ITP001 Axioms: ITP017^5.ax


%------------------------------------------------------------------------------
% File     : ITP017^5 : 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    : poset^2.ax [Gau20]
%          : HL4017^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   67 (   0 unt;  17 typ;   0 def)
%            Number of atoms       : 1237 (   9 equ;   0 cnn)
%            Maximal formula atoms :   76 (  18 avg)
%            Number of connectives : 2263 (   0   ~;   1   |;  75   &;2020   @)
%                                         (  16 <=>; 151  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   31 (  16 avg;2020 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :   19 (  19   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   32 (  31 usr;  14 con; 0-2 aty)
%            Number of variables   :  195 (   9   ^ 174   !;  12   ?; 195   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eposet_2Ebottom,type,
    c_2Eposet_2Ebottom: del > $i ).

thf(mem_c_2Eposet_2Ebottom,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Ebottom @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ A_27a @ bool ) ) ) ).

thf(tp_c_2Eposet_2Ecarrier,type,
    c_2Eposet_2Ecarrier: del > $i ).

thf(mem_c_2Eposet_2Ecarrier,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Ecarrier @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ A_27a @ bool ) ) ) ).

thf(tp_c_2Eposet_2Echain,type,
    c_2Eposet_2Echain: del > $i ).

thf(mem_c_2Eposet_2Echain,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Echain @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eposet_2Ecomplete,type,
    c_2Eposet_2Ecomplete: del > $i ).

thf(mem_c_2Eposet_2Ecomplete,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Ecomplete @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ bool ) ) ).

thf(tp_c_2Eposet_2Econtinuous,type,
    c_2Eposet_2Econtinuous: del > $i ).

thf(mem_c_2Eposet_2Econtinuous,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Econtinuous @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eposet_2Edown__continuous,type,
    c_2Eposet_2Edown__continuous: del > $i ).

thf(mem_c_2Eposet_2Edown__continuous,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Edown__continuous @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eposet_2Efunction,type,
    c_2Eposet_2Efunction: del > del > $i ).

thf(mem_c_2Eposet_2Efunction,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eposet_2Efunction @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ).

thf(tp_c_2Eposet_2Egfp,type,
    c_2Eposet_2Egfp: del > $i ).

thf(mem_c_2Eposet_2Egfp,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Egfp @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eposet_2Eglb,type,
    c_2Eposet_2Eglb: del > $i ).

thf(mem_c_2Eposet_2Eglb,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Eglb @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eposet_2Elfp,type,
    c_2Eposet_2Elfp: del > $i ).

thf(mem_c_2Eposet_2Elfp,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Elfp @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eposet_2Elub,type,
    c_2Eposet_2Elub: del > $i ).

thf(mem_c_2Eposet_2Elub,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Elub @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eposet_2Emonotonic,type,
    c_2Eposet_2Emonotonic: del > $i ).

thf(mem_c_2Eposet_2Emonotonic,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Emonotonic @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eposet_2Epointwise__lift,type,
    c_2Eposet_2Epointwise__lift: del > del > $i ).

thf(mem_c_2Eposet_2Epointwise__lift,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eposet_2Epointwise__lift @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ) ) ).

thf(tp_c_2Eposet_2Eposet,type,
    c_2Eposet_2Eposet: del > $i ).

thf(mem_c_2Eposet_2Eposet,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Eposet @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ bool ) ) ).

thf(tp_c_2Eposet_2Erelation,type,
    c_2Eposet_2Erelation: del > $i ).

thf(mem_c_2Eposet_2Erelation,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Erelation @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eposet_2Etop,type,
    c_2Eposet_2Etop: del > $i ).

thf(mem_c_2Eposet_2Etop,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Etop @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ A_27a @ bool ) ) ) ).

thf(tp_c_2Eposet_2Eup__continuous,type,
    c_2Eposet_2Eup__continuous: del > $i ).

thf(mem_c_2Eposet_2Eup__continuous,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eposet_2Eup__continuous @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ).

thf(ax_thm_2Eposet_2Efunction__def,axiom,
    ! [A_27a: del,A_27b: del,V0a: $i] :
      ( ( mem @ V0a @ ( arr @ A_27a @ bool ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ ( arr @ A_27b @ bool ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27b ) @ V0a ) @ V1b ) @ V2f ) )
              <=> ! [V3x: $i] :
                    ( ( mem @ V3x @ A_27a )
                   => ( ( p @ ( ap @ V0a @ V3x ) )
                     => ( p @ ( ap @ V1b @ ( ap @ V2f @ V3x ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Eposet__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
          <=> ( ? [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                  & ( p @ ( ap @ V0s @ V2x ) ) )
              & ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ( ( p @ ( ap @ V0s @ V3x ) )
                   => ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V3x ) ) ) )
              & ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ! [V5y: $i] :
                      ( ( mem @ V5y @ A_27a )
                     => ( ( ( p @ ( ap @ V0s @ V4x ) )
                          & ( p @ ( ap @ V0s @ V5y ) )
                          & ( p @ ( ap @ ( ap @ V1r @ V4x ) @ V5y ) )
                          & ( p @ ( ap @ ( ap @ V1r @ V5y ) @ V4x ) ) )
                       => ( V4x = V5y ) ) ) )
              & ! [V6x: $i] :
                  ( ( mem @ V6x @ A_27a )
                 => ! [V7y: $i] :
                      ( ( mem @ V7y @ A_27a )
                     => ! [V8z: $i] :
                          ( ( mem @ V8z @ A_27a )
                         => ( ( ( p @ ( ap @ V0s @ V6x ) )
                              & ( p @ ( ap @ V0s @ V7y ) )
                              & ( p @ ( ap @ V0s @ V8z ) )
                              & ( p @ ( ap @ ( ap @ V1r @ V6x ) @ V7y ) )
                              & ( p @ ( ap @ ( ap @ V1r @ V7y ) @ V8z ) ) )
                           => ( p @ ( ap @ ( ap @ V1r @ V6x ) @ V8z ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Ecarrier__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
            = V0s ) ) ) ).

thf(ax_thm_2Eposet_2Erelation__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ap @ ( c_2Eposet_2Erelation @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
            = V1r ) ) ) ).

thf(ax_thm_2Eposet_2Etop__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Etop @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2x ) )
              <=> ( ( p @ ( ap @ V0s @ V2x ) )
                  & ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( p @ ( ap @ V0s @ V3y ) )
                       => ( p @ ( ap @ ( ap @ V1r @ V3y ) @ V2x ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Ebottom__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Ebottom @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2x ) )
              <=> ( ( p @ ( ap @ V0s @ V2x ) )
                  & ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( p @ ( ap @ V0s @ V3y ) )
                       => ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V3y ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Echain__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2c: $i] :
              ( ( mem @ V2c @ ( arr @ A_27a @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Echain @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2c ) )
              <=> ! [V3x: $i] :
                    ( ( mem @ V3x @ A_27a )
                   => ! [V4y: $i] :
                        ( ( mem @ V4y @ A_27a )
                       => ( ( ( p @ ( ap @ V0s @ V3x ) )
                            & ( p @ ( ap @ V0s @ V4y ) )
                            & ( p @ ( ap @ V2c @ V3x ) )
                            & ( p @ ( ap @ V2c @ V4y ) ) )
                         => ( ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) )
                            | ( p @ ( ap @ ( ap @ V1r @ V4y ) @ V3x ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Elub__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2p: $i] :
              ( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2p ) @ V3x ) )
                  <=> ( ( p @ ( ap @ V0s @ V3x ) )
                      & ! [V4y: $i] :
                          ( ( mem @ V4y @ A_27a )
                         => ( ( ( p @ ( ap @ V0s @ V4y ) )
                              & ( p @ ( ap @ V2p @ V4y ) ) )
                           => ( p @ ( ap @ ( ap @ V1r @ V4y ) @ V3x ) ) ) )
                      & ! [V5z: $i] :
                          ( ( mem @ V5z @ A_27a )
                         => ( ( ( p @ ( ap @ V0s @ V5z ) )
                              & ! [V6y: $i] :
                                  ( ( mem @ V6y @ A_27a )
                                 => ( ( ( p @ ( ap @ V0s @ V6y ) )
                                      & ( p @ ( ap @ V2p @ V6y ) ) )
                                   => ( p @ ( ap @ ( ap @ V1r @ V6y ) @ V5z ) ) ) ) )
                           => ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V5z ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Eglb__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2p: $i] :
              ( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2p ) @ V3x ) )
                  <=> ( ( p @ ( ap @ V0s @ V3x ) )
                      & ! [V4y: $i] :
                          ( ( mem @ V4y @ A_27a )
                         => ( ( ( p @ ( ap @ V0s @ V4y ) )
                              & ( p @ ( ap @ V2p @ V4y ) ) )
                           => ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) ) ) )
                      & ! [V5z: $i] :
                          ( ( mem @ V5z @ A_27a )
                         => ( ( ( p @ ( ap @ V0s @ V5z ) )
                              & ! [V6y: $i] :
                                  ( ( mem @ V6y @ A_27a )
                                 => ( ( ( p @ ( ap @ V0s @ V6y ) )
                                      & ( p @ ( ap @ V2p @ V6y ) ) )
                                   => ( p @ ( ap @ ( ap @ V1r @ V5z ) @ V6y ) ) ) ) )
                           => ( p @ ( ap @ ( ap @ V1r @ V5z ) @ V3x ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Ecomplete__def,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ( ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
      <=> ! [V1c: $i] :
            ( ( mem @ V1c @ ( arr @ A_27a @ bool ) )
           => ( ? [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                  & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ V0p ) @ V1c ) @ V2x ) ) )
              & ? [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                  & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ V0p ) @ V1c ) @ V3x ) ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Eposet__nonempty,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ V0s @ V2x ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Eposet__refl,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
                  & ( p @ ( ap @ V0s @ V2x ) ) )
               => ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V2x ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Eposet__antisym,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
                      & ( p @ ( ap @ V0s @ V2x ) )
                      & ( p @ ( ap @ V0s @ V3y ) )
                      & ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V3y ) )
                      & ( p @ ( ap @ ( ap @ V1r @ V3y ) @ V2x ) ) )
                   => ( V2x = V3y ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Eposet__trans,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ! [V4z: $i] :
                      ( ( mem @ V4z @ A_27a )
                     => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
                          & ( p @ ( ap @ V0s @ V2x ) )
                          & ( p @ ( ap @ V0s @ V3y ) )
                          & ( p @ ( ap @ V0s @ V4z ) )
                          & ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V3y ) )
                          & ( p @ ( ap @ ( ap @ V1r @ V3y ) @ V4z ) ) )
                       => ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V4z ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Elub__pred,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2p: $i] :
              ( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ( ( p
                      @ ( ap
                        @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
                          @ ( lam @ A_27a
                            @ ^ [V4j: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0s @ V4j ) ) @ ( ap @ V2p @ V4j ) ) ) )
                        @ V3x ) )
                  <=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2p ) @ V3x ) ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Eglb__pred,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2p: $i] :
              ( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ( ( p
                      @ ( ap
                        @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
                          @ ( lam @ A_27a
                            @ ^ [V4j: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0s @ V4j ) ) @ ( ap @ V2p @ V4j ) ) ) )
                        @ V3x ) )
                  <=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2p ) @ V3x ) ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Ecomplete__up,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1c: $i] :
          ( ( mem @ V1c @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ V0p ) @ V1c ) @ V2x ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Ecomplete__down,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1c: $i] :
          ( ( mem @ V1c @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ V0p ) @ V1c ) @ V2x ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Ecomplete__top,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
          & ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) ) )
       => ? [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
            & ( p @ ( ap @ ( ap @ ( c_2Eposet_2Etop @ A_27a ) @ V0p ) @ V1x ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Ecomplete__bottom,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
          & ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) ) )
       => ? [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
            & ( p @ ( ap @ ( ap @ ( c_2Eposet_2Ebottom @ A_27a ) @ V0p ) @ V1x ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Epointwise__lift__def,axiom,
    ! [A_27a: del,A_27b: del,V0t: $i] :
      ( ( mem @ V0t @ ( arr @ A_27a @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
         => ! [V2r: $i] :
              ( ( mem @ V2r @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
             => ( ( ap @ ( ap @ ( c_2Eposet_2Epointwise__lift @ A_27a @ A_27b ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ V1s ) @ V2r ) )
                = ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27b ) @ V0t ) @ V1s ) )
                  @ ( lam @ ( arr @ A_27a @ A_27b )
                    @ ^ [V3f: $i] :
                        ( lam @ ( arr @ A_27a @ A_27b )
                        @ ^ [V4g: $i] :
                            ( ap @ ( c_2Ebool_2E_21 @ A_27a )
                            @ ( lam @ A_27a
                              @ ^ [V5x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V0t @ V5x ) ) @ ( ap @ ( ap @ V2r @ ( ap @ V3f @ V5x ) ) @ ( ap @ V4g @ V5x ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Ecomplete__pointwise,axiom,
    ! [A_27a: del,A_27b: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1t: $i] :
          ( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
         => ( ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
           => ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ ( arr @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eposet_2Epointwise__lift @ A_27b @ A_27a ) @ V1t ) @ V0p ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Emonotonic__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Emonotonic @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) )
              <=> ! [V3x: $i] :
                    ( ( mem @ V3x @ A_27a )
                   => ! [V4y: $i] :
                        ( ( mem @ V4y @ A_27a )
                       => ( ( ( p @ ( ap @ V0s @ V3x ) )
                            & ( p @ ( ap @ V0s @ V4y ) )
                            & ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) ) )
                         => ( p @ ( ap @ ( ap @ V1r @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Eup__continuous__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Eup__continuous @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) )
              <=> ! [V3c: $i] :
                    ( ( mem @ V3c @ ( arr @ A_27a @ bool ) )
                   => ! [V4x: $i] :
                        ( ( mem @ V4x @ A_27a )
                       => ( ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Echain @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V3c ) )
                            & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V3c ) @ V4x ) ) )
                         => ( p
                            @ ( ap
                              @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
                                @ ( lam @ A_27a
                                  @ ^ [V5y: $i] :
                                      ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                      @ ( lam @ A_27a
                                        @ ^ [V6z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0s @ V6z ) ) @ ( ap @ V3c @ V6z ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V5y ) @ ( ap @ V2f @ V6z ) ) ) ) ) ) )
                              @ ( ap @ V2f @ V4x ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Edown__continuous__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Edown__continuous @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) )
              <=> ! [V3c: $i] :
                    ( ( mem @ V3c @ ( arr @ A_27a @ bool ) )
                   => ! [V4x: $i] :
                        ( ( mem @ V4x @ A_27a )
                       => ( ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Echain @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V3c ) )
                            & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V3c ) @ V4x ) ) )
                         => ( p
                            @ ( ap
                              @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
                                @ ( lam @ A_27a
                                  @ ^ [V5y: $i] :
                                      ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                      @ ( lam @ A_27a
                                        @ ^ [V6z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0s @ V6z ) ) @ ( ap @ V3c @ V6z ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V5y ) @ ( ap @ V2f @ V6z ) ) ) ) ) ) )
                              @ ( ap @ V2f @ V4x ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Econtinuous__def,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Econtinuous @ A_27a ) @ V0p ) @ V1f ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Eup__continuous @ A_27a ) @ V0p ) @ V1f ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eposet_2Edown__continuous @ A_27a ) @ V0p ) @ V1f ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Elfp__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) @ V3x ) )
                  <=> ( ( p @ ( ap @ V0s @ V3x ) )
                      & ( ( ap @ V2f @ V3x )
                        = V3x )
                      & ! [V4y: $i] :
                          ( ( mem @ V4y @ A_27a )
                         => ( ( ( p @ ( ap @ V0s @ V4y ) )
                              & ( p @ ( ap @ ( ap @ V1r @ ( ap @ V2f @ V4y ) ) @ V4y ) ) )
                           => ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eposet_2Egfp__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) @ V3x ) )
                  <=> ( ( p @ ( ap @ V0s @ V3x ) )
                      & ( ( ap @ V2f @ V3x )
                        = V3x )
                      & ! [V4y: $i] :
                          ( ( mem @ V4y @ A_27a )
                         => ( ( ( p @ ( ap @ V0s @ V4y ) )
                              & ( p @ ( ap @ ( ap @ V1r @ V4y ) @ ( ap @ V2f @ V4y ) ) ) )
                           => ( p @ ( ap @ ( ap @ V1r @ V4y ) @ V3x ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Elfp__unique,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3x_27: $i] :
                  ( ( mem @ V3x_27 @ A_27a )
                 => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
                      & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) )
                      & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ V0p ) @ V1f ) @ V3x_27 ) ) )
                   => ( V2x = V3x_27 ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Egfp__unique,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3x_27: $i] :
                  ( ( mem @ V3x_27 @ A_27a )
                 => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
                      & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) )
                      & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ V0p ) @ V1f ) @ V3x_27 ) ) )
                   => ( V2x = V3x_27 ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Eknaster__tarski__lfp,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
              & ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
              & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27a ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ V1f ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eposet_2Emonotonic @ A_27a ) @ V0p ) @ V1f ) ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Eknaster__tarski__gfp,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
              & ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
              & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27a ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ V1f ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eposet_2Emonotonic @ A_27a ) @ V0p ) @ V1f ) ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) ) ) ) ) ) ).

thf(conj_thm_2Eposet_2Eknaster__tarski,axiom,
    ! [A_27a: del,V0p: $i] :
      ( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
              & ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
              & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27a ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ V1f ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eposet_2Emonotonic @ A_27a ) @ V0p ) @ V1f ) ) )
           => ( ? [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                  & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) ) )
              & ? [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                  & ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ V0p ) @ V1f ) @ V3x ) ) ) ) ) ) ) ).

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