ITP001 Axioms: ITP086^5.ax


%------------------------------------------------------------------------------
% File     : ITP086^5 : 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    : ordinal^2.ax [Gau20]
%          : HL4086^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  271 (  24 unt;  31 typ;   0 def)
%            Number of atoms       : 3755 ( 198 equ;   0 cnn)
%            Maximal formula atoms :   78 (  13 avg)
%            Number of connectives : 8119 (  75   ~;  10   |; 147   &;7307   @)
%                                         (  60 <=>; 520  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   34 (  12 avg;7307 nst)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   30 (  30   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  106 ( 103 usr;  76 con; 0-2 aty)
%            Number of variables   :  718 (  23   ^ 671   !;  24   ?; 718   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Eordinal_2Eordinal,type,
    ty_2Eordinal_2Eordinal: del > del ).

thf(tp_c_2Eordinal_2EallOrds,type,
    c_2Eordinal_2EallOrds: del > $i ).

thf(mem_c_2Eordinal_2EallOrds,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2EallOrds @ A_27a ) @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(tp_c_2Eordinal_2Edclose,type,
    c_2Eordinal_2Edclose: del > $i ).

thf(mem_c_2Eordinal_2Edclose,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Edclose @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eordinal_2Edownward__closed,type,
    c_2Eordinal_2Edownward__closed: del > $i ).

thf(mem_c_2Eordinal_2Edownward__closed,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) ) ).

thf(tp_c_2Eordinal_2Eepsilon0,type,
    c_2Eordinal_2Eepsilon0: del > $i ).

thf(mem_c_2Eordinal_2Eepsilon0,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ).

thf(tp_c_2Eordinal_2Eeval__poly,type,
    c_2Eordinal_2Eeval__poly: del > $i ).

thf(mem_c_2Eordinal_2Eeval__poly,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(tp_c_2Eordinal_2EfromNat,type,
    c_2Eordinal_2EfromNat: del > $i ).

thf(mem_c_2Eordinal_2EfromNat,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(tp_c_2Eordinal_2Eis__polyform,type,
    c_2Eordinal_2Eis__polyform: del > del > $i ).

thf(mem_c_2Eordinal_2Eis__polyform,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ bool ) ) ) ).

thf(tp_c_2Eordinal_2Eoleast,type,
    c_2Eordinal_2Eoleast: del > $i ).

thf(mem_c_2Eordinal_2Eoleast,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eoleast @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(tp_c_2Eordinal_2Eomax,type,
    c_2Eordinal_2Eomax: del > $i ).

thf(mem_c_2Eordinal_2Eomax,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ty_2Eoption_2Eoption @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(tp_c_2Eordinal_2Eomega,type,
    c_2Eordinal_2Eomega: del > $i ).

thf(mem_c_2Eordinal_2Eomega,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eomega @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ).

thf(tp_c_2Eordinal_2EordADD,type,
    c_2Eordinal_2EordADD: del > $i ).

thf(mem_c_2Eordinal_2EordADD,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(tp_c_2Eordinal_2EordDIV,type,
    c_2Eordinal_2EordDIV: del > $i ).

thf(mem_c_2Eordinal_2EordDIV,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordDIV @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(tp_c_2Eordinal_2EordEXP,type,
    c_2Eordinal_2EordEXP: del > $i ).

thf(mem_c_2Eordinal_2EordEXP,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(tp_c_2Eordinal_2EordMOD,type,
    c_2Eordinal_2EordMOD: del > $i ).

thf(mem_c_2Eordinal_2EordMOD,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordMOD @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(tp_c_2Eordinal_2EordMULT,type,
    c_2Eordinal_2EordMULT: del > $i ).

thf(mem_c_2Eordinal_2EordMULT,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(tp_c_2Eordinal_2EordSUC,type,
    c_2Eordinal_2EordSUC: del > $i ).

thf(mem_c_2Eordinal_2EordSUC,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(tp_c_2Eordinal_2Eordinal__ABS,type,
    c_2Eordinal_2Eordinal__ABS: del > $i ).

thf(mem_c_2Eordinal_2Eordinal__ABS,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(tp_c_2Eordinal_2Eordinal__ABS__CLASS,type,
    c_2Eordinal_2Eordinal__ABS__CLASS: del > $i ).

thf(mem_c_2Eordinal_2Eordinal__ABS__CLASS,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ ( arr @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(tp_c_2Eordinal_2Eordinal__REP,type,
    c_2Eordinal_2Eordinal__REP: del > $i ).

thf(mem_c_2Eordinal_2Eordinal__REP,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).

thf(tp_c_2Eordinal_2Eordinal__REP__CLASS,type,
    c_2Eordinal_2Eordinal__REP__CLASS: del > $i ).

thf(mem_c_2Eordinal_2Eordinal__REP__CLASS,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) ) ).

thf(tp_c_2Eordinal_2Eordlt,type,
    c_2Eordinal_2Eordlt: del > $i ).

thf(mem_c_2Eordinal_2Eordlt,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eordinal_2Epolyform,type,
    c_2Eordinal_2Epolyform: del > $i ).

thf(mem_c_2Eordinal_2Epolyform,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).

thf(tp_c_2Eordinal_2Epreds,type,
    c_2Eordinal_2Epreds: del > $i ).

thf(mem_c_2Eordinal_2Epreds,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eordinal_2Esup,type,
    c_2Eordinal_2Esup: del > $i ).

thf(mem_c_2Eordinal_2Esup,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eordinal_2Esup @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(ax_thm_2Eordinal_2Eordinal__TY__DEF,axiom,
    ! [A_27a: del] :
    ? [V0rep: $i] :
      ( ( mem @ V0rep @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) )
      & ( p
        @ ( ap
          @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
            @ ( lam @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool )
              @ ^ [V1c: $i] :
                  ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
                  @ ( lam @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) )
                    @ ^ [V2r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2r ) @ V2r ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) @ V1c ) @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2r ) ) ) ) ) ) )
          @ V0rep ) ) ) ).

thf(ax_thm_2Eordinal_2Eordinal__bijections,axiom,
    ! [A_27a: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ V0a ) )
            = V0a ) )
      & ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) )
         => ( ( p
              @ ( ap
                @ ( lam @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool )
                  @ ^ [V2c: $i] :
                      ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
                      @ ( lam @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) )
                        @ ^ [V3r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V3r ) @ V3r ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) @ V2c ) @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V3r ) ) ) ) ) )
                @ V1r ) )
          <=> ( ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ V1r ) )
              = V1r ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordinal__ABS__REP__CLASS,axiom,
    ! [A_27a: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ V0a ) )
            = V0a ) )
      & ! [V1c: $i] :
          ( ( mem @ V1c @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) )
         => ( ? [V2r: $i] :
                ( ( mem @ V2r @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
                & ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2r ) @ V2r ) )
                & ( V1c
                  = ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2r ) ) )
          <=> ( ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ V1c ) )
              = V1c ) ) ) ) ).

thf(ax_thm_2Eordinal_2Eordinal__REP__def,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) @ V0a )
        = ( ap @ ( c_2Emin_2E_40 @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ V0a ) ) ) ) ).

thf(ax_thm_2Eordinal_2Eordinal__ABS__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
     => ( ( ap @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ V0r )
        = ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0r ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordinal__QUOTIENT,axiom,
    ! [A_27a: del] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) ) @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) ) ) ).

thf(ax_thm_2Eordinal_2Eordlt__def,axiom,
    ! [A_27a: del,V0T1: $i] :
      ( ( mem @ V0T1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1T2: $i] :
          ( ( mem @ V1T2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0T1 ) @ V1T2 ) )
          <=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) @ V0T1 ) ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) @ V1T2 ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__REFL,axiom,
    ! [A_27a: del,V0w: $i] :
      ( ( mem @ V0w @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w ) @ V0w ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__TRANS,axiom,
    ! [A_27a: del,V0w1: $i] :
      ( ( mem @ V0w1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1w2: $i] :
          ( ( mem @ V1w2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2w3: $i] :
              ( ( mem @ V2w3 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w1 ) @ V1w2 ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1w2 ) @ V2w3 ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w1 ) @ V2w3 ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__WF,axiom,
    ! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2Eordlt @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__trichotomy,axiom,
    ! [A_27a: del,V0w2: $i] :
      ( ( mem @ V0w2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1w1: $i] :
          ( ( mem @ V1w1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1w1 ) @ V0w2 ) )
            | ( V1w1 = V0w2 )
            | ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w2 ) @ V1w1 ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2EallOrds__def,axiom,
    ! [A_27a: del] :
      ( ( c_2Eordinal_2EallOrds @ A_27a )
      = ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
        @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
          @ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) )
            @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
              @ ^ [V0x: $i] :
                  ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
                  @ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Ewellorder__allOrds,axiom,
    ! [A_27a: del] :
      ( p
      @ ( ap @ ( c_2Ewellorder_2Ewellorder @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
        @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
          @ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) )
            @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
              @ ^ [V0x: $i] :
                  ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
                  @ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EWIN__allOrds,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) ) )
          <=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EelsOf__allOrds,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Ewellorder_2EelsOf @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2EallOrds @ A_27a ) )
      = ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__mkOrdinal,axiom,
    ! [A_27a: del,V0o1: $i] :
      ( ( mem @ V0o1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1o2: $i] :
          ( ( mem @ V1o2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0o1 ) @ V1o2 ) )
          <=> ! [V2w1: $i] :
                ( ( mem @ V2w1 @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
               => ! [V3w2: $i] :
                    ( ( mem @ V3w2 @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
                   => ( ( ( ( ap @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ V2w1 )
                          = V0o1 )
                        & ( ( ap @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ V3w2 )
                          = V1o2 ) )
                     => ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2w1 ) @ V3w2 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eorderlt__iso__REFL,axiom,
    ! [A_27a: del,A_27b: del,V0w1: $i] :
      ( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
     => ! [V1w2: $i] :
          ( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
           => ~ ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eorderiso__wobound2,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1w: $i] :
          ( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V0x ) @ V1w ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V2y ) @ V1w ) ) )
               => ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Ewellorder__ordinal__isomorphism,axiom,
    ! [A_27a: del,V0w: $i] :
      ( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
     => ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0w ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ V0w ) ) @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2Epreds__def,axiom,
    ! [A_27a: del,V0w: $i] :
      ( ( mem @ V0w @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0w )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
            @ ^ [V1w0: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ V1w0 ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1w0 ) @ V0w ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EIN__preds,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1w: $i] :
          ( ( mem @ V1w @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1w ) ) )
          <=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1w ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__11,axiom,
    ! [A_27a: del,V0w1: $i] :
      ( ( mem @ V0w1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1w2: $i] :
          ( ( mem @ V1w2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0w1 )
              = ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1w2 ) )
          <=> ( V0w1 = V1w2 ) ) ) ) ).

thf(ax_thm_2Eordinal_2Edownward__closed__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0s ) )
      <=> ! [V1a: $i] :
            ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ! [V2b: $i] :
                ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
               => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ V0s ) )
                    & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ V1a ) ) )
                 => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__downward__closed,axiom,
    ! [A_27a: del,V0w: $i] :
      ( ( mem @ V0w @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0w ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__bij,axiom,
    ! [A_27a: del] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) @ ( c_2Eordinal_2Epreds @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) @ ( c_2Eordinal_2Edownward__closed @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__lt__PSUBSET,axiom,
    ! [A_27a: del,V0w1: $i] :
      ( ( mem @ V0w1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1w2: $i] :
          ( ( mem @ V1w2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w1 ) @ V1w2 ) )
          <=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0w1 ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1w2 ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__wobound,axiom,
    ! [A_27a: del,V0ord: $i] :
      ( ( mem @ V0ord @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0ord )
        = ( ap @ ( c_2Ewellorder_2EelsOf @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0ord ) @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__inj__univ,axiom,
    ! [A_27a: del,V0ord: $i] :
      ( ( mem @ V0ord @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0ord ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ).

thf(stp_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,type,
    tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone: $tType ).

thf(stp_inj_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,type,
    inj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone: tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone > $i ).

thf(stp_surj_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,type,
    surj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone: $i > tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone ).

thf(stp_inj_surj_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,axiom,
    ! [X: tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone] :
      ( ( surj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ ( inj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ X ) )
      = X ) ).

thf(stp_inj_mem_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,axiom,
    ! [X: tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone] : ( mem @ ( inj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ X ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) ) ).

thf(stp_iso_mem_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,axiom,
    ! [X: $i] :
      ( ( mem @ X @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) )
     => ( X
        = ( inj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ ( surj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ X ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eunitinf__univnum,axiom,
    p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ).

thf(stp_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,type,
    tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone: $tType ).

thf(stp_inj_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,type,
    inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone: tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone > $i ).

thf(stp_surj_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,type,
    surj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone: $i > tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone ).

thf(stp_inj_surj_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,axiom,
    ! [X: tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone] :
      ( ( surj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ ( inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ X ) )
      = X ) ).

thf(stp_inj_mem_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,axiom,
    ! [X: tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone] : ( mem @ ( inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ X ) @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) ) ).

thf(stp_iso_mem_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,axiom,
    ! [X: $i] :
      ( ( mem @ X @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) )
     => ( X
        = ( inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ ( surj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ X ) ) ) ) ).

thf(conj_thm_2Eordinal_2Ecord__countable__preds,axiom,
    ! [V0ord: tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone] : ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ V0ord ) ) ) ) ).

thf(conj_thm_2Eordinal_2Euniv__ord__greater__cardinal,axiom,
    ! [A_27a: del] :
      ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).

thf(conj_thm_2Eordinal_2Euniv__cord__uncountable,axiom,
    ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordle__lteq,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0b ) @ V1a ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) )
              | ( V1a = V0b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordle__ANTISYM,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0b ) @ V1a ) )
              & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) ) )
           => ( V1a = V0b ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordle__TRANS,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2z: $i] :
              ( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ V0x ) )
                  & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2z ) @ V1y ) ) )
               => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2z ) @ V0x ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlet__TRANS,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2z: $i] :
              ( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ V0x ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ V2z ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V2z ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlte__TRANS,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2z: $i] :
              ( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) )
                  & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2z ) @ V1y ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V2z ) ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2Eoleast__def,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eordinal_2Eoleast @ A_27a ) @ V0P )
        = ( ap @ ( c_2Emin_2E_40 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
            @ ^ [V1x: $i] :
                ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V1x ) )
                @ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
                    @ ^ [V2y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2y ) @ V1x ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ V0P @ V2y ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eoleast__intro,axiom,
    ! [A_27a: del,V0Q: $i] :
      ( ( mem @ V0Q @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1P: $i] :
          ( ( mem @ V1P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
         => ( ( ? [V2a: $i] :
                  ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  & ( p @ ( ap @ V1P @ V2a ) ) )
              & ! [V3a: $i] :
                  ( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( ! [V4b: $i] :
                          ( ( mem @ V4b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V4b ) @ V3a ) )
                           => ~ ( p @ ( ap @ V1P @ V4b ) ) ) )
                      & ( p @ ( ap @ V1P @ V3a ) ) )
                   => ( p @ ( ap @ V0Q @ V3a ) ) ) ) )
           => ( p @ ( ap @ V0Q @ ( ap @ ( c_2Eordinal_2Eoleast @ A_27a ) @ V1P ) ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2EordSUC__def,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a )
        = ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
            @ ^ [V1b: $i] : ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2EfromNat__def,axiom,
    ! [A_27a: del] :
      ( ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
        = ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
            @ ^ [V0a: $i] : c_2Ebool_2ET ) ) )
      & ! [V1n: tp__ty_2Enum_2Enum] :
          ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
          = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EfromNat__def__compute,axiom,
    ! [A_27a: del] :
      ( ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
        = ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
            @ ^ [V0a: $i] : c_2Ebool_2ET ) ) )
      & ! [V1n: tp__ty_2Enum_2Enum] :
          ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
          = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
      & ! [V2n: tp__ty_2Enum_2Enum] :
          ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
          = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EfromNat__SUC,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
      ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
      = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__ZERO,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__surj,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0x ) )
          & ( V0x
           != ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
       => ? [V1y: $i] :
            ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
            & ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1y )
              = V0x ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eno__maximal__ordinal,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ? [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
          & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__SUC,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordSUC__ZERO,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a )
       != ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__DISCRETE1,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ~ ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__SUC__DISCRETE,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1b ) ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
              | ( V0a = V1b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordSUC__MONO,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1b ) ) )
          <=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordSUC__11,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a )
              = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1b ) )
          <=> ( V0a = V1b ) ) ) ) ).

thf(ax_thm_2Eordinal_2Esup__def,axiom,
    ! [A_27a: del,V0ordset: $i] :
      ( ( mem @ V0ordset @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0ordset )
        = ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
            @ ^ [V1a: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) @ ( c_2Eordinal_2Epreds @ A_27a ) ) @ V0ordset ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eord__induction,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ! [V1min: $i] :
            ( ( mem @ V1min @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ! [V2b: $i] :
                  ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ V1min ) )
                   => ( p @ ( ap @ V0P @ V2b ) ) ) )
             => ( p @ ( ap @ V0P @ V1min ) ) ) )
       => ! [V3a: $i] :
            ( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( p @ ( ap @ V0P @ V3a ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__thm,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
       => ! [V1a: $i] :
            ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
            <=> ? [V2b: $i] :
                  ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esuple__thm,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
              & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0b ) @ V1s ) ) )
           => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) ) @ V0b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__eq__sup,axiom,
    ! [A_27a: del,V0s1: $i] :
      ( ( mem @ V0s1 @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1s2: $i] :
          ( ( mem @ V1s2 @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s1 ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
              & ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s2 ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
              & ! [V2a: $i] :
                  ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2a ) @ V0s1 ) )
                   => ? [V3b: $i] :
                        ( ( mem @ V3b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                        & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V3b ) @ V1s2 ) )
                        & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3b ) @ V2a ) ) ) ) )
              & ! [V4b: $i] :
                  ( ( mem @ V4b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V4b ) @ V1s2 ) )
                   => ? [V5a: $i] :
                        ( ( mem @ V5a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                        & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V5a ) @ V0s1 ) )
                        & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V5a ) @ V4b ) ) ) ) ) )
           => ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s1 )
              = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s2 ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EUnum__cle__Uinf,axiom,
    ! [A_27a: del] : ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).

thf(conj_thm_2Eordinal_2Ecsup__thm,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0s ) )
       => ! [V1b: $i] :
            ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
            <=> ? [V2d: $i] :
                  ( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2d ) @ V0s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epredimage__sup__thm,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) ) ) ) )
              <=> ? [V3d: $i] :
                    ( ( mem @ V3d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                    & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3d ) @ V1a ) )
                    & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ ( ap @ V0f @ V3d ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epredimage__suplt__ELIM,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) ) ) ) @ V2b ) )
               => ! [V3d: $i] :
                    ( ( mem @ V3d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3d ) @ V1a ) )
                     => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ ( ap @ V0f @ V3d ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esuppred__suplt__ELIM,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) @ V1b ) )
           => ! [V2d: $i] :
                ( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2d ) @ V0a ) )
                 => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__EMPTY,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
      = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__SING,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
        = V0a ) ) ).

thf(conj_thm_2Eordinal_2Esup__preds__SUC,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) )
        = V0a ) ) ).

thf(conj_thm_2Eordinal_2Epreds__ordSUC,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) )
        = ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) ) ).

thf(conj_thm_2Eordinal_2EcountableOrds__dclosed,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
              & ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1b ) ) ) )
           => ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2Eomax__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ V0s )
        = ( ap @ ( c_2Eoption_2Esome @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
            @ ^ [V1a: $i] :
                ( ap
                @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
                  @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0s )
                    @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
                      @ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) )
                        @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
                          @ ^ [V2x: $i] :
                              ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
                              @ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2x ) @ V3y ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3y ) @ V2x ) ) ) ) ) ) ) ) )
                @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomax__SOME,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ V0s )
              = ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ V0s ) )
              & ! [V2b: $i] :
                  ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) )
                   => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomax__NONE,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ V0s )
          = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
      <=> ! [V1a: $i] :
            ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ V0s ) )
             => ? [V2b: $i] :
                  ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomax__EMPTY,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
      = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__0,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
      = ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2Eordleq0,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
      <=> ( V0x
          = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__EQ__EMPTY,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0x )
          = ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
      <=> ( V0x
          = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomax__sup,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ V0s )
              = ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) )
           => ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s )
              = V1a ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__omax__SOME__SUC,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) )
              = ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1b ) )
          <=> ( V0a
              = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esimple__ord__induction,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
          & ! [V1a: $i] :
              ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ V0P @ V1a ) )
               => ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) ) ) )
          & ! [V2a: $i] :
              ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) )
                    = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
                  & ! [V3b: $i] :
                      ( ( mem @ V3b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3b ) @ V2a ) )
                       => ( p @ ( ap @ V0P @ V3b ) ) ) ) )
               => ( p @ ( ap @ V0P @ V2a ) ) ) ) )
       => ! [V4a: $i] :
            ( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( p @ ( ap @ V0P @ V4a ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__preds__omax__NONE,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) )
          = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
      <=> ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) )
          = V0a ) ) ) ).

thf(ax_thm_2Eordinal_2Edclose__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eordinal_2Edclose @ A_27a ) @ V0s )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
          @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
            @ ^ [V1x: $i] :
                ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ V1x )
                @ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
                    @ ^ [V2y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2y ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__sup,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
       => ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) )
          = ( ap @ ( c_2Eordinal_2Edclose @ A_27a ) @ V0s ) ) ) ) ).

thf(conj_thm_2Eordinal_2Ecsup__lesup,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0s ) )
       => ! [V1d: $i] :
            ( ( mem @ V1d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1d ) @ V0s ) )
             => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) @ V1d ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Ecsup__suple,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0s ) )
           => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
            <=> ! [V2d: $i] :
                  ( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2d ) @ V0s ) )
                   => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__sup__thm,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0s ) )
          & ( V0s
           != ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
       => ! [V1b: $i] :
            ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
            <=> ? [V2d: $i] :
                  ( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2d ) @ V0s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__lesup,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0s ) )
          & ( V0s
           != ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
       => ! [V1d: $i] :
            ( ( mem @ V1d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1d ) @ V0s ) )
             => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) @ V1d ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epreds__suple,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0s ) )
              & ( V0s
               != ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
           => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
            <=> ! [V2d: $i] :
                  ( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2d ) @ V0s ) )
                   => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EfromNat__11,axiom,
    ! [A_27a: del,V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
      ( ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) )
        = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) )
    <=> ( V0x = V1y ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__fromNat,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1x: $i] :
      ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
      <=> ? [V2m: tp__ty_2Enum_2Enum] :
            ( ( V1x
              = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
            & ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EfromNat__ordlt,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
    <=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ).

thf(ax_thm_2Eordinal_2Eomega__def,axiom,
    ! [A_27a: del] :
      ( ( c_2Eordinal_2Eomega @ A_27a )
      = ( ap @ ( c_2Eordinal_2Esup @ A_27a )
        @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ty_2Enum_2Enum )
          @ ( lam @ ty_2Enum_2Enum
            @ ^ [V0i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ V0i ) ) @ c_2Ebool_2ET ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Elt__omega,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) )
      <=> ? [V1m: tp__ty_2Enum_2Enum] :
            ( V0a
            = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EfromNat__lt__omega,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2EfromNat__eq__omega,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
      ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) )
     != ( c_2Eordinal_2Eomega @ A_27a ) ) ).

thf(conj_thm_2Eordinal_2Eord__RECURSION,axiom,
    ! [A_27a: del,A_27b: del,V0z: $i] :
      ( ( mem @ V0z @ A_27b )
     => ! [V1sf: $i] :
          ( ( mem @ V1sf @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ A_27b @ A_27b ) ) )
         => ! [V2lf: $i] :
              ( ( mem @ V2lf @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( arr @ A_27b @ bool ) @ A_27b ) ) )
             => ? [V3h: $i] :
                  ( ( mem @ V3h @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ A_27b ) )
                  & ( ( ap @ V3h @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
                    = V0z )
                  & ! [V4a: $i] :
                      ( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                     => ( ( ap @ V3h @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V4a ) )
                        = ( ap @ ( ap @ V1sf @ V4a ) @ ( ap @ V3h @ V4a ) ) ) )
                  & ! [V5a: $i] :
                      ( ( mem @ V5a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                     => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V5a ) )
                          & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V5a ) )
                            = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
                       => ( ( ap @ V3h @ V5a )
                          = ( ap @ ( ap @ V2lf @ V5a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ A_27b ) @ V3h ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V5a ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2EordADD__def,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
          = V0b )
        & ! [V1a: $i] :
            ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) )
              = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) @ V1a ) ) ) )
        & ! [V2a: $i] :
            ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
                & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) )
                  = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
             => ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) @ V2a )
                = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__0L,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a )
        = V0a ) ) ).

thf(conj_thm_2Eordinal_2Eubsup__thm,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ! [V2a: $i] :
                ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2a ) @ V0s ) )
                 => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2a ) @ V1b ) ) ) )
           => ! [V3c: $i] :
                ( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
                <=> ? [V4d: $i] :
                      ( ( mem @ V4d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V4d ) @ V0s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ V4d ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__fromNat,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
      ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
      = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomax__preds__omega,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) )
      = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2Eomega__islimit,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) )
      = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__fromNat__omega,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
      ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Eordinal_2Eomega @ A_27a ) )
      = ( c_2Eordinal_2Eomega @ A_27a ) ) ).

thf(conj_thm_2Eordinal_2Elt__suppreds,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) )
          <=> ? [V2d: $i] :
                ( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2d ) @ V0a ) )
                & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomax__preds__SUC,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0x ) ) )
        = ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) ) ) ).

thf(conj_thm_2Eordinal_2EORD__ONE,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
      = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordSUC__NUMERAL,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
      ( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
      = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordZERO__ltSUC,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0x ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__CANCEL__ADDR,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0b ) ) )
          <=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__CANCEL__ADDL,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V1b ) ) @ V0a ) )
          <=> $false ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__CANCEL1,axiom,
    ! [A_27a: del] :
      ( ! [V0c: $i] :
          ( ( mem @ V0c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V1a: $i] :
              ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( V1a
                  = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0c ) )
              <=> ( V0c
                  = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
      & ! [V2c: $i] :
          ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V3a: $i] :
              ( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V3a ) @ V2c )
                  = V3a )
              <=> ( V2c
                  = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__MONO,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2c ) @ V1a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2c ) @ V0b ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__CANCEL,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2c ) @ V1a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2c ) @ V0b ) ) )
              <=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__RIGHT__CANCEL,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0b )
                  = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V2c ) )
              <=> ( V0b = V2c ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EleqLEFT__CANCEL,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0x ) ) @ V0x ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordlt__EXISTS__ADD,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
          <=> ? [V2c: $i] :
                ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                & ( V2c
                 != ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
                & ( V1b
                  = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V2c ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordle__EXISTS__ADD,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V0a ) )
          <=> ? [V2c: $i] :
                ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                & ( V1b
                  = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V2c ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eordle__CANCEL__ADDR,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0x ) @ V1a ) ) @ V0x ) ) ) ) ).

thf(conj_thm_2Eordinal_2Edclose__BIGUNION,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eordinal_2Edclose @ A_27a ) @ V0s )
        = ( ap @ ( c_2Epred__set_2EBIGUNION @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) @ ( c_2Eordinal_2Epreds @ A_27a ) ) @ V0s ) ) ) ) ).

thf(conj_thm_2Eordinal_2EcountableOrds__uncountable,axiom,
    ! [A_27a: del] :
      ~ ( p
        @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
          @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
            @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
              @ ^ [V0a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ V0a ) @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Edclose__cardleq__univinf,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
       => ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Edclose @ A_27a ) @ V0s ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__lt__implies,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) @ V1a ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ V1a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__eq__max,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ! [V2b: $i] :
                  ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) )
                   => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) )
              & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ V0s ) ) )
           => ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s )
              = V1a ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__eq__SUC,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
              & ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s )
                = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) )
           => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) @ V0s ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Egeneric__continuity,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ( ( ! [V1a: $i] :
              ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1a ) )
                  & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) )
                    = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
               => ( ( ap @ V0f @ V1a )
                  = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) ) ) ) ) )
          & ! [V2x: $i] :
              ( ( mem @ V2x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3y ) @ V2x ) )
                   => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V3y ) ) @ ( ap @ V0f @ V2x ) ) ) ) ) ) )
       => ! [V4s: $i] :
            ( ( mem @ V4s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
           => ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V4s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
                & ( V4s
                 != ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
             => ( ( ap @ V0f @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V4s ) )
                = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ V4s ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eord__CASES,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( V0a
          = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        | ? [V1a0: $i] :
            ( ( mem @ V1a0 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
            & ( V0a
              = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a0 ) ) )
        | ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
          & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) )
            = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eislimit__0,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
      = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2Eordinal__IVT,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ! [V1a1: $i] :
          ( ( mem @ V1a1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2a2: $i] :
              ( ( mem @ V2a2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( ! [V4a: $i] :
                          ( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                         => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V4a ) )
                              & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V4a ) )
                                = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
                           => ( ( ap @ V0f @ V4a )
                              = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V4a ) ) ) ) ) )
                      & ! [V5x: $i] :
                          ( ( mem @ V5x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                         => ! [V6y: $i] :
                              ( ( mem @ V6y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                             => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V6y ) @ V5x ) )
                               => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V6y ) ) @ ( ap @ V0f @ V5x ) ) ) ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a1 ) @ V2a2 ) )
                      & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ V0f @ V1a1 ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ V0f @ V2a2 ) ) ) )
                   => ? [V7b: $i] :
                        ( ( mem @ V7b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                        & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V7b ) @ V1a1 ) )
                        & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V7b ) @ V2a2 ) )
                        & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ V0f @ V7b ) ) )
                        & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ V0f @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V7b ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__continuous,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
              & ( V1s
               != ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
           => ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) )
              = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__ASSOC,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1b ) @ V2c ) )
                = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V1b ) ) @ V2c ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EADD1R,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
        = ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__weak__MONO,axiom,
    ! [A_27a: del,V0c: $i] :
      ( ( mem @ V0c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) )
               => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2b ) @ V0c ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0c ) ) ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2EordMULT__def,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
          = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        & ! [V1a: $i] :
            ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) )
              = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ V1a ) ) @ V0b ) ) )
        & ! [V2a: $i] :
            ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
                & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) )
                  = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
             => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ V2a )
                = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__0L,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a )
        = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__0R,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__1L,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a )
        = V0a ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__1R,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
        = V0a ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__2R,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
        = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V0a ) ) ) ).

thf(conj_thm_2Eordinal_2Eislimit__SUC__lt,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0b ) )
                = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) ) )
           => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) @ V0b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__lt__MONO__R,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2c ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V0a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V1b ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__le__MONO__R,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V0a ) )
               => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V1b ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V0a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__lt__MONO__R__EQN,axiom,
    ! [A_27a: del,V0c: $i] :
      ( ( mem @ V0c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0c ) @ V1a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0c ) @ V2b ) ) )
              <=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0c ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__le__MONO__L,axiom,
    ! [A_27a: del,V0y: $i] :
      ( ( mem @ V0y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2z: $i] :
              ( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0y ) @ V1x ) )
               => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0y ) @ V2z ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1x ) @ V2z ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__le__MONO__L,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V0a ) )
               => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ V2c ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ V2c ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__CANCEL__R,axiom,
    ! [A_27a: del,V0z: $i] :
      ( ( mem @ V0z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0z ) @ V1x )
                  = ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0z ) @ V2y ) )
              <=> ( ( V0z
                    = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
                  | ( V1x = V2y ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__continuous,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
           => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) )
              = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__fromNat,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
      ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
      = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomega__MUL__fromNat,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Eordinal_2Eomega @ A_27a ) )
        = ( c_2Eordinal_2Eomega @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__LDISTRIB,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V1b ) )
                = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V0a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V1b ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__ASSOC,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ V2c ) )
                = ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ V1b ) ) @ V2c ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2EordDIVISION,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1b ) )
           => ( ( V0a
                = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordDIV @ A_27a ) @ V0a ) @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMOD @ A_27a ) @ V0a ) @ V1b ) ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMOD @ A_27a ) @ V0a ) @ V1b ) ) @ V1b ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordDIV__UNIQUE,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2q: $i] :
              ( ( mem @ V2q @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3r: $i] :
                  ( ( mem @ V3r @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1b ) )
                      & ( V0a
                        = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ V2q ) ) @ V3r ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3r ) @ V1b ) ) )
                   => ( ( ap @ ( ap @ ( c_2Eordinal_2EordDIV @ A_27a ) @ V0a ) @ V1b )
                      = V2q ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMOD__UNIQUE,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2q: $i] :
              ( ( mem @ V2q @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3r: $i] :
                  ( ( mem @ V3r @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1b ) )
                      & ( V0a
                        = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ V2q ) ) @ V3r ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3r ) @ V1b ) ) )
                   => ( ( ap @ ( ap @ ( c_2Eordinal_2EordMOD @ A_27a ) @ V0a ) @ V1b )
                      = V3r ) ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2EordEXP__def,axiom,
    ! [A_27a: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
            = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
      & ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2a_27: $i] :
              ( ( mem @ V2a_27 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V1a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V2a_27 ) )
                = ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V1a ) @ V2a_27 ) ) @ V1a ) ) ) )
      & ! [V3a: $i] :
          ( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V4a_27: $i] :
              ( ( mem @ V4a_27 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V4a_27 ) )
                  & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V4a_27 ) )
                    = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
               => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V3a ) @ V4a_27 )
                  = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V3a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V4a_27 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__1R,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
        = V0a ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__1L,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a )
        = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__2R,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
        = ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ V0a ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__fromNat,axiom,
    ! [A_27a: del,V0x: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
      = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__le__MONO__L,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ V1a ) )
               => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2b ) @ V0x ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V1a ) @ V0x ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EIFF__ZERO__lt,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ( V0x
           != ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        <=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) )
        & ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
        <=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eislimit__SUC,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0x ) ) )
          = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
      <=> $false ) ) ).

thf(conj_thm_2Eordinal_2Eislimit__fromNat,axiom,
    ! [A_27a: del,V0x: tp__ty_2Enum_2Enum] :
      ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) )
        = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
    <=> ( V0x = fo__c_2Enum_2E0 ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__ZERO__limit,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0x ) )
          = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
       => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x )
          = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__ZERO__nonlimit,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0x ) )
         != ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
       => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x )
          = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Esup__EQ__0,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
       => ( ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s )
            = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
        <=> ( ( V0s
              = ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
            | ( V0s
              = ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__EQ__0,axiom,
    ! [A_27a: del,V0y: $i] :
      ( ( mem @ V0y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1x ) @ V0y )
              = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
          <=> ( ( V1x
                = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
              & ( V0y
                = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EIMAGE__EQ__SING,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V0f ) @ V1s )
                  = ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
              <=> ( ? [V3y: $i] :
                      ( ( mem @ V3y @ A_27b )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1s ) ) )
                  & ! [V4y: $i] :
                      ( ( mem @ V4y @ A_27b )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V1s ) )
                       => ( ( ap @ V0f @ V4y )
                          = V2x ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMULT__EQ__0,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0x ) @ V1y )
              = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
          <=> ( ( V0x
                = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
              | ( V1y
                = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__EQ__0,axiom,
    ! [A_27a: del,V0y: $i] :
      ( ( mem @ V0y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V1x ) @ V0y )
              = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
          <=> ( ( V1x
                = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
              & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0y ) )
               != ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EZERO__lt__ordEXP__I,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
           => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V1x ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EZERO__lt__ordEXP,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V1x ) ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
              | ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1x ) )
                = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__lt__MONO__R,axiom,
    ! [A_27a: del,V0y: $i] :
      ( ( mem @ V0y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2a: $i] :
              ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2a ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ V0y ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V1x ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V0y ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__lt__IFF,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2a: $i] :
              ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2a ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V0x ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V1y ) ) )
                <=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__le__MONO__R,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2a: $i] :
              ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
                  & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ V0x ) ) )
               => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V0x ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__continuous,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
              & ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
              & ( V1s
               != ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
           => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) )
              = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__ADD,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2z: $i] :
              ( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
               => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1y ) @ V2z ) )
                  = ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ V2z ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__MUL,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2z: $i] :
              ( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
               => ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1y ) @ V2z ) )
                  = ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ V1y ) ) @ V2z ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Efixpoints__exist,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ( ( ! [V1s: $i] :
              ( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
             => ( ( ( V1s
                   != ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) )
               => ( ( ap @ V0f @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) )
                  = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ V1s ) ) ) ) )
          & ! [V2x: $i] :
              ( ( mem @ V2x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V2x ) ) @ V2x ) ) ) )
       => ! [V3a: $i] :
            ( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ? [V4b: $i] :
                ( ( mem @ V4b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                & ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V4b ) @ V3a ) )
                & ( ( ap @ V0f @ V4b )
                  = V4b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Ex__le__ordEXP__x,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
           => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V1x ) ) @ V1x ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2Eepsilon0__def,axiom,
    ! [A_27a: del] :
      ( ( c_2Eordinal_2Eepsilon0 @ A_27a )
      = ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
        @ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
          @ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0x ) ) @ V0x ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eepsilon0__fixpoint,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) )
      = ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ).

thf(conj_thm_2Eordinal_2Eepsilon0__least__fixpoint,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
       => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) )
          & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomega__lt__epsilon0,axiom,
    ! [A_27a: del] : ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2EfromNat__lt__epsilon0,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ).

thf(conj_thm_2Eordinal_2Eadd__nat__islimit,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1a: $i] :
      ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
       => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
            = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
        <=> $false ) ) ) ).

thf(conj_thm_2Eordinal_2Estrict__continuity__preserves__islimit,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ! [V2s: $i] :
                  ( ( mem @ V2s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
                      & ( V2s
                       != ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
                   => ( ( ap @ V0f @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V2s ) )
                      = ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ V2s ) ) ) ) )
              & ! [V3x: $i] :
                  ( ( mem @ V3x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ! [V4y: $i] :
                      ( ( mem @ V4y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3x ) @ V4y ) )
                       => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V3x ) ) @ ( ap @ V0f @ V4y ) ) ) ) ) )
              & ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) )
                = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
              & ( V1a
               != ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
           => ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ V0f @ V1a ) ) )
              = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eadd__omega__islimit,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) ) )
        = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eislimit__mul__R,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) )
              = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
           => ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ V1a ) ) )
              = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Emul__omega__islimit,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) )
        = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eomega__exp__islimit,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
       => ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) )
          = ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eexpbound__add,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EaddL__fixpoint__iff,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V1b )
              = V1b )
          <=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordADD__under__epsilon0,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) )
           => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0x ) @ V1y ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordMUL__under__epsilon0,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) )
           => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0x ) @ V1y ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordEXP__under__epsilon0,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) )
           => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V1b ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eeval__poly__ind,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ bool ) ) )
     => ( ( ! [V1a: $i] :
              ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( p @ ( ap @ ( ap @ V0P @ V1a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) )
          & ! [V2a: $i] :
              ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ! [V4e: $i] :
                      ( ( mem @ V4e @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                     => ! [V5t: $i] :
                          ( ( mem @ V5t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
                         => ( ( p @ ( ap @ ( ap @ V0P @ V2a ) @ V5t ) )
                           => ( p @ ( ap @ ( ap @ V0P @ V2a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V3c ) @ V4e ) ) @ V5t ) ) ) ) ) ) ) ) )
       => ! [V6v: $i] :
            ( ( mem @ V6v @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ! [V7v1: $i] :
                ( ( mem @ V7v1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
               => ( p @ ( ap @ ( ap @ V0P @ V6v ) @ V7v1 ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eeval__poly__def,axiom,
    ! [A_27a: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
            = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
      & ! [V1t: $i] :
          ( ( mem @ V1t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
         => ! [V2e: $i] :
              ( ( mem @ V2e @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ! [V4a: $i] :
                      ( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                     => ( ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V4a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V3c ) @ V2e ) ) @ V1t ) )
                        = ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V4a ) @ V2e ) ) @ V3c ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V4a ) @ V1t ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eis__polyform__ind,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ bool ) ) )
     => ( ( ! [V1a: $i] :
              ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( p @ ( ap @ ( ap @ V0P @ V1a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) ) ) )
          & ! [V2a: $i] :
              ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ! [V4e: $i] :
                      ( ( mem @ V4e @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
                     => ( p @ ( ap @ ( ap @ V0P @ V2a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V3c ) @ V4e ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) ) ) ) ) ) )
          & ! [V5a: $i] :
              ( ( mem @ V5a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V6c1: $i] :
                  ( ( mem @ V6c1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ! [V7e1: $i] :
                      ( ( mem @ V7e1 @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
                     => ! [V8c2: $i] :
                          ( ( mem @ V8c2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                         => ! [V9e2: $i] :
                              ( ( mem @ V9e2 @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
                             => ! [V10t: $i] :
                                  ( ( mem @ V10t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
                                 => ( ( p @ ( ap @ ( ap @ V0P @ V5a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V8c2 ) @ V9e2 ) ) @ V10t ) ) )
                                   => ( p @ ( ap @ ( ap @ V0P @ V5a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V6c1 ) @ V7e1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V8c2 ) @ V9e2 ) ) @ V10t ) ) ) ) ) ) ) ) ) ) ) )
       => ! [V11v: $i] :
            ( ( mem @ V11v @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ! [V12v1: $i] :
                ( ( mem @ V12v1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
               => ( p @ ( ap @ ( ap @ V0P @ V11v ) @ V12v1 ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eis__polyform__def,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V0a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) ) )
          <=> $true ) )
      & ! [V1e: $i] :
          ( ( mem @ V1e @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3a: $i] :
                  ( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V3a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V2c ) @ V1e ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) ) ) )
                  <=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2c ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2c ) @ V3a ) ) ) ) ) ) )
      & ! [V4t: $i] :
          ( ( mem @ V4t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
         => ! [V5e2: $i] :
              ( ( mem @ V5e2 @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
             => ! [V6e1: $i] :
                  ( ( mem @ V6e1 @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
                 => ! [V7c2: $i] :
                      ( ( mem @ V7c2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                     => ! [V8c1: $i] :
                          ( ( mem @ V8c1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                         => ! [V9a: $i] :
                              ( ( mem @ V9a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V9a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V8c1 ) @ V6e1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V7c2 ) @ V5e2 ) ) @ V4t ) ) ) )
                              <=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V8c1 ) )
                                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V8c1 ) @ V9a ) )
                                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27b ) @ V5e2 ) @ V6e1 ) )
                                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V9a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V7c2 ) @ V5e2 ) ) @ V4t ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eis__polyform__ELthm,axiom,
    ! [A_27a: del,A_27b: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1ces: $i] :
          ( ( mem @ V1ces @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V0a ) @ V1ces ) )
          <=> ( ! [V2i: tp__ty_2Enum_2Enum,V3j: tp__ty_2Enum_2Enum] :
                  ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) )
                    & ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ V1ces ) ) ) )
                 => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ V1ces ) ) ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1ces ) ) ) ) )
              & ! [V4c: $i] :
                  ( ( mem @ V4c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ! [V5e: $i] :
                      ( ( mem @ V5e @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V4c ) @ V5e ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ V1ces ) ) )
                       => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V4c ) )
                          & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V4c ) @ V0a ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epolyform__exists,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
           => ? [V2ces: $i] :
                ( ( mem @ V2ces @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
                & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ V2ces ) )
                & ( V1b
                  = ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ V2ces ) ) ) ) ) ) ).

thf(ax_thm_2Eordinal_2Epolyform__def,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ V1b ) ) )
              & ( V1b
                = ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ V1b ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2ECNF__thm,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0b ) ) )
        & ( V0b
          = ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0b ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epolyform__0,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
       => ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
          = ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epolyform__EQ__NIL,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
           => ( ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ V1x )
                = ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
            <=> ( V1x
                = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eis__polyform__CONS__E,axiom,
    ! [A_27a: del,A_27b: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1c: $i] :
          ( ( mem @ V1c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2e: $i] :
              ( ( mem @ V2e @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
             => ! [V3t: $i] :
                  ( ( mem @ V3t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V0a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V1c ) @ V2e ) ) @ V3t ) ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1c ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1c ) @ V0a ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V0a ) @ V3t ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eis__polyform__head__dominates__tail,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1c: $i] :
          ( ( mem @ V1c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2e: $i] :
              ( ( mem @ V2e @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ! [V3t: $i] :
                  ( ( mem @ V3t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1c ) @ V2e ) ) @ V3t ) ) ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V2e ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Ecx__lt__x,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1c: $i] :
          ( ( mem @ V1c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0x ) @ V1c ) ) @ V0x ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
              & ( V1c
                = ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epolyform__UNIQUE,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ! [V2ces: $i] :
              ( ( mem @ V2ces @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ V2ces ) )
                  & ( V1b
                    = ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ V2ces ) ) )
               => ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ V1b )
                  = V2ces ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Epolyform__eval__poly,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ V1b ) ) )
           => ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ V1b ) )
              = V1b ) ) ) ) ).

thf(conj_thm_2Eordinal_2ECNF__nat,axiom,
    ! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
      ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
      = ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2EordLOG__correct,axiom,
    ! [A_27a: del,V0b: $i] :
      ( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0b ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1x ) ) )
           => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0b ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0b ) @ V1x ) ) ) ) ) )
              & ! [V2a: $i] :
                  ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0b ) @ V1x ) ) ) ) @ V2a ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0b ) @ V2a ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eordinal_2Eolog__correct,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
       => ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0x ) ) ) ) ) )
          & ! [V1a: $i] :
              ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0x ) ) ) ) @ V1a ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V1a ) ) ) ) ) ) ) ) ).

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