(X_b,fun(X_a,bool),X_b),hAPP(fun(X_b,fun(X_a,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(X_b,fun(X_a,X_b)),fun(X_b,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),X_b),big_co1399186613setsum(X_a,X_b)),F)),S)),Xa),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),F_2,Xa)),S))) ) ) ) ).

fof(fact_4332_transfer__nat__int__sum__prod__closure_I1_J,axiom,
    ! [F,A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => ( ! [X_1] :
            ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,F,X_1))) )
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),F),A_2))) ) ) ).

fof(fact_4333_pos__summable,axiom,
    ! [Xa,F] :
      ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_1)))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_1))),Xa))
       => hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) ) ) ).

fof(fact_4334_setsum_Oneutral,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [G,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( ! [X_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
               => hAPP(X_a,X_b,G,X_1) = zero_zero(X_b) )
           => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = zero_zero(X_b) ) ) ) ).

fof(fact_4335_setsum__nonpos,axiom,
    ! [X_a,X_b] :
      ( ( comm_monoid_add(X_b)
        & ordere779506340up_add(X_b) )
     => ! [F,A_2] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),zero_zero(X_b))) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),zero_zero(X_b))) ) ) ).

fof(fact_4336_setsum__nonneg,axiom,
    ! [X_a,X_b] :
      ( ( comm_monoid_add(X_b)
        & ordere779506340up_add(X_b) )
     => ! [F,A_2] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2))) ) ) ).

fof(fact_4337_setsum__UNION__zero,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [F,S] :
          ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),S))
         => ( ! [X_1] :
                ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_1),S))
               => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_1)) )
           => ( ! [T1] :
                  ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),T1),S))
                 => ! [T2] :
                      ( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),T2),S))
                     => ( ti(fun(X_a,bool),T1) != ti(fun(X_a,bool),T2)
                       => ! [X_1] :
                            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),T1))
                           => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),T2))
                             => hAPP(X_a,X_b,F,X_1) = zero_zero(X_b) ) ) ) ) )
             => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),S)) = hAPP(fun(fun(X_a,bool),bool),X_b,hAPP(fun(fun(X_a,bool),X_b),fun(fun(fun(X_a,bool),bool),X_b),big_co1399186613setsum(fun(X_a,bool),X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F)),S) ) ) ) ) ).

fof(fact_4338_arith__series__int,axiom,
    ! [A_3,D_1,Na] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(int,int),fun(fun(nat,int),fun(nat,int)),combb(int,int,nat),hAPP(int,fun(int,int),plus_plus(int),A_3)),hAPP(int,fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(int,fun(nat,int)),combc(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),semiring_1_of_nat(int))),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))),D_1)))) ).

fof(fact_4339_BseqI2,axiom,
    ! [K_3,K,F] :
      ( ! [N_1] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),K),hAPP(nat,real,F,N_1)))
          & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),K_3)) )
     => hBOOL(hAPP(fun(nat,real),bool,bseq(real),F)) ) ).

fof(fact_4340_finite__lessThan,axiom,
    ! [K] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K))) ).

fof(fact_4341_lessThan__iff,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ! [I,K] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),K)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),I),K)) ) ) ).

fof(fact_4342_card__lessThan,axiom,
    ! [U] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),ord_lessThan(nat),U)) = U ).

fof(fact_4343_lessThan__eq__iff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Xa) = hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Ya)
        <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ).

fof(fact_4344_finite__nat__iff__bounded,axiom,
    ! [S] :
      ( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
    <=> ? [K_1] : hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K_1))) ) ).

fof(fact_4345_atLeast0LessThan,axiom,
    ! [Na] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na) = hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na) ).

fof(fact_4346_lessThan__subset__iff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Xa)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Ya)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) ) ) ).

fof(fact_4347_lessThan__strict__subset__iff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Ma,Na] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Ma)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Na)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ma),Na)) ) ) ).

fof(fact_4348_setsum__lessThan__Suc,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))),hAPP(nat,X_a,F,Na)) ) ).

fof(fact_4349_arith__series__nat,axiom,
    ! [A_3,D_1,Na] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),plus_plus(nat),A_3)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),D_1)))) ).

fof(fact_4350_image__atLeastZeroLessThan__int,axiom,
    ! [U] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U))
     => hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(int,nat,nat_1,U))) ) ).

fof(fact_4351_arith__series__general,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [A_3,D_1,Na] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3)),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))),D_1)))) ) ).

fof(fact_4352_finite__nat__bounded,axiom,
    ! [S] :
      ( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
     => ? [K_1] : hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K_1))) ) ).

fof(fact_4353_sum__diff__distrib,axiom,
    ! [Na,Q_1,P_1] :
      ( ! [X_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,Q_1,X_1)),hAPP(nat,nat,P_1,X_1)))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),P_1),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),Q_1),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),minus_minus(nat)),P_1)),Q_1)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na)) ) ).

fof(fact_4354_Sup__lessThan,axiom,
    ! [Xa] : hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),ord_lessThan(real),Xa)) = Xa ).

fof(fact_4355_suminf__eq__lim,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & metric_space(X_a) )
     => ! [F] : hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(fun(nat,X_a),X_a,lim(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))) ) ).

fof(fact_4356_cond__split__eta,axiom,
    ! [X_b,X_c,X_a,F,G] :
      ( ! [X_1,Y_1] : hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,X_1),Y_1) = hAPP(product_prod(X_a,X_b),X_c,G,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1))
     => hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F) = ti(fun(product_prod(X_a,X_b),X_c),G) ) ).

fof(fact_4357_def__nat__rec__Suc,axiom,
    ! [X_a,Na,F,C,H] :
      ( ! [N_1] : hAPP(nat,X_a,F,N_1) = hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),C),H),N_1)
     => hAPP(nat,X_a,F,hAPP(nat,nat,suc,Na)) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),H,Na),hAPP(nat,X_a,F,Na)) ) ).

fof(fact_4358_pigeonhole__infinite,axiom,
    ! [X_b,X_a,F,A_2] :
      ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
     => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)))
       => ? [X_1] :
            ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
            & ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,bool))),combb(X_b,fun(X_b,bool),X_a),fequal(X_b)),F)),hAPP(X_a,X_b,F,X_1)))))) ) ) ) ).

fof(fact_4359_def__nat__rec__0,axiom,
    ! [X_a,F,C,H] :
      ( ! [N_1] : hAPP(nat,X_a,F,N_1) = hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),C),H),N_1)
     => hAPP(nat,X_a,F,zero_zero(nat)) = ti(X_a,C) ) ).

fof(fact_4360_setsum__same__function__zcong,axiom,
    ! [X_a,F,G,Ma,S] :
      ( ! [X_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(X_a,int,F,X_1)),hAPP(X_a,int,G,X_1)),Ma)) )
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),S)),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),G),S)),Ma)) ) ).

fof(fact_4361_transfer__nat__int__set__cong,axiom,
    ! [P_2,P_1] :
      ( ! [X_1] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
         => ( hBOOL(hAPP(int,bool,P_1,X_1))
          <=> hBOOL(hAPP(int,bool,P_2,X_1)) ) )
     => hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1)) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_2)) ) ).

fof(fact_4362_pred__subset__eq2,axiom,
    ! [X_a,X_b,R_2,S] :
      ( hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)))),combb(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),X_a),combc(X_b,fun(product_prod(X_a,X_b),bool),bool)),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),X_a),hAPP(fun(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),combb(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),X_b),member(product_prod(X_a,X_b)))),product_Pair(X_a,X_b)))),R_2)),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)))),combb(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),X_a),combc(X_b,fun(product_prod(X_a,X_b),bool),bool)),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),X_a),hAPP(fun(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),combb(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),X_b),member(product_prod(X_a,X_b)))),product_Pair(X_a,X_b)))),S)))
    <=> hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(product_prod(X_a,X_b),bool),bool),ord_less_eq(fun(product_prod(X_a,X_b),bool)),R_2),S)) ) ).

fof(fact_4363_predicate1D,axiom,
    ! [X_a,Xa,P_1,Q_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),P_1),Q_1))
     => ( hBOOL(hAPP(X_a,bool,P_1,Xa))
       => hBOOL(hAPP(X_a,bool,Q_1,Xa)) ) ) ).

fof(fact_4364_rev__predicate1D,axiom,
    ! [X_a,Q_1,P_1,Xa] :
      ( hBOOL(hAPP(X_a,bool,P_1,Xa))
     => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),P_1),Q_1))
       => hBOOL(hAPP(X_a,bool,Q_1,Xa)) ) ) ).

fof(fact_4365_pred__equals__eq2,axiom,
    ! [X_a,X_b,S,R_2] :
      ( ! [X_1,Xa_1] :
          ( hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Xa_1)),R_2))
        <=> hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Xa_1)),S)) )
    <=> ti(fun(product_prod(X_a,X_b),bool),R_2) = ti(fun(product_prod(X_a,X_b),bool),S) ) ).

fof(fact_4366_pred__subset__eq,axiom,
    ! [X_a,R_2,S] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),R_2)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S)))
    <=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),R_2),S)) ) ).

fof(fact_4367_predicate2D,axiom,
    ! [X_a,X_b,Xa,Ya,P_1,Q_1] :
      ( hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1))
     => ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Xa),Ya))
       => hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,Xa),Ya)) ) ) ).

fof(fact_4368_rev__predicate2D,axiom,
    ! [X_b,X_a,Q_1,P_1,Xa,Ya] :
      ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Xa),Ya))
     => ( hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1))
       => hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,Xa),Ya)) ) ) ).

fof(fact_4369_predicate2I,axiom,
    ! [X_b,X_a,Q_1,P_1] :
      ( ! [X_1,Y_1] :
          ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,X_1),Y_1))
         => hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,X_1),Y_1)) )
     => hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1)) ) ).

fof(fact_4370_setsum__mono,axiom,
    ! [X_a,X_b] :
      ( ( comm_monoid_add(X_b)
        & ordere779506340up_add(X_b) )
     => ! [F,G,K_3] :
          ( ! [I_2] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),K_3))
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I_2)),hAPP(X_a,X_b,G,I_2))) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),K_3)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),K_3))) ) ) ).

fof(fact_4371_normalize__def,axiom,
    ! [P_5] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)))
       => hAPP(product_prod(int,int),product_prod(int,int),normalize,P_5) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)))) )
      & ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)))
       => ( ( hAPP(product_prod(int,int),int,product_snd(int,int),P_5) = zero_zero(int)
           => hAPP(product_prod(int,int),product_prod(int,int),normalize,P_5) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) )
          & ( hAPP(product_prod(int,int),int,product_snd(int,int),P_5) != zero_zero(int)
           => hAPP(product_prod(int,int),product_prod(int,int),normalize,P_5) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5))))) ) ) ) ) ).

fof(fact_4372_setsum__mono__zero__cong__right,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [G,F,S,T_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S)))
                 => hAPP(X_a,X_b,F,X_1) = zero_zero(X_b) )
             => ( ! [X_1] :
                    ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S))
                   => hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) )
               => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),S) ) ) ) ) ) ).

fof(fact_4373_gcd__dvd2__int,axiom,
    ! [X,Y] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)),Y)) ).

fof(fact_4374_gcd__dvd1__int,axiom,
    ! [X,Y] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)),X)) ).

fof(fact_4375_coprime__exp2__int,axiom,
    ! [N,M,A_1,B] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),M)) = one_one(int) ) ).

fof(fact_4376_divides__mult__int,axiom,
    ! [N,M,R_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),R_3))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),R_3))
       => ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) = one_one(int)
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)),R_3)) ) ) ) ).

fof(fact_4377_coprime__dvd__mult__iff__int,axiom,
    ! [Ma,K,Na] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K),Na) = one_one(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),Na)))
      <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Ma)) ) ) ).

fof(fact_4378_coprime__dvd__mult__int,axiom,
    ! [M,K_4,N] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K_4),N) = one_one(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M)) ) ) ).

fof(fact_4379_div__gcd__coprime__int,axiom,
    ! [B,A_1] :
      ( ( ti(int,A_1) != zero_zero(int)
        | ti(int,B) != zero_zero(int) )
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))) = one_one(int) ) ).

fof(fact_4380_Fract__coprime,axiom,
    ! [A_1,B] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ).

fof(fact_4381_gcd__0__int,axiom,
    ! [X] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),zero_zero(int)) = hAPP(int,int,abs_abs(int),X) ).

fof(fact_4382_gcd__0__left__int,axiom,
    ! [X] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),zero_zero(int)),X) = hAPP(int,int,abs_abs(int),X) ).

fof(fact_4383_gcd__add__mult__int,axiom,
    ! [M,K_4,N] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) ).

fof(fact_4384_gcd__mult__distrib__int,axiom,
    ! [K_4,M,N] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)) ).

fof(fact_4385_gcd__add1__int,axiom,
    ! [M,N] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),N)),N) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) ).

fof(fact_4386_gcd__add2__int,axiom,
    ! [M,N] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),N)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) ).

fof(fact_4387_gcd__proj1__if__dvd__int,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X),Y))
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,abs_abs(int),X) ) ).

fof(fact_4388_gcd__proj2__if__dvd__int,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Y),X))
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,abs_abs(int),Y) ) ).

fof(fact_4389_abs__gcd__int,axiom,
    ! [X,Y] : hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) ).

fof(fact_4390_gcd__idem__int,axiom,
    ! [X] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),X) = hAPP(int,int,abs_abs(int),X) ).

fof(fact_4391_gcd__abs__int,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,abs_abs(int),X)),hAPP(int,int,abs_abs(int),Y)) ).

fof(fact_4392_gcd__abs2__int,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),hAPP(int,int,abs_abs(int),Y)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) ).

fof(fact_4393_gcd__abs1__int,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,abs_abs(int),X)),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) ).

fof(fact_4394_coprime__plus__one__int,axiom,
    ! [N] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int))),N) = one_one(int) ).

fof(fact_4395_coprime__minus__one__int,axiom,
    ! [N] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N),one_one(int))),N) = one_one(int) ).

fof(fact_4396_gcd__1__int,axiom,
    ! [M] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),one_one(int)) = one_one(int) ).

fof(fact_4397_coprime__divisors__nat,axiom,
    ! [E_1,B,D_3,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),A_1))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E_1),B))
       => ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = one_one(int)
         => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),E_1) = one_one(int) ) ) ) ).

fof(fact_4398_coprime__exp__int,axiom,
    ! [N,D_3,A_1] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),A_1) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)) = one_one(int) ) ).

fof(fact_4399_gcd__red__int,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Y),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)) ).

fof(fact_4400_gcd__exp__int,axiom,
    ! [A_1,N,B] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),N) ).

fof(fact_4401_dvd__gcd__D2__int,axiom,
    ! [I_1,M,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),N)) ) ).

fof(fact_4402_dvd__gcd__D1__int,axiom,
    ! [I_1,M,N] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N)))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),M)) ) ).

fof(fact_4403_gcd__greatest__int,axiom,
    ! [N,K_4,M] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N))) ) ) ).

fof(fact_4404_gcd__greatest__iff__int,axiom,
    ! [K,Ma,Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Ma),Na)))
    <=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Ma))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Na)) ) ) ).

fof(fact_4405_gcd__commute__int,axiom,
    ! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),A_1) ).

fof(fact_4406_gcd__int_Oleft__commute,axiom,
    ! [B,A_1,C_1] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),C_1)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),C_1)) ).

fof(fact_4407_gcd__assoc__int,axiom,
    ! [A_1,B,C_1] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),C_1) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),C_1)) ).

fof(fact_4408_gcd__neg2__int,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),hAPP(int,int,uminus_uminus(int),Y)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) ).

fof(fact_4409_gcd__neg1__int,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,uminus_uminus(int),X)),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) ).

fof(fact_4410_gcd__zero__int,axiom,
    ! [Ma,Na] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Ma),Na) = zero_zero(int)
    <=> ( ti(int,Ma) = zero_zero(int)
        & ti(int,Na) = zero_zero(int) ) ) ).

fof(fact_4411_gcd__pos__int,axiom,
    ! [Ma,Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Ma),Na)))
    <=> ( ti(int,Ma) != zero_zero(int)
        | ti(int,Na) != zero_zero(int) ) ) ).

fof(fact_4412_coprime__mul__eq__int,axiom,
    ! [D_1,A_3,B_2] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),B_2)) = one_one(int)
    <=> ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),A_3) = one_one(int)
        & hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),B_2) = one_one(int) ) ) ).

fof(fact_4413_gcd__mult__cancel__int,axiom,
    ! [M,K_4,N] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K_4),N) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) ) ).

fof(fact_4414_coprime__mult__int,axiom,
    ! [B,D_3,A_1] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),A_1) = one_one(int)
     => ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),B) = one_one(int)
       => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) = one_one(int) ) ) ).

fof(fact_4415_coprime__lmult__int,axiom,
    ! [D_3,A_1,B] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),A_1) = one_one(int) ) ).

fof(fact_4416_coprime__rmult__int,axiom,
    ! [D_3,A_1,B] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),B) = one_one(int) ) ).

fof(fact_4417_transfer__int__nat__gcd_I1_J,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) ).

fof(fact_4418_gcd__coprime__int,axiom,
    ! [B_5,A_6,A_1,B] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) != zero_zero(int)
     => ( ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_6),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))
       => ( ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))
         => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_6),B_5) = one_one(int) ) ) ) ).

fof(fact_4419_gcd__non__0__int,axiom,
    ! [X,Y] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Y),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)) ) ).

fof(fact_4420_invertible__coprime__int,axiom,
    ! [X,Y,M] :
      ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),M) = one_one(int)
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),M) = one_one(int) ) ).

fof(fact_4421_coprime__crossproduct__int,axiom,
    ! [B_2,C,A_3,D_1] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),D_1) = one_one(int)
     => ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B_2),C) = one_one(int)
       => ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),A_3)),hAPP(int,int,abs_abs(int),C)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),B_2)),hAPP(int,int,abs_abs(int),D_1))
        <=> ( hAPP(int,int,abs_abs(int),A_3) = hAPP(int,int,abs_abs(int),B_2)
            & hAPP(int,int,abs_abs(int),C) = hAPP(int,int,abs_abs(int),D_1) ) ) ) ) ).

fof(fact_4422_gcd__unique__int,axiom,
    ! [B_2,A_3,D_1] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D_1))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),A_3))
        & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),B_2))
        & ! [E] :
            ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),A_3))
              & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),B_2)) )
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),D_1)) ) )
    <=> ti(int,D_1) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),B_2) ) ).

fof(fact_4423_gcd__le1__int,axiom,
    ! [B,A_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),A_1)) ) ).

fof(fact_4424_gcd__le2__int,axiom,
    ! [A_1,B] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),B)) ) ).

fof(fact_4425_gcd__ge__0__int,axiom,
    ! [X,Y] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y))) ).

fof(fact_4426_transfer__nat__int__gcd__closures_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y))) ) ) ).

fof(fact_4427_gcd__cases__int,axiom,
    ! [P_1,Ya,Xa] :
      ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
         => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Xa),Ya))) ) )
     => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ya),zero_zero(int)))
           => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Xa),hAPP(int,int,uminus_uminus(int),Ya)))) ) )
       => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),zero_zero(int)))
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya))
             => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,uminus_uminus(int),Xa)),Ya))) ) )
         => ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),zero_zero(int)))
             => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ya),zero_zero(int)))
               => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,uminus_uminus(int),Xa)),hAPP(int,int,uminus_uminus(int),Ya)))) ) )
           => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Xa),Ya))) ) ) ) ) ).

fof(fact_4428_gcd__code__int,axiom,
    ! [K,L] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K),L) = hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),L),zero_zero(int))),K),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),L),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K)),hAPP(int,int,abs_abs(int),L))))) ).

fof(fact_4429_coprime__common__divisor__int,axiom,
    ! [X,A_1,B] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = one_one(int)
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X),A_1))
       => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X),B))
         => hAPP(int,int,abs_abs(int),X) = one_one(int) ) ) ) ).

fof(fact_4430_quotient__of__coprime,axiom,
    ! [R_3,P_5,Q_3] :
      ( hAPP(rat,product_prod(int,int),quotient_of,R_3) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_5),Q_3) = one_one(int) ) ).

fof(fact_4431_normalize__coprime,axiom,
    ! [R_3,P_5,Q_3] :
      ( hAPP(product_prod(int,int),product_prod(int,int),normalize,R_3) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_5),Q_3) = one_one(int) ) ).

fof(fact_4432_coprime__int,axiom,
    ! [A_3,B_2] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),B_2) = one_one(int)
    <=> ! [D_2] :
          ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D_2))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_2),A_3))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_2),B_2)) )
        <=> ti(int,D_2) = one_one(int) ) ) ).

fof(fact_4433_transfer__nat__int__gcd_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)) ) ) ).

fof(fact_4434_gcd__int__def,axiom,
    ! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y)))) ).

fof(fact_4435_normalize__stable,axiom,
    ! [P_5,Q_3] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_3))
     => ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_5),Q_3) = one_one(int)
       => hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3) ) ) ).

fof(fact_4436_quotient__of__def,axiom,
    ! [Xa] : hAPP(rat,product_prod(int,int),quotient_of,Xa) = hAPP(fun(product_prod(int,int),bool),product_prod(int,int),the(product_prod(int,int)),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combs(product_prod(int,int),bool,bool),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),rat),fun(product_prod(int,int),bool),hAPP(fun(rat,bool),fun(fun(product_prod(int,int),rat),fun(product_prod(int,int),bool)),combb(rat,bool,product_prod(int,int)),hAPP(rat,fun(rat,bool),fequal(rat),Xa)),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),rat),hAPP(fun(product_prod(int,int),fun(int,rat)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),rat)),combs(product_prod(int,int),int,rat),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,rat)),hAPP(fun(int,fun(int,rat)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,rat))),combb(int,fun(int,rat),product_prod(int,int)),fract),product_fst(int,int))),product_snd(int,int))))),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combs(product_prod(int,int),bool,bool),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),bool),hAPP(fun(int,bool),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool)),combb(int,bool,product_prod(int,int)),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int))),product_snd(int,int)))),hAPP(int,fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(int,bool)),fun(int,fun(product_prod(int,int),bool)),combc(product_prod(int,int),int,bool),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int)),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),int),hAPP(fun(product_prod(int,int),fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int)),combs(product_prod(int,int),int,int),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int))),combb(int,fun(int,int),product_prod(int,int)),gcd_gcd(int)),product_fst(int,int))),product_snd(int,int)))),one_one(int))))) ).

fof(fact_4437_quotient__of__unique,axiom,
    ! [R_3] :
    ? [X_1] :
      ( R_3 = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),X_1)),hAPP(product_prod(int,int),int,product_snd(int,int),X_1))
      & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),X_1)))
      & hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),X_1)),hAPP(product_prod(int,int),int,product_snd(int,int),X_1)) = one_one(int)
      & ! [Y_1] :
          ( ( R_3 = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),Y_1)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_1))
            & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_1)))
            & hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),Y_1)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_1)) = one_one(int) )
         => Y_1 = X_1 ) ) ).

fof(fact_4438_Rat__cases__nonzero,axiom,
    ! [Q_3] :
      ( ! [A_4,B_4] :
          ( Q_3 = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_4),B_4)
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
           => ( ti(int,A_4) != zero_zero(int)
             => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_4),B_4) != one_one(int) ) ) )
     => Q_3 = zero_zero(rat) ) ).

fof(fact_4439_Rat__induct,axiom,
    ! [Q_2,P_1] :
      ( ! [A_4,B_4] :
          ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
         => ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_4),B_4) = one_one(int)
           => hBOOL(hAPP(rat,bool,P_1,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_4),B_4))) ) )
     => hBOOL(hAPP(rat,bool,P_1,Q_2)) ) ).

fof(fact_4440_Rat__cases,axiom,
    ! [Q_3] :
      ~ ! [A_4,B_4] :
          ( Q_3 = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_4),B_4)
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
           => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_4),B_4) != one_one(int) ) ) ).

fof(fact_4441_setprod__coprime__int,axiom,
    ! [X_a,F,Xa,A_2] :
      ( ! [I_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2))
         => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(X_a,int,F,I_2)),Xa) = one_one(int) )
     => hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2)),Xa) = one_one(int) ) ).

fof(fact_4442_setsum__mono__zero__cong__left,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [F,G,S,T_3] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3))
         => ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3))
           => ( ! [X_1] :
                  ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S)))
                 => hAPP(X_a,X_b,G,X_1) = zero_zero(X_b) )
             => ( ! [X_1] :
                    ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S))
                   => hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) )
               => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),T_3) ) ) ) ) ) ).

fof(fact_4443_card__setsum__aux,axiom,
    ! [Na,S] :
      ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,finite_finite_1(fun(int,bool)),S))
     => ( ! [X_1] :
            ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),S))
           => hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_1)) )
       => ( ! [X_1] :
              ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),S))
             => hAPP(fun(int,bool),nat,finite_card(int),X_1) = Na )
         => hAPP(fun(fun(int,bool),bool),nat,hAPP(fun(fun(int,bool),nat),fun(fun(fun(int,bool),bool),nat),big_co1399186613setsum(fun(int,bool),nat),finite_card(int)),S) = hAPP(fun(fun(int,bool),bool),nat,hAPP(fun(fun(int,bool),nat),fun(fun(fun(int,bool),bool),nat),big_co1399186613setsum(fun(int,bool),nat),hAPP(nat,fun(fun(int,bool),nat),combk(nat,fun(int,bool)),Na)),S) ) ) ) ).

fof(fact_4444_setprod__same__function__zcong,axiom,
    ! [X_a,F,G,Ma,S] :
      ( ! [X_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S))
         => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(X_a,int,F,X_1)),hAPP(X_a,int,G,X_1)),Ma)) )
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),S)),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),G),S)),Ma)) ) ).

fof(fact_4445_Nitpick_Ofrac__def,axiom,
    ! [X_a,A_1,B] : hAPP(int,X_a,hAPP(int,fun(int,X_a),frac(X_a),A_1),B) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B)) ).

fof(fact_4446_zOdd__def,axiom,
    zOdd = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,fun(int,bool)),fun(int,bool),hAPP(fun(fun(int,bool),bool),fun(fun(int,fun(int,bool)),fun(int,bool)),combb(fun(int,bool),bool,int),fEx(int)),hAPP(fun(int,int),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,int),fun(int,bool))),fun(fun(int,int),fun(int,fun(int,bool))),combc(int,fun(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),hAPP(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int))),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),plus_plus(int)),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(int))))) ).

fof(fact_4447_finite__image__set,axiom,
    ! [X_b,X_a,F,P_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)))
     => hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_b,fun(X_a,bool)),fun(X_b,bool)),combb(fun(X_a,bool),bool,X_b),fEx(X_a)),hAPP(fun(X_a,bool),fun(X_b,fun(X_a,bool)),hAPP(fun(X_b,fun(fun(X_a,bool),fun(X_a,bool))),fun(fun(X_a,bool),fun(X_b,fun(X_a,bool))),combc(X_b,fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_b,fun(X_a,fun(bool,bool))),fun(X_b,fun(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool))),fun(fun(X_b,fun(X_a,fun(bool,bool))),fun(X_b,fun(fun(X_a,bool),fun(X_a,bool)))),combb(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),X_b),combs(X_a,bool,bool)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,fun(bool,bool))),hAPP(fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,fun(bool,bool)))),combb(fun(X_a,bool),fun(X_a,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj)),hAPP(fun(X_a,X_b),fun(X_b,fun(X_a,bool)),hAPP(fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool))),fun(fun(X_a,X_b),fun(X_b,fun(X_a,bool))),combc(X_b,fun(X_a,X_b),fun(X_a,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_a,X_b),fun(X_a,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool)))),combb(fun(X_b,bool),fun(fun(X_a,X_b),fun(X_a,bool)),X_b),combb(X_b,bool,X_a)),fequal(X_b))),F)))),P_1))))) ) ).

fof(fact_4448_finite__Collect__bounded__ex,axiom,
    ! [X_b,X_a,Q_1,P_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)))
     => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_b,fun(X_a,bool)),fun(X_b,bool)),combb(fun(X_a,bool),bool,X_b),fEx(X_a)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),P_1))),Q_1)))))
      <=> ! [Y_1] :
            ( hBOOL(hAPP(X_a,bool,P_1,Y_1))
           => hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_a,fun(X_b,bool),hAPP(fun(X_b,fun(X_a,bool)),fun(X_a,fun(X_b,bool)),combc(X_b,X_a,bool),Q_1),Y_1)))) ) ) ) ).

fof(fact_4449_Rats__eq__int__div__int,axiom,
    field_char_0_Rats(real) = hAPP(fun(real,bool),fun(real,bool),collect(real),hAPP(fun(real,fun(int,bool)),fun(real,bool),hAPP(fun(fun(int,bool),bool),fun(fun(real,fun(int,bool)),fun(real,bool)),combb(fun(int,bool),bool,real),fEx(int)),hAPP(fun(real,fun(int,fun(int,bool))),fun(real,fun(int,bool)),hAPP(fun(fun(int,fun(int,bool)),fun(int,bool)),fun(fun(real,fun(int,fun(int,bool))),fun(real,fun(int,bool))),combb(fun(int,fun(int,bool)),fun(int,bool),real),hAPP(fun(fun(int,bool),bool),fun(fun(int,fun(int,bool)),fun(int,bool)),combb(fun(int,bool),bool,int),fEx(int))),hAPP(fun(int,bool),fun(real,fun(int,fun(int,bool))),hAPP(fun(real,fun(fun(int,bool),fun(int,fun(int,bool)))),fun(fun(int,bool),fun(real,fun(int,fun(int,bool)))),combc(real,fun(int,bool),fun(int,fun(int,bool))),hAPP(fun(real,fun(int,fun(fun(int,bool),fun(int,bool)))),fun(real,fun(fun(int,bool),fun(int,fun(int,bool)))),hAPP(fun(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,bool),fun(int,fun(int,bool)))),fun(fun(real,fun(int,fun(fun(int,bool),fun(int,bool)))),fun(real,fun(fun(int,bool),fun(int,fun(int,bool))))),combb(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,bool),fun(int,fun(int,bool))),real),combc(int,fun(int,bool),fun(int,bool))),hAPP(fun(real,fun(int,fun(int,fun(bool,bool)))),fun(real,fun(int,fun(fun(int,bool),fun(int,bool)))),hAPP(fun(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool)))),fun(fun(real,fun(int,fun(int,fun(bool,bool)))),fun(real,fun(int,fun(fun(int,bool),fun(int,bool))))),combb(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool))),real),hAPP(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),int),combs(int,bool,bool))),hAPP(fun(real,fun(int,fun(int,bool))),fun(real,fun(int,fun(int,fun(bool,bool)))),hAPP(fun(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool)))),fun(fun(real,fun(int,fun(int,bool))),fun(real,fun(int,fun(int,fun(bool,bool))))),combb(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool))),real),hAPP(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),int),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj))),hAPP(fun(int,fun(int,real)),fun(real,fun(int,fun(int,bool))),hAPP(fun(real,fun(fun(int,fun(int,real)),fun(int,fun(int,bool)))),fun(fun(int,fun(int,real)),fun(real,fun(int,fun(int,bool)))),combc(real,fun(int,fun(int,real)),fun(int,fun(int,bool))),hAPP(fun(real,fun(fun(int,real),fun(int,bool))),fun(real,fun(fun(int,fun(int,real)),fun(int,fun(int,bool)))),hAPP(fun(fun(fun(int,real),fun(int,bool)),fun(fun(int,fun(int,real)),fun(int,fun(int,bool)))),fun(fun(real,fun(fun(int,real),fun(int,bool))),fun(real,fun(fun(int,fun(int,real)),fun(int,fun(int,bool))))),combb(fun(fun(int,real),fun(int,bool)),fun(fun(int,fun(int,real)),fun(int,fun(int,bool))),real),combb(fun(int,real),fun(int,bool),int)),hAPP(fun(real,fun(real,bool)),fun(real,fun(fun(int,real),fun(int,bool))),hAPP(fun(fun(real,bool),fun(fun(int,real),fun(int,bool))),fun(fun(real,fun(real,bool)),fun(real,fun(fun(int,real),fun(int,bool)))),combb(fun(real,bool),fun(fun(int,real),fun(int,bool)),real),combb(real,bool,int)),fequal(real)))),hAPP(fun(int,real),fun(int,fun(int,real)),hAPP(fun(int,fun(fun(int,real),fun(int,real))),fun(fun(int,real),fun(int,fun(int,real))),combc(int,fun(int,real),fun(int,real)),hAPP(fun(int,fun(real,real)),fun(int,fun(fun(int,real),fun(int,real))),hAPP(fun(fun(real,real),fun(fun(int,real),fun(int,real))),fun(fun(int,fun(real,real)),fun(int,fun(fun(int,real),fun(int,real)))),combb(fun(real,real),fun(fun(int,real),fun(int,real)),int),combb(real,real,int)),hAPP(fun(int,real),fun(int,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(int,real),fun(int,fun(real,real))),combb(real,fun(real,real),int),inverse_divide(real)),real_1(int)))),real_1(int))))))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),combb(bool,bool,int),fNot),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))))) ).

fof(fact_4450_zEven__def,axiom,
    zEven = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,fun(int,bool)),fun(int,bool),hAPP(fun(fun(int,bool),bool),fun(fun(int,fun(int,bool)),fun(int,bool)),combb(fun(int,bool),bool,int),fEx(int)),hAPP(fun(int,int),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,int),fun(int,bool))),fun(fun(int,int),fun(int,fun(int,bool))),combc(int,fun(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),hAPP(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int))),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).

fof(fact_4451_summable__Leibniz_I5_J,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3))
       => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),sequentially)) ) ) ).

fof(fact_4452_setprod__dvd__setprod,axiom,
    ! [X_a,X_b] :
      ( ( comm_monoid_mult(X_b)
        & dvd(X_b) )
     => ! [F,G,A_2] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2))) ) ) ).

fof(fact_4453_Pow__Compl,axiom,
    ! [X_a,A_2] : hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)) = hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool),hAPP(fun(fun(fun(X_a,bool),bool),bool),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool)),combb(fun(fun(X_a,bool),bool),bool,fun(X_a,bool)),fEx(fun(X_a,bool))),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combs(fun(X_a,bool),bool,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool)),fun(X_a,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),combb(bool,fun(bool,bool),fun(X_a,bool)),fconj)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combb(fun(X_a,bool),bool,fun(X_a,bool))),fequal(fun(X_a,bool)))),uminus_uminus(fun(X_a,bool)))))),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool),hAPP(fun(fun(fun(X_a,bool),bool),bool),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool)),combb(fun(fun(X_a,bool),bool),bool,fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2)),pow(X_a))))) ).

fof(fact_4454_lim__def,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2] : hAPP(fun(nat,X_a),X_a,lim(X_a),X_2) = hAPP(fun(X_a,bool),X_a,the(X_a),hAPP(filter(nat),fun(X_a,bool),hAPP(fun(X_a,fun(filter(nat),bool)),fun(filter(nat),fun(X_a,bool)),combc(X_a,filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2)),sequentially)) ) ).

fof(fact_4455_limI,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2,L_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially))
         => hAPP(fun(nat,X_a),X_a,lim(X_a),X_2) = ti(X_a,L_2) ) ) ).

fof(fact_4456_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [Ra] : hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),Ra)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc))))),Ra),sequentially)) ).

fof(fact_4457_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(real,fun(nat,real),power_power(real),Xa))),zero_zero(real)),sequentially)) ) ).

fof(fact_4458_Bseq__inverse,axiom,
    ! [X_a] :
      ( real_n1866405975lgebra(X_a)
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => ( ti(X_a,A_3) != zero_zero(X_a)
           => hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),inverse_inverse(X_a)),X_2))) ) ) ) ).

fof(fact_4459_LIMSEQ__imp__Cauchy,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) ) ) ).

fof(fact_4460_LIMSEQ__power__zero,axiom,
    ! [X_a] :
      ( real_n2089651433ebra_1(X_a)
     => ! [Xa] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),one_one(real)))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),zero_zero(X_a)),sequentially)) ) ) ).

fof(fact_4461_LIMSEQ__divide__realpow__zero,axiom,
    ! [A_3,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),A_3)),hAPP(real,fun(nat,real),power_power(real),Xa))),zero_zero(real)),sequentially)) ) ).

fof(fact_4462_LIMSEQ__rabs__realpow__zero,axiom,
    ! [C] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),C)),one_one(real)))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,abs_abs(real),C))),zero_zero(real)),sequentially)) ) ).

fof(fact_4463_LIMSEQ__rabs__realpow__zero2,axiom,
    ! [C] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),C)),one_one(real)))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),C)),zero_zero(real)),sequentially)) ) ).

fof(fact_4464_LIMSEQ__unique,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [B_2,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),B_2),sequentially))
           => ti(X_a,A_3) = ti(X_a,B_2) ) ) ) ).

fof(fact_4465_tendsto__divide,axiom,
    ! [X_a,X_b] :
      ( real_normed_field(X_b)
     => ! [G,B_2,F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2))
           => ( ti(X_b,B_2) != zero_zero(X_b)
             => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),A_3),B_2)),A_2)) ) ) ) ) ).

fof(fact_4466_LIMSEQ__divide,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Y_2,B_2,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially))
           => ( ti(X_a,B_2) != zero_zero(X_a)
             => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),sequentially)) ) ) ) ) ).

fof(fact_4467_LIMSEQ__linear,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [L,X_2,Xa] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),Xa),sequentially))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),L))
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_2),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),L))),Xa),sequentially)) ) ) ) ).

fof(fact_4468_LIMSEQ__inverse__real__of__nat,axiom,
    hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc))),zero_zero(real)),sequentially)) ).

fof(fact_4469_LIMSEQ__minus,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),sequentially)) ) ) ).

fof(fact_4470_LIMSEQ__minus__cancel,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) ) ) ).

fof(fact_4471_LIMSEQ__inverse,axiom,
    ! [X_a] :
      ( real_n1866405975lgebra(X_a)
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => ( ti(X_a,A_3) != zero_zero(X_a)
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),inverse_inverse(X_a)),X_2)),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),sequentially)) ) ) ) ).

fof(fact_4472_LIMSEQ__norm__zero,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),X_2)),zero_zero(real)),sequentially))
        <=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),zero_zero(X_a)),sequentially)) ) ) ).

fof(fact_4473_tendsto__norm__zero__iff,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [F,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2)) ) ) ).

fof(fact_4474_tendsto__norm__zero,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [F,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2)) ) ) ).

fof(fact_4475_tendsto__norm__zero__cancel,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [F,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2)) ) ) ).

fof(fact_4476_tendsto__inverse,axiom,
    ! [X_a,X_b] :
      ( real_n1866405975lgebra(X_b)
     => ! [F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2))
         => ( ti(X_b,A_3) != zero_zero(X_b)
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),hAPP(X_b,X_b,inverse_inverse(X_b),A_3)),A_2)) ) ) ) ).

fof(fact_4477_tendsto__minus,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),A_3)),A_2)) ) ) ).

fof(fact_4478_tendsto__minus__cancel,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),A_3)),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)) ) ) ).

fof(fact_4479_mult__right_Otendsto,axiom,
    ! [X_a,X_b] :
      ( real_normed_algebra(X_b)
     => ! [Xa,G,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),A_3),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa),A_3)),A_2)) ) ) ).

fof(fact_4480_mult__left_Otendsto,axiom,
    ! [X_a,X_b] :
      ( real_normed_algebra(X_b)
     => ! [Ya,G,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),A_3),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Ya)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_3),Ya)),A_2)) ) ) ).

fof(fact_4481_mult_Otendsto,axiom,
    ! [X_a,X_b] :
      ( real_normed_algebra(X_b)
     => ! [G,B_2,F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_3),B_2)),A_2)) ) ) ) ).

fof(fact_4482_mult__right_OLIMSEQ,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),A_3)),sequentially)) ) ) ).

fof(fact_4483_mult__left_OLIMSEQ,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Ya,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),Ya)),sequentially)) ) ) ).

fof(fact_4484_LIMSEQ__mult,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Y_2,B_2,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially))
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),sequentially)) ) ) ) ).

fof(fact_4485_tendsto__dist,axiom,
    ! [X_a,X_b] :
      ( metric_space(X_b)
     => ! [G,Ma,F,L,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ma),A_2))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_a,fun(X_b,real)),fun(fun(X_a,X_b),fun(X_a,real)),combs(X_a,X_b,real),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,real)),hAPP(fun(X_b,fun(X_b,real)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,real))),combb(X_b,fun(X_b,real),X_a),dist_dist(X_b)),F)),G)),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),L),Ma)),A_2)) ) ) ) ).

fof(fact_4486_tendsto__const,axiom,
    ! [X_a,X_b] :
      ( topological_space(X_b)
     => ! [K,A_2] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),K),A_2)) ) ).

fof(fact_4487_tendsto__norm,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),hAPP(X_b,real,norm_norm(X_b),A_3)),A_2)) ) ) ).

fof(fact_4488_of__real_Otendsto,axiom,
    ! [X_a,X_b] :
      ( ( real_algebra_1(X_b)
        & real_normed_vector(X_b) )
     => ! [G,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),A_3),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),hAPP(real,X_b,of_real(X_b),A_3)),A_2)) ) ) ).

fof(fact_4489_of__real_OLIMSEQ,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2)),hAPP(real,X_a,of_real(X_a),A_3)),sequentially)) ) ) ).

fof(fact_4490_LIMSEQ__pow,axiom,
    ! [X_a] :
      ( ( power(X_a)
        & real_normed_algebra(X_a) )
     => ! [Ma,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(nat,fun(nat,X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),combc(nat,nat,X_a),hAPP(fun(nat,X_a),fun(nat,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(nat,X_a),fun(nat,fun(nat,X_a))),combb(X_a,fun(nat,X_a),nat),power_power(X_a)),X_2)),Ma)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Ma)),sequentially)) ) ) ).

fof(fact_4491_LIMSEQ__const__iff,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [K,L] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),K)),L),sequentially))
        <=> ti(X_a,K) = ti(X_a,L) ) ) ).

fof(fact_4492_LIMSEQ__const,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [K] : hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),K)),K),sequentially)) ) ).

fof(fact_4493_LIMSEQ__imp__Suc,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,L] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially)) ) ) ).

fof(fact_4494_LIMSEQ__Suc,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,L] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially)) ) ) ).

fof(fact_4495_LIMSEQ__Suc__iff,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,L] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially))
        <=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially)) ) ) ).

fof(fact_4496_LIMSEQ__norm,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),X_2)),hAPP(X_a,real,norm_norm(X_a),A_3)),sequentially)) ) ) ).

fof(fact_4497_divide_OLIMSEQ,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Ya,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),Ya)),sequentially)) ) ) ).

fof(fact_4498_divide_Otendsto,axiom,
    ! [X_a,X_b] :
      ( real_normed_field(X_b)
     => ! [Ya,G,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),A_3),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Ya)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),A_3),Ya)),A_2)) ) ) ).

fof(fact_4499_LIMSEQ__imp__rabs,axiom,
    ! [F,L] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),L),sequentially))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)),hAPP(real,real,abs_abs(real),L)),sequentially)) ) ).

fof(fact_4500_tendsto__add,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [G,B_2,F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_3),B_2)),A_2)) ) ) ) ).

fof(fact_4501_tendsto__diff,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [G,B_2,F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_3),B_2)),A_2)) ) ) ) ).

fof(fact_4502_LIMSEQ__diff,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Y_2,B_2,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially))
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),sequentially)) ) ) ) ).

fof(fact_4503_LIMSEQ__diff__const,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [B_2,F,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),B_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),sequentially)) ) ) ).

fof(fact_4504_LIMSEQ__offset,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,K,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),A_3),sequentially)) ) ) ).

fof(fact_4505_LIMSEQ__add,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Y_2,B_2,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially))
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)),sequentially)) ) ) ) ).

fof(fact_4506_LIMSEQ__add__const,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [B_2,F,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),F)),B_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)),sequentially)) ) ) ).

fof(fact_4507_LIMSEQ__ignore__initial__segment,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [K,F,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),A_3),sequentially)) ) ) ).

fof(fact_4508_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [Ra] : hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),Ra)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),Ra),sequentially)) ).

fof(fact_4509_LIMSEQ__add__minus,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Y_2,B_2,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially))
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),Y_2))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),B_2))),sequentially)) ) ) ) ).

fof(fact_4510_LIMSEQ__diff__approach__zero,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [F,G,L_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),G),L_2),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),G)),zero_zero(X_a)),sequentially))
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L_2),sequentially)) ) ) ) ).

fof(fact_4511_LIMSEQ__diff__approach__zero2,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [G,F,L_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L_2),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),G)),zero_zero(X_a)),sequentially))
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),G),L_2),sequentially)) ) ) ) ).

fof(fact_4512_LIMSEQ__rabs__zero,axiom,
    ! [F] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)),zero_zero(real)),sequentially))
    <=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),zero_zero(real)),sequentially)) ) ).

fof(fact_4513_LIMSEQ__neg__powr,axiom,
    ! [Sb] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Sb))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),powr),real_1(nat))),hAPP(real,real,uminus_uminus(real),Sb))),zero_zero(real)),sequentially)) ) ).

fof(fact_4514_summable__LIMSEQ__zero,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),zero_zero(X_a)),sequentially)) ) ) ).

fof(fact_4515_monoseq__le,axiom,
    ! [Xa,A_3] :
      ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3))
     => ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),Xa),sequentially))
       => ( ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,N_1)),Xa))
            & ! [M_2,N_1] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1))
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,M_2)),hAPP(nat,real,A_3,N_1))) ) )
          | ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(nat,real,A_3,N_1)))
            & ! [M_2,N_1] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1))
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,N_1)),hAPP(nat,real,A_3,M_2))) ) ) ) ) ) ).

fof(fact_4516_tendsto__mono,axiom,
    ! [X_a,X_b] :
      ( topological_space(X_b)
     => ! [F,L,A_2,A_7] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),A_2),A_7))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_7))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2)) ) ) ) ).

fof(fact_4517_LIMSEQ__le__const,axiom,
    ! [A_3,X_2,Xa] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),Xa),sequentially))
     => ( ? [N_2] :
          ! [N_1] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),hAPP(nat,real,X_2,N_1))) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),Xa)) ) ) ).

fof(fact_4518_LIMSEQ__le__const2,axiom,
    ! [A_3,X_2,Xa] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),Xa),sequentially))
     => ( ? [N_2] :
          ! [N_1] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,X_2,N_1)),A_3)) )
       => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),A_3)) ) ) ).

fof(fact_4519_LIMSEQ__le,axiom,
    ! [Y_2,Ya,X_2,Xa] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),Xa),sequentially))
     => ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Y_2),Ya),sequentially))
       => ( ? [N_2] :
            ! [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1))
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,X_2,N_1)),hAPP(nat,real,Y_2,N_1))) )
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) ) ) ) ).

fof(fact_4520_LIMSEQ__realpow__zero,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real)))
       => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),Xa)),zero_zero(real)),sequentially)) ) ) ).

fof(fact_4521_summable__sumr__LIMSEQ__suminf,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & t2_space(X_a) )
     => ! [F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),sequentially)) ) ) ).

fof(fact_4522_sums__def,axiom,
    ! [X_a] :
      ( ( comm_monoid_add(X_a)
        & topological_space(X_a) )
     => ! [F,Sb] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Sb))
        <=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),Sb),sequentially)) ) ) ).

fof(fact_4523_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [Ra] : hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),Ra)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),one_one(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))))),Ra),sequentially)) ).

fof(fact_4524_summable__Leibniz_I1_J,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3))
       => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))) ) ) ).

fof(fact_4525_zeroseq__arctan__series,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real)))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),zero_zero(real)),sequentially)) ) ).

fof(fact_4526_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = hAPP(fun(real,bool),fun(real,bool),collect(real),hAPP(fun(real,fun(int,bool)),fun(real,bool),hAPP(fun(fun(int,bool),bool),fun(fun(real,fun(int,bool)),fun(real,bool)),combb(fun(int,bool),bool,real),fEx(int)),hAPP(fun(real,fun(int,fun(nat,bool))),fun(real,fun(int,bool)),hAPP(fun(fun(int,fun(nat,bool)),fun(int,bool)),fun(fun(real,fun(int,fun(nat,bool))),fun(real,fun(int,bool))),combb(fun(int,fun(nat,bool)),fun(int,bool),real),hAPP(fun(fun(nat,bool),bool),fun(fun(int,fun(nat,bool)),fun(int,bool)),combb(fun(nat,bool),bool,int),fEx(nat))),hAPP(fun(nat,bool),fun(real,fun(int,fun(nat,bool))),hAPP(fun(real,fun(fun(nat,bool),fun(int,fun(nat,bool)))),fun(fun(nat,bool),fun(real,fun(int,fun(nat,bool)))),combc(real,fun(nat,bool),fun(int,fun(nat,bool))),hAPP(fun(real,fun(int,fun(fun(nat,bool),fun(nat,bool)))),fun(real,fun(fun(nat,bool),fun(int,fun(nat,bool)))),hAPP(fun(fun(int,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,bool),fun(int,fun(nat,bool)))),fun(fun(real,fun(int,fun(fun(nat,bool),fun(nat,bool)))),fun(real,fun(fun(nat,bool),fun(int,fun(nat,bool))))),combb(fun(int,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,bool),fun(int,fun(nat,bool))),real),combc(int,fun(nat,bool),fun(nat,bool))),hAPP(fun(real,fun(int,fun(nat,fun(bool,bool)))),fun(real,fun(int,fun(fun(nat,bool),fun(nat,bool)))),hAPP(fun(fun(int,fun(nat,fun(bool,bool))),fun(int,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(real,fun(int,fun(nat,fun(bool,bool)))),fun(real,fun(int,fun(fun(nat,bool),fun(nat,bool))))),combb(fun(int,fun(nat,fun(bool,bool))),fun(int,fun(fun(nat,bool),fun(nat,bool))),real),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(int,fun(nat,fun(bool,bool))),fun(int,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),int),combs(nat,bool,bool))),hAPP(fun(real,fun(int,fun(nat,bool))),fun(real,fun(int,fun(nat,fun(bool,bool)))),hAPP(fun(fun(int,fun(nat,bool)),fun(int,fun(nat,fun(bool,bool)))),fun(fun(real,fun(int,fun(nat,bool))),fun(real,fun(int,fun(nat,fun(bool,bool))))),combb(fun(int,fun(nat,bool)),fun(int,fun(nat,fun(bool,bool))),real),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(int,fun(nat,bool)),fun(int,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),int),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj))),hAPP(fun(int,fun(nat,real)),fun(real,fun(int,fun(nat,bool))),hAPP(fun(real,fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool)))),fun(fun(int,fun(nat,real)),fun(real,fun(int,fun(nat,bool)))),combc(real,fun(int,fun(nat,real)),fun(int,fun(nat,bool))),hAPP(fun(real,fun(fun(nat,real),fun(nat,bool))),fun(real,fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool)))),hAPP(fun(fun(fun(nat,real),fun(nat,bool)),fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool)))),fun(fun(real,fun(fun(nat,real),fun(nat,bool))),fun(real,fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool))))),combb(fun(fun(nat,real),fun(nat,bool)),fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool))),real),combb(fun(nat,real),fun(nat,bool),int)),hAPP(fun(real,fun(real,bool)),fun(real,fun(fun(nat,real),fun(nat,bool))),hAPP(fun(fun(real,bool),fun(fun(nat,real),fun(nat,bool))),fun(fun(real,fun(real,bool)),fun(real,fun(fun(nat,real),fun(nat,bool)))),combb(fun(real,bool),fun(fun(nat,real),fun(nat,bool)),real),combb(real,bool,nat)),fequal(real)))),hAPP(fun(nat,real),fun(int,fun(nat,real)),hAPP(fun(int,fun(fun(nat,real),fun(nat,real))),fun(fun(nat,real),fun(int,fun(nat,real))),combc(int,fun(nat,real),fun(nat,real)),hAPP(fun(int,fun(real,real)),fun(int,fun(fun(nat,real),fun(nat,real))),hAPP(fun(fun(real,real),fun(fun(nat,real),fun(nat,real))),fun(fun(int,fun(real,real)),fun(int,fun(fun(nat,real),fun(nat,real)))),combb(fun(real,real),fun(fun(nat,real),fun(nat,real)),int),combb(real,real,nat)),hAPP(fun(int,real),fun(int,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(int,real),fun(int,fun(real,real))),combb(real,fun(real,real),int),inverse_divide(real)),real_1(int)))),real_1(nat))))))),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),fequal(nat)),zero_zero(nat))))))) ).

fof(fact_4527_summable__Leibniz_I4_J,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3))
       => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),sequentially)) ) ) ).

fof(fact_4528_summable__Leibniz_H_I4_J,axiom,
    ! [Na,A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))
       => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)),one_one(nat)))))) ) ) ) ).

fof(fact_4529_summable__Leibniz_H_I5_J,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))
       => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))
         => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),sequentially)) ) ) ) ).

fof(fact_4530_tendsto__Complex,axiom,
    ! [X_a,G,B_2,F,A_3,Net] :
      ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),A_3),Net))
     => ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),B_2),Net))
       => hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,real),fun(X_a,complex),hAPP(fun(X_a,fun(real,complex)),fun(fun(X_a,real),fun(X_a,complex)),combs(X_a,real,complex),hAPP(fun(X_a,real),fun(X_a,fun(real,complex)),hAPP(fun(real,fun(real,complex)),fun(fun(X_a,real),fun(X_a,fun(real,complex))),combb(real,fun(real,complex),X_a),complex_1),F)),G)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2)),Net)) ) ) ).

fof(fact_4531_cnj_Otendsto,axiom,
    ! [X_a,G,A_3,A_2] :
      ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),A_3),A_2))
     => hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),hAPP(complex,complex,cnj,A_3)),A_2)) ) ).

fof(fact_4532_cnj_OLIMSEQ,axiom,
    ! [X_2,A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_2),A_3),sequentially))
     => hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2)),hAPP(complex,complex,cnj,A_3)),sequentially)) ) ).

fof(fact_4533_Im_Otendsto,axiom,
    ! [X_a,G,A_3,A_2] :
      ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),A_3),A_2))
     => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),hAPP(complex,real,im,A_3)),A_2)) ) ).

fof(fact_4534_Re_Otendsto,axiom,
    ! [X_a,G,A_3,A_2] :
      ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),A_3),A_2))
     => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),hAPP(complex,real,re,A_3)),A_2)) ) ).

fof(fact_4535_less__filter__def,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less(filter(X_a)),A_2),B_1))
    <=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),A_2),B_1))
        & ~ hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),B_1),A_2)) ) ) ).

fof(fact_4536_Re_OLIMSEQ,axiom,
    ! [X_2,A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_2),A_3),sequentially))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2)),hAPP(complex,real,re,A_3)),sequentially)) ) ).

fof(fact_4537_Im_OLIMSEQ,axiom,
    ! [X_2,A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_2),A_3),sequentially))
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2)),hAPP(complex,real,im,A_3)),sequentially)) ) ).

fof(fact_4538_LIMSEQ__Complex,axiom,
    ! [Y_2,B_2,X_2,A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),A_3),sequentially))
     => ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Y_2),B_2),sequentially))
       => hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,real),fun(nat,complex),hAPP(fun(nat,fun(real,complex)),fun(fun(nat,real),fun(nat,complex)),combs(nat,real,complex),hAPP(fun(nat,real),fun(nat,fun(real,complex)),hAPP(fun(real,fun(real,complex)),fun(fun(nat,real),fun(nat,fun(real,complex))),combb(real,fun(real,complex),nat),complex_1),X_2)),Y_2)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2)),sequentially)) ) ) ).

fof(fact_4539_Bolzano__nest__unique,axiom,
    ! [P_1,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_fst(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2)))),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_snd(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2)))),zero_zero(real)),sequentially))
           => ? [L_3] :
                ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))),L_3))
                & hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_fst(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2))),L_3),sequentially))
                & ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),L_3),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))))
                & hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_snd(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2))),L_3),sequentially)) ) ) ) ) ) ).

fof(fact_4540_summable__Leibniz_H_I3_J,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))
       => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))
         => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),sequentially)) ) ) ) ).

fof(fact_4541_summable__Leibniz_H_I2_J,axiom,
    ! [Na,A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))
       => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)))) ) ) ) ).

fof(fact_4542_LIMSEQ__iff__nz,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2,L_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially))
        <=> ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [No] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),No))
                  & ! [N_1] :
                      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1))
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,N_1)),L_2)),R_1)) ) ) ) ) ) ).

fof(fact_4543_LIMSEQ__D,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra,X_2,L_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra))
           => ? [No] :
              ! [N_1] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1))
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,N_1)),L_2))),Ra)) ) ) ) ) ).

fof(fact_4544_LIMSEQ__iff,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,L_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially))
        <=> ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [No] :
                ! [N_1] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,N_1)),L_2))),R_1)) ) ) ) ) ).

fof(fact_4545_LIMSEQ__def,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2,L_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially))
        <=> ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [No] :
                ! [N_1] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,N_1)),L_2)),R_1)) ) ) ) ) ).

fof(fact_4546_metric__LIMSEQ__D,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [Ra,X_2,L_2] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra))
           => ? [No] :
              ! [N_1] :
                ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1))
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,N_1)),L_2)),Ra)) ) ) ) ) ).

fof(fact_4547_summable__Leibniz_I2_J,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,A_3,zero_zero(nat))))
         => ! [N_1] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)),one_one(nat))))))) ) ) ) ).

fof(fact_4548_summable__Leibniz_I3_J,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3))
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,A_3,zero_zero(nat))),zero_zero(real)))
         => ! [N_1] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)),one_one(nat))))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)))))) ) ) ) ).

fof(fact_4549_Sup__atLeastAtMost,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),Xa))
     => hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),Ya),Xa)) = Xa ) ).

fof(fact_4550_atLeastatMost__subset__iff,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C),D_1)))
        <=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))
            | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) ) ) ) ) ).

fof(fact_4551_image__uminus__atLeastAtMost,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [Xa,Ya] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),uminus_uminus(X_a)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),Xa),Ya)) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Ya)),hAPP(X_a,X_a,uminus_uminus(X_a),Xa)) ) ).

fof(fact_4552_atLeastatMost__psubset__iff,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C),D_1)))
        <=> ( ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))
              | ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))
                & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1))
                & ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),A_3))
                  | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),D_1)) ) ) )
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1)) ) ) ) ).

fof(fact_4553_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [X_a] :
      ( dense_linorder(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C),D_1)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) ) ) ) ) ).

fof(fact_4554_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [X_a] :
      ( dense_linorder(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),D_1)) ) ) ) ) ).

fof(fact_4555_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [X_a] :
      ( dense_linorder(X_a)
     => ! [A_3,B_2,C,D_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C),D_1)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))
           => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))
              & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) ) ) ) ) ).

fof(fact_4556_summable,axiom,
    ! [A_3] :
      ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))
       => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))
         => hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))) ) ) ) ).

fof(fact_4557_termdiffs__aux,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [Xa,C,K_3] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3))))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),hAPP(X_a,real,norm_norm(X_a),K_3)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,X_a),hAPP(fun(fun(nat,X_a),X_a),fun(fun(X_a,fun(nat,X_a)),fun(X_a,X_a)),combb(fun(nat,X_a),X_a,X_a),suminf(X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C))),hAPP(fun(nat,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(nat,X_a),fun(X_a,fun(nat,X_a))),combc(X_a,fun(nat,X_a),fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),X_a),combs(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a))),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(X_a,fun(nat,X_a))),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combs(X_a,X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(X_a,fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(X_a,fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),X_a),combc(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a))),hAPP(fun(nat,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(nat,X_a),fun(X_a,fun(nat,X_a))),combc(X_a,fun(nat,X_a),fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),X_a),combs(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a))),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))),combi(X_a))))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))))))),zero_zero(X_a)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ) ).

fof(fact_4558_finite__atLeastAtMost,axiom,
    ! [L,U] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U))) ).

fof(fact_4559_finite__atLeastAtMost__int,axiom,
    ! [L,U] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U))) ).

fof(fact_4560_ex__nat__less,axiom,
    ! [P_1,Na] :
      ( ? [M_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),Na))
          & hBOOL(hAPP(nat,bool,P_1,M_2)) )
    <=> ? [X_1] :
          ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),Na)))
          & hBOOL(hAPP(nat,bool,P_1,X_1)) ) ) ).

fof(fact_4561_all__nat__less,axiom,
    ! [P_1,Na] :
      ( ! [M_2] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),Na))
         => hBOOL(hAPP(nat,bool,P_1,M_2)) )
    <=> ! [X_1] :
          ( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),Na)))
         => hBOOL(hAPP(nat,bool,P_1,X_1)) ) ) ).

fof(fact_4562_SetInterval_Otransfer__nat__int__set__function__closures,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))
     => hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),Xa),Ya))) ) ).

fof(fact_4563_card__atLeastAtMost,axiom,
    ! [L,U] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,U)),L) ).

fof(fact_4564_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L,U] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),hAPP(nat,nat,suc,U)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U) ).

fof(fact_4565_fact__altdef__nat,axiom,
    ! [Na] : hAPP(nat,nat,fact_fact(nat),Na) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co302546470etprod(nat,nat),combi(nat)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),Na)) ).

fof(fact_4566_image__add__atLeastAtMost,axiom,
    ! [K,I,J] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

fof(fact_4567_SetInterval_Otransfer__nat__int__set__functions_I2_J,axiom,
    ! [Ma,Na] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na))) ).

fof(fact_4568_image__Suc__atLeastAtMost,axiom,
    ! [I,J] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),suc),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,I)),hAPP(nat,nat,suc,J)) ).

fof(fact_4569_setsum__shift__bounds__cl__Suc__ivl,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Ma,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,Ma)),hAPP(nat,nat,suc,Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)) ) ).

fof(fact_4570_setsum__shift__bounds__cl__nat__ivl,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Ma,K,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)) ) ).

fof(fact_4571_Re_Ocont,axiom,
    ! [A_3] : hBOOL(hAPP(filter(complex),bool,hAPP(real,fun(filter(complex),bool),hAPP(fun(complex,real),fun(real,fun(filter(complex),bool)),tendsto(complex,real),re),hAPP(complex,real,re,A_3)),hAPP(complex,filter(complex),at(complex),A_3))) ).

fof(fact_4572_Im_Ocont,axiom,
    ! [A_3] : hBOOL(hAPP(filter(complex),bool,hAPP(real,fun(filter(complex),bool),hAPP(fun(complex,real),fun(real,fun(filter(complex),bool)),tendsto(complex,real),im),hAPP(complex,real,im,A_3)),hAPP(complex,filter(complex),at(complex),A_3))) ).

fof(fact_4573_cnj_Ocont,axiom,
    ! [A_3] : hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),cnj),hAPP(complex,complex,cnj,A_3)),hAPP(complex,filter(complex),at(complex),A_3))) ).

fof(fact_4574_cnj_OLIM,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),hAPP(complex,complex,cnj,L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4575_LIM__add,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,M_1,F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_1),hAPP(X_a,filter(X_a),at(X_a),A_3)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),L_2),M_1)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4576_LIM__diff,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,Ma,F,L,Xa] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Xa)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ma),hAPP(X_a,filter(X_a),at(X_a),Xa)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),L),Ma)),hAPP(X_a,filter(X_a),at(X_a),Xa))) ) ) ) ).

fof(fact_4577_LIM__rabs,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),hAPP(real,real,abs_abs(real),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4578_divide_Ocont,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Ya,A_3] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),inverse_divide(X_a)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),Ya)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ).

fof(fact_4579_divide_OLIM,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_field(X_b) )
     => ! [Ya,G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Ya)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),L),Ya)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4580_LIM__const,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & topological_space(X_b) )
     => ! [K,Xa] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),K),hAPP(X_a,filter(X_a),at(X_a),Xa))) ) ).

fof(fact_4581_LIM__ident,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [A_3] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),combi(X_a)),A_3),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ).

fof(fact_4582_LIM__const__not__eq,axiom,
    ! [X_a,X_b] :
      ( ( real_n2089651433ebra_1(X_b)
        & metric_space(X_a) )
     => ! [A_3,K,L_2] :
          ( ti(X_a,K) != ti(X_a,L_2)
         => ~ hBOOL(hAPP(filter(X_b),bool,hAPP(X_a,fun(filter(X_b),bool),hAPP(fun(X_b,X_a),fun(X_a,fun(filter(X_b),bool)),tendsto(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),K)),L_2),hAPP(X_b,filter(X_b),at(X_b),A_3))) ) ) ).

fof(fact_4583_LIM__compose,axiom,
    ! [X_a,X_b,X_c] :
      ( ( topological_space(X_c)
        & topological_space(X_b)
        & topological_space(X_a) )
     => ! [F,A_3,G,L] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),hAPP(X_a,X_b,G,L)),hAPP(X_a,filter(X_a),at(X_a),L)))
         => ( hBOOL(hAPP(filter(X_c),bool,hAPP(X_a,fun(filter(X_c),bool),hAPP(fun(X_c,X_a),fun(X_a,fun(filter(X_c),bool)),tendsto(X_c,X_a),F),L),hAPP(X_c,filter(X_c),at(X_c),A_3)))
           => hBOOL(hAPP(filter(X_c),bool,hAPP(X_b,fun(filter(X_c),bool),hAPP(fun(X_c,X_b),fun(X_b,fun(filter(X_c),bool)),tendsto(X_c,X_b),hAPP(fun(X_c,X_a),fun(X_c,X_b),hAPP(fun(X_a,X_b),fun(fun(X_c,X_a),fun(X_c,X_b)),combb(X_a,X_b,X_c),G),F)),hAPP(X_a,X_b,G,L)),hAPP(X_c,filter(X_c),at(X_c),A_3))) ) ) ) ).

fof(fact_4584_LIM__const__eq,axiom,
    ! [X_a,X_b] :
      ( ( metric_space(X_b)
        & real_n2089651433ebra_1(X_a) )
     => ! [K,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ti(X_b,K) = ti(X_b,L_2) ) ) ).

fof(fact_4585_LIM__power,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & power(X_b)
        & real_normed_algebra(X_b) )
     => ! [Na,F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(nat,fun(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(nat,X_b)),hAPP(fun(X_b,fun(nat,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(nat,X_b))),combb(X_b,fun(nat,X_b),X_a),power_power(X_b)),F)),Na)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),L),Na)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4586_of__real_Ocont,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [A_3] : hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),of_real(X_a)),hAPP(real,X_a,of_real(X_a),A_3)),hAPP(real,filter(real),at(real),A_3))) ) ).

fof(fact_4587_mult__right_Ocont,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa,A_3] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ).

fof(fact_4588_mult__left_Ocont,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Ya,A_3] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),times_times(X_a)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),Ya)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ).

fof(fact_4589_mult__right_OLIM,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [Xa,G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4590_mult__left_OLIM,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [Ya,G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Ya)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),L),Ya)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4591_LIM__mult,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [G,M_1,F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_1),hAPP(X_a,filter(X_a),at(X_a),A_3)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),L_2),M_1)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4592_LIM__not__zero,axiom,
    ! [X_a,X_b] :
      ( ( real_n2089651433ebra_1(X_b)
        & zero(X_a)
        & metric_space(X_a) )
     => ! [A_3,K] :
          ( ti(X_a,K) != zero_zero(X_a)
         => ~ hBOOL(hAPP(filter(X_b),bool,hAPP(X_a,fun(filter(X_b),bool),hAPP(fun(X_b,X_a),fun(X_a,fun(filter(X_b),bool)),tendsto(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),K)),zero_zero(X_a)),hAPP(X_b,filter(X_b),at(X_b),A_3))) ) ) ).

fof(fact_4593_LIM__minus,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),L_2)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4594_LIM__inverse__fun,axiom,
    ! [X_a] :
      ( real_n1866405975lgebra(X_a)
     => ! [A_3] :
          ( ti(X_a,A_3) != zero_zero(X_a)
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),inverse_inverse(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4595_LIM__unique,axiom,
    ! [X_a,X_b] :
      ( ( metric_space(X_b)
        & real_n2089651433ebra_1(X_a) )
     => ! [M_1,F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),M_1),hAPP(X_a,filter(X_a),at(X_a),A_3)))
           => ti(X_b,L_2) = ti(X_b,M_1) ) ) ) ).

fof(fact_4596_LIM__cong__limit,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & topological_space(X_b) )
     => ! [K_3,F,L_2,Xa] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Xa)))
         => ( ti(X_b,K_3) = ti(X_b,L_2)
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),K_3),hAPP(X_a,filter(X_a),at(X_a),Xa))) ) ) ) ).

fof(fact_4597_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L,U] : hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),U),one_one(int))) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U) ).

fof(fact_4598_LIM__mult__zero,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4599_mult__left_OLIM__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [Ya,G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Ya)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4600_LIM__mult__left__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [C,F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),C)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4601_mult__right_OLIM__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [Xa,G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4602_LIM__mult__right__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [C,F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),C)),F)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4603_LIM__isCont__iff,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & topological_space(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),hAPP(X_a,X_b,F,A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3))),hAPP(X_a,X_b,F,A_3)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).

fof(fact_4604_LIM__add__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4605_LIM__offset__zero,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & topological_space(X_b) )
     => ! [F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3))),L_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).

fof(fact_4606_LIM__offset__zero__cancel,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & topological_space(X_b) )
     => ! [F,A_3,L_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3))),L_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a))))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4607_LIM__zero__iff,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4608_LIM__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4609_LIM__zero__cancel,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4610_LIM__offset,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & topological_space(X_b) )
     => ! [K,F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),plus_plus(X_a)),K))),L_2),hAPP(X_a,filter(X_a),at(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),K)))) ) ) ).

fof(fact_4611_divide_OLIM__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_field(X_b) )
     => ! [Ya,G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Ya)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4612_LIM__add__minus,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,Ma,F,L,Xa] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Xa)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ma),hAPP(X_a,filter(X_a),at(X_a),Xa)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),G))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),L),hAPP(X_b,X_b,uminus_uminus(X_b),Ma))),hAPP(X_a,filter(X_a),at(X_a),Xa))) ) ) ) ).

fof(fact_4613_LIM__inverse,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_n1866405975lgebra(X_b) )
     => ! [F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( ti(X_b,L_2) != zero_zero(X_b)
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),hAPP(X_b,X_b,inverse_inverse(X_b),L_2)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4614_setsum__shift__lb__Suc0__0,axiom,
    ! [K,F] :
      ( hAPP(nat,nat,F,zero_zero(nat)) = zero_zero(nat)
     => hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,zero_zero(nat))),K)) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),K)) ) ).

fof(fact_4615_LIM__sgn,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( ti(X_b,L) != zero_zero(X_b)
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),sgn_sgn(X_b)),F)),hAPP(X_b,X_b,sgn_sgn(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4616_LIM__norm,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),hAPP(X_b,real,norm_norm(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4617_LIM__rabs__zero__cancel,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4618_LIM__rabs__zero,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4619_LIM__rabs__zero__iff,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4620_LIM__of__real,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_algebra_1(X_b)
        & real_normed_vector(X_b) )
     => ! [G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),hAPP(real,X_b,of_real(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4621_Re_OLIM,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),hAPP(complex,real,re,L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4622_Im_OLIM,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),hAPP(complex,real,im,L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4623_setsum__head__Suc,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Ma,Na] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
         => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,Ma)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,Ma)),Na))) ) ) ).

fof(fact_4624_cnj_OLIM__zero,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4625_LIM__trans,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & real_normed_vector(X_b) )
     => ! [L,F,G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),G))),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4626_DERIV__iff2,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),hAPP(X_a,X_a,F,Xa)))),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),minus_minus(X_a)),Xa))),D),hAPP(X_a,filter(X_a),at(X_a),Xa))) ) ) ).

fof(fact_4627_metric__LIM__compose2,axiom,
    ! [X_c,X_b,X_a] :
      ( ( metric_space(X_a)
        & topological_space(X_b)
        & topological_space(X_c) )
     => ! [G,C,F,B_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),B_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),C),hAPP(X_b,filter(X_b),at(X_b),B_2)))
           => ( ? [D_2] :
                  ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
                  & ! [X_1] :
                      ( ( ti(X_a,X_1) != ti(X_a,A_3)
                        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),D_2)) )
                     => hAPP(X_a,X_b,F,X_1) != ti(X_b,B_2) ) )
             => hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),C),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ) ).

fof(fact_4628_setsum__cl__ivl__Suc,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Na,Ma] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),Ma))
           => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,suc,Na))) = zero_zero(X_a) )
          & ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),Ma))
           => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na))),hAPP(nat,X_a,F,hAPP(nat,nat,suc,Na))) ) ) ) ).

fof(fact_4629_setsum__ub__add__nat,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,P_4,Ma,Na] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
         => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),P_4))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),P_4)))) ) ) ).

fof(fact_4630_card__atLeastAtMost__int,axiom,
    ! [L,U] : hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ).

fof(fact_4631_setsum__cl__ivl__Suc2,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Na,Ma] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),Ma))
           => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,suc,Na))) = zero_zero(X_a) )
          & ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),Ma))
           => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,Ma)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na))) ) ) ) ).

fof(fact_4632_DERIV__LIM__iff,axiom,
    ! [X_a] :
      ( ( inverse(X_a)
        & real_normed_vector(X_a) )
     => ! [F,A_3,D] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3)))),hAPP(X_a,X_a,F,A_3)))),combi(X_a))),D),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a))))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),hAPP(X_a,X_a,F,A_3)))),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),minus_minus(X_a)),A_3))),D),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4633_LIM__compose2,axiom,
    ! [X_c,X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & topological_space(X_b)
        & topological_space(X_c) )
     => ! [G,C,F,B_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),B_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),C),hAPP(X_b,filter(X_b),at(X_b),B_2)))
           => ( ? [D_2] :
                  ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
                  & ! [X_1] :
                      ( ( ti(X_a,X_1) != ti(X_a,A_3)
                        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),D_2)) )
                     => hAPP(X_a,X_b,F,X_1) != ti(X_b,B_2) ) )
             => hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),C),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ) ).

fof(fact_4634_LIM__norm__zero__cancel,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4635_LIM__norm__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4636_LIM__norm__zero__iff,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4637_fact__div__fact,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),Ma))
     => hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,fact_fact(nat),Ma)),hAPP(nat,nat,fact_fact(nat),Na)) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co302546470etprod(nat,nat),combi(nat)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),Ma)) ) ).

fof(fact_4638_of__real_OLIM__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_algebra_1(X_b)
        & real_normed_vector(X_b) )
     => ! [G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4639_fact__altdef__int,axiom,
    ! [Na] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na))
     => hAPP(int,int,fact_fact(int),Na) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),Na)) ) ).

fof(fact_4640_Re_OLIM__zero,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4641_Im_OLIM__zero,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4642_gauss__sum,axiom,
    ! [X_a] :
      ( comm_semiring_1(X_a)
     => ! [Na] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),semiring_1_of_nat(X_a)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),one_one(X_a))) ) ).

fof(fact_4643_setsum__natinterval__difff,axiom,
    ! [X_a] :
      ( ab_group_add(X_a)
     => ! [F,Ma,Na] :
          ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
           => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,F,Ma)),hAPP(nat,X_a,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) )
          & ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))
           => hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)) = zero_zero(X_a) ) ) ) ).

fof(fact_4644_DERIV__D,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa)))),hAPP(X_a,X_a,F,Xa)))),combi(X_a))),D),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).

fof(fact_4645_DERIV__iff,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa)))),hAPP(X_a,X_a,F,Xa)))),combi(X_a))),D),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).

fof(fact_4646_Cauchy__product,axiom,
    ! [X_a] :
      ( ( real_normed_algebra(X_a)
        & banach(X_a) )
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),A_3)))
         => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),B_2)))
           => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),A_3)),hAPP(fun(nat,X_a),X_a,suminf(X_a),B_2)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),nat),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),A_3))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),B_2)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))) ) ) ) ).

fof(fact_4647_setsum__triangle__reindex,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Na] : hAPP(fun(product_prod(nat,nat),bool),X_a,hAPP(fun(product_prod(nat,nat),X_a),fun(fun(product_prod(nat,nat),bool),X_a),big_co1399186613setsum(product_prod(nat,nat),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(product_prod(nat,nat),X_a),product_prod_case(nat,nat,X_a),F)),hAPP(fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),collect(product_prod(nat,nat)),hAPP(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_prod_case(nat,nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool))),combc(nat,nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),fun(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool)))),combb(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),nat),combc(nat,nat,bool)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(fun(nat,nat),fun(nat,fun(nat,bool))),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,fun(nat,bool)))),combb(fun(nat,nat),fun(nat,fun(nat,bool)),nat),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less(nat))),plus_plus(nat)))),Na)))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,fun(nat,X_a)),fun(fun(nat,nat),fun(nat,X_a)),combs(nat,nat,X_a),F)),minus_minus(nat)))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) ) ).

fof(fact_4648_Cauchy__product__sums,axiom,
    ! [X_a] :
      ( ( real_normed_algebra(X_a)
        & banach(X_a) )
     => ! [B_2,A_3] :
          ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),A_3)))
         => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),B_2)))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),nat),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),A_3))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),B_2)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),A_3)),hAPP(fun(nat,X_a),X_a,suminf(X_a),B_2)))) ) ) ) ).

fof(fact_4649_LIM__cos__div__sin,axiom,
    hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),cos)),sin)),zero_zero(real)),hAPP(real,filter(real),at(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).

fof(fact_4650_lemma__termdiff5,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_b) )
     => ! [G,F,K] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
           => ( ! [H_1,N_1] :
                  ( ti(X_a,H_1) != zero_zero(X_a)
                 => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),H_1)),K))
                   => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(nat,X_b,hAPP(X_a,fun(nat,X_b),G,H_1),N_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,F,N_1)),hAPP(X_a,real,norm_norm(X_a),H_1)))) ) )
             => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(X_a,X_b),hAPP(fun(fun(nat,X_b),X_b),fun(fun(X_a,fun(nat,X_b)),fun(X_a,X_b)),combb(fun(nat,X_b),X_b,X_a),suminf(X_b)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ) ) ).

fof(fact_4651_lemma__termdiff4,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_field(X_a)
        & real_normed_vector(X_b) )
     => ! [F,K_3,K] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( ! [H_1] :
                ( ti(X_a,H_1) != zero_zero(X_a)
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),H_1)),K))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,F,H_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),K_3),hAPP(X_a,real,norm_norm(X_a),H_1)))) ) )
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ) ).

fof(fact_4652_LIM__fun__less__zero,axiom,
    ! [F,L,C] :
      ( hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [R_1] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
            & ! [X_1] :
                ( ( X_1 != C
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),X_1))),R_1)) )
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,X_1)),zero_zero(real))) ) ) ) ) ).

fof(fact_4653_LIM__fun__gt__zero,axiom,
    ! [F,L,C] :
      ( hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C)))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [R_1] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
            & ! [X_1] :
                ( ( X_1 != C
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),X_1))),R_1)) )
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,F,X_1))) ) ) ) ) ).

fof(fact_4654_LIM__fun__not__zero,axiom,
    ! [F,L,C] :
      ( hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C)))
     => ( L != zero_zero(real)
       => ? [R_1] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
            & ! [X_1] :
                ( ( X_1 != C
                  & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),X_1))),R_1)) )
               => hAPP(real,real,F,X_1) != zero_zero(real) ) ) ) ) ).

fof(fact_4655_aset_I8_J,axiom,
    ! [Ta,A_2,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ! [X_1] :
          ( ! [Xa_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
             => ! [Xb] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                 => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ta),X_1))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ta),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D))) ) ) ) ).

fof(fact_4656_aset_I6_J,axiom,
    ! [Ta,A_2,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A_2))
       => ! [X_1] :
            ( ! [Xa_1] :
                ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
               => ! [Xb] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                   => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),Ta))
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)),Ta)) ) ) ) ) ).

fof(fact_4657_bset_I8_J,axiom,
    ! [Ta,B_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ta),one_one(int))),B_1))
       => ! [X_1] :
            ( ! [Xa_1] :
                ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
               => ! [Xb] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                   => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ta),X_1))
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ta),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D))) ) ) ) ) ).

fof(fact_4658_bset_I6_J,axiom,
    ! [Ta,B_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ! [X_1] :
          ( ! [Xa_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
             => ! [Xb] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                 => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),Ta))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)),Ta)) ) ) ) ).

fof(fact_4659_LIMSEQ__SEQ__conv,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2,L_2,A_3] :
          ( ! [S_2] :
              ( ( ! [N_1] : hAPP(nat,real,S_2,N_1) != A_3
                & hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),S_2),A_3),sequentially)) )
             => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),X_2),S_2)),L_2),sequentially)) )
        <=> hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),X_2),L_2),hAPP(real,filter(real),at(real),A_3))) ) ) ).

fof(fact_4660_bset_I3_J,axiom,
    ! [Ta,B_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ta),one_one(int))),B_1))
       => ! [X_1] :
            ( ! [Xa_1] :
                ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
               => ! [Xb] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                   => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
           => ( ti(int,X_1) = ti(int,Ta)
             => hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D) = ti(int,Ta) ) ) ) ) ).

fof(fact_4661_aset_I3_J,axiom,
    ! [Ta,A_2,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A_2))
       => ! [X_1] :
            ( ! [Xa_1] :
                ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
               => ! [Xb] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                   => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
           => ( ti(int,X_1) = ti(int,Ta)
             => hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D) = ti(int,Ta) ) ) ) ) ).

fof(fact_4662_aset_I7_J,axiom,
    ! [Ta,A_2,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ! [X_1] :
          ( ! [Xa_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
             => ! [Xb] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                 => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ta),X_1))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ta),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D))) ) ) ) ).

fof(fact_4663_bset_I5_J,axiom,
    ! [Ta,B_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ! [X_1] :
          ( ! [Xa_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
             => ! [Xb] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                 => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Ta))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)),Ta)) ) ) ) ).

fof(fact_4664_aset_I5_J,axiom,
    ! [Ta,A_2,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ta),A_2))
       => ! [X_1] :
            ( ! [Xa_1] :
                ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
               => ! [Xb] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                   => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Ta))
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)),Ta)) ) ) ) ) ).

fof(fact_4665_bset_I7_J,axiom,
    ! [Ta,B_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ta),B_1))
       => ! [X_1] :
            ( ! [Xa_1] :
                ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
               => ! [Xb] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                   => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
           => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ta),X_1))
             => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ta),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D))) ) ) ) ) ).

fof(fact_4666_aset_I4_J,axiom,
    ! [Ta,A_2,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ta),A_2))
       => ! [X_1] :
            ( ! [Xa_1] :
                ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
               => ! [Xb] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                   => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
           => ( ti(int,X_1) != ti(int,Ta)
             => hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D) != ti(int,Ta) ) ) ) ) ).

fof(fact_4667_bset_I4_J,axiom,
    ! [Ta,B_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ta),B_1))
       => ! [X_1] :
            ( ! [Xa_1] :
                ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
               => ! [Xb] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                   => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
           => ( ti(int,X_1) != ti(int,Ta)
             => hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D) != ti(int,Ta) ) ) ) ) ).

fof(fact_4668_periodic__finite__ex,axiom,
    ! [P_1,D_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
     => ( ! [X_1,K_1] :
            ( hBOOL(hAPP(int,bool,P_1,X_1))
          <=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))) )
       => ( ? [X1] : hBOOL(hAPP(int,bool,P_1,X1))
        <=> ? [X_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_1)))
              & hBOOL(hAPP(int,bool,P_1,X_1)) ) ) ) ) ).

fof(fact_4669_LIM__eq,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & real_normed_vector(X_b) )
     => ! [F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
        <=> ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [S_1] :
                  ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
                  & ! [X_1] :
                      ( ( ti(X_a,X_1) != ti(X_a,A_3)
                        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),S_1)) )
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_1)),L_2))),R_1)) ) ) ) ) ) ).

fof(fact_4670_LIM__D,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & real_normed_vector(X_b) )
     => ! [Ra,F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra))
           => ? [S_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
                & ! [X_1] :
                    ( ( ti(X_a,X_1) != ti(X_a,A_3)
                      & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),S_1)) )
                   => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_1)),L_2))),Ra)) ) ) ) ) ) ).

fof(fact_4671_LIM__equal2,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & topological_space(X_b) )
     => ! [L,F,G,A_3,R_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
         => ( ! [X_1] :
                ( ti(X_a,X_1) != ti(X_a,A_3)
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),R_2))
                 => hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) ) )
           => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
             => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ) ).

fof(fact_4672_aset_I10_J,axiom,
    ! [Ta,A_2,D_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D))
     => ! [X_1] :
          ( ! [Xa_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
             => ! [Xb] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                 => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Ta)))
           => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)),Ta))) ) ) ) ).

fof(fact_4673_bset_I10_J,axiom,
    ! [Ta,B_1,D_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D))
     => ! [X_1] :
          ( ! [Xa_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
             => ! [Xb] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                 => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
         => ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Ta)))
           => ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)),Ta))) ) ) ) ).

fof(fact_4674_bset_I9_J,axiom,
    ! [Ta,B_1,D_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D))
     => ! [X_1] :
          ( ! [Xa_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
             => ! [Xb] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                 => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Ta)))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)),Ta))) ) ) ) ).

fof(fact_4675_aset_I9_J,axiom,
    ! [Ta,A_2,D_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D))
     => ! [X_1] :
          ( ! [Xa_1] :
              ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
             => ! [Xb] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                 => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
         => ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Ta)))
           => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)),Ta))) ) ) ) ).

fof(fact_4676_LIM__def,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & metric_space(X_b) )
     => ! [F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
        <=> ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [S_1] :
                  ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
                  & ! [X_1] :
                      ( ( ti(X_a,X_1) != ti(X_a,A_3)
                        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),S_1)) )
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_1)),L_2)),R_1)) ) ) ) ) ) ).

fof(fact_4677_metric__LIM__equal2,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & topological_space(X_b) )
     => ! [L,F,G,A_3,R_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
         => ( ! [X_1] :
                ( ti(X_a,X_1) != ti(X_a,A_3)
               => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),R_2))
                 => hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) ) )
           => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
             => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ) ).

fof(fact_4678_metric__LIM__D,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & metric_space(X_b) )
     => ! [Ra,F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra))
           => ? [S_1] :
                ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
                & ! [X_1] :
                    ( ( ti(X_a,X_1) != ti(X_a,A_3)
                      & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),S_1)) )
                   => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_1)),L_2)),Ra)) ) ) ) ) ) ).

fof(fact_4679_LIM__imp__LIM,axiom,
    ! [X_b,X_c,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_c)
        & real_normed_vector(X_b) )
     => ! [G,Ma,F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( ! [X_1] :
                ( ti(X_a,X_1) != ti(X_a,A_3)
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,norm_norm(X_c),hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_a,X_c,G,X_1)),Ma))),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_1)),L)))) )
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),G),Ma),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4680_LIMSEQ__SEQ__conv1,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & metric_space(X_b) )
     => ! [X_2,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),X_2),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ! [S_2] :
              ( ( ! [N_1] : hAPP(nat,X_a,S_2,N_1) != ti(X_a,A_3)
                & hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),S_2),A_3),sequentially)) )
             => hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(nat,X_a),fun(nat,X_b),hAPP(fun(X_a,X_b),fun(fun(nat,X_a),fun(nat,X_b)),combb(X_a,X_b,nat),X_2),S_2)),L_2),sequentially)) ) ) ) ).

fof(fact_4681_metric__LIM__imp__LIM,axiom,
    ! [X_b,X_c,X_a] :
      ( ( topological_space(X_a)
        & metric_space(X_c)
        & metric_space(X_b) )
     => ! [G,Ma,F,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( ! [X_1] :
                ( ti(X_a,X_1) != ti(X_a,A_3)
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,hAPP(X_c,fun(X_c,real),dist_dist(X_c),hAPP(X_a,X_c,G,X_1)),Ma)),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_1)),L))) )
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),G),Ma),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4682_LIMSEQ__setprod,axiom,
    ! [X_a,X_b] :
      ( ( real_normed_algebra(X_b)
        & comm_ring_1(X_b) )
     => ! [X_2,L_2,S] :
          ( ! [N_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_1),S))
             => hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(X_a,fun(nat,X_b),X_2,N_1)),hAPP(X_a,X_b,L_2,N_1)),sequentially)) )
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(X_a,bool),fun(nat,X_b),hAPP(fun(nat,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(nat,X_b)),combc(nat,fun(X_a,bool),X_b),hAPP(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),nat),big_co302546470etprod(X_a,X_b)),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),X_2))),S)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),L_2),S)),sequentially)) ) ) ).

fof(fact_4683_LIMSEQ__setsum,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [X_2,L_2,S] :
          ( ! [N_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_1),S))
             => hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(X_a,fun(nat,X_b),X_2,N_1)),hAPP(X_a,X_b,L_2,N_1)),sequentially)) )
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(X_a,bool),fun(nat,X_b),hAPP(fun(nat,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(nat,X_b)),combc(nat,fun(X_a,bool),X_b),hAPP(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),nat),big_co1399186613setsum(X_a,X_b)),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),X_2))),S)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),L_2),S)),sequentially)) ) ) ).

fof(fact_4684_LIMSEQ__I,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X_2,L_2] :
          ( ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [No] :
                ! [N_1] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,N_1)),L_2))),R_1)) ) )
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)) ) ) ).

fof(fact_4685_LIMSEQ__inverse__zero,axiom,
    ! [X_2] :
      ( ! [R_1] :
        ? [N_2] :
        ! [N_1] :
          ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_1),hAPP(nat,real,X_2,N_1))) )
     => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),X_2)),zero_zero(real)),sequentially)) ) ).

fof(fact_4686_LIMSEQ__SEQ__conv2,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2,L_2,A_3] :
          ( ! [S_2] :
              ( ( ! [N_1] : hAPP(nat,real,S_2,N_1) != A_3
                & hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),S_2),A_3),sequentially)) )
             => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),X_2),S_2)),L_2),sequentially)) )
         => hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),X_2),L_2),hAPP(real,filter(real),at(real),A_3))) ) ) ).

fof(fact_4687_metric__LIMSEQ__I,axiom,
    ! [X_a] :
      ( metric_space(X_a)
     => ! [X_2,L_2] :
          ( ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [No] :
                ! [N_1] :
                  ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1))
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,N_1)),L_2)),R_1)) ) )
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)) ) ) ).

fof(fact_4688_LIM__I,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & real_normed_vector(X_b) )
     => ! [F,L_2,A_3] :
          ( ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [S_1] :
                  ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
                  & ! [X_1] :
                      ( ( ti(X_a,X_1) != ti(X_a,A_3)
                        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),S_1)) )
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_1)),L_2))),R_1)) ) ) )
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4689_exp__series__add,axiom,
    ! [X_a] :
      ( real_field(X_a)
     => ! [Na,Xa,Ya] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa),Ya)),Na)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),Na)) ) ).

fof(fact_4690_scaleR__minus1__left,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(int,real,number_number_of(real),min)),X) = hAPP(X_a,X_a,uminus_uminus(X_a),X) ) ).

fof(fact_4691_scaleR_Oprod__diff__prod,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X,Y,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Y)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),A_1)),B))),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))) ) ).

fof(fact_4692_sgn__scaleR,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [R_3,X] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,sgn_sgn(real),R_3)),hAPP(X_a,X_a,sgn_sgn(X_a),X)) ) ).

fof(fact_4693_real__vector_Oscale__left__distrib,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,B,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) ) ).

fof(fact_4694_scaleR_Oadd__left,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,A_6,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),A_6)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_6),B)) ) ).

fof(fact_4695_scaleR__left_Oadd,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [X,Y,Xa_2] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),Xa_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Y),Xa_2)) ) ).

fof(fact_4696_scaleR__left_Odiff,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [X,Y,Xa_2] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)),Xa_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Y),Xa_2)) ) ).

fof(fact_4697_scaleR_Odiff__left,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,A_6,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),A_6)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_6),B)) ) ).

fof(fact_4698_scaleR__left__diff__distrib,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,B,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) ) ).

fof(fact_4699_scaleR_Oadd__right,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B,B_5] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),B_5)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B_5)) ) ).

fof(fact_4700_scaleR__right_Oadd,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,X,Y] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)) ) ).

fof(fact_4701_scaleR__right_Odiff,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,X,Y] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)) ) ).

fof(fact_4702_scaleR_Odiff__right,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B,B_5] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),B_5)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B_5)) ) ).

fof(fact_4703_real__vector_Oscale__one,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),one_one(real)),X) = ti(X_a,X) ) ).

fof(fact_4704_divide_OscaleR,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [R_3,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) ) ).

fof(fact_4705_of__real__def,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [R_3] : hAPP(real,X_a,of_real(X_a),R_3) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),one_one(X_a)) ) ).

fof(fact_4706_Re_OscaleR,axiom,
    ! [R_3,X] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),hAPP(complex,real,re,X)) ).

fof(fact_4707_Im_OscaleR,axiom,
    ! [R_3,X] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),hAPP(complex,real,im,X)) ).

fof(fact_4708_cnj_OscaleR,axiom,
    ! [R_3,X] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),hAPP(complex,complex,cnj,X)) ).

fof(fact_4709_scaleR__left_OscaleR,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [R_3,X,Xa_2] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),X)),Xa_2) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_2)) ) ).

fof(fact_4710_scaleR_OscaleR__left,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [R_3,A_1,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),A_1)),B) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) ) ).

fof(fact_4711_scaleR__right_OscaleR,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra_1,R_3,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra_1),X)) ) ).

fof(fact_4712_scaleR_OscaleR__right,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,R_3,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) ) ).

fof(fact_4713_scaleR__left__commute,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,B,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) ) ).

fof(fact_4714_of__real_OscaleR,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [R_3,X] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(real,X_a,of_real(X_a),X)) ) ).

fof(fact_4715_scaleR__conv__of__real,axiom,
    ! [X_a] :
      ( real_algebra_1(X_a)
     => ! [R_3,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(real,X_a,of_real(X_a),R_3)),X) ) ).

fof(fact_4716_real__vector_Oscale__scale,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,B,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)),X) ) ).

fof(fact_4717_real__scaleR__def,axiom,
    ! [A_1,X] : hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),X) ).

fof(fact_4718_mult_OscaleR__right,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [A_1,R_3,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) ) ).

fof(fact_4719_mult__scaleR__right,axiom,
    ! [X_a] :
      ( real_algebra(X_a)
     => ! [X,A_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) ) ).

fof(fact_4720_mult__right_OscaleR,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa_2,R_3,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),X)) ) ).

fof(fact_4721_mult__scaleR__left,axiom,
    ! [X_a] :
      ( real_algebra(X_a)
     => ! [A_1,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) ) ).

fof(fact_4722_mult_OscaleR__left,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [R_3,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),A_1)),B) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) ) ).

fof(fact_4723_mult__left_OscaleR,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [R_3,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) ) ).

fof(fact_4724_complex__scaleR,axiom,
    ! [R_3,A_1,B] : hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),B)) ).

fof(fact_4725_complex__Im__scaleR,axiom,
    ! [R_3,X] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(complex,real,im,X)) ).

fof(fact_4726_complex__Re__scaleR,axiom,
    ! [R_3,X] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(complex,real,re,X)) ).

fof(fact_4727_inverse__scaleR__distrib,axiom,
    ! [X_a] :
      ( ( divisi14063676e_zero(X_a)
        & real_div_algebra(X_a) )
     => ! [A_1,X] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),X)) ) ).

fof(fact_4728_scaleR__cancel__left,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_3,Xa,Ya] :
          ( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Ya)
        <=> ( ti(X_a,Xa) = ti(X_a,Ya)
            | A_3 = zero_zero(real) ) ) ) ).

fof(fact_4729_scaleR__left__imp__eq,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [X,Y,A_1] :
          ( A_1 != zero_zero(real)
         => ( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)
           => ti(X_a,X) = ti(X_a,Y) ) ) ) ).

fof(fact_4730_scaleR__right_Ozero,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_4731_scaleR_Ozero__right,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).

fof(fact_4732_scaleR__cancel__right,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_3,Xa,B_2] :
          ( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B_2),Xa)
        <=> ( A_3 = B_2
            | ti(X_a,Xa) = zero_zero(X_a) ) ) ) ).

fof(fact_4733_scaleR__right__imp__eq,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,B,X] :
          ( ti(X_a,X) != zero_zero(X_a)
         => ( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)
           => A_1 = B ) ) ) ).

fof(fact_4734_scaleR__minus__right,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) ) ).

fof(fact_4735_scaleR_Ominus__right,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) ) ).

fof(fact_4736_scaleR__eq__0__iff,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_3,Xa] :
          ( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa) = zero_zero(X_a)
        <=> ( A_3 = zero_zero(real)
            | ti(X_a,Xa) = zero_zero(X_a) ) ) ) ).

fof(fact_4737_scaleR__left_Ozero,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),zero_zero(real)),X) = zero_zero(X_a) ) ).

fof(fact_4738_scaleR_Ozero__left,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),zero_zero(real)),B) = zero_zero(X_a) ) ).

fof(fact_4739_scaleR__left_Ominus,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [X,Xa_2] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),X)),Xa_2) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_2)) ) ).

fof(fact_4740_scaleR__minus__left,axiom,
    ! [X_a] :
      ( real_vector(X_a)
     => ! [A_1,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),A_1)),X) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) ) ).

fof(fact_4741_scaleR_Ominus__left,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),A_1)),B) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) ) ).

fof(fact_4742_scaleR__left_Osummable,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa,X_2] :
          ( hBOOL(hAPP(fun(nat,real),bool,summable(real),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa))) ) ) ).

fof(fact_4743_scaleR__right_Osummable,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2))) ) ) ).

fof(fact_4744_scaleR__right_Osuminf,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2))
         => hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2)) ) ) ).

fof(fact_4745_scaleR__right_OCauchy,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra,X_2] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2))) ) ) ).

fof(fact_4746_scaleR__right_Osums,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra,X_2,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),A_3))) ) ) ).

fof(fact_4747_scaleR__left_Osums,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa,X_2,A_3] :
          ( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),X_2),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa))) ) ) ).

fof(fact_4748_scaleR__left_OCauchy,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa,X_2] :
          ( hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_2))
         => hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa))) ) ) ).

fof(fact_4749_norm__scaleR,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [A_1,X] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,abs_abs(real),A_1)),hAPP(X_a,real,norm_norm(X_a),X)) ) ).

fof(fact_4750_sgn__div__norm,axiom,
    ! [X_a] :
      ( sgn_div_norm(X_a)
     => ! [X] : hAPP(X_a,X_a,sgn_sgn(X_a),X) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),X))),X) ) ).

fof(fact_4751_scaleR__right_Otendsto,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [Ra,G,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),A_3),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra),A_3)),A_2)) ) ) ).

fof(fact_4752_scaleR__left_Otendsto,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [Xa,G,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),A_3),A_2))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),Xa)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),A_3),Xa)),A_2)) ) ) ).

fof(fact_4753_scaleR_Otendsto,axiom,
    ! [X_a,X_b] :
      ( real_normed_vector(X_b)
     => ! [G,B_2,F,A_3,A_2] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),A_3),A_2))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),A_3),B_2)),A_2)) ) ) ) ).

fof(fact_4754_scaleR__right_OLIMSEQ,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),A_3)),sequentially)) ) ) ).

fof(fact_4755_scaleR__left_Osetsum,axiom,
    ! [X_b,X_a] :
      ( real_vector(X_a)
     => ! [G,A_2,Xa] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),G),A_2)),Xa) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,real),fun(X_b,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(X_b,real),fun(X_b,fun(X_a,X_a))),combb(real,fun(X_a,X_a),X_b),scaleR_scaleR(X_a)),G)),Xa)),A_2) ) ).

fof(fact_4756_scaleR__right_Osetsum,axiom,
    ! [X_b,X_a] :
      ( real_vector(X_a)
     => ! [A_3,G,A_2] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3)),G)),A_2) ) ).

fof(fact_4757_scaleR__right_OLIM,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [Ra,G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4758_scaleR__left_Ocont,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa,A_3] : hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),hAPP(X_a,fun(real,X_a),hAPP(fun(real,fun(X_a,X_a)),fun(X_a,fun(real,X_a)),combc(real,X_a,X_a),scaleR_scaleR(X_a)),Xa)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa)),hAPP(real,filter(real),at(real),A_3))) ) ).

fof(fact_4759_scaleR__right_Ocont,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra,A_3] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ).

fof(fact_4760_scaleR_OLIM__right__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [C,F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),C)),F)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4761_scaleR__right_OLIM__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [Ra,G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4762_LIM__scaleR,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,M_1,F,L_2,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_1),hAPP(X_a,filter(X_a),at(X_a),A_3)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),L_2),M_1)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4763_scaleR__left_OLIM,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [Xa,G,L,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),Xa)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),L),Xa)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4764_scaleR__left_OLIMSEQ,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),A_3),sequentially))
         => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa)),sequentially)) ) ) ).

fof(fact_4765_scaleR_OLIMSEQ,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Y_2,B_2,X_2,A_3] :
          ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),A_3),sequentially))
         => ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially))
           => hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),B_2)),sequentially)) ) ) ) ).

fof(fact_4766_scaleR__left_Osuminf,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa,X_2] :
          ( hBOOL(hAPP(fun(nat,real),bool,summable(real),X_2))
         => hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(fun(nat,real),real,suminf(real),X_2)),Xa) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa)) ) ) ).

fof(fact_4767_complex__sgn__def,axiom,
    ! [X] : hAPP(complex,complex,sgn_sgn(complex),X) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(real,real,inverse_inverse(real),hAPP(complex,real,norm_norm(complex),X))),X) ).

fof(fact_4768_nonzero__inverse__scaleR__distrib,axiom,
    ! [X_a] :
      ( real_div_algebra(X_a)
     => ! [X,A_1] :
          ( A_1 != zero_zero(real)
         => ( ti(X_a,X) != zero_zero(X_a)
           => hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),X)) ) ) ) ).

fof(fact_4769_complex__scaleR__def,axiom,
    ! [R_3,X] : hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(complex,real,re,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(complex,real,im,X))) ).

fof(fact_4770_scaleR_OLIM__prod__zero,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
           => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4771_scaleR__left_OLIM__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [Xa,G,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),Xa)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4772_scaleR_OLIM__left__zero,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [C,F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),C)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4773_summable__exp__generic,axiom,
    ! [X_a] :
      ( ( real_n2089651433ebra_1(X_a)
        & banach(X_a) )
     => ! [Xa] : hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))) ) ).

fof(fact_4774_summable__norm__exp,axiom,
    ! [X_a] :
      ( ( real_n2089651433ebra_1(X_a)
        & banach(X_a) )
     => ! [Xa] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))) ) ).

fof(fact_4775_lemma__exp__ext,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [X_1] : hAPP(X_a,X_a,exp(X_a),X_1) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))) ) ).

fof(fact_4776_exp__def,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [Xa] : hAPP(X_a,X_a,exp(X_a),Xa) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))) ) ).

fof(fact_4777_exp__converges,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [Xa] : hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(X_a,X_a,exp(X_a),Xa))) ) ).

fof(fact_4778_scaleR_Opos__bounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
          & ! [A_4,B_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2))) ) ) ).

fof(fact_4779_scaleR__left_Opos__bounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_1),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2))) ) ) ).

fof(fact_4780_scaleR_Ononneg__bounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
          & ! [A_4,B_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2))) ) ) ).

fof(fact_4781_scaleR__left_Ononneg__bounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_1),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2))) ) ) ).

fof(fact_4782_scaleR__right_Opos__bounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [R_3] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ) ).

fof(fact_4783_scaleR__left_Obounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [X] :
        ? [K_2] :
        ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_1),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2))) ) ).

fof(fact_4784_scaleR_Obounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ? [K_2] :
        ! [A_4,B_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2))) ) ).

fof(fact_4785_scaleR__right_Ononneg__bounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [R_3] :
        ? [K_2] :
          ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
          & ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ) ).

fof(fact_4786_scaleR__right_Obounded,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [R_3] :
        ? [K_2] :
        ! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2))) ) ).

fof(fact_4787_LIM__cong,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & topological_space(X_b) )
     => ! [L,Ma,F,G,A_3,B_2] :
          ( ti(X_a,A_3) = ti(X_a,B_2)
         => ( ! [X_1] :
                ( ti(X_a,X_1) != ti(X_a,B_2)
               => hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) )
           => ( ti(X_b,L) = ti(X_b,Ma)
             => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
              <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ma),hAPP(X_a,filter(X_a),at(X_a),B_2))) ) ) ) ) ) ).

fof(fact_4788_tendsto__setsum,axiom,
    ! [X_b,X_a,X_c] :
      ( real_normed_vector(X_c)
     => ! [F,A_3,A_2,S] :
          ( ! [I_2] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),S))
             => hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),hAPP(X_a,fun(X_b,X_c),F,I_2)),hAPP(X_a,X_c,A_3,I_2)),A_2)) )
         => hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),hAPP(fun(X_a,bool),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_a,bool),fun(X_b,X_c)),combc(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),S)),hAPP(fun(X_a,bool),X_c,hAPP(fun(X_a,X_c),fun(fun(X_a,bool),X_c),big_co1399186613setsum(X_a,X_c),A_3),S)),A_2)) ) ) ).

fof(fact_4789_metric__LIM__I,axiom,
    ! [X_b,X_a] :
      ( ( metric_space(X_a)
        & metric_space(X_b) )
     => ! [F,L_2,A_3] :
          ( ! [R_1] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
             => ? [S_1] :
                  ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
                  & ! [X_1] :
                      ( ( ti(X_a,X_1) != ti(X_a,A_3)
                        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),S_1)) )
                     => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_1)),L_2)),R_1)) ) ) )
         => hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4790_trivial__limit__sequentially,axiom,
    ~ hBOOL(hAPP(filter(nat),bool,trivial_limit(nat),sequentially)) ).

fof(fact_4791_LIM__equal,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & topological_space(X_b) )
     => ! [L,F,G,A_3] :
          ( ! [X_1] :
              ( ti(X_a,X_1) != ti(X_a,A_3)
             => hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) )
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))
          <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ).

fof(fact_4792_tendsto__unique,axiom,
    ! [X_a,X_b] :
      ( t2_space(X_b)
     => ! [L_1,F,L,A_2] :
          ( ~ hBOOL(hAPP(filter(X_a),bool,trivial_limit(X_a),A_2))
         => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2))
           => ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_1),A_2))
             => ti(X_b,L) = ti(X_b,L_1) ) ) ) ) ).

fof(fact_4793_pred__nat__def,axiom,
    pred_nat = hAPP(fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),collect(product_prod(nat,nat)),hAPP(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_prod_case(nat,nat,bool),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),fequal(nat))),suc))) ).

fof(fact_4794_setprod_Ocong,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [G,H,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( ! [X_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
               => hAPP(X_a,X_b,G,X_1) = hAPP(X_a,X_b,H,X_1) )
           => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),H),A_2) ) ) ) ).

fof(fact_4795_splitE2,axiom,
    ! [X_a,X_b,X_c,Q_1,P_1,Z] :
      ( hBOOL(hAPP(X_a,bool,Q_1,hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),P_1),Z)))
     => ~ ! [X_1,Y_1] :
            ( ti(product_prod(X_b,X_c),Z) = hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),X_1),Y_1)
           => ~ hBOOL(hAPP(X_a,bool,Q_1,hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),P_1,X_1),Y_1))) ) ) ).

fof(fact_4796_split__cong,axiom,
    ! [X_c,X_b,X_a,P_4,F,G,Q_2] :
      ( ! [X_1,Y_1] :
          ( hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1) = ti(product_prod(X_a,X_b),Q_2)
         => hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,X_1),Y_1) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),G,X_1),Y_1) )
     => ( ti(product_prod(X_a,X_b),P_4) = ti(product_prod(X_a,X_b),Q_2)
       => hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F),P_4) = hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),G),Q_2) ) ) ).

fof(fact_4797_setprod__1_H,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_mult(X_b)
     => ! [F,F_1] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),F_1))
             => hAPP(X_a,X_b,F,X_1) = one_one(X_b) )
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),F_1) = one_one(X_b) ) ) ).

fof(fact_4798_dvd__setsum,axiom,
    ! [X_a,X_b] :
      ( comm_semiring_1(X_b)
     => ! [D_1,F,A_2] :
          ( ! [I_2] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2))
             => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),D_1),hAPP(X_a,X_b,F,I_2))) )
         => hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),D_1),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2))) ) ) ).

fof(fact_4799_setsum_Ocong,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [G,H,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( ! [X_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
               => hAPP(X_a,X_b,G,X_1) = hAPP(X_a,X_b,H,X_1) )
           => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),H),A_2) ) ) ) ).

fof(fact_4800_setsum__0_H,axiom,
    ! [X_a,X_b] :
      ( comm_monoid_add(X_b)
     => ! [F,A_2] :
          ( ! [X_1] :
              ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
             => hAPP(X_a,X_b,F,X_1) = zero_zero(X_b) )
         => hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = zero_zero(X_b) ) ) ).

fof(fact_4801_real__LIM__sandwich__zero,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3)))
         => ( ! [X_1] :
                ( ti(X_a,X_1) != ti(X_a,A_3)
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,G,X_1))) )
           => ( ! [X_1] :
                  ( ti(X_a,X_1) != ti(X_a,A_3)
                 => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,G,X_1)),hAPP(X_a,real,F,X_1))) )
             => hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ) ).

fof(fact_4802_gcd__coprime__exists__int,axiom,
    ! [A_1,B] :
      ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) != zero_zero(int)
     => ? [A_5,B_3] :
          ( ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_5),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))
          & ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_3),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))
          & hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_5),B_3) = one_one(int) ) ) ).

fof(fact_4803_exE__some,axiom,
    ! [X_a,C,P_1] :
      ( ? [X1] : hBOOL(hAPP(X_a,bool,P_1,X1))
     => ( ti(X_a,C) = hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)
       => hBOOL(hAPP(X_a,bool,P_1,C)) ) ) ).

fof(fact_4804_setprod_Orelated,axiom,
    ! [X_b,X_a] :
      ( comm_monoid_mult(X_a)
     => ! [H,G,S,R_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,one_one(X_a)),one_one(X_a)))
         => ( ! [X1,Y1,X2,Y2] :
                ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,X1),X2))
                  & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,Y1),Y2)) )
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X1),Y1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X2),Y2))) )
           => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S))
             => ( ! [X_1] :
                    ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),S))
                   => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(X_b,X_a,H,X_1)),hAPP(X_b,X_a,G,X_1))) )
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),H),S)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),G),S))) ) ) ) ) ) ).

fof(fact_4805_Eps__split__eq,axiom,
    ! [X_a,X_b,Xa,Ya] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(X_a,fun(X_a,bool),fequal(X_a),Xa)))),hAPP(X_b,fun(X_b,bool),fequal(X_b),Ya)))) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),Ya) ).

fof(fact_4806_split__paired__Eps,axiom,
    ! [X_a,X_b,P_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),P_1) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool),X_a),hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),combb(product_prod(X_a,X_b),bool,X_b),P_1)),product_Pair(X_a,X_b)))) ).

fof(fact_4807_Eps__split,axiom,
    ! [X_a,X_b,P_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),P_1)) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool),hAPP(fun(product_prod(X_a,X_b),fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool)),combs(product_prod(X_a,X_b),X_b,bool),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool))),combb(X_a,fun(X_b,bool),product_prod(X_a,X_b)),P_1),product_fst(X_a,X_b))),product_snd(X_a,X_b))) ).

fof(fact_4808_someI,axiom,
    ! [X_a,P_1,Xa] :
      ( hBOOL(hAPP(X_a,bool,P_1,Xa))
     => hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) ) ).

fof(fact_4809_tfl__some,axiom,
    ! [X_a,P_3,X_1] :
      ( hBOOL(hAPP(X_a,bool,P_3,X_1))
     => hBOOL(hAPP(X_a,bool,P_3,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_3))) ) ).

fof(fact_4810_some__sym__eq__trivial,axiom,
    ! [X_a,Xa] : hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),hAPP(X_a,fun(X_a,bool),fequal(X_a),Xa)) = ti(X_a,Xa) ).

fof(fact_4811_some__eq__trivial,axiom,
    ! [X_a,Xa] : hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),Xa)) = ti(X_a,Xa) ).

fof(fact_4812_some__eq__ex,axiom,
    ! [X_a,P_1] :
      ( hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)))
    <=> ? [X1] : hBOOL(hAPP(X_a,bool,P_1,X1)) ) ).

fof(fact_4813_someI__ex,axiom,
    ! [X_a,P_1] :
      ( ? [X1] : hBOOL(hAPP(X_a,bool,P_1,X1))
     => hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) ) ).

fof(fact_4814_some__equality,axiom,
    ! [X_a,P_1,A_3] :
      ( hBOOL(hAPP(X_a,bool,P_1,A_3))
     => ( ! [X_1] :
            ( hBOOL(hAPP(X_a,bool,P_1,X_1))
           => ti(X_a,X_1) = ti(X_a,A_3) )
       => hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) = ti(X_a,A_3) ) ) ).

fof(fact_4815_dvd_Ostrict__monoD,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Xa,Ya,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F))
         => ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
              & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) )
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,Xa)),hAPP(nat,X_a,F,Ya))) ) ) ) ).

fof(fact_4816_dvd_Ostrict__monoI,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [F] :
          ( ! [X_1,Y_1] :
              ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
                & ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) )
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,X_1)),hAPP(nat,X_a,F,Y_1))) )
         => hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F)) ) ) ).

fof(fact_4817_dvd_Ostrict__mono__mono,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F))
         => hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F)) ) ) ).

fof(fact_4818_dvd_OmonoD,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Xa,Ya,F] :
          ( hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F))
         => ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,Xa)),hAPP(nat,X_a,F,Ya))) ) ) ) ).

fof(fact_4819_dvd_OmonoI,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [F] :
          ( ! [X_1,Y_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
             => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,X_1)),hAPP(nat,X_a,F,Y_1))) )
         => hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F)) ) ) ).

fof(fact_4820_setsum_Orelated,axiom,
    ! [X_b,X_a] :
      ( comm_monoid_add(X_a)
     => ! [H,G,S,R_2] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,zero_zero(X_a)),zero_zero(X_a)))
         => ( ! [X1,Y1,X2,Y2] :
                ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,X1),X2))
                  & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,Y1),Y2)) )
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X1),Y1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X2),Y2))) )
           => ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S))
             => ( ! [X_1] :
                    ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),S))
                   => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(X_b,X_a,H,X_1)),hAPP(X_b,X_a,G,X_1))) )
               => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),H),S)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),S))) ) ) ) ) ) ).

fof(fact_4821_min__number__of__Suc,axiom,
    ! [Na,Va] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,Na)),hAPP(int,nat,number_number_of(nat),Va)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),zero_zero(nat)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),Na),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))))) ).

fof(fact_4822_min__Suc__number__of,axiom,
    ! [Va,Na] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),zero_zero(nat)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na))) ).

fof(fact_4823_min__less__iff__disj,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya,Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Xa),Ya)),Z))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Z))
            | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Z)) ) ) ) ).

fof(fact_4824_min__less__iff__conj,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Z,Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Xa),Ya)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Xa))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Ya)) ) ) ) ).

fof(fact_4825_min__max_Oless__infI2,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_1,B,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),X))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)) ) ) ).

fof(fact_4826_min__max_Oless__infI1,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [B,A_1,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),X))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)) ) ) ).

fof(fact_4827_min__0R,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat) ).

fof(fact_4828_min__0L,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat) ).

fof(fact_4829_min__Suc__Suc,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M),N)) ).

fof(fact_4830_min__max_Oinf_Oidem,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),A_1) = ti(X_a,A_1) ) ).

fof(fact_4831_min__max_Oinf_Ocommute,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),A_1) ) ).

fof(fact_4832_min__max_Oinf__commute,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),X) ) ).

fof(fact_4833_min__max_Oinf_Oleft__idem,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B) ) ).

fof(fact_4834_min__max_Oinf__left__idem,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) ) ).

fof(fact_4835_min__max_Oinf_Oleft__commute,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [B,A_1,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),C_1)) ) ).

fof(fact_4836_min__max_Oinf__left__commute,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z_1)) ) ).

fof(fact_4837_min__max_Oinf_Oassoc,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_1,B,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),C_1)) ) ).

fof(fact_4838_min__max_Oinf__assoc,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)) ) ).

fof(fact_4839_min__diff,axiom,
    ! [M,I_1,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),I_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M),N)),I_1) ).

fof(fact_4840_min__diff__distrib__left,axiom,
    ! [X_a] :
      ( ordered_ab_group_add(X_a)
     => ! [X,Y,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z_1)) ) ).

fof(fact_4841_min__add__distrib__left,axiom,
    ! [X_a] :
      ( ordere236663937imp_le(X_a)
     => ! [X,Y,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z_1)) ) ).

fof(fact_4842_Min_Oidem,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),X) = ti(X_a,X) ) ).

fof(fact_4843_min__max_Oinf__le1,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),X)) ) ).

fof(fact_4844_min__max_Oinf__le2,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Y)) ) ).

fof(fact_4845_min__max_Ole__iff__inf,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
        <=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Xa),Ya) = ti(X_a,Xa) ) ) ).

fof(fact_4846_min__max_Ole__inf__iff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya,Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Ya),Z)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Z)) ) ) ) ).

fof(fact_4847_min__max_Ole__infI1,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [B,A_1,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)) ) ) ).

fof(fact_4848_min__max_Ole__infI2,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_1,B,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)) ) ) ).

fof(fact_4849_min__max_Oinf__absorb1,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = ti(X_a,X) ) ) ).

fof(fact_4850_min__max_Oinf__absorb2,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Y,X] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = ti(X_a,Y) ) ) ).

fof(fact_4851_min__max_Ole__infI,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [B,X,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B))) ) ) ) ).

fof(fact_4852_min__max_Oinf__greatest,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Z_1,X,Y] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z_1))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1))) ) ) ) ).

fof(fact_4853_min__max_Oinf__mono,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [B,D_3,A_1,C_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C_1))
         => ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),D_3))
           => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),C_1),D_3))) ) ) ) ).

fof(fact_4854_min__max_Ole__infE,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,A_1,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)))
         => ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
             => ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B)) ) ) ) ).

fof(fact_4855_min__le__iff__disj,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya,Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Xa),Ya)),Z))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Z))
            | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Z)) ) ) ) ).

fof(fact_4856_min__number__of,axiom,
    ! [X_b] :
      ( ( number(X_b)
        & ord(X_b) )
     => ! [U_1,V] :
          ( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U_1)),hAPP(int,X_b,number_number_of(X_b),V)))
           => hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(int,X_b,number_number_of(X_b),U_1)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),U_1) )
          & ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U_1)),hAPP(int,X_b,number_number_of(X_b),V)))
           => hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(int,X_b,number_number_of(X_b),U_1)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),V) ) ) ) ).

fof(fact_4857_min__Suc2,axiom,
    ! [Ma,Na] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),Ma),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),ord_min(nat)),Na))),Ma) ).

fof(fact_4858_min__Suc1,axiom,
    ! [Na,Ma] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,Na)),Ma) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),ord_min(nat),Na))),Ma) ).

fof(fact_4859_dual__max,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),max(X_a),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),ord_less_eq(X_a))) = ord_min(X_a) ) ).

fof(fact_4860_min__leastR,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [X,Least] :
          ( ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_1))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Least) = ti(X_a,Least) ) ) ).

fof(fact_4861_min__leastL,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ! [X,Least] :
          ( ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_1))
         => hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Least),X) = ti(X_a,Least) ) ) ).

fof(fact_4862_min__ord__min,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ord_min(X_a) = hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),min_1(X_a),ord_less_eq(X_a)) ) ).

fof(fact_4863_vanishes__mult__bounded,axiom,
    ! [Y_2,X_2] :
      ( ? [A_4] :
          ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A_4))
          & ! [N_1] : hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_2,N_1))),A_4)) )
     => ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_2))
       => hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),X_2)),Y_2))) ) ) ).

fof(fact_4864_SetInterval_Otransfer__int__nat__set__functions,axiom,
    ! [Na,Ma] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,Ma))
     => ( hBOOL(hAPP(int,bool,nat_is_nat,Na))
       => hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),Ma),Na) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(int,nat,nat_1,Ma)),hAPP(int,nat,nat_1,Na))) ) ) ).

fof(fact_4865_is__nat__def,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,Xa))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) ) ).

fof(fact_4866_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,X))
     => ( hBOOL(hAPP(int,bool,nat_is_nat,Y))
       => hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) ) ) ).

fof(fact_4867_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
    hBOOL(hAPP(int,bool,nat_is_nat,one_one(int))) ).

fof(fact_4868_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
    ! [Xa,A_2] :
      ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
     => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),A_2))
       => hBOOL(hAPP(int,bool,nat_is_nat,Xa)) ) ) ).

fof(fact_4869_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,
    ! [N,X] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,X))
     => hBOOL(hAPP(int,bool,nat_is_nat,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N))) ) ).

fof(fact_4870_Divides_Otransfer__int__nat__function__closures_I2_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,X))
     => ( hBOOL(hAPP(int,bool,nat_is_nat,Y))
       => hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y))) ) ) ).

fof(fact_4871_Divides_Otransfer__int__nat__function__closures_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,X))
     => ( hBOOL(hAPP(int,bool,nat_is_nat,Y))
       => hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y))) ) ) ).

fof(fact_4872_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,X))
     => ( hBOOL(hAPP(int,bool,nat_is_nat,Y))
       => hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y))) ) ) ).

fof(fact_4873_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,X))
     => ( hBOOL(hAPP(int,bool,nat_is_nat,Y))
       => hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) ) ) ).

fof(fact_4874_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
    hBOOL(hAPP(int,bool,nat_is_nat,zero_zero(int))) ).

fof(fact_4875_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,
    ! [Z_1] : hBOOL(hAPP(int,bool,nat_is_nat,hAPP(nat,int,semiring_1_of_nat(int),Z_1))) ).

fof(fact_4876_vanishes__minus,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2))
     => hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(rat,rat),fun(fun(nat,rat),fun(nat,rat)),combb(rat,rat,nat),uminus_uminus(rat)),X_2))) ) ).

fof(fact_4877_vanishes__const,axiom,
    ! [C] :
      ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(rat,fun(nat,rat),combk(rat,nat),C)))
    <=> C = zero_zero(rat) ) ).

fof(fact_4878_vanishes__diff,axiom,
    ! [Y_2,X_2] :
      ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2))
     => ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_2))
       => hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),minus_minus(rat)),X_2)),Y_2))) ) ) ).

fof(fact_4879_vanishes__add,axiom,
    ! [Y_2,X_2] :
      ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2))
     => ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_2))
       => hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),plus_plus(rat)),X_2)),Y_2))) ) ) ).

fof(fact_4880_SetInterval_Otransfer__int__nat__set__function__closures,axiom,
    ! [Ya,Xa] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,Xa))
     => hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),Xa),Ya))) ) ).

fof(fact_4881_Nat__Transfer_Otransfer__int__nat__function__closures_I8_J,axiom,
    hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).

fof(fact_4882_transfer__int__nat__gcd__closures_I1_J,axiom,
    ! [Y,X] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,X))
     => ( hBOOL(hAPP(int,bool,nat_is_nat,Y))
       => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y))) ) ) ).

fof(fact_4883_transfer__int__nat__factorial__closure,axiom,
    ! [X] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,X))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,fact_fact(int),X))) ) ).

fof(fact_4884_transfer__int__nat__set__relations_I2_J,axiom,
    ! [A_2,Xa] :
      ( hBOOL(hAPP(int,bool,nat_is_nat,Xa))
     => ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
       => ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),A_2))
        <=> hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),hAPP(int,nat,nat_1,Xa)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2))) ) ) ) ).

fof(fact_4885_Nat__Transfer_Otransfer__int__nat__function__closures_I7_J,axiom,
    hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).

fof(fact_4886_transfer__int__nat__sum__prod2_I2_J,axiom,
    ! [X_a,F,A_2] :
      ( ! [X_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
         => hBOOL(hAPP(int,bool,nat_is_nat,hAPP(X_a,int,F,X_1))) )
     => hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),hAPP(fun(X_a,int),fun(X_a,nat),hAPP(fun(int,nat),fun(fun(X_a,int),fun(X_a,nat)),combb(int,nat,X_a),nat_1),F)),A_2)) ) ).

fof(fact_4887_transfer__int__nat__sum__prod2_I1_J,axiom,
    ! [X_a,F,A_2] :
      ( ! [X_1] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
         => hBOOL(hAPP(int,bool,nat_is_nat,hAPP(X_a,int,F,X_1))) )
     => hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,int),fun(X_a,nat),hAPP(fun(int,nat),fun(fun(X_a,int),fun(X_a,nat)),combb(int,nat,X_a),nat_1),F)),A_2)) ) ).

fof(fact_4888_vanishesD,axiom,
    ! [Ra,X_2] :
      ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2))
     => ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),Ra))
       => ? [K_1] :
          ! [N_1] :
            ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1))
           => hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_2,N_1))),Ra)) ) ) ) ).

fof(fact_4889_vanishes__def,axiom,
    ! [X_2] :
      ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2))
    <=> ! [R_1] :
          ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_1))
         => ? [K_1] :
            ! [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1))
             => hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_2,N_1))),R_1)) ) ) ) ).

fof(fact_4890_vanishesI,axiom,
    ! [X_2] :
      ( ! [R_1] :
          ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_1))
         => ? [K_1] :
            ! [N_1] :
              ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1))
             => hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_2,N_1))),R_1)) ) )
     => hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2)) ) ).

fof(fact_4891_EulerFermat_Oaux,axiom,
    ! [X_a,Q_1,P_1] :
      ( hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)))
     => ( hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),Q_1)))
       => ( hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) = hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),Q_1)
         => ? [A_4] :
              ( hBOOL(hAPP(X_a,bool,P_1,A_4))
              & hBOOL(hAPP(X_a,bool,Q_1,A_4)) ) ) ) ) ).

fof(fact_4892_someI2__ex,axiom,
    ! [X_a,Q_1,P_1] :
      ( ? [X1] : hBOOL(hAPP(X_a,bool,P_1,X1))
     => ( ! [X_1] :
            ( hBOOL(hAPP(X_a,bool,P_1,X_1))
           => hBOOL(hAPP(X_a,bool,Q_1,X_1)) )
       => hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) ) ) ).

fof(fact_4893_some1__equality,axiom,
    ! [X_a,A_3,P_1] :
      ( ? [X_1] :
          ( hBOOL(hAPP(X_a,bool,P_1,X_1))
          & ! [Y_1] :
              ( hBOOL(hAPP(X_a,bool,P_1,Y_1))
             => ti(X_a,Y_1) = ti(X_a,X_1) ) )
     => ( hBOOL(hAPP(X_a,bool,P_1,A_3))
       => hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) = ti(X_a,A_3) ) ) ).

fof(fact_4894_someI2,axiom,
    ! [X_a,Q_1,P_1,A_3] :
      ( hBOOL(hAPP(X_a,bool,P_1,A_3))
     => ( ! [X_1] :
            ( hBOOL(hAPP(X_a,bool,P_1,X_1))
           => hBOOL(hAPP(X_a,bool,Q_1,X_1)) )
       => hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) ) ) ).

fof(fact_4895_Powp__mono,axiom,
    ! [X_a,A_2,B_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
     => hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),A_2)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),B_1))) ) ).

fof(fact_4896_Powp__Pow__eq,axiom,
    ! [X_a,A_2,X_1] :
      ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2)),X_1))
    <=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_1),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2))) ) ).

fof(fact_4897_zcongm__def,axiom,
    ! [Ma,X_1,Xa_1] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcongm,Ma),X_1),Xa_1))
    <=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X_1),Xa_1),Ma)) ) ).

fof(fact_4898_increasing__LIMSEQ,axiom,
    ! [L,F] :
      ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),hAPP(nat,real,F,hAPP(nat,nat,suc,N_1))))
     => ( ! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),L))
       => ( ! [E] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),L),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,F,N_1)),E))) )
         => hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),L),sequentially)) ) ) ) ).

fof(fact_4899_bijzcong__zcong__prod,axiom,
    ! [A_2,B_1,Ma] :
      ( hBOOL(hAPP(fun(product_prod(fun(int,bool),fun(int,bool)),bool),bool,hAPP(product_prod(fun(int,bool),fun(int,bool)),fun(fun(product_prod(fun(int,bool),fun(int,bool)),bool),bool),member(product_prod(fun(int,bool),fun(int,bool))),hAPP(fun(int,bool),product_prod(fun(int,bool),fun(int,bool)),hAPP(fun(int,bool),fun(fun(int,bool),product_prod(fun(int,bool),fun(int,bool))),product_Pair(fun(int,bool),fun(int,bool)),A_2),B_1)),hAPP(fun(int,fun(int,bool)),fun(product_prod(fun(int,bool),fun(int,bool)),bool),bijR(int,int),hAPP(int,fun(int,fun(int,bool)),zcongm,Ma))))
     => hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),A_2)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),B_1)),Ma)) ) ).

fof(fact_4900_isCont__LIM__compose2,axiom,
    ! [X_c,X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & topological_space(X_b)
        & topological_space(X_c) )
     => ! [G,L,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),L),hAPP(X_b,filter(X_b),at(X_b),hAPP(X_a,X_b,F,A_3))))
           => ( ? [D_2] :
                  ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
                  & ! [X_1] :
                      ( ( ti(X_a,X_1) != ti(X_a,A_3)
                        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),D_2)) )
                     => hAPP(X_a,X_b,F,X_1) != hAPP(X_a,X_b,F,A_3) ) )
             => hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ) ).

fof(fact_4901_isCont__LIM__compose,axiom,
    ! [X_a,X_b,X_c] :
      ( ( topological_space(X_c)
        & topological_space(X_b)
        & topological_space(X_a) )
     => ! [F,A_3,G,L] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),L))
         => ( hBOOL(hAPP(filter(X_c),bool,hAPP(X_a,fun(filter(X_c),bool),hAPP(fun(X_c,X_a),fun(X_a,fun(filter(X_c),bool)),tendsto(X_c,X_a),F),L),hAPP(X_c,filter(X_c),at(X_c),A_3)))
           => hBOOL(hAPP(filter(X_c),bool,hAPP(X_b,fun(filter(X_c),bool),hAPP(fun(X_c,X_b),fun(X_b,fun(filter(X_c),bool)),tendsto(X_c,X_b),hAPP(fun(X_c,X_a),fun(X_c,X_b),hAPP(fun(X_a,X_b),fun(fun(X_c,X_a),fun(X_c,X_b)),combb(X_a,X_b,X_c),G),F)),hAPP(X_a,X_b,G,L)),hAPP(X_c,filter(X_c),at(X_c),A_3))) ) ) ) ).

fof(fact_4902_isCont__def,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & topological_space(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),hAPP(X_a,X_b,F,A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ).

fof(fact_4903_CARAT__DERIV,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F,Xa,L] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),L))
        <=> ? [G_1] :
              ( ! [Z_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,F,Z_2)),hAPP(X_a,X_a,F,Xa)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,G_1,Z_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Z_2),Xa))
              & hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),G_1),Xa))
              & hAPP(X_a,X_a,G_1,Xa) = ti(X_a,L) ) ) ) ).

fof(fact_4904_isCont__real__root,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,Na)),Xa)) ) ).

fof(fact_4905_isCont__ln,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
     => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),ln),Xa)) ) ).

fof(fact_4906_isCont__tan,axiom,
    ! [Xa] :
      ( hAPP(real,real,cos,Xa) != zero_zero(real)
     => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),tan),Xa)) ) ).

fof(fact_4907_isCont__abs,axiom,
    ! [A_3] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),abs_abs(real)),A_3)) ).

fof(fact_4908_isCont__minus,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),A_3)) ) ) ).

fof(fact_4909_isCont__sgn,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hAPP(X_a,X_b,F,A_3) != zero_zero(X_b)
           => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),sgn_sgn(X_b)),F)),A_3)) ) ) ) ).

fof(fact_4910_isCont__inverse,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_n1866405975lgebra(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hAPP(X_a,X_b,F,A_3) != zero_zero(X_b)
           => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),A_3)) ) ) ) ).

fof(fact_4911_mult_OisCont,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),A_3)) ) ) ) ).

fof(fact_4912_isCont__mult,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_algebra(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),A_3)) ) ) ) ).

fof(fact_4913_mult__left_OisCont,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Ya,A_3] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),times_times(X_a)),Ya)),A_3)) ) ).

fof(fact_4914_mult__right_OisCont,axiom,
    ! [X_a] :
      ( real_normed_algebra(X_a)
     => ! [Xa,A_3] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),A_3)) ) ).

fof(fact_4915_isCont__of__real,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_n2089651433ebra_1(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),F)),A_3)) ) ) ).

fof(fact_4916_of__real_OisCont,axiom,
    ! [X_a] :
      ( ( real_algebra_1(X_a)
        & real_normed_vector(X_a) )
     => ! [A_3] : hBOOL(hAPP(real,bool,hAPP(fun(real,X_a),fun(real,bool),isCont(real,X_a),of_real(X_a)),A_3)) ) ).

fof(fact_4917_isCont__power,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & power(X_b)
        & real_normed_algebra(X_b) )
     => ! [Na,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(nat,fun(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(nat,X_b)),hAPP(fun(X_b,fun(nat,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(nat,X_b))),combb(X_b,fun(nat,X_b),X_a),power_power(X_b)),F)),Na)),A_3)) ) ) ).

fof(fact_4918_isCont__norm,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),A_3)) ) ) ).

fof(fact_4919_isCont__o2,axiom,
    ! [X_b,X_c,X_a] :
      ( ( topological_space(X_a)
        & topological_space(X_c)
        & topological_space(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),G),hAPP(X_a,X_b,F,A_3)))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_c),fun(X_a,bool),isCont(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),A_3)) ) ) ) ).

fof(fact_4920_isCont__ident,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [A_3] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),combi(X_a)),A_3)) ) ).

fof(fact_4921_isCont__const,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & topological_space(X_b) )
     => ! [K,A_3] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),A_3)) ) ).

fof(fact_4922_divide_OisCont,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [Ya,A_3] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),inverse_divide(X_a)),Ya)),A_3)) ) ).

fof(fact_4923_isCont__rabs,axiom,
    ! [X_a] :
      ( topological_space(X_a)
     => ! [F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),A_3))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),A_3)) ) ) ).

fof(fact_4924_isCont__diff,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),A_3)) ) ) ) ).

fof(fact_4925_isCont__add,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),A_3)) ) ) ) ).

fof(fact_4926_DERIV__isCont,axiom,
    ! [X_a] :
      ( real_normed_field(X_a)
     => ! [F,Xa,D] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D))
         => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),F),Xa)) ) ) ).

fof(fact_4927_isCont__sin,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),sin),Xa)) ).

fof(fact_4928_isCont__cos,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),cos),Xa)) ).

fof(fact_4929_isCont__exp,axiom,
    ! [X_a] :
      ( ( real_normed_field(X_a)
        & banach(X_a) )
     => ! [Xa] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),exp(X_a)),Xa)) ) ).

fof(fact_4930_isCont__real__sqrt,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),sqrt),Xa)) ).

fof(fact_4931_isCont__arctan,axiom,
    ! [Xa] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arctan),Xa)) ).

fof(fact_4932_cnj_OisCont,axiom,
    ! [A_3] : hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),cnj),A_3)) ).

fof(fact_4933_Im_OisCont,axiom,
    ! [A_3] : hBOOL(hAPP(complex,bool,hAPP(fun(complex,real),fun(complex,bool),isCont(complex,real),im),A_3)) ).

fof(fact_4934_Re_OisCont,axiom,
    ! [A_3] : hBOOL(hAPP(complex,bool,hAPP(fun(complex,real),fun(complex,bool),isCont(complex,real),re),A_3)) ).

fof(fact_4935_isCont__scaleR,axiom,
    ! [X_b,X_a] :
      ( ( topological_space(X_a)
        & real_normed_vector(X_b) )
     => ! [G,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),A_3))
         => ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3))
           => hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),A_3)) ) ) ) ).

fof(fact_4936_scaleR__left_OisCont,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Xa,A_3] : hBOOL(hAPP(real,bool,hAPP(fun(real,X_a),fun(real,bool),isCont(real,X_a),hAPP(X_a,fun(real,X_a),hAPP(fun(real,fun(X_a,X_a)),fun(X_a,fun(real,X_a)),combc(real,X_a,X_a),scaleR_scaleR(X_a)),Xa)),A_3)) ) ).

fof(fact_4937_scaleR__right_OisCont,axiom,
    ! [X_a] :
      ( real_normed_vector(X_a)
     => ! [Ra,A_3] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),A_3)) ) ).

fof(fact_4938_isCont__root__zero,axiom,
    ! [Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,Na)),zero_zero(real))) ) ).

fof(fact_4939_isCont__root__pos,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa))
       => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,Na)),Xa)) ) ) ).

fof(fact_4940_isCont__root__neg,axiom,
    ! [Xa,Na] :
      ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real)))
       => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,Na)),Xa)) ) ) ).

fof(fact_4941_isCont__iff,axiom,
    ! [X_b,X_a] :
      ( ( real_normed_vector(X_a)
        & topological_space(X_b) )
     => ! [F,Xa] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Xa))
        <=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa))),hAPP(X_a,X_b,F,Xa)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).

fof(fact_4942_isCont__arccos,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arccos),Xa)) ) ) ).

fof(fact_4943_isCont__arcsin,axiom,
    ! [Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Xa))
     => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real)))
       => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arcsin),Xa)) ) ) ).

fof(fact_4944_metric__isCont__LIM__compose2,axiom,
    ! [X_c,X_b,X_a] :
      ( ( metric_space(X_a)
        & topological_space(X_b)
        & topological_space(X_c) )
     => ! [G,L,F,A_3] :
          ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3))
         => ( hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),L),hAPP(X_b,filter(X_b),at(X_b),hAPP(X_a,X_b,F,A_3))))
           => ( ? [D_2] :
                  ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))
                  & ! [X_1] :
                      ( ( ti(X_a,X_1) != ti(X_a,A_3)
                        & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),D_2)) )
                     => hAPP(X_a,X_b,F,X_1) != hAPP(X_a,X_b,F,A_3) ) )
             => hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) ) ) ) ) ).

fof(fact_4945_LIM__less__bound,axiom,
    ! [F,B_2,Xa] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B_2),Xa))
     => ( ! [X_1] :
            ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),B_2),Xa)))
           => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,F,X_1))) )
       => ( hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Xa))
         => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,F,Xa))) ) ) ) ).

fof(fact_4946_DERIV__inverse__function,axiom,
    ! [B_2,A_3,F,G,Xa,D] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),hAPP(real,real,G,Xa)),D))
     => ( D != zero_zero(real)
       => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),Xa))
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),B_2))
           => ( ! [Y_1] :
                  ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),Y_1))
                    & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),B_2)) )
                 => hAPP(real,real,F,hAPP(real,real,G,Y_1)) = Y_1 )
             => ( hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),G),Xa))
               => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),Xa),hAPP(real,real,inverse_inverse(real),D))) ) ) ) ) ) ) ).

fof(fact_4947_isCont__setsum,axiom,
    ! [X_a,X_c,X_b] :
      ( ( topological_space(X_b)
        & real_normed_vector(X_c) )
     => ! [F,Xa,A_2] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
         => ( ! [X_1] :
                ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
               => hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),hAPP(X_a,fun(X_b,X_c),F,X_1)),Xa)) )
           => hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),hAPP(fun(X_a,bool),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_a,bool),fun(X_b,X_c)),combc(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),A_2)),Xa)) ) ) ) ).

fof(fact_4948_fin__bijRl,axiom,
    ! [X_a,X_b,A_2,B_1,P_1] :
      ( hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_b,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_b,bool))),hAPP(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,bool),fun(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool))),product_Pair(fun(X_a,bool),fun(X_b,bool)),A_2),B_1)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bijR(X_a,X_b),P_1)))
     => hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).

fof(fact_4949_fin__bijRr,axiom,
    ! [X_a,X_b,A_2,B_1,P_1] :
      ( hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_b,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_b,bool))),hAPP(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,bool),fun(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool))),product_Pair(fun(X_a,bool),fun(X_b,bool)),A_2),B_1)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bijR(X_a,X_b),P_1)))
     => hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) ) ).

fof(fact_4950_isCont__inv__fun,axiom,
    ! [G,F,Xa,D_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
     => ( ! [Z_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
           => hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2 )
       => ( ! [Z_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
             => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)) )
         => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),G),hAPP(real,real,F,Xa))) ) ) ) ).

fof(fact_4951_DERIV__isconst1,axiom,
    ! [F,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2))
     => ( ! [X_1] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
           => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_1)) )
       => ( ! [X_1] :
              ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),X_1))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),B_2)) )
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real))) )
         => ! [X_1] :
              ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
             => hAPP(real,real,F,X_1) = hAPP(real,real,F,A_3) ) ) ) ) ).

fof(fact_4952_DERIV__isconst2,axiom,
    ! [Xa,F,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2))
     => ( ! [X_1] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
           => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_1)) )
       => ( ! [X_1] :
              ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),X_1))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),B_2)) )
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real))) )
         => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),Xa))
           => ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),B_2))
             => hAPP(real,real,F,Xa) = hAPP(real,real,F,A_3) ) ) ) ) ) ).

fof(fact_4953_DERIV__isconst__end,axiom,
    ! [F,A_3,B_2] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2))
     => ( ! [X_1] :
            ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) )
           => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_1)) )
       => ( ! [X_1] :
              ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),X_1))
                & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),B_2)) )
             => hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real))) )
         => hAPP(real,real,F,B_2) = hAPP(real,real,F,A_3) ) ) ) ).

fof(fact_4954_isCont__inv__fun__inv,axiom,
    ! [G,F,Xa,D_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
     => ( ! [Z_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
           => hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2 )
       => ( ! [Z_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
             => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)) )
         => ? [E] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
              & ! [Y_1] :
                  ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y_1),hAPP(real,real,F,Xa)))))
                    & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y_1),hAPP(real,real,F,Xa)))),E)) )
                 => hAPP(real,real,F,hAPP(real,real,G,Y_1)) = Y_1 ) ) ) ) ) ).

fof(fact_4955_lemma__isCont__inj,axiom,
    ! [G,F,Xa,D_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
     => ( ! [Z_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
           => hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2 )
       => ( ! [Z_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
             => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)) )
         => ? [Z_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Xa)),hAPP(real,real,F,Z_2))) ) ) ) ) ).

fof(fact_4956_lemma__isCont__inj2,axiom,
    ! [G,F,Xa,D_1] :
      ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
     => ( ! [Z_2] :
            ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
           => hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2 )
       => ( ! [Z_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
             => hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)) )
         => ? [Z_2] :
              ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1))
              & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Z_2)),hAPP(real,real,F,Xa))) ) ) ) ) ).

fof(fact_4957_cpmi,axiom,
    ! [B_1,P_2,P_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( ? [Z_2] :
          ! [X_1] :
            ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Z_2))
           => ( hBOOL(hAPP(int,bool,P_1,X_1))
            <=> hBOOL(hAPP(int,bool,P_2,X_1)) ) )
       => ( ! [X_1] :
              ( ! [Xa_1] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
                 => ! [Xb] :
                      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
                     => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) ) )
             => ( hBOOL(hAPP(int,bool,P_1,X_1))
               => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D))) ) )
         => ( ! [X_1,K_1] :
                ( hBOOL(hAPP(int,bool,P_2,X_1))
              <=> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D)))) )
           => ( ? [X1] : hBOOL(hAPP(int,bool,P_1,X1))
            <=> ( ? [X_1] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
                    & hBOOL(hAPP(int,bool,P_2,X_1)) )
                | ? [X_1] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
                    & ? [Xa_1] :
                        ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),B_1))
                        & hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa_1),X_1))) ) ) ) ) ) ) ) ) ).

fof(fact_4958_cppi,axiom,
    ! [A_2,P_2,P_1,D] :
      ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
     => ( ? [Z_2] :
          ! [X_1] :
            ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),X_1))
           => ( hBOOL(hAPP(int,bool,P_1,X_1))
            <=> hBOOL(hAPP(int,bool,P_2,X_1)) ) )
       => ( ! [X_1] :
              ( ! [Xa_1] :
                  ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
                 => ! [Xb] :
                      ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
                     => ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) ) )
             => ( hBOOL(hAPP(int,bool,P_1,X_1))
               => hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D))) ) )
         => ( ! [X_1,K_1] :
                ( hBOOL(hAPP(int,bool,P_2,X_1))
              <=> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D)))) )
           => ( ? [X1] : hBOOL(hAPP(int,bool,P_1,X1))
            <=> ( ? [X_1] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
                    & hBOOL(hAPP(int,bool,P_2,X_1)) )
                | ? [X_1] :
                    ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D)))
                    & ? [Xa_1] :
                        ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),A_2))
                        & hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa_1),X_1))) ) ) ) ) ) ) ) ) ).

fof(fact_4959_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,
    ! [Na] : hAPP(nat,fun(nat,bool),ord_atMost(nat),Na) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Na))) ).

fof(fact_4960_finite__atMost,axiom,
    ! [K] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),ord_atMost(nat),K))) ).

fof(fact_4961_atMost__iff,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ! [I,K] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),K)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),I),K)) ) ) ).

fof(fact_4962_atMost__subset__iff,axiom,
    ! [X_a] :
      ( order(X_a)
     => ! [Xa,Ya] :
          ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Xa)),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Ya)))
        <=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) ) ) ).

fof(fact_4963_finite__nat__iff__bounded__le,axiom,
    ! [S] :
      ( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
    <=> ? [K_1] : hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_atMost(nat),K_1))) ) ).

fof(fact_4964_lessThan__Suc__atMost,axiom,
    ! [K] : hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(nat,nat,suc,K)) = hAPP(nat,fun(nat,bool),ord_atMost(nat),K) ).

fof(fact_4965_atMost__eq__iff,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya] :
          ( hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Xa) = hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Ya)
        <=> ti(X_a,Xa) = ti(X_a,Ya) ) ) ).

fof(fact_4966_card__atMost,axiom,
    ! [U] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),ord_atMost(nat),U)) = hAPP(nat,nat,suc,U) ).

fof(fact_4967_atLeast0AtMost,axiom,
    ! [Na] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),Na) = hAPP(nat,fun(nat,bool),ord_atMost(nat),Na) ).

fof(fact_4968_setsum__atMost__Suc,axiom,
    ! [X_a] :
      ( comm_monoid_add(X_a)
     => ! [F,Na] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_atMost(nat),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_atMost(nat),Na))),hAPP(nat,X_a,F,hAPP(nat,nat,suc,Na))) ) ).

fof(fact_4969_max__number__of__Suc,axiom,
    ! [Na,Va] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,suc,Na)),hAPP(int,nat,number_number_of(nat),Va)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,nat,suc,Na)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),Na),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))))) ).

fof(fact_4970_max__Suc__number__of,axiom,
    ! [Va,Na] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,nat,suc,Na)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na))) ).

fof(fact_4971_Sup__atMost,axiom,
    ! [Xa] : hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),ord_atMost(real),Xa)) = Xa ).

fof(fact_4972_minus__max__eq__min,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).

fof(fact_4973_minus__min__eq__max,axiom,
    ! [X_a] :
      ( linord219039673up_add(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).

fof(fact_4974_min__max_Osup__inf__absorb,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = ti(X_a,X) ) ).

fof(fact_4975_min__max_Oinf__sup__absorb,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = ti(X_a,X) ) ).

fof(fact_4976_min__max_Osup__inf__distrib1,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z_1)) ) ).

fof(fact_4977_min__max_Oinf__sup__distrib1,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z_1)) ) ).

fof(fact_4978_min__max_Osup__inf__distrib2,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Y,Z_1,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Z_1),X)) ) ).

fof(fact_4979_min__max_Oinf__sup__distrib2,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Y,Z_1,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z_1)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Z_1),X)) ) ).

fof(fact_4980_min__max_Odistrib__sup__le,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y,Z_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z_1)))) ) ).

fof(fact_4981_min__max_Odistrib__inf__le,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [X,Y,Z_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z_1))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z_1)))) ) ).

fof(fact_4982_max__ord__max,axiom,
    ! [X_a] :
      ( ord(X_a)
     => ord_max(X_a) = hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),max(X_a),ord_less_eq(X_a)) ) ).

fof(fact_4983_max__less__iff__conj,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Xa,Ya,Z] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Xa),Ya)),Z))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Z))
            & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Z)) ) ) ) ).

fof(fact_4984_less__max__iff__disj,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [Z,Xa,Ya] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Xa),Ya)))
        <=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Xa))
            | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Ya)) ) ) ) ).

fof(fact_4985_min__max_Oless__supI2,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [A_1,X,B] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),B))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B))) ) ) ).

fof(fact_4986_min__max_Oless__supI1,axiom,
    ! [X_a] :
      ( linorder(X_a)
     => ! [B,X,A_1] :
          ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),A_1))
         => hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B))) ) ) ).

fof(fact_4987_max__Suc2,axiom,
    ! [Ma,Na] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),Ma),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),hAPP(nat,nat,suc,Na)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),ord_max(nat)),Na))),Ma) ).

fof(fact_4988_max__Suc1,axiom,
    ! [Na,Ma] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,suc,Na)),Ma) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),hAPP(nat,nat,suc,Na)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),ord_max(nat),Na))),Ma) ).

fof(fact_4989_max__0R,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N ).

fof(fact_4990_max__0L,axiom,
    ! [N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N ).

fof(fact_4991_max__Suc__Suc,axiom,
    ! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),M),N)) ).

%----Arities (446)
fof(arity_fun___Complete__Lattice_Ocomplete__lattice,axiom,
    ! [T_2,T_1] :
      ( comple44059174attice(T_1)
     => comple44059174attice(fun(T_2,T_1)) ) ).

fof(arity_fun___Lattices_Oboolean__algebra,axiom,
    ! [T_2,T_1] :
      ( boolean_algebra(T_1)
     => boolean_algebra(fun(T_2,T_1)) ) ).

fof(arity_fun___Orderings_Opreorder,axiom,
    ! [T_2,T_1] :
      ( preorder(T_1)
     => preorder(fun(T_2,T_1)) ) ).

fof(arity_fun___Finite__Set_Ofinite,axiom,
    ! [T_2,T_1] :
      ( ( finite_finite(T_1)
        & finite_finite(T_2) )
     => finite_finite(fun(T_2,T_1)) ) ).

fof(arity_fun___Orderings_Oorder,axiom,
    ! [T_2,T_1] :
      ( order(T_1)
     => order(fun(T_2,T_1)) ) ).

fof(arity_fun___Orderings_Oord,axiom,
    ! [T_2,T_1] :
      ( ord(T_1)
     => ord(fun(T_2,T_1)) ) ).

fof(arity_fun___Groups_Ouminus,axiom,
    ! [T_2,T_1] :
      ( uminus(T_1)
     => uminus(fun(T_2,T_1)) ) ).

fof(arity_fun___Groups_Ominus,axiom,
    ! [T_2,T_1] :
      ( minus(T_1)
     => minus(fun(T_2,T_1)) ) ).

fof(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
    semiri456707255roduct(int) ).

fof(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere223160158up_add(int) ).

fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere236663937imp_le(int) ).

fof(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord893533164strict(int) ).

fof(arity_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord626643107strict(int) ).

fof(arity_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord20386208strict(int) ).

fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere779506340up_add(int) ).

fof(arity_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere142940540dd_abs(int) ).

fof(arity_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere216010020id_add(int) ).

fof(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord219039673up_add(int) ).

fof(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel146912293up_add(int) ).

fof(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_11004092258visors(int) ).

fof(arity_Int_Oint___Rings_Oordered__cancel__semiring,axiom,
    ordere453448008miring(int) ).

fof(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord581940658strict(int) ).

fof(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
    ring_n68954251visors(int) ).

fof(arity_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere1490568538miring(int) ).

fof(arity_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord1278240602ring_1(int) ).

fof(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(int) ).

fof(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(int) ).

fof(arity_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring(int) ).

fof(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

fof(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

fof(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

fof(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

fof(arity_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

fof(arity_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

fof(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
    no_zero_divisors(int) ).

fof(arity_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

fof(arity_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

fof(arity_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

fof(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

fof(arity_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

fof(arity_Int_Oint___Divides_Osemiring__div,axiom,
    semiring_div(int) ).

fof(arity_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

fof(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

fof(arity_Int_Oint___Int_Onumber__semiring,axiom,
    number_semiring(int) ).

fof(arity_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

fof(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

fof(arity_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

fof(arity_Int_Oint___Orderings_Opreorder,axiom,
    preorder(int) ).

fof(arity_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

fof(arity_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

fof(arity_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

fof(arity_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

fof(arity_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

fof(arity_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

fof(arity_Int_Oint___Groups_Ogroup__add,axiom,
    group_add(int) ).

fof(arity_Int_Oint___Divides_Oring__div,axiom,
    ring_div(int) ).

fof(arity_Int_Oint___Rings_Omult__zero,axiom,
    mult_zero(int) ).

fof(arity_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

fof(arity_Int_Oint___Orderings_Oorder,axiom,
    order(int) ).

fof(arity_Int_Oint___Int_Oring__char__0,axiom,
    ring_char_0(int) ).

fof(arity_Int_Oint___Int_Onumber__ring,axiom,
    number_ring(int) ).

fof(arity_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

fof(arity_Int_Oint___Orderings_Oord,axiom,
    ord(int) ).

fof(arity_Int_Oint___Groups_Ouminus,axiom,
    uminus(int) ).

fof(arity_Int_Oint___Groups_Osgn__if,axiom,
    sgn_if(int) ).

fof(arity_Int_Oint___Groups_Oabs__if,axiom,
    abs_if(int) ).

fof(arity_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

fof(arity_Int_Oint___Groups_Ominus,axiom,
    minus(int) ).

fof(arity_Int_Oint___Power_Opower,axiom,
    power(int) ).

fof(arity_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

fof(arity_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

fof(arity_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

fof(arity_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

fof(arity_Int_Oint___Int_Onumber,axiom,
    number(int) ).

fof(arity_Int_Oint___Groups_Oone,axiom,
    one(int) ).

fof(arity_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

fof(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
    semiri456707255roduct(nat) ).

fof(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere223160158up_add(nat) ).

fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere236663937imp_le(nat) ).

fof(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict,axiom,
    linord893533164strict(nat) ).

fof(arity_Nat_Onat___Rings_Olinordered__semiring__strict,axiom,
    linord20386208strict(nat) ).

fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
    ordere779506340up_add(nat) ).

fof(arity_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
    ordere216010020id_add(nat) ).

fof(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel146912293up_add(nat) ).

fof(arity_Nat_Onat___Rings_Oordered__cancel__semiring,axiom,
    ordere453448008miring(nat) ).

fof(arity_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
    ordere1490568538miring(nat) ).

fof(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(nat) ).

fof(arity_Nat_Onat___Rings_Olinordered__semiring,axiom,
    linordered_semiring(nat) ).

fof(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
    linordered_semidom(nat) ).

fof(arity_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(nat) ).

fof(arity_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(nat) ).

fof(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(nat) ).

fof(arity_Nat_Onat___Rings_Oordered__semiring,axiom,
    ordered_semiring(nat) ).

fof(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
    no_zero_divisors(nat) ).

fof(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(nat) ).

fof(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(nat) ).

fof(arity_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(nat) ).

fof(arity_Nat_Onat___Divides_Osemiring__div,axiom,
    semiring_div(nat) ).

fof(arity_Nat_Onat___Rings_Ocomm__semiring,axiom,
    comm_semiring(nat) ).

fof(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
    semiring_char_0(nat) ).

fof(arity_Nat_Onat___Int_Onumber__semiring,axiom,
    number_semiring(nat) ).

fof(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
    zero_neq_one(nat) ).

fof(arity_Nat_Onat___Orderings_Opreorder,axiom,
    preorder(nat) ).

fof(arity_Nat_Onat___Orderings_Olinorder,axiom,
    linorder(nat) ).

fof(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
    monoid_mult(nat) ).

fof(arity_Nat_Onat___Groups_Omonoid__add,axiom,
    monoid_add(nat) ).

fof(arity_Nat_Onat___Rings_Osemiring__1,axiom,
    semiring_1(nat) ).

fof(arity_Nat_Onat___Rings_Osemiring__0,axiom,
    semiring_0(nat) ).

fof(arity_Nat_Onat___Rings_Omult__zero,axiom,
    mult_zero(nat) ).

fof(arity_Nat_Onat___Orderings_Oorder,axiom,
    order(nat) ).

fof(arity_Nat_Onat___Rings_Osemiring,axiom,
    semiring(nat) ).

fof(arity_Nat_Onat___Orderings_Oord,axiom,
    ord(nat) ).

fof(arity_Nat_Onat___Groups_Ominus,axiom,
    minus(nat) ).

fof(arity_Nat_Onat___Power_Opower,axiom,
    power(nat) ).

fof(arity_Nat_Onat___Groups_Ozero,axiom,
    zero(nat) ).

fof(arity_Nat_Onat___Groups_Oplus,axiom,
    plus(nat) ).

fof(arity_Nat_Onat___Int_Onumber,axiom,
    number(nat) ).

fof(arity_Nat_Onat___Groups_Oone,axiom,
    one(nat) ).

fof(arity_Nat_Onat___Rings_Odvd,axiom,
    dvd(nat) ).

fof(arity_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
    semiri456707255roduct(rat) ).

fof(arity_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere223160158up_add(rat) ).

fof(arity_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere236663937imp_le(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__comm__semiring__strict,axiom,
    linord893533164strict(rat) ).

fof(arity_Rat_Orat___Fields_Olinordered__field__inverse__zero,axiom,
    linord1117847801e_zero(rat) ).

fof(arity_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim1804426504_field(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__semiring__1__strict,axiom,
    linord626643107strict(rat) ).

fof(arity_Rat_Orat___Fields_Odivision__ring__inverse__zero,axiom,
    divisi14063676e_zero(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__semiring__strict,axiom,
    linord20386208strict(rat) ).

fof(arity_Rat_Orat___Groups_Oordered__ab__semigroup__add,axiom,
    ordere779506340up_add(rat) ).

fof(arity_Rat_Orat___Groups_Oordered__ab__group__add__abs,axiom,
    ordere142940540dd_abs(rat) ).

fof(arity_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim1727834104eiling(rat) ).

fof(arity_Rat_Orat___Groups_Oordered__comm__monoid__add,axiom,
    ordere216010020id_add(rat) ).

fof(arity_Rat_Orat___Groups_Olinordered__ab__group__add,axiom,
    linord219039673up_add(rat) ).

fof(arity_Rat_Orat___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel146912293up_add(rat) ).

fof(arity_Rat_Orat___Rings_Oring__1__no__zero__divisors,axiom,
    ring_11004092258visors(rat) ).

fof(arity_Rat_Orat___Rings_Oordered__cancel__semiring,axiom,
    ordere453448008miring(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__ring__strict,axiom,
    linord581940658strict(rat) ).

fof(arity_Rat_Orat___Rings_Oring__no__zero__divisors,axiom,
    ring_n68954251visors(rat) ).

fof(arity_Rat_Orat___Rings_Oordered__comm__semiring,axiom,
    ordere1490568538miring(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__semiring__1,axiom,
    linord1278240602ring_1(rat) ).

fof(arity_Rat_Orat___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(rat) ).

fof(arity_Rat_Orat___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__semiring,axiom,
    linordered_semiring(rat) ).

fof(arity_Rat_Orat___Fields_Ofield__inverse__zero,axiom,
    field_inverse_zero(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__semidom,axiom,
    linordered_semidom(rat) ).

fof(arity_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

fof(arity_Rat_Orat___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(rat) ).

fof(arity_Rat_Orat___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(rat) ).

fof(arity_Rat_Orat___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(rat) ).

fof(arity_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

fof(arity_Rat_Orat___Rings_Oordered__semiring,axiom,
    ordered_semiring(rat) ).

fof(arity_Rat_Orat___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(rat) ).

fof(arity_Rat_Orat___Rings_Ono__zero__divisors,axiom,
    no_zero_divisors(rat) ).

fof(arity_Rat_Orat___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__ring,axiom,
    linordered_ring(rat) ).

fof(arity_Rat_Orat___Rings_Olinordered__idom,axiom,
    linordered_idom(rat) ).

fof(arity_Rat_Orat___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(rat) ).

fof(arity_Rat_Orat___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(rat) ).

fof(arity_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

fof(arity_Rat_Orat___Rings_Ocomm__semiring,axiom,
    comm_semiring(rat) ).

fof(arity_Rat_Orat___Nat_Osemiring__char__0,axiom,
    semiring_char_0(rat) ).

fof(arity_Rat_Orat___Int_Onumber__semiring,axiom,
    number_semiring(rat) ).

fof(arity_Rat_Orat___Groups_Oab__group__add,axiom,
    ab_group_add(rat) ).

fof(arity_Rat_Orat___Rings_Ozero__neq__one,axiom,
    zero_neq_one(rat) ).

fof(arity_Rat_Orat___Rings_Oordered__ring,axiom,
    ordered_ring(rat) ).

fof(arity_Rat_Orat___Orderings_Opreorder,axiom,
    preorder(rat) ).

fof(arity_Rat_Orat___Orderings_Olinorder,axiom,
    linorder(rat) ).

fof(arity_Rat_Orat___Groups_Omonoid__mult,axiom,
    monoid_mult(rat) ).

fof(arity_Rat_Orat___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(rat) ).

fof(arity_Rat_Orat___Groups_Omonoid__add,axiom,
    monoid_add(rat) ).

fof(arity_Rat_Orat___Rings_Osemiring__1,axiom,
    semiring_1(rat) ).

fof(arity_Rat_Orat___Rings_Osemiring__0,axiom,
    semiring_0(rat) ).

fof(arity_Rat_Orat___Rat_Ofield__char__0,axiom,
    field_char_0(rat) ).

fof(arity_Rat_Orat___Groups_Ogroup__add,axiom,
    group_add(rat) ).

fof(arity_Rat_Orat___Rings_Omult__zero,axiom,
    mult_zero(rat) ).

fof(arity_Rat_Orat___Rings_Ocomm__ring,axiom,
    comm_ring(rat) ).

fof(arity_Rat_Orat___Orderings_Oorder,axiom,
    order(rat) ).

fof(arity_Rat_Orat___Int_Oring__char__0,axiom,
    ring_char_0(rat) ).

fof(arity_Rat_Orat___Int_Onumber__ring,axiom,
    number_ring(rat) ).

fof(arity_Rat_Orat___Rings_Osemiring,axiom,
    semiring(rat) ).

fof(arity_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

fof(arity_Rat_Orat___Orderings_Oord,axiom,
    ord(rat) ).

fof(arity_Rat_Orat___Groups_Ouminus,axiom,
    uminus(rat) ).

fof(arity_Rat_Orat___Groups_Osgn__if,axiom,
    sgn_if(rat) ).

fof(arity_Rat_Orat___Groups_Oabs__if,axiom,
    abs_if(rat) ).

fof(arity_Rat_Orat___Rings_Oring__1,axiom,
    ring_1(rat) ).

fof(arity_Rat_Orat___Groups_Ominus,axiom,
    minus(rat) ).

fof(arity_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

fof(arity_Rat_Orat___Power_Opower,axiom,
    power(rat) ).

fof(arity_Rat_Orat___Groups_Ozero,axiom,
    zero(rat) ).

fof(arity_Rat_Orat___Groups_Oplus,axiom,
    plus(rat) ).

fof(arity_Rat_Orat___Rings_Oring,axiom,
    ring(rat) ).

fof(arity_Rat_Orat___Rings_Oidom,axiom,
    idom(rat) ).

fof(arity_Rat_Orat___Int_Onumber,axiom,
    number(rat) ).

fof(arity_Rat_Orat___Groups_Oone,axiom,
    one(rat) ).

fof(arity_Rat_Orat___Rings_Odvd,axiom,
    dvd(rat) ).

fof(arity_HOL_Obool___Complete__Lattice_Ocomplete__lattice,axiom,
    comple44059174attice(bool) ).

fof(arity_HOL_Obool___Lattices_Oboolean__algebra,axiom,
    boolean_algebra(bool) ).

fof(arity_HOL_Obool___Orderings_Opreorder,axiom,
    preorder(bool) ).

fof(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
    finite_finite(bool) ).

fof(arity_HOL_Obool___Orderings_Oorder,axiom,
    order(bool) ).

fof(arity_HOL_Obool___Orderings_Oord,axiom,
    ord(bool) ).

fof(arity_HOL_Obool___Groups_Ouminus,axiom,
    uminus(bool) ).

fof(arity_HOL_Obool___Groups_Ominus,axiom,
    minus(bool) ).

fof(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp,axiom,
    semiri456707255roduct(real) ).

fof(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere223160158up_add(real) ).

fof(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere236663937imp_le(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict,axiom,
    linord893533164strict(real) ).

fof(arity_RealDef_Oreal___Fields_Olinordered__field__inverse__zero,axiom,
    linord1117847801e_zero(real) ).

fof(arity_RealDef_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
    archim1804426504_field(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict,axiom,
    linord626643107strict(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__normed__div__algebra,axiom,
    real_n1866405975lgebra(real) ).

fof(arity_RealDef_Oreal___Fields_Odivision__ring__inverse__zero,axiom,
    divisi14063676e_zero(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict,axiom,
    linord20386208strict(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra__1,axiom,
    real_n2089651433ebra_1(real) ).

fof(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
    ordere779506340up_add(real) ).

fof(arity_RealDef_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
    ordere142940540dd_abs(real) ).

fof(arity_RealDef_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
    archim1727834104eiling(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra,axiom,
    real_normed_algebra(real) ).

fof(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add,axiom,
    ordere216010020id_add(real) ).

fof(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
    linord219039673up_add(real) ).

fof(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel146912293up_add(real) ).

fof(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
    ring_11004092258visors(real) ).

fof(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring,axiom,
    ordere453448008miring(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__normed__vector,axiom,
    real_normed_vector(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__ring__strict,axiom,
    linord581940658strict(real) ).

fof(arity_RealDef_Oreal___RealVector_Otopological__space,axiom,
    topological_space(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__normed__field,axiom,
    real_normed_field(real) ).

fof(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors,axiom,
    ring_n68954251visors(real) ).

fof(arity_RealDef_Oreal___Rings_Oordered__comm__semiring,axiom,
    ordere1490568538miring(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__1,axiom,
    linord1278240602ring_1(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__div__algebra,axiom,
    real_div_algebra(real) ).

fof(arity_RealDef_Oreal___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(real) ).

fof(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__semiring,axiom,
    linordered_semiring(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__algebra__1,axiom,
    real_algebra_1(real) ).

fof(arity_RealDef_Oreal___Fields_Ofield__inverse__zero,axiom,
    field_inverse_zero(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
    linordered_semidom(real) ).

fof(arity_RealDef_Oreal___Orderings_Odense__linorder,axiom,
    dense_linorder(real) ).

fof(arity_RealDef_Oreal___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(real) ).

fof(arity_RealDef_Oreal___RealVector_Osgn__div__norm,axiom,
    sgn_div_norm(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__algebra,axiom,
    real_algebra(real) ).

fof(arity_RealDef_Oreal___RealVector_Ometric__space,axiom,
    metric_space(real) ).

fof(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(real) ).

fof(arity_RealDef_Oreal___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(real) ).

fof(arity_RealDef_Oreal___Fields_Olinordered__field,axiom,
    linordered_field(real) ).

fof(arity_RealDef_Oreal___Rings_Oordered__semiring,axiom,
    ordered_semiring(real) ).

fof(arity_RealDef_Oreal___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(real) ).

fof(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
    no_zero_divisors(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__vector,axiom,
    real_vector(real) ).

fof(arity_RealDef_Oreal___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__ring,axiom,
    linordered_ring(real) ).

fof(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
    linordered_idom(real) ).

fof(arity_RealDef_Oreal___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(real) ).

fof(arity_RealDef_Oreal___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(real) ).

fof(arity_RealDef_Oreal___RealVector_Oreal__field,axiom,
    real_field(real) ).

fof(arity_RealDef_Oreal___RealVector_Odist__norm,axiom,
    dist_norm(real) ).

fof(arity_RealDef_Oreal___Fields_Odivision__ring,axiom,
    division_ring(real) ).

fof(arity_RealDef_Oreal___Rings_Ocomm__semiring,axiom,
    comm_semiring(real) ).

fof(arity_RealDef_Oreal___RealVector_Ot2__space,axiom,
    t2_space(real) ).

fof(arity_RealDef_Oreal___Nat_Osemiring__char__0,axiom,
    semiring_char_0(real) ).

fof(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
    number_semiring(real) ).

fof(arity_RealDef_Oreal___Groups_Oab__group__add,axiom,
    ab_group_add(real) ).

fof(arity_RealDef_Oreal___Rings_Ozero__neq__one,axiom,
    zero_neq_one(real) ).

fof(arity_RealDef_Oreal___Rings_Oordered__ring,axiom,
    ordered_ring(real) ).

fof(arity_RealDef_Oreal___Orderings_Opreorder,axiom,
    preorder(real) ).

fof(arity_RealDef_Oreal___Orderings_Olinorder,axiom,
    linorder(real) ).

fof(arity_RealDef_Oreal___Groups_Omonoid__mult,axiom,
    monoid_mult(real) ).

fof(arity_RealDef_Oreal___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(real) ).

fof(arity_RealDef_Oreal___Groups_Omonoid__add,axiom,
    monoid_add(real) ).

fof(arity_RealDef_Oreal___Rings_Osemiring__1,axiom,
    semiring_1(real) ).

fof(arity_RealDef_Oreal___Rings_Osemiring__0,axiom,
    semiring_0(real) ).

fof(arity_RealDef_Oreal___Rat_Ofield__char__0,axiom,
    field_char_0(real) ).

fof(arity_RealDef_Oreal___Groups_Ogroup__add,axiom,
    group_add(real) ).

fof(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
    mult_zero(real) ).

fof(arity_RealDef_Oreal___Rings_Ocomm__ring,axiom,
    comm_ring(real) ).

fof(arity_RealDef_Oreal___Orderings_Oorder,axiom,
    order(real) ).

fof(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
    ring_char_0(real) ).

fof(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
    number_ring(real) ).

fof(arity_RealDef_Oreal___Rings_Osemiring,axiom,
    semiring(real) ).

fof(arity_RealDef_Oreal___Fields_Oinverse,axiom,
    inverse(real) ).

fof(arity_RealDef_Oreal___Orderings_Oord,axiom,
    ord(real) ).

fof(arity_RealDef_Oreal___Groups_Ouminus,axiom,
    uminus(real) ).

fof(arity_RealDef_Oreal___Groups_Osgn__if,axiom,
    sgn_if(real) ).

fof(arity_RealDef_Oreal___Groups_Oabs__if,axiom,
    abs_if(real) ).

fof(arity_RealDef_Oreal___Rings_Oring__1,axiom,
    ring_1(real) ).

fof(arity_RealDef_Oreal___Groups_Ominus,axiom,
    minus(real) ).

fof(arity_RealDef_Oreal___Fields_Ofield,axiom,
    field(real) ).

fof(arity_RealDef_Oreal___Power_Opower,axiom,
    power(real) ).

fof(arity_RealDef_Oreal___Groups_Ozero,axiom,
    zero(real) ).

fof(arity_RealDef_Oreal___Groups_Oplus,axiom,
    plus(real) ).

fof(arity_RealDef_Oreal___SEQ_Obanach,axiom,
    banach(real) ).

fof(arity_RealDef_Oreal___Rings_Oring,axiom,
    ring(real) ).

fof(arity_RealDef_Oreal___Rings_Oidom,axiom,
    idom(real) ).

fof(arity_RealDef_Oreal___Int_Onumber,axiom,
    number(real) ).

fof(arity_RealDef_Oreal___Groups_Oone,axiom,
    one(real) ).

fof(arity_RealDef_Oreal___Rings_Odvd,axiom,
    dvd(real) ).

fof(arity_sum___Finite__Set_Ofinite,axiom,
    ! [T_2,T_1] :
      ( ( finite_finite(T_1)
        & finite_finite(T_2) )
     => finite_finite(sum_sum(T_2,T_1)) ) ).

fof(arity_Limits_Ofilter___Complete__Lattice_Ocomplete__lattice,axiom,
    ! [T_1] : comple44059174attice(filter(T_1)) ).

fof(arity_Limits_Ofilter___Orderings_Opreorder,axiom,
    ! [T_1] : preorder(filter(T_1)) ).

fof(arity_Limits_Ofilter___Orderings_Oorder,axiom,
    ! [T_1] : order(filter(T_1)) ).

fof(arity_Limits_Ofilter___Orderings_Oord,axiom,
    ! [T_1] : ord(filter(T_1)) ).

fof(arity_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__cro,axiom,
    semiri456707255roduct(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__div__algebra,axiom,
    real_n1866405975lgebra(complex) ).

fof(arity_Complex_Ocomplex___Fields_Odivision__ring__inverse__zero,axiom,
    divisi14063676e_zero(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra__1,axiom,
    real_n2089651433ebra_1(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra,axiom,
    real_normed_algebra(complex) ).

fof(arity_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel146912293up_add(complex) ).

fof(arity_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors,axiom,
    ring_11004092258visors(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__vector,axiom,
    real_normed_vector(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Otopological__space,axiom,
    topological_space(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__field,axiom,
    real_normed_field(complex) ).

fof(arity_Complex_Ocomplex___Rings_Oring__no__zero__divisors,axiom,
    ring_n68954251visors(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__div__algebra,axiom,
    real_div_algebra(complex) ).

fof(arity_Complex_Ocomplex___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__algebra__1,axiom,
    real_algebra_1(complex) ).

fof(arity_Complex_Ocomplex___Fields_Ofield__inverse__zero,axiom,
    field_inverse_zero(complex) ).

fof(arity_Complex_Ocomplex___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Osgn__div__norm,axiom,
    sgn_div_norm(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__algebra,axiom,
    real_algebra(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Ometric__space,axiom,
    metric_space(complex) ).

fof(arity_Complex_Ocomplex___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(complex) ).

fof(arity_Complex_Ocomplex___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(complex) ).

fof(arity_Complex_Ocomplex___Rings_Ono__zero__divisors,axiom,
    no_zero_divisors(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__vector,axiom,
    real_vector(complex) ).

fof(arity_Complex_Ocomplex___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(complex) ).

fof(arity_Complex_Ocomplex___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(complex) ).

fof(arity_Complex_Ocomplex___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Oreal__field,axiom,
    real_field(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Odist__norm,axiom,
    dist_norm(complex) ).

fof(arity_Complex_Ocomplex___Fields_Odivision__ring,axiom,
    division_ring(complex) ).

fof(arity_Complex_Ocomplex___Rings_Ocomm__semiring,axiom,
    comm_semiring(complex) ).

fof(arity_Complex_Ocomplex___RealVector_Ot2__space,axiom,
    t2_space(complex) ).

fof(arity_Complex_Ocomplex___Nat_Osemiring__char__0,axiom,
    semiring_char_0(complex) ).

fof(arity_Complex_Ocomplex___Int_Onumber__semiring,axiom,
    number_semiring(complex) ).

fof(arity_Complex_Ocomplex___Groups_Oab__group__add,axiom,
    ab_group_add(complex) ).

fof(arity_Complex_Ocomplex___Rings_Ozero__neq__one,axiom,
    zero_neq_one(complex) ).

fof(arity_Complex_Ocomplex___Groups_Omonoid__mult,axiom,
    monoid_mult(complex) ).

fof(arity_Complex_Ocomplex___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(complex) ).

fof(arity_Complex_Ocomplex___Groups_Omonoid__add,axiom,
    monoid_add(complex) ).

fof(arity_Complex_Ocomplex___Rings_Osemiring__1,axiom,
    semiring_1(complex) ).

fof(arity_Complex_Ocomplex___Rings_Osemiring__0,axiom,
    semiring_0(complex) ).

fof(arity_Complex_Ocomplex___Rat_Ofield__char__0,axiom,
    field_char_0(complex) ).

fof(arity_Complex_Ocomplex___Groups_Ogroup__add,axiom,
    group_add(complex) ).

fof(arity_Complex_Ocomplex___Rings_Omult__zero,axiom,
    mult_zero(complex) ).

fof(arity_Complex_Ocomplex___Rings_Ocomm__ring,axiom,
    comm_ring(complex) ).

fof(arity_Complex_Ocomplex___Int_Oring__char__0,axiom,
    ring_char_0(complex) ).

fof(arity_Complex_Ocomplex___Int_Onumber__ring,axiom,
    number_ring(complex) ).

fof(arity_Complex_Ocomplex___Rings_Osemiring,axiom,
    semiring(complex) ).

fof(arity_Complex_Ocomplex___Fields_Oinverse,axiom,
    inverse(complex) ).

fof(arity_Complex_Ocomplex___Groups_Ouminus,axiom,
    uminus(complex) ).

fof(arity_Complex_Ocomplex___Rings_Oring__1,axiom,
    ring_1(complex) ).

fof(arity_Complex_Ocomplex___Groups_Ominus,axiom,
    minus(complex) ).

fof(arity_Complex_Ocomplex___Fields_Ofield,axiom,
    field(complex) ).

fof(arity_Complex_Ocomplex___Power_Opower,axiom,
    power(complex) ).

fof(arity_Complex_Ocomplex___Groups_Ozero,axiom,
    zero(complex) ).

fof(arity_Complex_Ocomplex___Groups_Oplus,axiom,
    plus(complex) ).

fof(arity_Complex_Ocomplex___SEQ_Obanach,axiom,
    banach(complex) ).

fof(arity_Complex_Ocomplex___Rings_Oring,axiom,
    ring(complex) ).

fof(arity_Complex_Ocomplex___Rings_Oidom,axiom,
    idom(complex) ).

fof(arity_Complex_Ocomplex___Int_Onumber,axiom,
    number(complex) ).

fof(arity_Complex_Ocomplex___Groups_Oone,axiom,
    one(complex) ).

fof(arity_Complex_Ocomplex___Rings_Odvd,axiom,
    dvd(complex) ).

fof(arity_prod___Finite__Set_Ofinite,axiom,
    ! [T_2,T_1] :
      ( ( finite_finite(T_1)
        & finite_finite(T_2) )
     => finite_finite(product_prod(T_2,T_1)) ) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere223160158up_add(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__ab__semigroup__add__imp__l,axiom,
    ordere236663937imp_le(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__comm__semiring__strict,axiom,
    linord893533164strict(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__semiring__strict,axiom,
    linord20386208strict(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__ab__semigroup__add,axiom,
    ordere779506340up_add(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__comm__monoid__add,axiom,
    ordere216010020id_add(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel146912293up_add(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Oordered__cancel__semiring,axiom,
    ordere453448008miring(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Oordered__comm__semiring,axiom,
    ordere1490568538miring(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__semiring,axiom,
    linordered_semiring(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__semidom,axiom,
    linordered_semidom(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Oordered__semiring,axiom,
    ordered_semiring(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Ono__zero__divisors,axiom,
    no_zero_divisors(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Divides_Osemiring__div,axiom,
    semiring_div(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Ocomm__semiring,axiom,
    comm_semiring(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Nat_Osemiring__char__0,axiom,
    semiring_char_0(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Ozero__neq__one,axiom,
    zero_neq_one(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Orderings_Opreorder,axiom,
    preorder(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Orderings_Olinorder,axiom,
    linorder(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Omonoid__mult,axiom,
    monoid_mult(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Omonoid__add,axiom,
    monoid_add(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Osemiring__1,axiom,
    semiring_1(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Osemiring__0,axiom,
    semiring_0(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Omult__zero,axiom,
    mult_zero(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Orderings_Oorder,axiom,
    order(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Osemiring,axiom,
    semiring(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Orderings_Oord,axiom,
    ord(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Ominus,axiom,
    minus(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Power_Opower,axiom,
    power(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Ozero,axiom,
    zero(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Oplus,axiom,
    plus(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Int_Onumber,axiom,
    number(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Groups_Oone,axiom,
    one(code_code_numeral) ).

fof(arity_Code__Numeral_Ocode__numeral___Rings_Odvd,axiom,
    dvd(code_code_numeral) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__cancel__ab__semigroup_,axiom,
    ordere223160158up_add(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__ab__semigroup__add__im,axiom,
    ordere236663937imp_le(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__comm__semiring__stri,axiom,
    linord893533164strict(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__semiring__strict,axiom,
    linord20386208strict(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__ab__semigroup__add,axiom,
    ordere779506340up_add(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__comm__monoid__add,axiom,
    ordere216010020id_add(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel146912293up_add(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Oordered__cancel__semiring,axiom,
    ordere453448008miring(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Oordered__comm__semiring,axiom,
    ordere1490568538miring(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__semiring,axiom,
    linordered_semiring(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__semidom,axiom,
    linordered_semidom(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Oordered__semiring,axiom,
    ordered_semiring(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ono__zero__divisors,axiom,
    no_zero_divisors(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Divides_Osemiring__div,axiom,
    semiring_div(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ocomm__semiring,axiom,
    comm_semiring(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Nat_Osemiring__char__0,axiom,
    semiring_char_0(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ozero__neq__one,axiom,
    zero_neq_one(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Opreorder,axiom,
    preorder(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Olinorder,axiom,
    linorder(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Omonoid__mult,axiom,
    monoid_mult(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Omonoid__add,axiom,
    monoid_add(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Osemiring__1,axiom,
    semiring_1(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Osemiring__0,axiom,
    semiring_0(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Omult__zero,axiom,
    mult_zero(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Oorder,axiom,
    order(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Osemiring,axiom,
    semiring(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Oord,axiom,
    ord(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ominus,axiom,
    minus(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Power_Opower,axiom,
    power(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ozero,axiom,
    zero(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oplus,axiom,
    plus(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Int_Onumber,axiom,
    number(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oone,axiom,
    one(quickcheck_code_int) ).

fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Odvd,axiom,
    dvd(quickcheck_code_int) ).

%----Helper facts (27)
fof(help_ti_idem,axiom,
    ! [T,A] : ti(T,ti(T,A)) = ti(T,A) ).

fof(help_If_1_1_T,axiom,
    ! [X_a,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fTrue),X),Y) = ti(X_a,X) ).

fof(help_If_2_1_T,axiom,
    ! [X_a,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fFalse),X),Y) = ti(X_a,Y) ).

fof(help_If_3_1_T,axiom,
    ! [P] :
      ( ti(bool,P) = fTrue
      | ti(bool,P) = fFalse ) ).

fof(help_fEx_1_1_U,axiom,
    ! [X_a,P,X] :
      ( ~ hBOOL(hAPP(X_a,bool,P,X))
      | hBOOL(hAPP(fun(X_a,bool),bool,fEx(X_a),P)) ) ).

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_COMBB_1_1_U,axiom,
    ! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),P),Q),R) = hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R)) ).

fof(help_COMBC_1_1_U,axiom,
    ! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),Q) ).

fof(help_COMBI_1_1_U,axiom,
    ! [X_a,P] : hAPP(X_a,X_a,combi(X_a),P) = ti(X_a,P) ).

fof(help_COMBK_1_1_U,axiom,
    ! [X_b,X_a,P,Q] : hAPP(X_b,X_a,hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),P),Q) = ti(X_a,P) ).

fof(help_COMBS_1_1_U,axiom,
    ! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c)),combs(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R)) ).

fof(help_fTrue_1_1_U,axiom,
    hBOOL(fTrue) ).

fof(help_fTrue_1_1_T,axiom,
    ! [P] :
      ( ti(bool,P) = fTrue
      | ti(bool,P) = fFalse ) ).

fof(help_fconj_1_1_U,axiom,
    ! [Q,P] :
      ( ~ hBOOL(P)
      | ~ hBOOL(Q)
      | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) ) ).

fof(help_fconj_2_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
      | hBOOL(P) ) ).

fof(help_fconj_3_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
      | hBOOL(Q) ) ).

fof(help_fdisj_1_1_U,axiom,
    ! [Q,P] :
      ( ~ hBOOL(P)
      | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).

fof(help_fdisj_2_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).

fof(help_fdisj_3_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,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] :
      ( ti(bool,P) = fTrue
      | ti(bool,P) = fFalse ) ).

fof(help_fequal_1_1_T,axiom,
    ! [X_a,X,Y] :
      ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y))
      | ti(X_a,X) = ti(X_a,Y) ) ).

fof(help_fequal_2_1_T,axiom,
    ! [X_a,X,Y] :
      ( ti(X_a,X) != ti(X_a,Y)
      | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y)) ) ).

fof(help_fimplies_1_1_U,axiom,
    ! [Q,P] :
      ( hBOOL(P)
      | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) ) ).

fof(help_fimplies_2_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(Q)
      | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) ) ).

fof(help_fimplies_3_1_U,axiom,
    ! [P,Q] :
      ( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q))
      | ~ hBOOL(P)
      | hBOOL(Q) ) ).

%----Conjectures (1)
fof(conj_0,conjecture,
    hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(int) ).

%------------------------------------------------------------------------------