ITP001 Axioms: ITP131^5.ax


%------------------------------------------------------------------------------
% File     : ITP131^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : powser^2.ax [Gau20]
%          : HL4131^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   18 (   4 unt;   1 typ;   0 def)
%            Number of atoms       : 1128 (  11 equ;   0 cnn)
%            Maximal formula atoms :  112 (  62 avg)
%            Number of connectives : 1331 (   2   ~;   0   |;  12   &;1296   @)
%                                         (   0 <=>;  21  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   27 (  11 avg;1296 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   38 (  37 usr;  37 con; 0-2 aty)
%            Number of variables   :   77 (  30   ^  47   !;   0   ?;  77   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Epowser_2Ediffs,type,
    c_2Epowser_2Ediffs: $i ).

thf(mem_c_2Epowser_2Ediffs,axiom,
    mem @ c_2Epowser_2Ediffs @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ).

thf(conj_thm_2Epowser_2EPOWDIFF__LEMMA,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
      ( ( surj__ty_2Erealax_2Ereal
        @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
          @ ( lam @ ty_2Enum_2Enum
            @ ^ [V3p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3p ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V3p ) ) ) ) ) )
      = ( surj__ty_2Erealax_2Ereal
        @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2y ) )
          @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
            @ ( lam @ ty_2Enum_2Enum
              @ ^ [V4p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V4p ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V4p ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2EPOWDIFF,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
      ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
      = ( surj__ty_2Erealax_2Ereal
        @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
          @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
            @ ( lam @ ty_2Enum_2Enum
              @ ^ [V3p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3p ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3p ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2EPOWREV,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
      ( ( surj__ty_2Erealax_2Ereal
        @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
          @ ( lam @ ty_2Enum_2Enum
            @ ^ [V3p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3p ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3p ) ) ) ) ) )
      = ( surj__ty_2Erealax_2Ereal
        @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
          @ ( lam @ ty_2Enum_2Enum
            @ ^ [V4p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V4p ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V4p ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2EPOWSER__INSIDEA,axiom,
    ! [V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ! [V1x: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
          ( ( ( p
              @ ( ap @ c_2Eseq_2Esummable
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3n ) ) ) ) )
            & ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) )
         => ( p
            @ ( ap @ c_2Eseq_2Esummable
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V4n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ V4n ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2EPOWSER__INSIDE,axiom,
    ! [V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ! [V1x: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
          ( ( ( p
              @ ( ap @ c_2Eseq_2Esummable
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3n ) ) ) ) )
            & ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) )
         => ( p
            @ ( ap @ c_2Eseq_2Esummable
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ V4n ) ) ) ) ) ) ) ).

thf(ax_thm_2Epowser_2Ediffs,axiom,
    ! [V0c: $i] :
      ( ( mem @ V0c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ( ( ap @ c_2Epowser_2Ediffs @ V0c )
        = ( lam @ ty_2Enum_2Enum
          @ ^ [V1n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Enum_2ESUC @ V1n ) ) ) @ ( ap @ V0c @ ( ap @ c_2Enum_2ESUC @ V1n ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2EDIFFS__NEG,axiom,
    ! [V0c: $i] :
      ( ( mem @ V0c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ( ( ap @ c_2Epowser_2Ediffs
          @ ( lam @ ty_2Enum_2Enum
            @ ^ [V1n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0c @ V1n ) ) ) )
        = ( lam @ ty_2Enum_2Enum
          @ ^ [V2n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V2n ) ) ) ) ) ).

thf(conj_thm_2Epowser_2EDIFFS__LEMMA,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
      ( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ! [V2x: tp__ty_2Erealax_2Ereal] :
          ( ( surj__ty_2Erealax_2Ereal
            @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V1c ) @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V3n ) ) ) ) )
          = ( surj__ty_2Erealax_2Ereal
            @ ( ap
              @ ( ap @ c_2Erealax_2Ereal__add
                @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
                  @ ( lam @ ty_2Enum_2Enum
                    @ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ V4n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V4n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
              @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2EDIFFS__LEMMA2,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
      ( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ! [V2x: tp__ty_2Erealax_2Ereal] :
          ( ( surj__ty_2Erealax_2Ereal
            @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ V3n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V3n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
          = ( surj__ty_2Erealax_2Ereal
            @ ( ap
              @ ( ap @ c_2Ereal_2Ereal__sub
                @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
                  @ ( lam @ ty_2Enum_2Enum
                    @ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V1c ) @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V4n ) ) ) ) )
              @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2EDIFFS__EQUIV,axiom,
    ! [V0c: $i] :
      ( ( mem @ V0c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ! [V1x: tp__ty_2Erealax_2Ereal] :
          ( ( p
            @ ( ap @ c_2Eseq_2Esummable
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V2n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V2n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2n ) ) ) ) )
         => ( p
            @ ( ap
              @ ( ap @ c_2Eseq_2Esums
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ V3n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0c @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V3n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
              @ ( ap @ c_2Eseq_2Esuminf
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V4n ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA1,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1z: tp__ty_2Erealax_2Ereal,V2h: tp__ty_2Erealax_2Ereal] :
      ( ( surj__ty_2Erealax_2Ereal
        @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
          @ ( lam @ ty_2Enum_2Enum
            @ ^ [V3p: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ V3p ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ V3p ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) )
      = ( surj__ty_2Erealax_2Ereal
        @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
          @ ( lam @ ty_2Enum_2Enum
            @ ^ [V4p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ V4p ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ V4p ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ V4p ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA2,axiom,
    ! [V0z: tp__ty_2Erealax_2Ereal,V1h: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
      ( ( V1h
       != ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
     => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
        = ( surj__ty_2Erealax_2Ereal
          @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1h ) )
            @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V3p: $i] :
                    ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ V3p ) )
                    @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3p ) ) )
                      @ ( lam @ ty_2Enum_2Enum
                        @ ^ [V4q: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ V4q ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3p ) ) @ V4q ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA3,axiom,
    ! [V0z: tp__ty_2Erealax_2Ereal,V1h: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum,V3k_27: tp__ty_2Erealax_2Ereal] :
      ( ( ( V1h
         != ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
        & ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3k_27 ) ) )
        & ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3k_27 ) ) ) )
     => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V3k_27 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA4,axiom,
    ! [V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
     => ! [V1k_27: tp__ty_2Erealax_2Ereal,V2k: tp__ty_2Erealax_2Ereal] :
          ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k ) ) )
            & ! [V3h: tp__ty_2Erealax_2Ereal] :
                ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) )
                  & ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k ) ) ) )
               => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) ) ) ) )
         => ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA5,axiom,
    ! [V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) )
         => ! [V2k: tp__ty_2Erealax_2Ereal] :
              ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k ) ) )
                & ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
                & ! [V3h: tp__ty_2Erealax_2Ereal] :
                    ( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) )
                      & ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k ) ) ) )
                   => ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) ) ) ) )
             => ( p
                @ ( ap
                  @ ( ap
                    @ ( ap @ c_2Elim_2Etends__real__real
                      @ ( lam @ ty_2Erealax_2Ereal
                        @ ^ [V5h: $i] : ( ap @ c_2Eseq_2Esuminf @ ( ap @ V1g @ V5h ) ) ) )
                    @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
                  @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).

thf(conj_thm_2Epowser_2ETERMDIFF,axiom,
    ! [V0c: $i] :
      ( ( mem @ V0c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ! [V1k_27: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
          ( ( ( p
              @ ( ap @ c_2Eseq_2Esummable
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0c @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) @ V3n ) ) ) ) )
            & ( p
              @ ( ap @ c_2Eseq_2Esummable
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) @ V4n ) ) ) ) )
            & ( p
              @ ( ap @ c_2Eseq_2Esummable
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V5n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ ( ap @ c_2Epowser_2Ediffs @ V0c ) ) @ V5n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) @ V5n ) ) ) ) )
            & ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) ) ) )
         => ( p
            @ ( ap
              @ ( ap
                @ ( ap @ c_2Elim_2Ediffl
                  @ ( lam @ ty_2Erealax_2Ereal
                    @ ^ [V6x: $i] :
                        ( ap @ c_2Eseq_2Esuminf
                        @ ( lam @ ty_2Enum_2Enum
                          @ ^ [V7n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0c @ V7n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ V6x ) @ V7n ) ) ) ) ) )
                @ ( ap @ c_2Eseq_2Esuminf
                  @ ( lam @ ty_2Enum_2Enum
                    @ ^ [V8n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V8n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V8n ) ) ) ) )
              @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).

%------------------------------------------------------------------------------