ITP001 Axioms: ITP033^5.ax


%------------------------------------------------------------------------------
% File     : ITP033^5 : 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    : set_relation^2.ax [Gau20]
%          : HL4033^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  233 (   4 unt;  31 typ;   0 def)
%            Number of atoms       : 5470 (  96 equ;   0 cnn)
%            Maximal formula atoms :  154 (  23 avg)
%            Number of connectives : 10096 (  23   ~;   8   |; 188   &;9214   @)
%                                         (  52 <=>; 611  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   63 (  16 avg;9214 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :   39 (  39   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  104 ( 103 usr;  72 con; 0-3 aty)
%            Number of variables   :  821 (  64   ^ 730   !;  27   ?; 821   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eset__relation_2ERREFL__EXP,type,
    c_2Eset__relation_2ERREFL__EXP: del > $i ).

thf(mem_c_2Eset__relation_2ERREFL__EXP,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ) ).

thf(tp_c_2Eset__relation_2ERRUNIV,type,
    c_2Eset__relation_2ERRUNIV: del > $i ).

thf(mem_c_2Eset__relation_2ERRUNIV,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eset__relation_2Eacyclic,type,
    c_2Eset__relation_2Eacyclic: del > $i ).

thf(mem_c_2Eset__relation_2Eacyclic,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).

thf(tp_c_2Eset__relation_2Eall__choices,type,
    c_2Eset__relation_2Eall__choices: del > $i ).

thf(mem_c_2Eset__relation_2Eall__choices,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eall__choices @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Eantisym,type,
    c_2Eset__relation_2Eantisym: del > $i ).

thf(mem_c_2Eset__relation_2Eantisym,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).

thf(tp_c_2Eset__relation_2Echain,type,
    c_2Eset__relation_2Echain: del > $i ).

thf(mem_c_2Eset__relation_2Echain,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Echain @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Edomain,type,
    c_2Eset__relation_2Edomain: del > del > $i ).

thf(mem_c_2Eset__relation_2Edomain,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Efchains,type,
    c_2Eset__relation_2Efchains: del > $i ).

thf(mem_c_2Eset__relation_2Efchains,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Efchains @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Efinite__prefixes,type,
    c_2Eset__relation_2Efinite__prefixes: del > del > $i ).

thf(mem_c_2Eset__relation_2Efinite__prefixes,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Eget__min,type,
    c_2Eset__relation_2Eget__min: del > $i ).

thf(mem_c_2Eset__relation_2Eget__min,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).

thf(tp_c_2Eset__relation_2Eirreflexive,type,
    c_2Eset__relation_2Eirreflexive: del > $i ).

thf(mem_c_2Eset__relation_2Eirreflexive,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Elinear__order,type,
    c_2Eset__relation_2Elinear__order: del > $i ).

thf(mem_c_2Eset__relation_2Elinear__order,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Emaximal__elements,type,
    c_2Eset__relation_2Emaximal__elements: del > $i ).

thf(mem_c_2Eset__relation_2Emaximal__elements,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eset__relation_2Eminimal__elements,type,
    c_2Eset__relation_2Eminimal__elements: del > $i ).

thf(mem_c_2Eset__relation_2Eminimal__elements,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eset__relation_2Enth__min,type,
    c_2Eset__relation_2Enth__min: del > $i ).

thf(mem_c_2Eset__relation_2Enth__min,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ).

thf(tp_c_2Eset__relation_2Enum__order,type,
    c_2Eset__relation_2Enum__order: del > $i ).

thf(mem_c_2Eset__relation_2Enum__order,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Enum__order @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ) ).

thf(tp_c_2Eset__relation_2Epartial__order,type,
    c_2Eset__relation_2Epartial__order: del > $i ).

thf(mem_c_2Eset__relation_2Epartial__order,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Eper,type,
    c_2Eset__relation_2Eper: del > $i ).

thf(mem_c_2Eset__relation_2Eper,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eper @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Eper__restrict,type,
    c_2Eset__relation_2Eper__restrict: del > $i ).

thf(mem_c_2Eset__relation_2Eper__restrict,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eper__restrict @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).

thf(tp_c_2Eset__relation_2Erange,type,
    c_2Eset__relation_2Erange: del > del > $i ).

thf(mem_c_2Eset__relation_2Erange,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Ercomp,type,
    c_2Eset__relation_2Ercomp: del > del > del > $i ).

thf(mem_c_2Eset__relation_2Ercomp,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ) ).

thf(tp_c_2Eset__relation_2Ereflexive,type,
    c_2Eset__relation_2Ereflexive: del > $i ).

thf(mem_c_2Eset__relation_2Ereflexive,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Erel__to__reln,type,
    c_2Eset__relation_2Erel__to__reln: del > del > $i ).

thf(mem_c_2Eset__relation_2Erel__to__reln,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Ereln__to__rel,type,
    c_2Eset__relation_2Ereln__to__rel: del > del > $i ).

thf(mem_c_2Eset__relation_2Ereln__to__rel,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ).

thf(tp_c_2Eset__relation_2Errestrict,type,
    c_2Eset__relation_2Errestrict: del > $i ).

thf(mem_c_2Eset__relation_2Errestrict,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ) ).

thf(tp_c_2Eset__relation_2Estrict,type,
    c_2Eset__relation_2Estrict: del > $i ).

thf(mem_c_2Eset__relation_2Estrict,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Estrict__linear__order,type,
    c_2Eset__relation_2Estrict__linear__order: del > $i ).

thf(mem_c_2Eset__relation_2Estrict__linear__order,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Etc,type,
    c_2Eset__relation_2Etc: del > $i ).

thf(mem_c_2Eset__relation_2Etc,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Etransitive,type,
    c_2Eset__relation_2Etransitive: del > $i ).

thf(mem_c_2Eset__relation_2Etransitive,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).

thf(tp_c_2Eset__relation_2Euniv__reln,type,
    c_2Eset__relation_2Euniv__reln: del > $i ).

thf(mem_c_2Eset__relation_2Euniv__reln,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ).

thf(tp_c_2Eset__relation_2Eupper__bounds,type,
    c_2Eset__relation_2Eupper__bounds: del > del > $i ).

thf(mem_c_2Eset__relation_2Eupper__bounds,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Erextension,axiom,
    ! [A_27a: del,A_27b: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1t: $i] :
          ( ( mem @ V1t @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
         => ( ( V0s = V1t )
          <=> ! [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
               => ! [V3y: $i] :
                    ( ( mem @ V3y @ A_27b )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) @ V0s ) )
                    <=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) @ V1t ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Edomain__def,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V0r )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
          @ ( lam @ A_27a
            @ ^ [V1x: $i] :
                ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1x )
                @ ( ap @ ( c_2Ebool_2E_3F @ A_27b )
                  @ ( lam @ A_27b
                    @ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ V0r ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Erange__def,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V0r )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
          @ ( lam @ A_27a
            @ ^ [V1y: $i] :
                ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1y )
                @ ( ap @ ( c_2Ebool_2E_3F @ A_27b )
                  @ ( lam @ A_27b
                    @ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V2x ) @ V1y ) ) @ V0r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ein__domain,axiom,
    ! [A_27a: del,A_27b: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V1r ) ) )
          <=> ? [V2y: $i] :
                ( ( mem @ V2y @ A_27b )
                & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V2y ) ) @ V1r ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ein__range,axiom,
    ! [A_27a: del,A_27b: del,V0y: $i] :
      ( ( mem @ V0y @ A_27a )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0y ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V1r ) ) )
          <=> ? [V2x: $i] :
                ( ( mem @ V2x @ A_27b )
                & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V2x ) @ V0y ) ) @ V1r ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ein__dom__rg,axiom,
    ! [A_27a: del,A_27b: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ A_27b )
         => ! [V2r: $i] :
              ( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y ) ) @ V2r ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V2r ) ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V1y ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a ) @ V2r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Edomain__mono,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1r_27: $i] :
          ( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0r ) @ V1r_27 ) )
           => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V1r_27 ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Erange__mono,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1r_27: $i] :
          ( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0r ) @ V1r_27 ) )
           => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a ) @ V1r_27 ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Errestrict__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s )
            = ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
              @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                @ ( lam @ A_27a
                  @ ^ [V2x: $i] :
                      ( lam @ A_27a
                      @ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ein__rrestrict,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ A_27a )
         => ! [V2r: $i] :
              ( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V3s: $i] :
                  ( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V2r ) @ V3s ) ) )
                  <=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ V2r ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V3s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1y ) @ V3s ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ein__rrestrict__alt,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2s: $i] :
              ( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r ) @ V2s ) ) )
              <=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0x ) @ V1r ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27a ) @ V0x ) ) @ V2s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27a ) @ V0x ) ) @ V2s ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Errestrict__SUBSET,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) @ V0r ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Errestrict__union,axiom,
    ! [A_27a: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2s: $i] :
              ( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
             => ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r1 ) @ V1r2 ) ) @ V2s )
                = ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r1 ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r2 ) @ V2s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Errestrict__rrestrict,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ ( arr @ A_27a @ bool ) )
             => ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1x ) ) @ V2y )
                = ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Edomain__rrestrict__SUBSET,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) @ V1s ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Erange__rrestrict__SUBSET,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) @ V1s ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Ercomp__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c ) @ V0r1 ) @ V1r2 )
            = ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
              @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
                @ ( lam @ A_27a
                  @ ^ [V2x: $i] :
                      ( lam @ A_27b
                      @ ^ [V3y: $i] :
                          ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) )
                          @ ( ap @ ( c_2Ebool_2E_3F @ A_27c )
                            @ ( lam @ A_27c
                              @ ^ [V4z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2x ) @ V4z ) ) @ V0r1 ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27b ) @ V4z ) @ V3y ) ) @ V1r2 ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Estrict__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
          @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
            @ ( lam @ A_27a
              @ ^ [V1x: $i] :
                  ( lam @ A_27a
                  @ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__rrestrict,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) )
            = ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Euniv__reln__def,axiom,
    ! [A_27a: del,V0xs: $i] :
      ( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ V0xs )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
          @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
            @ ( lam @ A_27a
              @ ^ [V1x1: $i] :
                  ( lam @ A_27a
                  @ ^ [V2x2: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x1 ) @ V2x2 ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x1 ) @ V0xs ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x2 ) @ V0xs ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Efinite__prefixes__def,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
          <=> ! [V2e: $i] :
                ( ( mem @ V2e @ A_27b )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2e ) @ V1s ) )
                 => ( p
                    @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
                      @ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V3e_27: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3e_27 ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3e_27 ) @ V2e ) ) @ V0r ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__s,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27b @ bool ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V2s_27 ) @ V1s ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V2s_27 ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__r,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1r_27: $i] :
          ( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
         => ! [V2s: $i] :
              ( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V2s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1r_27 ) @ V0r ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V1r_27 ) @ V2s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__rs,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
         => ! [V2r_27: $i] :
              ( ( mem @ V2r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
             => ! [V3s_27: $i] :
                  ( ( mem @ V3s_27 @ ( arr @ A_27b @ bool ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2r_27 ) @ V0r ) )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V3s_27 ) @ V1s ) )
                       => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V2r_27 ) @ V3s_27 ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V0r ) @ V1s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s_27 ) @ V1s ) ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V0r ) @ V2s_27 ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V2s_27 ) ) @ V2s_27 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefixes__union,axiom,
    ! [A_27a: del,A_27b: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
         => ! [V2s1: $i] :
              ( ( mem @ V2s1 @ ( arr @ A_27b @ bool ) )
             => ! [V3s2: $i] :
                  ( ( mem @ V3s2 @ ( arr @ A_27b @ bool ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r1 ) @ V2s1 ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V1r2 ) @ V3s2 ) ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0r1 ) @ V1r2 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V2s1 ) @ V3s2 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefixes__comp,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) @ bool ) )
         => ! [V2s1: $i] :
              ( ( mem @ V2s1 @ ( arr @ A_27b @ bool ) )
             => ! [V3s2: $i] :
                  ( ( mem @ V3s2 @ ( arr @ A_27c @ bool ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r1 ) @ V2s1 ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27b @ A_27c ) @ V1r2 ) @ V3s2 ) )
                      & ( p
                        @ ( ap
                          @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b )
                            @ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
                              @ ( lam @ A_27b
                                @ ^ [V4x: $i] :
                                    ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V4x )
                                    @ ( ap @ ( c_2Ebool_2E_3F @ A_27c )
                                      @ ( lam @ A_27c
                                        @ ^ [V5y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27c ) @ V5y ) @ V3s2 ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ V4x ) @ V5y ) ) @ V1r2 ) ) ) ) ) ) ) )
                          @ V2s1 ) ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27c @ A_27b ) @ V0r1 ) @ V1r2 ) ) @ V3s2 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefixes__inj__image,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2s: $i] :
              ( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
             => ( ( ! [V3x: $i] :
                      ( ( mem @ V3x @ A_27a )
                     => ! [V4y: $i] :
                          ( ( mem @ V4y @ A_27a )
                         => ( ( ( ap @ V0f @ V3x )
                              = ( ap @ V0f @ V4y ) )
                           => ( V3x = V4y ) ) ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V1r ) @ V2s ) ) )
               => ( p
                  @ ( ap
                    @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27b @ A_27b )
                      @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
                        @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) @ bool ) )
                          @ ( lam @ A_27a
                            @ ^ [V5x: $i] :
                                ( lam @ A_27a
                                @ ^ [V6y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ ( ap @ V0f @ V6y ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V5x ) @ V6y ) ) @ V1r ) ) ) ) ) ) )
                    @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefixes__range,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
         => ! [V2t: $i] :
              ( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27b ) @ V2t ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a ) @ V0r ) ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Etc__def,axiom,
    ! [A_27a: del] :
      ( ( c_2Eset__relation_2Etc @ A_27a )
      = ( lam @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool )
        @ ^ [V0r: $i] :
            ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27a )
            @ ^ [V1a0: $i] :
                ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                @ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool )
                  @ ^ [V2tc_27: $i] :
                      ( ap
                      @ ( ap @ c_2Emin_2E_3D_3D_3E
                        @ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
                          @ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27a )
                            @ ^ [V3a0: $i] :
                                ( ap
                                @ ( ap @ c_2Emin_2E_3D_3D_3E
                                  @ ( ap
                                    @ ( ap @ c_2Ebool_2E_5C_2F
                                      @ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                        @ ( lam @ A_27a
                                          @ ^ [V4x: $i] :
                                              ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                              @ ( lam @ A_27a
                                                @ ^ [V5y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V3a0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) ) @ ( ap @ V0r @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) ) ) ) ) ) )
                                    @ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                      @ ( lam @ A_27a
                                        @ ^ [V6x: $i] :
                                            ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                            @ ( lam @ A_27a
                                              @ ^ [V7y: $i] :
                                                  ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V3a0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6x ) @ V7y ) ) )
                                                  @ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
                                                    @ ( lam @ A_27a
                                                      @ ^ [V8z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V2tc_27 @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6x ) @ V8z ) ) ) @ ( ap @ V2tc_27 @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V8z ) @ V7y ) ) ) ) ) ) ) ) ) ) ) )
                                @ ( ap @ V2tc_27 @ V3a0 ) ) ) ) )
                      @ ( ap @ V2tc_27 @ V1a0 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__rules,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ! [V2y: $i] :
                ( ( mem @ V2y @ A_27a )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
                 => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) )
        & ! [V3x: $i] :
            ( ( mem @ V3x @ A_27a )
           => ! [V4y: $i] :
                ( ( mem @ V4y @ A_27a )
               => ( ? [V5z: $i] :
                      ( ( mem @ V5z @ A_27a )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V5z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V5z ) @ V4y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) )
                 => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V4y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__cases,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
              <=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
                  | ? [V3z: $i] :
                      ( ( mem @ V3z @ A_27a )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3z ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__ind,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1tc_27: $i] :
          ( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
              & ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ! [V5y: $i] :
                      ( ( mem @ V5y @ A_27a )
                     => ( ? [V6z: $i] :
                            ( ( mem @ V6z @ A_27a )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V6z ) )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V6z ) @ V5y ) ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
           => ! [V7x: $i] :
                ( ( mem @ V7x @ A_27a )
               => ! [V8y: $i] :
                    ( ( mem @ V8y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                     => ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__closure,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1s ) ) )
           => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Esubset__tc,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__idemp,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) )
        = ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__mono,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ V1s ) )
           => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__strongind,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1tc_27: $i] :
          ( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
              & ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ! [V5y: $i] :
                      ( ( mem @ V5y @ A_27a )
                     => ( ? [V6z: $i] :
                            ( ( mem @ V6z @ A_27a )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V6z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V6z ) )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6z ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V6z ) @ V5y ) ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
           => ! [V7x: $i] :
                ( ( mem @ V7x @ A_27a )
               => ! [V8y: $i] :
                    ( ( mem @ V8y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                     => ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__cases__right,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
              <=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
                  | ? [V3z: $i] :
                      ( ( mem @ V3z @ A_27a )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3z ) @ V2y ) ) @ V0r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__cases__left,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
              <=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
                  | ? [V3z: $i] :
                      ( ( mem @ V3z @ A_27a )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) @ V0r ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3z ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__ind__left,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1tc_27: $i] :
          ( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
              & ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ! [V5y: $i] :
                      ( ( mem @ V5y @ A_27a )
                     => ( ? [V6z: $i] :
                            ( ( mem @ V6z @ A_27a )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V6z ) ) @ V0r ) )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V6z ) @ V5y ) ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
           => ! [V7x: $i] :
                ( ( mem @ V7x @ A_27a )
               => ! [V8y: $i] :
                    ( ( mem @ V8y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                     => ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__strongind__left,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1tc_27: $i] :
          ( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
              & ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ! [V5y: $i] :
                      ( ( mem @ V5y @ A_27a )
                     => ( ? [V6z: $i] :
                            ( ( mem @ V6z @ A_27a )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V6z ) ) @ V0r ) )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6z ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V6z ) @ V5y ) ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
           => ! [V7x: $i] :
                ( ( mem @ V7x @ A_27a )
               => ! [V8y: $i] :
                    ( ( mem @ V8y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                     => ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__ind__right,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1tc_27: $i] :
          ( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
              & ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ! [V5y: $i] :
                      ( ( mem @ V5y @ A_27a )
                     => ( ? [V6z: $i] :
                            ( ( mem @ V6z @ A_27a )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V6z ) )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6z ) @ V5y ) ) @ V0r ) ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
           => ! [V7x: $i] :
                ( ( mem @ V7x @ A_27a )
               => ! [V8y: $i] :
                    ( ( mem @ V8y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                     => ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ertc__ind__right,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1tc_27: $i] :
          ( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) )
                      | ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) )
                   => ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V2x ) ) ) )
              & ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ! [V4y: $i] :
                      ( ( mem @ V4y @ A_27a )
                     => ( ? [V5z: $i] :
                            ( ( mem @ V5z @ A_27a )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V3x ) @ V5z ) )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V5z ) @ V4y ) ) @ V0r ) ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V3x ) @ V4y ) ) ) ) ) )
           => ! [V6x: $i] :
                ( ( mem @ V6x @ A_27a )
               => ! [V7y: $i] :
                    ( ( mem @ V7y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6x ) @ V7y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                     => ( p @ ( ap @ ( ap @ V1tc_27 @ V6x ) @ V7y ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__strongind__right,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1tc_27: $i] :
          ( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ( ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
              & ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ! [V5y: $i] :
                      ( ( mem @ V5y @ A_27a )
                     => ( ? [V6z: $i] :
                            ( ( mem @ V6z @ A_27a )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V6z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                            & ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V6z ) )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6z ) @ V5y ) ) @ V0r ) ) )
                       => ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
           => ! [V7x: $i] :
                ( ( mem @ V7x @ A_27a )
               => ! [V8y: $i] :
                    ( ( mem @ V8y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                     => ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__union,axiom,
    ! [A_27a: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r1 ) ) )
               => ! [V3r2: $i] :
                    ( ( mem @ V3r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r1 ) @ V3r2 ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__implication,axiom,
    ! [A_27a: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ! [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
               => ! [V3y: $i] :
                    ( ( mem @ V3y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r1 ) )
                     => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V1r2 ) ) ) ) )
           => ! [V4x: $i] :
                ( ( mem @ V4x @ A_27a )
               => ! [V5y: $i] :
                    ( ( mem @ V5y @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r1 ) ) )
                     => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1r2 ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__empty,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ A_27a )
         => ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__empty__eqn,axiom,
    ! [A_27a: del] :
      ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) )
      = ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__domain__range,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Errestrict__tc,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
         => ! [V2e: $i] :
              ( ( mem @ V2e @ A_27a )
             => ! [V3e_27: $i] :
                  ( ( mem @ V3e_27 @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2e ) @ V3e_27 ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1x ) ) ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2e ) @ V3e_27 ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__SWAP,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) ) @ V0r ) )
        = ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eacyclic__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
      <=> ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V1x ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eacyclic__subset,axiom,
    ! [A_27a: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r1 ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V1r2 ) @ V0r1 ) ) )
           => ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r2 ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eacyclic__rrestrict,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
           => ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eacyclic__irreflexive,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
           => ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V1x ) ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eacyclic__SWAP,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) ) @ V0r ) ) )
      <=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eacyclic__bigunion,axiom,
    ! [A_27a: del,V0rs: $i] :
      ( ( mem @ V0rs @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) )
     => ( ( ! [V1r: $i] :
              ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V2r_27: $i] :
                  ( ( mem @ V2r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V1r ) @ V0rs ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V2r_27 ) @ V0rs ) )
                      & ( V1r != V2r_27 ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V2r_27 ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V2r_27 ) ) ) ) ) ) )
          & ! [V3r: $i] :
              ( ( mem @ V3r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3r ) @ V0rs ) )
               => ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V3r ) ) ) ) )
       => ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0rs ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eacyclic__union,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1r_27: $i] :
          ( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r_27 ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r_27 ) ) ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r_27 ) ) )
           => ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ V1r_27 ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Ereflexive__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ V1s ) )
          <=> ! [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
                 => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) @ V0r ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eirreflexive__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V0r ) @ V1s ) )
          <=> ! [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
                 => ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) @ V0r ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Etransitive__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
      <=> ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ! [V2y: $i] :
                ( ( mem @ V2y @ A_27a )
               => ! [V3z: $i] :
                    ( ( mem @ V3z @ A_27a )
                   => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
                        & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V3z ) ) @ V0r ) ) )
                     => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) @ V0r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etransitive__tc,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
       => ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r )
          = V0r ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__transitive,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eantisym__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
      <=> ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ! [V2y: $i] :
                ( ( mem @ V2y @ A_27a )
               => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V1x ) ) @ V0r ) ) )
                 => ( V1x = V2y ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Epartial__order__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ V1s ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eantisym__subset,axiom,
    ! [A_27a: del,V0t: $i] :
      ( ( mem @ V0t @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0t ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V1s ) @ V0t ) )
             => ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Epartial__order__dom__rng,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Epartial__order__subset,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s_27 ) @ V1s ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V2s_27 ) ) @ V2s_27 ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__partial__order,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) @ V1s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) @ V1s ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__partial__order__acyclic,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
           => ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Elinear__order__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
              & ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ! [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                     => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
                          & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) )
                       => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                          | ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) @ V0r ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__subset,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s_27 ) @ V1s ) ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V2s_27 ) ) @ V2s_27 ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Epartial__order__linear__order,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
           => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Estrict__linear__order__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ V1s ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
              & ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) @ V0r ) ) )
              & ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ! [V4y: $i] :
                      ( ( mem @ V4y @ A_27a )
                     => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
                          & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V1s ) )
                          & ( V3x != V4y ) )
                       => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V4y ) ) @ V0r ) )
                          | ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4y ) @ V3x ) ) @ V0r ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__linear__order__dom__rng,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ V1s ) ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__dom__rng,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Ereln__to__rel__def,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r )
        = ( lam @ A_27a
          @ ^ [V1x: $i] :
              ( lam @ A_27b
              @ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ V0r ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Erel__to__reln__def,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V0R )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
          @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
            @ ( lam @ A_27a
              @ ^ [V1x: $i] :
                  ( lam @ A_27b
                  @ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2ERRUNIV__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V0s )
        = ( lam @ A_27a
          @ ^ [V1x: $i] :
              ( lam @ A_27a
              @ ^ [V2y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V0s ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2ERREFL__EXP__def,axiom,
    ! [A_27a: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ V0R ) @ V1s )
            = ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27a ) @ V0R )
              @ ( lam @ A_27a
                @ ^ [V2x: $i] :
                    ( lam @ A_27a
                    @ ^ [V3y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2ERREFL__EXP__RSUBSET,axiom,
    ! [A_27a: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ V0R ) @ V1s ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2ERREFL__EXP__UNIV,axiom,
    ! [A_27a: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ( ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ V0R ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
        = V0R ) ) ).

thf(conj_thm_2Eset__relation_2EREL__RESTRICT__UNIV,axiom,
    ! [A_27a: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ( ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V0R ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
        = V0R ) ) ).

thf(conj_thm_2Eset__relation_2Ein__rel__to__reln,axiom,
    ! [A_27a: del,A_27b: del,V0xy: $i] :
      ( ( mem @ V0xy @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
     => ! [V1R: $i] :
          ( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0xy ) @ ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R ) ) )
          <=> ( p @ ( ap @ ( ap @ V1R @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0xy ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0xy ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ereln__to__rel__app,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27b )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r ) @ V1x ) @ V2y ) )
              <=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ V0r ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Erel__to__reln__IS__UNCURRY,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b )
      = ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) ) ).

thf(conj_thm_2Eset__relation_2Ereln__to__rel__IS__CURRY,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b )
      = ( c_2Epair_2ECURRY @ A_27a @ A_27b @ bool ) ) ).

thf(conj_thm_2Eset__relation_2Erel__to__reln__inv,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V0R ) )
        = V0R ) ) ).

thf(conj_thm_2Eset__relation_2Ereln__to__rel__inv,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r ) )
        = V0r ) ) ).

thf(conj_thm_2Eset__relation_2Ereln__to__rel__11,axiom,
    ! [A_27a: del,A_27b: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
         => ( ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r1 )
              = ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V1r2 ) )
          <=> ( V0r1 = V1r2 ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Erel__to__reln__11,axiom,
    ! [A_27a: del,A_27b: del,V0R1: $i] :
      ( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ! [V1R2: $i] :
          ( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
         => ( ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V0R1 )
              = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R2 ) )
          <=> ( V0R1 = V1R2 ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Erel__to__reln__swap,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ! [V1R: $i] :
          ( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
         => ( ( V0r
              = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R ) )
          <=> ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r )
              = V1R ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Edomain__to__rel__conv,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V0r )
        = ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Erange__to__rel__conv,axiom,
    ! [A_27a: del,A_27b: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V0r )
        = ( ap @ ( c_2Erelation_2ERRANGE @ A_27b @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27b @ A_27a ) @ V0r ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r )
        = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Errestrict__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s )
            = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ercomp__to__rel__conv,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c ) @ V0r1 ) @ V1r2 )
            = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27c @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27c @ A_27b ) @ V1r2 ) ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27c ) @ V0r1 ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Euniv__reln__to__rel__conv,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ V0s )
        = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V0s ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etc__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r )
        = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eacyclic__reln__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
      <=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eirreflexive__reln__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V0r ) @ V1s ) )
          <=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eirreflexive__reln__to__rel__conv__UNIV,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
      <=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ereflexive__reln__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ V1s ) )
          <=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ereflexive__reln__to__rel__conv__UNIV,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
      <=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Etransitive__reln__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
      <=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eantisym__reln__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
      <=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Epartial__order__reln__to__rel__conv,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V1s ) ) )
              & ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Epartial__order__reln__to__rel__conv__UNIV,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
      <=> ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__reln__to__rel__conv__UNIV,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
      <=> ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__linear__order__reln__to__rel__conv__UNIV,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
      <=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ereln__rel__conv__thms,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0xy: $i] :
      ( ( mem @ V0xy @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
     => ! [V1R: $i] :
          ( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
         => ! [V2r: $i] :
              ( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ! [V4y: $i] :
                      ( ( mem @ V4y @ A_27b )
                     => ! [V5r1: $i] :
                          ( ( mem @ V5r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
                         => ! [V6r2: $i] :
                              ( ( mem @ V6r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
                             => ! [V7R1: $i] :
                                  ( ( mem @ V7R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
                                 => ! [V8R2: $i] :
                                      ( ( mem @ V8R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
                                     => ! [V9R: $i] :
                                          ( ( mem @ V9R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
                                         => ! [V10r: $i] :
                                              ( ( mem @ V10r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
                                             => ! [V11r: $i] :
                                                  ( ( mem @ V11r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                                                 => ! [V12s: $i] :
                                                      ( ( mem @ V12s @ ( arr @ A_27a @ bool ) )
                                                     => ! [V13r1: $i] :
                                                          ( ( mem @ V13r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) )
                                                         => ! [V14r2: $i] :
                                                              ( ( mem @ V14r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ bool ) )
                                                             => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0xy ) @ ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R ) ) )
                                                                <=> ( p @ ( ap @ ( ap @ V1R @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0xy ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0xy ) ) ) )
                                                                & ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V2r ) @ V3x ) @ V4y ) )
                                                                <=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4y ) ) @ V2r ) ) )
                                                                & ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R ) )
                                                                  = V1R )
                                                                & ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V2r ) )
                                                                  = V2r )
                                                                & ( ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V5r1 )
                                                                    = ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V6r2 ) )
                                                                <=> ( V5r1 = V6r2 ) )
                                                                & ( ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V7R1 )
                                                                    = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V8R2 ) )
                                                                <=> ( V7R1 = V8R2 ) )
                                                                & ( ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ V9R ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
                                                                  = V9R )
                                                                & ( ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V9R ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
                                                                  = V9R )
                                                                & ( ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V2r )
                                                                  = ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V2r ) ) )
                                                                & ( ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V10r )
                                                                  = ( ap @ ( c_2Erelation_2ERRANGE @ A_27b @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27b @ A_27a ) @ V10r ) ) )
                                                                & ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V11r )
                                                                  = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
                                                                & ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V11r ) @ V12s )
                                                                  = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) @ V12s ) ) )
                                                                & ( ( ap @ ( ap @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c ) @ V13r1 ) @ V14r2 )
                                                                  = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27c @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27c @ A_27b ) @ V14r2 ) ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27c ) @ V13r1 ) ) ) )
                                                                & ( ( ap @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ V12s )
                                                                  = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V12s ) ) )
                                                                & ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V11r )
                                                                  = ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
                                                                & ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V11r ) )
                                                                <=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) ) )
                                                                & ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V11r ) @ V12s ) )
                                                                <=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) @ V12s ) ) ) )
                                                                & ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V11r ) @ V12s ) )
                                                                <=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) @ V12s ) ) ) )
                                                                & ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V11r ) )
                                                                <=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
                                                                & ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V11r ) )
                                                                <=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
                                                                & ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V11r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
                                                                <=> ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
                                                                & ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V11r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
                                                                <=> ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) )
                                                                & ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V11r ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
                                                                <=> ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V11r ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eacyclic__WF,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
              & ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) ) )
           => ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V1r ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Emaximal__elements__def,axiom,
    ! [A_27a: del,V0xs: $i] :
      ( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0xs ) @ V1r )
            = ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
              @ ( lam @ A_27a
                @ ^ [V2x: $i] :
                    ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x )
                    @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0xs ) )
                      @ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V3x_27: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x_27 ) @ V0xs ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3x_27 ) ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V3x_27 ) ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eminimal__elements__def,axiom,
    ! [A_27a: del,V0xs: $i] :
      ( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0xs ) @ V1r )
            = ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
              @ ( lam @ A_27a
                @ ^ [V2x: $i] :
                    ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x )
                    @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0xs ) )
                      @ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V3x_27: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x_27 ) @ V0xs ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x_27 ) @ V2x ) ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V3x_27 ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eminimal__elements__subset,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1lo: $i] :
          ( ( mem @ V1lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1lo ) ) @ V0s ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eminimal__elements__SWAP,axiom,
    ! [A_27a: del,V0xs: $i] :
      ( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0xs ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) ) @ V1r ) )
            = ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0xs ) @ V1r ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Emaximal__union,axiom,
    ! [A_27a: del,V0e: $i] :
      ( ( mem @ V0e @ A_27a )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2r1: $i] :
              ( ( mem @ V2r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V3r2: $i] :
                  ( ( mem @ V3r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V2r1 ) @ V3r2 ) ) ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V1s ) @ V2r1 ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V1s ) @ V3r2 ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eminimal__union,axiom,
    ! [A_27a: del,V0e: $i] :
      ( ( mem @ V0e @ A_27a )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2r1: $i] :
              ( ( mem @ V2r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V3r2: $i] :
                  ( ( mem @ V3r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V2r1 ) @ V3r2 ) ) ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1s ) @ V2r1 ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1s ) @ V3r2 ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eminimal__elements__mono,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1r_27: $i] :
          ( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2xs: $i] :
              ( ( mem @ V2xs @ ( arr @ A_27a @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ V1r_27 ) )
               => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2xs ) @ V1r_27 ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2xs ) @ V0r ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eminimal__elements__rrestrict,axiom,
    ! [A_27a: del,V0xs: $i] :
      ( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0xs ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r ) @ V0xs ) )
            = ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0xs ) @ V1r ) ) ) ) ).

thf(conj_thm_2Eset__relation_2EWF__has__minimal__path,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2s: $i] :
              ( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
             => ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a ) @ V0r ) ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2s ) )
                 => ? [V3y: $i] :
                      ( ( mem @ V3y @ A_27a )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2s ) @ V0r ) ) )
                      & ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V1x ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
                        | ( V3y = V1x ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__acyclic__has__maximal,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
       => ( ( V0s
           != ( c_2Epred__set_2EEMPTY @ A_27a ) )
         => ! [V1r: $i] :
              ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
               => ? [V2x: $i] :
                    ( ( mem @ V2x @ A_27a )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__acyclic__has__minimal,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
       => ( ( V0s
           != ( c_2Epred__set_2EEMPTY @ A_27a ) )
         => ! [V1r: $i] :
              ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
               => ? [V2x: $i] :
                    ( ( mem @ V2x @ A_27a )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Emaximal__TC,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) ) )
           => ( ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1r ) )
              = ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eminimal__TC,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r ) ) @ V0s ) ) )
           => ( ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1r ) )
              = ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__acyclic__has__minimal__path,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
                  & ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
                  & ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) )
               => ? [V3y: $i] :
                    ( ( mem @ V3y @ A_27a )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__acyclic__has__maximal__path,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
                  & ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
                  & ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) )
               => ? [V3y: $i] :
                    ( ( mem @ V3y @ A_27a )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefix__po__has__minimal__path,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3s_27: $i] :
                  ( ( mem @ V3s_27 @ ( arr @ A_27a @ bool ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V0r ) @ V1s ) )
                      & ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V3s_27 ) @ V0r ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V3s_27 ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3s_27 ) @ V1s ) ) )
                   => ? [V4x_27: $i] :
                        ( ( mem @ V4x_27 @ A_27a )
                        & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x_27 ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V3s_27 ) @ V0r ) ) )
                        & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x_27 ) @ V2x ) ) @ V0r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eempty__strict__linear__order,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
      <=> ( V0r
          = ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eempty__linear__order,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
      <=> ( V0r
          = ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__restrict,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r ) @ V2s_27 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V2s_27 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__linear__order__restrict,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V1r ) @ V0s ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r ) @ V2s_27 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V2s_27 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__dom__rg,axiom,
    ! [A_27a: del,V0lo: $i] :
      ( ( mem @ V0lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1X: $i] :
          ( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0lo ) @ V1X ) )
           => ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0lo ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0lo ) )
              = V1X ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__refl,axiom,
    ! [A_27a: del,V0lo: $i] :
      ( ( mem @ V0lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1X: $i] :
          ( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0lo ) @ V1X ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1X ) )
                 => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) @ V0lo ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__in__set,axiom,
    ! [A_27a: del,V0lo: $i] :
      ( ( mem @ V0lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1X: $i] :
          ( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0lo ) @ V1X ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0lo ) )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1X ) )
                        & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1X ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2EIN__MIN__LO,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1X: $i] :
          ( ( mem @ V1X @ ( arr @ A_27a @ bool ) )
         => ! [V2lo: $i] :
              ( ( mem @ V2lo @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1X ) )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V2lo ) @ V1X ) )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1X ) @ V2lo ) ) )
                       => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V0x ) ) @ V2lo ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eextend__linear__order,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) ) )
               => ( p
                  @ ( ap
                    @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a )
                      @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r )
                        @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a )
                          @ ( lam @ A_27a
                            @ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) )
                    @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__linear__order__acyclic,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ V1s ) )
           => ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__linear__order__union__acyclic,axiom,
    ! [A_27a: del,V0r1: $i] :
      ( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1r2: $i] :
          ( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2s: $i] :
              ( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r1 ) @ V2s ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r2 ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r2 ) ) ) @ V2s ) ) )
               => ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r1 ) @ V1r2 ) ) )
                <=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V1r2 ) @ V0r1 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Estrict__linear__order,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
           => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ V1s ) )
           => ( p
              @ ( ap
                @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a )
                  @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r )
                    @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a )
                      @ ( lam @ A_27a
                        @ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) ) ) ) )
                @ V1s ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__strict__linear__order__has__maximal,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V1r ) @ V0s ) )
              & ( V0s
               != ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__strict__linear__order__has__minimal,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V1r ) @ V0s ) )
              & ( V0s
               != ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__linear__order__has__maximal,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
              & ( V0s
               != ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__linear__order__has__minimal,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
              & ( V0s
               != ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Emaximal__linear__order,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) @ V1r ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eminimal__linear__order,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ! [V3y: $i] :
                  ( ( mem @ V3y @ A_27a )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V0s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V0s ) @ V1r ) ) ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V1r ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eminimal__linear__order__unique,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ! [V4y: $i] :
                      ( ( mem @ V4y @ A_27a )
                     => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
                          & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2s_27 ) @ V0r ) ) )
                          & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V2s_27 ) @ V0r ) ) )
                          & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s_27 ) @ V1s ) ) )
                       => ( V3x = V4y ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Efinite__prefix__linear__order__has__unique__minimal,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ! [V2s: $i] :
              ( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
             => ! [V3s_27: $i] :
                  ( ( mem @ V3s_27 @ ( arr @ A_27a @ bool ) )
                 => ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V1r ) @ V2s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V1r ) @ V2s ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V3s_27 ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3s_27 ) @ V2s ) ) )
                   => ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V3s_27 ) @ V1r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Enat__order__iso__thm,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
                  ( ( ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) )
                      = ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
                    & ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) )
                     != ( c_2Eoption_2ENONE @ A_27a ) ) )
                 => ( V3m = V2n ) )
              & ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) )
                   => ? [V5m: tp__ty_2Enum_2Enum] :
                        ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5m ) )
                        = ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V4x ) ) ) )
              & ! [V6m: tp__ty_2Enum_2Enum,V7x: $i] :
                  ( ( mem @ V7x @ A_27a )
                 => ( ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V6m ) )
                      = ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V7x ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V1s ) ) ) ) )
           => ( ( p
                @ ( ap
                  @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a )
                    @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
                      @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                        @ ( lam @ A_27a
                          @ ^ [V8x: $i] :
                              ( lam @ A_27a
                              @ ^ [V9y: $i] :
                                  ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V8x ) @ V9y ) )
                                  @ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
                                    @ ( lam @ ty_2Enum_2Enum
                                      @ ^ [V10m: $i] :
                                          ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
                                          @ ( lam @ ty_2Enum_2Enum
                                            @ ^ [V11n: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V10m ) @ V11n ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0f @ V10m ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V8x ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0f @ V11n ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) )
                  @ V1s ) )
              & ( p
                @ ( ap
                  @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a )
                    @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
                      @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                        @ ( lam @ A_27a
                          @ ^ [V12x: $i] :
                              ( lam @ A_27a
                              @ ^ [V13y: $i] :
                                  ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V12x ) @ V13y ) )
                                  @ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
                                    @ ( lam @ ty_2Enum_2Enum
                                      @ ^ [V14m: $i] :
                                          ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
                                          @ ( lam @ ty_2Enum_2Enum
                                            @ ^ [V15n: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V14m ) @ V15n ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0f @ V14m ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V12x ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0f @ V15n ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V13y ) ) ) ) ) ) ) ) ) ) ) ) ) )
                  @ V1s ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Echain__def,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Echain @ A_27a ) @ V0s ) @ V1r ) )
          <=> ! [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
               => ! [V3y: $i] :
                    ( ( mem @ V3y @ A_27a )
                   => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
                        & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0s ) ) )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V1r ) )
                        | ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) @ V1r ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eupper__bounds__def,axiom,
    ! [A_27a: del,A_27b: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27b @ bool ) )
     => ! [V1r: $i] :
          ( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27b ) @ V0s ) @ V1r )
            = ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
              @ ( lam @ A_27a
                @ ^ [V2x: $i] :
                    ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x )
                    @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V1r ) ) )
                      @ ( ap @ ( c_2Ebool_2E_21 @ A_27b )
                        @ ( lam @ A_27b
                          @ ^ [V3y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V3y ) @ V2x ) ) @ V1r ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eupper__bounds__lem,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2x1: $i] :
              ( ( mem @ V2x1 @ A_27a )
             => ! [V3x2: $i] :
                  ( ( mem @ V3x2 @ A_27a )
                 => ( ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x1 ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V1s ) @ V0r ) ) )
                      & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x1 ) @ V3x2 ) ) @ V0r ) ) )
                   => ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x2 ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V1s ) @ V0r ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Efchains__def,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Efchains @ A_27a ) @ V0r )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
          @ ( lam @ ( arr @ A_27a @ bool )
            @ ^ [V1k: $i] :
                ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1k )
                @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Eset__relation_2Echain @ A_27a ) @ V1k ) @ V0r ) )
                  @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V1k ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
                    @ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ bool ) )
                      @ ( lam @ ( arr @ A_27a @ bool )
                        @ ^ [V2C: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Eset__relation_2Echain @ A_27a ) @ V2C ) @ V0r ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2C ) @ V1k ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V2C ) @ V0r ) ) @ V2C ) ) @ V1k ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V2C ) @ V0r ) ) @ V2C ) ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V2C ) @ V0r ) ) @ V2C ) ) @ V1k ) ) @ V0r ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ezorns__lemma,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ( V1s
               != ( c_2Epred__set_2EEMPTY @ A_27a ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
              & ! [V2t: $i] :
                  ( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Echain @ A_27a ) @ V2t ) @ V0r ) )
                   => ( ( ap @ ( ap @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a ) @ V2t ) @ V0r )
                     != ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) )
           => ? [V3x: $i] :
                ( ( mem @ V3x @ A_27a )
                & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ V1s ) @ V0r ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eper__def,axiom,
    ! [A_27a: del,V0xs: $i] :
      ( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
     => ! [V1xss: $i] :
          ( ( mem @ V1xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V0xs ) @ V1xss ) )
          <=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1xss ) ) @ V0xs ) )
              & ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1xss ) )
              & ! [V2xs1: $i] :
                  ( ( mem @ V2xs1 @ ( arr @ A_27a @ bool ) )
                 => ! [V3xs2: $i] :
                      ( ( mem @ V3xs2 @ ( arr @ A_27a @ bool ) )
                     => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2xs1 ) @ V1xss ) )
                          & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3xs2 ) @ V1xss ) )
                          & ( V2xs1 != V3xs2 ) )
                       => ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V2xs1 ) @ V3xs2 ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eper__restrict__def,axiom,
    ! [A_27a: del,V0xss: $i] :
      ( ( mem @ V0xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
     => ! [V1xs: $i] :
          ( ( mem @ V1xs @ ( arr @ A_27a @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Eper__restrict @ A_27a ) @ V0xss ) @ V1xs )
            = ( ap
              @ ( ap @ ( c_2Epred__set_2EDELETE @ ( arr @ A_27a @ bool ) )
                @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
                  @ ( lam @ ( arr @ A_27a @ bool )
                    @ ^ [V2xs_27: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2xs_27 ) @ V1xs ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2xs_27 ) @ V0xss ) ) ) ) )
              @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eper__delete,axiom,
    ! [A_27a: del,V0xs: $i] :
      ( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
     => ! [V1xss: $i] :
          ( ( mem @ V1xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
         => ! [V2e: $i] :
              ( ( mem @ V2e @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V0xs ) @ V1xss ) )
               => ( p
                  @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0xs ) @ V2e ) )
                    @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
                      @ ( lam @ ( arr @ A_27a @ bool )
                        @ ^ [V3es: $i] :
                            ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V3es )
                            @ ( ap
                              @ ( ap @ c_2Ebool_2E_2F_5C
                                @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3es )
                                  @ ( ap
                                    @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
                                      @ ( lam @ ( arr @ A_27a @ bool )
                                        @ ^ [V4es: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V4es ) @ V2e ) ) )
                                    @ V1xss ) ) )
                              @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V3es ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eper__restrict__per,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V1s ) @ V0r ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V2s_27 ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper__restrict @ A_27a ) @ V0r ) @ V2s_27 ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Ecountable__per,axiom,
    ! [A_27a: del,V0xs: $i] :
      ( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
     => ! [V1xss: $i] :
          ( ( mem @ V1xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0xs ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eper @ A_27a ) @ V0xs ) @ V1xss ) ) )
           => ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V1xss ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eall__choices__def,axiom,
    ! [A_27a: del,V0xss: $i] :
      ( ( mem @ V0xss @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
     => ( ( ap @ ( c_2Eset__relation_2Eall__choices @ A_27a ) @ V0xss )
        = ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ A_27a ) )
          @ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ A_27a )
            @ ^ [V1choice: $i] :
                ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ A_27a ) @ V1choice ) @ V0xss ) )
                @ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ bool ) )
                  @ ( lam @ ( arr @ A_27a @ bool )
                    @ ^ [V2xs: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2xs ) @ V0xss ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V1choice @ V2xs ) ) @ V2xs ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Eall__choices__thm,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V0x ) @ ( ap @ ( c_2Eset__relation_2Eall__choices @ A_27a ) @ V1s ) ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V0x ) ) )
               => ? [V3z: $i] :
                    ( ( mem @ V3z @ ( arr @ A_27a @ bool ) )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3z ) @ V1s ) )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V3z ) ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Enum__order__def,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ap @ ( ap @ ( c_2Eset__relation_2Enum__order @ A_27a ) @ V0f ) @ V1s )
            = ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
              @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                @ ( lam @ A_27a
                  @ ^ [V2x: $i] :
                      ( lam @ A_27a
                      @ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V2x ) ) @ ( ap @ V0f @ V3y ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__num__order,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2t: $i] :
              ( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1s ) @ V2t ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Enum__order @ A_27a ) @ V0f ) @ V1s ) ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Enum__order__finite__prefix,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2t: $i] :
              ( ( mem @ V2t @ ( arr @ ty_2Enum_2Enum @ bool ) )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1s ) @ V2t ) )
               => ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Enum__order @ A_27a ) @ V0f ) @ V1s ) ) @ V1s ) ) ) ) ) ) ).

thf(ax_thm_2Eset__relation_2Eget__min__def,axiom,
    ! [A_27a: del,V0r_27: $i] :
      ( ( mem @ V0r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2r: $i] :
              ( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ( ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V0r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V1s ) @ V2r ) )
                = ( ap
                  @ ( ap @ ( c_2Ebool_2ELET @ ( arr @ A_27a @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
                    @ ( lam @ ( arr @ A_27a @ bool )
                      @ ^ [V3mins: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V3mins ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V3mins ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) )
                  @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ V1s ) @ V2r ) ) @ V0r_27 ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Enth__min__ind,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
     => ( ( ! [V1r_27: $i] :
              ( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V2s: $i] :
                  ( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
                 => ! [V3r: $i] :
                      ( ( mem @ V3r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                     => ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V2s ) @ V3r ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
          & ! [V4r_27: $i] :
              ( ( mem @ V4r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V5s: $i] :
                  ( ( mem @ V5s @ ( arr @ A_27a @ bool ) )
                 => ! [V6r: $i] :
                      ( ( mem @ V6r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                     => ! [V7n: tp__ty_2Enum_2Enum] :
                          ( ! [V8min: $i] :
                              ( ( mem @ V8min @ ( ty_2Eoption_2Eoption @ A_27a ) )
                             => ( ( ( V8min
                                    = ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V5s ) @ V6r ) ) )
                                  & ( V8min
                                   != ( c_2Eoption_2ENONE @ A_27a ) ) )
                               => ( p @ ( ap @ ( ap @ ( ap @ V0P @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V5s ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ V8min ) ) ) @ V6r ) ) @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) )
                         => ( p @ ( ap @ ( ap @ ( ap @ V0P @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V5s ) @ V6r ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) ) ) ) ) ) )
       => ! [V9v: $i] :
            ( ( mem @ V9v @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
           => ! [V10v1: $i] :
                ( ( mem @ V10v1 @ ( arr @ A_27a @ bool ) )
               => ! [V11v2: $i] :
                    ( ( mem @ V11v2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                   => ! [V12v3: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ V0P @ V9v ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V10v1 ) @ V11v2 ) ) @ ( inj__ty_2Enum_2Enum @ V12v3 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Enth__min__def,axiom,
    ! [A_27a: del] :
      ( ! [V0s: $i] :
          ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
         => ! [V1r_27: $i] :
              ( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V2r: $i] :
                  ( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                 => ( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V0s ) @ V2r ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
                    = ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V0s ) @ V2r ) ) ) ) ) )
      & ! [V3s: $i] :
          ( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
         => ! [V4r_27: $i] :
              ( ( mem @ V4r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V5r: $i] :
                  ( ( mem @ V5r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                 => ! [V6n: tp__ty_2Enum_2Enum] :
                      ( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3s ) @ V5r ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
                      = ( ap
                        @ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
                          @ ( lam @ ( ty_2Eoption_2Eoption @ A_27a )
                            @ ^ [V7min: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V7min ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V3s ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ V7min ) ) ) @ V5r ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) )
                        @ ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3s ) @ V5r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Enth__min__def__compute,axiom,
    ! [A_27a: del] :
      ( ! [V0s: $i] :
          ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
         => ! [V1r_27: $i] :
              ( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V2r: $i] :
                  ( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                 => ( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V0s ) @ V2r ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
                    = ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V1r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V0s ) @ V2r ) ) ) ) ) )
      & ! [V3s: $i] :
          ( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
         => ! [V4r_27: $i] :
              ( ( mem @ V4r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V5r: $i] :
                  ( ( mem @ V5r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                 => ! [V6n: tp__ty_2Enum_2Enum] :
                      ( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3s ) @ V5r ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
                      = ( ap
                        @ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
                          @ ( lam @ ( ty_2Eoption_2Eoption @ A_27a )
                            @ ^ [V7min: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V7min ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V3s ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ V7min ) ) ) @ V5r ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
                        @ ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V4r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3s ) @ V5r ) ) ) ) ) ) )
      & ! [V8s: $i] :
          ( ( mem @ V8s @ ( arr @ A_27a @ bool ) )
         => ! [V9r_27: $i] :
              ( ( mem @ V9r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
             => ! [V10r: $i] :
                  ( ( mem @ V10r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                 => ! [V11n: tp__ty_2Enum_2Enum] :
                      ( ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V9r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V8s ) @ V10r ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V11n ) ) ) )
                      = ( ap
                        @ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
                          @ ( lam @ ( ty_2Eoption_2Eoption @ A_27a )
                            @ ^ [V12min: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V12min ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ V9r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V8s ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ V12min ) ) ) @ V10r ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V11n ) ) ) ) ) ) )
                        @ ( ap @ ( ap @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ V9r_27 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V8s ) @ V10r ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eset__relation_2Elinear__order__of__countable__po,axiom,
    ! [A_27a: del,V0r: $i] :
      ( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
              & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V0r ) @ V1s ) ) )
           => ? [V2r_27: $i] :
                ( ( mem @ V2r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
                & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V2r_27 ) @ V1s ) )
                & ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V2r_27 ) @ V1s ) )
                & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ V2r_27 ) ) ) ) ) ) ).

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