%------------------------------------------------------------------------------
% File     : SCT116+1 : TPTP v8.2.0. Released v5.2.0.
% Domain   : Social Choice Theory
% Problem  : Arrow's Impossibility Theorem 432728, 500 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : arrow_432728.500.p [Bla11]

% Status   : Theorem
% Rating   : 1.00 v7.0.0, 0.97 v6.4.0, 1.00 v5.2.0
% Syntax   : Number of formulae    :  542 ( 165 unt;   0 def)
%            Number of atoms       : 1235 ( 371 equ)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives :  797 ( 104   ~;  27   |;  50   &)
%                                         ( 102 <=>; 514  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   6 avg)
%            Maximal term depth    :   25 (   2 avg)
%            Number of predicates  :   36 (  35 usr;   0 prp; 1-7 aty)
%            Number of functors    :   57 (  57 usr;  13 con; 0-5 aty)
%            Number of variables   : 2036 (2022   !;  14   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-03-01 11:09:54
%          : Renamed from SWW118+1 
%------------------------------------------------------------------------------
%----Relevant facts (493)
fof(fact_ext,axiom,
    ! [V_g_2,V_f_2] :
      ( ! [B_x] : hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)
     => V_f_2 = V_g_2 ) ).

fof(fact_above__def,axiom,
    ! [V_b_2,V_a_2,V_La_2] : c_Arrow__Order__Mirabelle_Oabove(V_La_2,V_a_2,V_b_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_Oprod_Oprod__case(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,hAPP(hAPP(c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),V_La_2)))),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2))))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2))))))),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_a_2),V_b_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))))),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_HOL_Obool,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_HOL_OEx(tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),c_fequal)),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),V_b_2))))),hAPP(hAPP(c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt))),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),V_a_2))),V_La_2))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2)))))))),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_HOL_Obool,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_HOL_OEx(tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),c_fequal)),hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_b_2))))),hAPP(hAPP(c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt))),hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_a_2))),V_La_2))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2))))))) ).

fof(fact_below__def,axiom,
    ! [V_b_2,V_a_2,V_La_2] : c_Arrow__Order__Mirabelle_Obelow(V_La_2,V_a_2,V_b_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_Oprod_Oprod__case(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,hAPP(hAPP(c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),V_La_2)))),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_a_2))))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_a_2))))))),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_a_2),V_b_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))))),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_HOL_Obool,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_HOL_OEx(tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),c_fequal)),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),V_a_2))))),hAPP(hAPP(c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt))),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),V_b_2))),V_La_2))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_a_2)))))))),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_HOL_Obool,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_HOL_OEx(tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),c_fequal)),hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_a_2))))),hAPP(hAPP(c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt))),hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_b_2))),V_La_2))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_a_2))))))) ).

fof(fact_mkbot__def,axiom,
    ! [V_b_2,V_La_2] : c_Arrow__Order__Mirabelle_Omkbot(V_La_2,V_b_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_Oprod_Oprod__case(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,hAPP(hAPP(c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),V_La_2)))),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2))))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2))))))),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_HOL_Obool,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_HOL_OEx(tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool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).

fof(fact_mktop__def,axiom,
    ! [V_b_2,V_La_2] : c_Arrow__Order__Mirabelle_Omktop(V_La_2,V_b_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_Oprod_Oprod__case(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,hAPP(hAPP(c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),V_La_2)))),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2))))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2))))))),c_Set_OCollect(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_HOL_Obool,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_HOL_OEx(tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fconj)),hAPP(hAPP(c_COMBC(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt)),c_fequal)),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),V_b_2))))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_fNot),hAPP(hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),c_fequal),V_b_2)))))) ).

fof(fact_CollectI,axiom,
    ! [T_a,V_a_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,V_a_2))
     => hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Set_OCollect(T_a,V_P_2))) ) ).

fof(fact_splitI,axiom,
    ! [T_b,T_a,V_b_2,V_a_2,V_f_2] :
      ( hBOOL(hAPP(hAPP(V_f_2,V_a_2),V_b_2))
     => hBOOL(hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_HOL_Obool,V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2))) ) ).

fof(fact_prod__caseI,axiom,
    ! [T_b,T_a,V_b_2,V_a_2,V_f1_2] :
      ( hBOOL(hAPP(hAPP(V_f1_2,V_a_2),V_b_2))
     => hBOOL(hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_HOL_Obool,V_f1_2),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2))) ) ).

fof(fact_UnE,axiom,
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
     => ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2)) ) ) ).

fof(fact_UnCI,axiom,
    ! [V_A_2,V_B_2,V_c_2,T_a] :
      ( ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2)) )
     => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2))) ) ).

fof(fact_UNIV__I,axiom,
    ! [V_xa_2,T_a] : hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) ).

fof(fact_insertE,axiom,
    ! [V_A_2,V_b_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),V_A_2)))
     => ( V_a_2 != V_b_2
       => hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2)) ) ) ).

fof(fact_insertCI,axiom,
    ! [V_b_2,V_B_2,V_a_2,T_a] :
      ( ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_B_2))
       => V_a_2 = V_b_2 )
     => hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),V_B_2))) ) ).

fof(fact_equalityCE,axiom,
    ! [V_c_2,T_a,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => ( ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
         => ~ hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2)) )
       => ~ ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
           => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2)) ) ) ) ).

fof(fact_emptyE,axiom,
    ! [V_a_2,T_a] : ~ hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) ).

fof(fact_eq__mem__trans,axiom,
    ! [V_A_2,T_a,V_b_2,V_a_2] :
      ( V_a_2 = V_b_2
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2)) ) ) ).

fof(fact_eqelem__imp__iff,axiom,
    ! [V_A_2,T_a,V_ya_2,V_xa_2] :
      ( V_xa_2 = V_ya_2
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
      <=> hBOOL(hAPP(hAPP(c_member(T_a),V_ya_2),V_A_2)) ) ) ).

fof(fact_eqset__imp__iff,axiom,
    ! [V_xa_2,T_a,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
      <=> hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_B_2)) ) ) ).

fof(fact_mem__def,axiom,
    ! [V_A_2,V_xa_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
    <=> hBOOL(hAPP(V_A_2,V_xa_2)) ) ).

fof(fact_Pair__inject,axiom,
    ! [V_b_H,V_a_H,V_b,V_a,T_b,T_a] :
      ( hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a),V_b) = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_H),V_b_H)
     => ~ ( V_a = V_a_H
         => V_b != V_b_H ) ) ).

fof(fact_Pair__eq,axiom,
    ! [V_b_H_2,V_a_H_2,V_b_2,V_a_2,T_b,T_a] :
      ( hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2) = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_H_2),V_b_H_2)
    <=> ( V_a_2 = V_a_H_2
        & V_b_2 = V_b_H_2 ) ) ).

fof(fact_split__paired__All,axiom,
    ! [T_b,T_a,V_P_2] :
      ( ! [B_x1] : hBOOL(hAPP(V_P_2,B_x1))
    <=> ! [B_a,B_b] : hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_a),B_b))) ) ).

fof(fact_insert__code,axiom,
    ! [V_xa_2,V_A_2,V_ya_2,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Set_Oinsert(T_a),V_ya_2),V_A_2),V_xa_2))
    <=> ( V_ya_2 = V_xa_2
        | hBOOL(hAPP(V_A_2,V_xa_2)) ) ) ).

fof(fact_insert__commute,axiom,
    ! [V_A_2,V_ya_2,V_xa_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_ya_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_ya_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),V_A_2)) ).

fof(fact_insert__absorb2,axiom,
    ! [V_A_2,V_xa_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),V_A_2) ).

fof(fact_Collect__def,axiom,
    ! [V_P_2,T_a] : c_Set_OCollect(T_a,V_P_2) = V_P_2 ).

fof(fact_ball__Un,axiom,
    ! [V_P_2,V_B_2,V_A_2,T_a] :
      ( ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
         => hBOOL(hAPP(V_P_2,B_x)) )
    <=> ( ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
           => hBOOL(hAPP(V_P_2,B_x)) )
        & ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_B_2))
           => hBOOL(hAPP(V_P_2,B_x)) ) ) ) ).

fof(fact_bex__Un,axiom,
    ! [V_P_2,V_B_2,V_A_2,T_a] :
      ( ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
          & hBOOL(hAPP(V_P_2,B_x)) )
    <=> ( ? [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
            & hBOOL(hAPP(V_P_2,B_x)) )
        | ? [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_B_2))
            & hBOOL(hAPP(V_P_2,B_x)) ) ) ) ).

fof(fact_Un__assoc,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) ).

fof(fact_Un__left__commute,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_C_2)) ).

fof(fact_Un__left__absorb,axiom,
    ! [V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) ).

fof(fact_Un__commute,axiom,
    ! [V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_A_2) ).

fof(fact_Un__absorb,axiom,
    ! [V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2 ).

fof(fact_split__weak__cong,axiom,
    ! [V_c_2,T_c,T_b,T_a,V_q_2,V_p_2] :
      ( V_p_2 = V_q_2
     => hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_b,T_c,V_c_2),V_p_2) = hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_b,T_c,V_c_2),V_q_2) ) ).

fof(fact_split__twice,axiom,
    ! [V_p_2,V_g_2,T_e,T_d,V_f_2,T_a,T_c,T_b] : hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_a,V_f_2),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,tc_prod(T_b,T_c),V_g_2),V_p_2)) = hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_a,hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_b,T_c)),tc_fun(T_e,T_a),T_d),hAPP(c_COMBB(tc_prod(T_b,T_c),T_a,T_e),c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_a,V_f_2))),V_g_2)),V_p_2) ).

fof(fact_equals0D,axiom,
    ! [V_a_2,T_a,V_A_2] :
      ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
     => ~ hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2)) ) ).

fof(fact_empty__iff,axiom,
    ! [V_c_2,T_a] : ~ hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) ).

fof(fact_ex__in__conv,axiom,
    ! [V_A_2,T_a] :
      ( ? [B_x] : hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
    <=> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ).

fof(fact_all__not__in__conv,axiom,
    ! [V_A_2,T_a] :
      ( ! [B_x] : ~ hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ).

fof(fact_insert__absorb,axiom,
    ! [V_A_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
     => hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_A_2) = V_A_2 ) ).

fof(fact_insertI2,axiom,
    ! [V_b_2,V_B_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),V_B_2))) ) ).

fof(fact_insert__ident,axiom,
    ! [V_B_2,V_A_2,V_xa_2,T_a] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
     => ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_B_2))
       => ( hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),V_B_2)
        <=> V_A_2 = V_B_2 ) ) ) ).

fof(fact_insert__iff,axiom,
    ! [V_A_2,V_b_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),V_A_2)))
    <=> ( V_a_2 = V_b_2
        | hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2)) ) ) ).

fof(fact_insertI1,axiom,
    ! [V_B_2,V_a_2,T_a] : hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2))) ).

fof(fact_UnI2,axiom,
    ! [V_A_2,V_B_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2))) ) ).

fof(fact_UnI1,axiom,
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
     => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2))) ) ).

fof(fact_Un__iff,axiom,
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
    <=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
        | hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2)) ) ) ).

fof(fact_singleton__inject,axiom,
    ! [V_b_2,V_a_2,T_a] :
      ( hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))
     => V_a_2 = V_b_2 ) ).

fof(fact_doubleton__eq__iff,axiom,
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
      ( hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = hAPP(hAPP(c_Set_Oinsert(T_a),V_c_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_d_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
    <=> ( ( V_a_2 = V_c_2
          & V_b_2 = V_d_2 )
        | ( V_a_2 = V_d_2
          & V_b_2 = V_c_2 ) ) ) ).

fof(fact_insert__not__empty,axiom,
    ! [V_A_2,V_a_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_empty__not__insert,axiom,
    ! [V_A_2,V_a_2,T_a] : c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) != hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_A_2) ).

fof(fact_Collect__empty__eq,axiom,
    ! [V_P_2,T_a] :
      ( c_Set_OCollect(T_a,V_P_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> ! [B_x] : ~ hBOOL(hAPP(V_P_2,B_x)) ) ).

fof(fact_empty__Collect__eq,axiom,
    ! [V_P_2,T_a] :
      ( c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Set_OCollect(T_a,V_P_2)
    <=> ! [B_x] : ~ hBOOL(hAPP(V_P_2,B_x)) ) ).

fof(fact_UNIV__not__empty,axiom,
    ! [T_a] : c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_ball__empty,axiom,
    ! [V_P_2,T_a,B_x] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
     => hBOOL(hAPP(V_P_2,B_x)) ) ).

fof(fact_Un__empty,axiom,
    ! [V_B_2,V_A_2,T_a] :
      ( hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
        & V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ).

fof(fact_Un__empty__right,axiom,
    ! [V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = V_A_2 ).

fof(fact_Un__empty__left,axiom,
    ! [V_B_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_B_2) = V_B_2 ).

fof(fact_Un__insert__left,axiom,
    ! [V_C_2,V_B_2,V_a_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) ).

fof(fact_Un__insert__right,axiom,
    ! [V_B_2,V_a_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) ).

fof(fact_Un__UNIV__right,axiom,
    ! [V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_Un__UNIV__left,axiom,
    ! [V_B_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))),V_B_2) = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_prod_Osimps_I2_J,axiom,
    ! [V_b_2,V_a_2,V_f1_2,T_a,T_c,T_b] : hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_a,V_f1_2),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2) ).

fof(fact_split__conv,axiom,
    ! [V_b_2,V_a_2,V_f_2,T_a,T_c,T_b] : hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_a,V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)) = hAPP(hAPP(V_f_2,V_a_2),V_b_2) ).

fof(fact_splitD,axiom,
    ! [V_b_2,V_a_2,V_f_2,T_b,T_a] :
      ( hBOOL(hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_HOL_Obool,V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2)))
     => hBOOL(hAPP(hAPP(V_f_2,V_a_2),V_b_2)) ) ).

fof(fact_in__mktop,axiom,
    ! [V_z_2,V_La_2,V_ya_2,V_xa_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),c_Arrow__Order__Mirabelle_Omktop(V_La_2,V_z_2)))
    <=> ( V_xa_2 != V_z_2
        & ( V_ya_2 = V_z_2
         => V_xa_2 != V_ya_2 )
        & ( V_ya_2 != V_z_2
         => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),V_La_2)) ) ) ) ).

fof(fact_in__mkbot,axiom,
    ! [V_z_2,V_La_2,V_ya_2,V_xa_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),c_Arrow__Order__Mirabelle_Omkbot(V_La_2,V_z_2)))
    <=> ( V_ya_2 != V_z_2
        & ( V_xa_2 = V_z_2
         => V_xa_2 != V_ya_2 )
        & ( V_xa_2 != V_z_2
         => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),V_La_2)) ) ) ) ).

fof(fact_CollectE,axiom,
    ! [V_P_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Set_OCollect(T_a,V_P_2)))
     => hBOOL(hAPP(V_P_2,V_a_2)) ) ).

fof(fact_CollectD,axiom,
    ! [V_P_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Set_OCollect(T_a,V_P_2)))
     => hBOOL(hAPP(V_P_2,V_a_2)) ) ).

fof(fact_mem__Collect__eq,axiom,
    ! [V_P_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Set_OCollect(T_a,V_P_2)))
    <=> hBOOL(hAPP(V_P_2,V_a_2)) ) ).

fof(fact_Collect__mem__eq,axiom,
    ! [V_A_2,T_a] : c_Set_OCollect(T_a,hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_A_2)) = V_A_2 ).

fof(fact_empty__def,axiom,
    ! [T_a] : c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Set_OCollect(T_a,c_COMBK(tc_HOL_Obool,T_a,c_fFalse)) ).

fof(fact_insert__Collect,axiom,
    ! [V_P_2,V_a_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Set_OCollect(T_a,V_P_2)) = c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_a),c_fNot),hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_a_2)))),V_P_2)) ).

fof(fact_UNIV__def,axiom,
    ! [T_a] : c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) = c_Set_OCollect(T_a,c_COMBK(tc_HOL_Obool,T_a,c_fTrue)) ).

fof(fact_Collect__disj__eq,axiom,
    ! [V_Q_2,V_P_2,T_a] : c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fdisj),V_P_2)),V_Q_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),c_Set_OCollect(T_a,V_P_2)),c_Set_OCollect(T_a,V_Q_2)) ).

fof(fact_split__eta,axiom,
    ! [V_f_2,T_c,T_b,T_a] : c_Product__Type_Oprod_Oprod__case(T_a,T_b,T_c,hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,T_c),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),T_c,T_b),V_f_2)),c_Product__Type_OPair(T_a,T_b))) = V_f_2 ).

fof(fact_singletonE,axiom,
    ! [V_a_2,V_b_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
     => V_b_2 = V_a_2 ) ).

fof(fact_singleton__iff,axiom,
    ! [V_a_2,V_b_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
    <=> V_b_2 = V_a_2 ) ).

fof(fact_insert__is__Un,axiom,
    ! [V_A_2,V_a_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_A_2) ).

fof(fact_ball__UNIV,axiom,
    ! [V_P_2,T_a] :
      ( ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))))
         => hBOOL(hAPP(V_P_2,B_x)) )
    <=> ! [B_x1] : hBOOL(hAPP(V_P_2,B_x1)) ) ).

fof(fact_insert__compr__raw,axiom,
    ! [V_B_2,V_a_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2) = c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fdisj),hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_a_2))),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_B_2))) ).

fof(fact_insert__compr,axiom,
    ! [V_B_2,V_a_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2) = c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fdisj),hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_a_2))),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_B_2))) ).

fof(fact_Un__def,axiom,
    ! [V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fdisj),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_A_2))),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_B_2))) ).

fof(fact_Collect__conv__if,axiom,
    ! [T_a,V_a_2,V_P_2] :
      ( ( hBOOL(hAPP(V_P_2,V_a_2))
       => c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_a_2))),V_P_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) )
      & ( ~ hBOOL(hAPP(V_P_2,V_a_2))
       => c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_a_2))),V_P_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ).

fof(fact_Collect__conv__if2,axiom,
    ! [T_a,V_a_2,V_P_2] :
      ( ( hBOOL(hAPP(V_P_2,V_a_2))
       => c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),hAPP(c_fequal,V_a_2))),V_P_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) )
      & ( ~ hBOOL(hAPP(V_P_2,V_a_2))
       => c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),hAPP(c_fequal,V_a_2))),V_P_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ).

fof(fact_singleton__conv,axiom,
    ! [V_a_2,T_a] : c_Set_OCollect(T_a,hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_a_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).

fof(fact_singleton__conv2,axiom,
    ! [V_a_2,T_a] : c_Set_OCollect(T_a,hAPP(c_fequal,V_a_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).

fof(fact_insert__def,axiom,
    ! [V_B_2,V_a_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),c_Set_OCollect(T_a,hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_a_2))),V_B_2) ).

fof(fact_in__above,axiom,
    ! [V_ya_2,V_xa_2,V_La_2,V_b_2,V_a_2] :
      ( V_a_2 != V_b_2
     => ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_La_2),c_Arrow__Order__Mirabelle_OLin))
       => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),c_Arrow__Order__Mirabelle_Oabove(V_La_2,V_a_2,V_b_2)))
        <=> ( V_xa_2 != V_ya_2
            & ( V_xa_2 = V_b_2
             => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_a_2),V_ya_2)),V_La_2)) )
            & ( V_xa_2 != V_b_2
             => ( ( V_ya_2 = V_b_2
                 => ( V_xa_2 = V_a_2
                    | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_a_2)),V_La_2)) ) )
                & ( V_ya_2 != V_b_2
                 => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),V_La_2)) ) ) ) ) ) ) ) ).

fof(fact_in__below,axiom,
    ! [V_ya_2,V_xa_2,V_La_2,V_b_2,V_a_2] :
      ( V_a_2 != V_b_2
     => ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_La_2),c_Arrow__Order__Mirabelle_OLin))
       => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),c_Arrow__Order__Mirabelle_Obelow(V_La_2,V_a_2,V_b_2)))
        <=> ( V_xa_2 != V_ya_2
            & ( V_ya_2 = V_a_2
             => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_b_2)),V_La_2)) )
            & ( V_ya_2 != V_a_2
             => ( ( V_xa_2 = V_a_2
                 => ( V_ya_2 = V_b_2
                    | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_b_2),V_ya_2)),V_La_2)) ) )
                & ( V_xa_2 != V_a_2
                 => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),V_La_2)) ) ) ) ) ) ) ) ).

fof(fact_sup1CI,axiom,
    ! [T_a,V_A_2,V_xa_2,V_B_2] :
      ( ( ~ hBOOL(hAPP(V_B_2,V_xa_2))
       => hBOOL(hAPP(V_A_2,V_xa_2)) )
     => hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_xa_2)) ) ).

fof(fact_sup1E,axiom,
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_xa_2))
     => ( ~ hBOOL(hAPP(V_A_2,V_xa_2))
       => hBOOL(hAPP(V_B_2,V_xa_2)) ) ) ).

fof(fact_top1I,axiom,
    ! [V_xa_2,T_a] : hBOOL(hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),V_xa_2)) ).

fof(fact_the__elem__eq,axiom,
    ! [V_xa_2,T_a] : c_Set_Othe__elem(T_a,hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_xa_2 ).

fof(fact_sup__fun__def,axiom,
    ! [V_xa_2,V_g_2,V_f_2,T_b,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,T_a)),V_f_2),V_g_2),V_xa_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(V_f_2,V_xa_2)),hAPP(V_g_2,V_xa_2)) ) ).

fof(fact_sup__Un__eq,axiom,
    ! [V_xa_2,V_S_2,V_R_2,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_R_2)),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_S_2)),V_xa_2))
    <=> hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_R_2),V_S_2))) ) ).

fof(fact_bot__fun__def,axiom,
    ! [V_xa_2,T_b,T_a] :
      ( class_Orderings_Obot(T_a)
     => hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_xa_2) = c_Orderings_Obot__class_Obot(T_a) ) ).

fof(fact_pred__equals__eq2,axiom,
    ! [V_S_2,V_R_2,T_b,T_a] :
      ( hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_R_2) = hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_S_2)
    <=> V_R_2 = V_S_2 ) ).

fof(fact_bot__empty__eq,axiom,
    ! [V_xa_2,T_a] :
      ( hBOOL(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_xa_2))
    <=> hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) ) ).

fof(fact_mem__splitI,axiom,
    ! [T_c,T_b,V_b_2,V_a_2,V_c_2,V_z_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_z_2),hAPP(hAPP(V_c_2,V_a_2),V_b_2)))
     => hBOOL(hAPP(hAPP(c_member(T_a),V_z_2),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_fun(T_a,tc_HOL_Obool),V_c_2),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)))) ) ).

fof(fact_splitD_H,axiom,
    ! [V_c_2,V_b_2,V_a_2,V_R_2,T_c,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_fun(T_c,tc_HOL_Obool),V_R_2),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2)),V_c_2))
     => hBOOL(hAPP(hAPP(hAPP(V_R_2,V_a_2),V_b_2),V_c_2)) ) ).

fof(fact_sup__Un__eq2,axiom,
    ! [V_ya_2,V_xa_2,V_S_2,V_R_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_R_2)),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_S_2)),V_xa_2),V_ya_2))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_xa_2),V_ya_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),V_R_2),V_S_2))) ) ).

fof(fact_bot__empty__eq2,axiom,
    ! [V_ya_2,V_xa_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_xa_2),V_ya_2))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_xa_2),V_ya_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)))) ) ).

fof(fact_eq__mem,axiom,
    ! [V_ya_2,V_xa_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),hAPP(c_fequal,V_ya_2)))
    <=> V_xa_2 = V_ya_2 ) ).

fof(fact_sup_Oidem,axiom,
    ! [V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_a) = V_a ) ).

fof(fact_sup__idem,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_x) = V_x ) ).

fof(fact_sup_Ocommute,axiom,
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_a) ) ).

fof(fact_inf__sup__aci_I5_J,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x) ) ).

fof(fact_sup__commute,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x) ) ).

fof(fact_sup_Oleft__idem,axiom,
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b) ) ).

fof(fact_inf__sup__aci_I8_J,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) ) ).

fof(fact_sup__left__idem,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) ) ).

fof(fact_sup_Oleft__commute,axiom,
    ! [V_c,V_a,V_b,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_c)) ) ).

fof(fact_inf__sup__aci_I7_J,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)) ) ).

fof(fact_sup__left__commute,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)) ) ).

fof(fact_sup_Oassoc,axiom,
    ! [V_c,V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_c)) ) ).

fof(fact_inf__sup__aci_I6_J,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) ) ).

fof(fact_sup__assoc,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) ) ).

fof(fact_sup1I1,axiom,
    ! [V_B_2,T_a,V_xa_2,V_A_2] :
      ( hBOOL(hAPP(V_A_2,V_xa_2))
     => hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_xa_2)) ) ).

fof(fact_sup1I2,axiom,
    ! [V_A_2,T_a,V_xa_2,V_B_2] :
      ( hBOOL(hAPP(V_B_2,V_xa_2))
     => hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_xa_2)) ) ).

fof(fact_pred__equals__eq,axiom,
    ! [V_S_2,V_R_2,T_a] :
      ( hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_R_2) = hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_S_2)
    <=> V_R_2 = V_S_2 ) ).

fof(fact_below__Lin,axiom,
    ! [V_La_2,V_ya_2,V_xa_2] :
      ( V_xa_2 != V_ya_2
     => ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_La_2),c_Arrow__Order__Mirabelle_OLin))
       => hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Arrow__Order__Mirabelle_Obelow(V_La_2,V_xa_2,V_ya_2)),c_Arrow__Order__Mirabelle_OLin)) ) ) ).

fof(fact_mkbot__Lin,axiom,
    ! [V_xa_2,V_La_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_La_2),c_Arrow__Order__Mirabelle_OLin))
     => hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Arrow__Order__Mirabelle_Omkbot(V_La_2,V_xa_2)),c_Arrow__Order__Mirabelle_OLin)) ) ).

fof(fact_mktop__Lin,axiom,
    ! [V_xa_2,V_La_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_La_2),c_Arrow__Order__Mirabelle_OLin))
     => hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Arrow__Order__Mirabelle_Omktop(V_La_2,V_xa_2)),c_Arrow__Order__Mirabelle_OLin)) ) ).

fof(fact_notin__Lin__iff,axiom,
    ! [V_ya_2,V_xa_2,V_La_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_La_2),c_Arrow__Order__Mirabelle_OLin))
     => ( V_xa_2 != V_ya_2
       => ( ~ hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_xa_2),V_ya_2)),V_La_2))
        <=> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_ya_2),V_xa_2)),V_La_2)) ) ) ) ).

fof(fact_Lin__irrefl,axiom,
    ! [V_b_2,V_a_2,V_La_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_La_2),c_Arrow__Order__Mirabelle_OLin))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_a_2),V_b_2)),V_La_2))
       => ~ hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),V_b_2),V_a_2)),V_La_2)) ) ) ).

fof(fact_bot__apply,axiom,
    ! [V_xa_2,T_b,T_a] :
      ( class_Orderings_Obot(T_a)
     => hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_xa_2) = c_Orderings_Obot__class_Obot(T_a) ) ).

fof(fact_sup__bot__left,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x) = V_x ) ).

fof(fact_sup__bot__right,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)) = V_x ) ).

fof(fact_sup__eq__bot__iff,axiom,
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => ( hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_xa_2),V_ya_2) = c_Orderings_Obot__class_Obot(T_a)
      <=> ( V_xa_2 = c_Orderings_Obot__class_Obot(T_a)
          & V_ya_2 = c_Orderings_Obot__class_Obot(T_a) ) ) ) ).

fof(fact_sup__top__left,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__top(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),c_Orderings_Otop__class_Otop(T_a)),V_x) = c_Orderings_Otop__class_Otop(T_a) ) ).

fof(fact_sup__top__right,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__top(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),c_Orderings_Otop__class_Otop(T_a)) = c_Orderings_Otop__class_Otop(T_a) ) ).

fof(fact_sup__apply,axiom,
    ! [V_xa_2,V_g_2,V_f_2,T_b,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,T_a)),V_f_2),V_g_2),V_xa_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(V_f_2,V_xa_2)),hAPP(V_g_2,V_xa_2)) ) ).

fof(fact_top__apply,axiom,
    ! [V_xa_2,T_b,T_a] :
      ( class_Orderings_Otop(T_a)
     => hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_b,T_a)),V_xa_2) = c_Orderings_Otop__class_Otop(T_a) ) ).

fof(fact_Lin__def,axiom,
    c_Arrow__Order__Mirabelle_OLin = c_Set_OCollect(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Order__Relation_Ostrict__linear__order__on(tc_Arrow__Order__Mirabelle_Oalt,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)))) ).

fof(fact_Nitpick_Oprod__def,axiom,
    ! [V_B_2,V_A_2,T_b,T_a] : c_Nitpick_Oprod(T_a,T_b,V_A_2,V_B_2) = c_Set_OCollect(tc_prod(T_a,T_b),c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_HOL_Obool,hAPP(hAPP(c_COMBC(T_a,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2)))) ).

fof(fact_prod_Orecs,axiom,
    ! [V_b_2,V_a_2,V_f1_2,T_a,T_c,T_b] : c_Product__Type_Oprod_Oprod__rec(T_b,T_c,T_a,V_f1_2,hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2) ).

fof(fact_iso__tuple__UNIV__I,axiom,
    ! [V_xa_2,T_a] : hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) ).

fof(fact_Chain__def,axiom,
    ! [V_r_2,T_a] : c_Zorn_OChain(T_a,V_r_2) = c_Set_OCollect(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_a,tc_HOL_Obool)),c_HOL_OAll(T_a)),hAPP(hAPP(c_COMBS(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fimplies)),hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a))))),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool,T_a),c_HOL_OAll(T_a))),hAPP(hAPP(c_COMBC(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_fun(tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool))),tc_fun(T_a,tc_HOL_Obool)),c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),T_a)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fimplies)),hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)))))),hAPP(hAPP(c_COMBS(T_a,tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fdisj)),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_prod(T_a,T_a)),tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_a),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool),T_a),c_member(tc_prod(T_a,T_a)))),c_Product__Type_OPair(T_a,T_a)))),V_r_2)))),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_prod(T_a,T_a)),tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_a),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool),T_a),c_member(tc_prod(T_a,T_a)))),hAPP(c_COMBC(T_a,T_a,tc_prod(T_a,T_a)),c_Product__Type_OPair(T_a,T_a))))),V_r_2))))))) ).

fof(fact_the__elem__def,axiom,
    ! [V_X_2,T_a] : c_Set_Othe__elem(T_a,V_X_2) = c_HOL_OThe(T_a,hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool,T_a),hAPP(c_fequal,V_X_2)),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Set_Oinsert(T_a)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) ).

fof(fact_inv__image__def,axiom,
    ! [V_f_2,V_r_2,T_a,T_b] : hAPP(hAPP(c_Relation_Oinv__image(T_b,T_a),V_r_2),V_f_2) = c_Set_OCollect(tc_prod(T_a,T_a),c_Product__Type_Oprod_Oprod__case(T_a,T_a,tc_HOL_Obool,hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBC(T_a,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_prod(T_b,T_b)),tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_a),c_member(tc_prod(T_b,T_b)))),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,T_b),tc_fun(T_a,tc_prod(T_b,T_b))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(tc_fun(T_a,T_b),tc_fun(T_a,tc_prod(T_b,T_b))),T_a),c_COMBB(T_b,tc_prod(T_b,T_b),T_a)),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,tc_prod(T_b,T_b)),T_a),c_Product__Type_OPair(T_b,T_b)),V_f_2))),V_f_2)))),V_r_2))) ).

fof(fact_sup2E,axiom,
    ! [V_ya_2,V_xa_2,V_B_2,V_A_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_A_2),V_B_2),V_xa_2),V_ya_2))
     => ( ~ hBOOL(hAPP(hAPP(V_A_2,V_xa_2),V_ya_2))
       => hBOOL(hAPP(hAPP(V_B_2,V_xa_2),V_ya_2)) ) ) ).

fof(fact_sup2CI,axiom,
    ! [T_b,T_a,V_A_2,V_ya_2,V_xa_2,V_B_2] :
      ( ( ~ hBOOL(hAPP(hAPP(V_B_2,V_xa_2),V_ya_2))
       => hBOOL(hAPP(hAPP(V_A_2,V_xa_2),V_ya_2)) )
     => hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_A_2),V_B_2),V_xa_2),V_ya_2)) ) ).

fof(fact_bot2E,axiom,
    ! [V_ya_2,V_xa_2,T_b,T_a] : ~ hBOOL(hAPP(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_xa_2),V_ya_2)) ).

fof(fact_sup2I1,axiom,
    ! [V_B_2,T_b,T_a,V_ya_2,V_xa_2,V_A_2] :
      ( hBOOL(hAPP(hAPP(V_A_2,V_xa_2),V_ya_2))
     => hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_A_2),V_B_2),V_xa_2),V_ya_2)) ) ).

fof(fact_sup2I2,axiom,
    ! [V_A_2,T_b,T_a,V_ya_2,V_xa_2,V_B_2] :
      ( hBOOL(hAPP(hAPP(V_B_2,V_xa_2),V_ya_2))
     => hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_A_2),V_B_2),V_xa_2),V_ya_2)) ) ).

fof(fact_in__inv__image,axiom,
    ! [V_f_2,V_r_2,T_b,V_ya_2,V_xa_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),hAPP(hAPP(c_Relation_Oinv__image(T_b,T_a),V_r_2),V_f_2)))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(V_f_2,V_xa_2)),hAPP(V_f_2,V_ya_2))),V_r_2)) ) ).

fof(fact_The__split__eq,axiom,
    ! [V_ya_2,V_xa_2,T_b,T_a] : c_HOL_OThe(tc_prod(T_a,T_b),c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_HOL_Obool,hAPP(hAPP(c_COMBC(T_a,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),hAPP(c_fequal,V_xa_2)))),hAPP(c_fequal,V_ya_2)))) = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_xa_2),V_ya_2) ).

fof(fact_split__paired__The,axiom,
    ! [V_P_2,T_b,T_a] : c_HOL_OThe(tc_prod(T_a,T_b),V_P_2) = c_HOL_OThe(tc_prod(T_a,T_b),c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_HOL_Obool,hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_HOL_Obool,T_b),V_P_2)),c_Product__Type_OPair(T_a,T_b)))) ).

fof(fact_Nitpick_Orefl_H__def,axiom,
    ! [V_r_2,T_a] :
      ( c_Nitpick_Orefl_H(T_a,V_r_2)
    <=> ! [B_x] : hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_x),B_x)),V_r_2)) ) ).

fof(fact_irrefl__def,axiom,
    ! [V_r_2,T_a] :
      ( c_Relation_Oirrefl(T_a,V_r_2)
    <=> ! [B_x] : ~ hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_x),B_x)),V_r_2)) ) ).

fof(fact_Field__insert,axiom,
    ! [V_r_2,V_b_2,V_a_2,T_a] : c_Relation_OField(T_a,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_b_2)),V_r_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))),c_Relation_OField(T_a,V_r_2)) ).

fof(fact_rel__comp__def,axiom,
    ! [V_s_2,V_r_2,T_b,T_c,T_a] : hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_r_2),V_s_2) = c_Set_OCollect(tc_prod(T_a,T_b),c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_HOL_Obool,hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_c))),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),T_a),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_a),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj)),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_a),c_COMBC(T_c,tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_a,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_c),tc_fun(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_a,T_c)))),c_Product__Type_OPair(T_a,T_c)))),V_r_2))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_b)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_c,T_b),tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_c,T_b)))),hAPP(c_COMBC(T_c,T_b,tc_prod(T_c,T_b)),c_Product__Type_OPair(T_c,T_b))))),V_s_2))))) ).

fof(fact_Range__Collect__split,axiom,
    ! [V_P_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Set_OCollect(tc_prod(T_b,T_a),c_Product__Type_Oprod_Oprod__case(T_b,T_a,tc_HOL_Obool,V_P_2))) = c_Set_OCollect(T_a,hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_a),c_HOL_OEx(T_b)),hAPP(c_COMBC(T_b,T_a,tc_HOL_Obool),V_P_2))) ).

fof(fact_Domain__Collect__split,axiom,
    ! [V_P_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Set_OCollect(tc_prod(T_a,T_b),c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_HOL_Obool,V_P_2))) = c_Set_OCollect(T_a,hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_a),c_HOL_OEx(T_b)),V_P_2)) ).

fof(fact_rel__compI,axiom,
    ! [V_s_2,V_c_2,T_c,V_r_2,V_b_2,V_a_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2)),V_r_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_b_2),V_c_2)),V_s_2))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_b,T_c),V_r_2),V_s_2))) ) ) ).

fof(fact_DomainI,axiom,
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2)),V_r_2))
     => hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Relation_ODomain(T_a,T_b,V_r_2))) ) ).

fof(fact_RangeI,axiom,
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2)),V_r_2))
     => hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Relation_ORange(T_a,T_b,V_r_2))) ) ).

fof(fact_Field__def,axiom,
    ! [V_r_2,T_a] : c_Relation_OField(T_a,V_r_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),c_Relation_ODomain(T_a,T_a,V_r_2)),c_Relation_ORange(T_a,T_a,V_r_2)) ).

fof(fact_O__assoc,axiom,
    ! [V_T_2,V_S_2,V_R_2,T_d,T_b,T_c,T_a] : hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_d,T_c),V_R_2),V_S_2)),V_T_2) = hAPP(hAPP(c_Relation_Orel__comp(T_a,T_d,T_b),V_R_2),hAPP(hAPP(c_Relation_Orel__comp(T_d,T_c,T_b),V_S_2),V_T_2)) ).

fof(fact_rel__comp__empty2,axiom,
    ! [V_R_2,T_b,T_c,T_a] : hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_R_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) ).

fof(fact_rel__comp__empty1,axiom,
    ! [V_R_2,T_b,T_c,T_a] : hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool))),V_R_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) ).

fof(fact_rel__comp__distrib2,axiom,
    ! [V_R_2,V_T_2,V_S_2,T_b,T_c,T_a] : hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool)),V_S_2),V_T_2)),V_R_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_S_2),V_R_2)),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_T_2),V_R_2)) ).

fof(fact_rel__comp__distrib,axiom,
    ! [V_T_2,V_S_2,V_R_2,T_b,T_c,T_a] : hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_S_2),V_T_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_R_2),V_S_2)),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_R_2),V_T_2)) ).

fof(fact_Domain__empty,axiom,
    ! [T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_Domain__empty__iff,axiom,
    ! [V_r_2,T_b,T_a] :
      ( c_Relation_ODomain(T_a,T_b,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) ) ).

fof(fact_Field__empty,axiom,
    ! [T_a] : c_Relation_OField(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_Domain__Un__eq,axiom,
    ! [V_B_2,V_A_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),c_Relation_ODomain(T_a,T_b,V_A_2)),c_Relation_ODomain(T_a,T_b,V_B_2)) ).

fof(fact_Range__empty,axiom,
    ! [T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_Range__empty__iff,axiom,
    ! [V_r_2,T_a,T_b] :
      ( c_Relation_ORange(T_b,T_a,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)) ) ).

fof(fact_Field__Un,axiom,
    ! [V_s_2,V_r_2,T_a] : c_Relation_OField(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),c_Relation_OField(T_a,V_r_2)),c_Relation_OField(T_a,V_s_2)) ).

fof(fact_Range__Un__eq,axiom,
    ! [V_B_2,V_A_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),c_Relation_ORange(T_b,T_a,V_A_2)),c_Relation_ORange(T_b,T_a,V_B_2)) ).

fof(fact_Domain__iff,axiom,
    ! [V_r_2,T_b,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Relation_ODomain(T_a,T_b,V_r_2)))
    <=> ? [B_y] : hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),B_y)),V_r_2)) ) ).

fof(fact_Range__iff,axiom,
    ! [V_r_2,T_b,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Relation_ORange(T_b,T_a,V_r_2)))
    <=> ? [B_y] : hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),B_y),V_a_2)),V_r_2)) ) ).

fof(fact_Domain__insert,axiom,
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2)),V_r_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Relation_ODomain(T_a,T_b,V_r_2)) ).

fof(fact_Range__insert,axiom,
    ! [V_r_2,V_b_2,V_a_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_a_2),V_b_2)),V_r_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Relation_ORange(T_b,T_a,V_r_2)) ).

fof(fact_union__comp__emptyR,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] :
      ( hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
     => ( hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),V_A_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
       => hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),V_B_2),V_C_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) ) ) ).

fof(fact_union__comp__emptyL,axiom,
    ! [V_B_2,V_C_2,V_A_2,T_a] :
      ( hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),V_A_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
     => ( hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),V_B_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
       => hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) ) ) ).

fof(fact_RangeP__Range__eq,axiom,
    ! [V_xa_2,V_r_2,T_b,T_a] :
      ( c_Predicate_ORangeP(T_a,T_b,hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_r_2),V_xa_2)
    <=> hBOOL(hAPP(hAPP(c_member(T_b),V_xa_2),c_Relation_ORange(T_a,T_b,V_r_2))) ) ).

fof(fact_DomainP__Domain__eq,axiom,
    ! [V_xa_2,V_r_2,T_b,T_a] :
      ( c_Predicate_ODomainP(T_a,T_b,hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_r_2),V_xa_2)
    <=> hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),c_Relation_ODomain(T_a,T_b,V_r_2))) ) ).

fof(fact_RangeE,axiom,
    ! [V_r_2,T_b,V_b_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),c_Relation_ORange(T_b,T_a,V_r_2)))
     => ~ ! [B_x] : ~ hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),B_x),V_b_2)),V_r_2)) ) ).

fof(fact_DomainE,axiom,
    ! [V_r_2,T_b,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Relation_ODomain(T_a,T_b,V_r_2)))
     => ~ ! [B_y] : ~ hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),B_y)),V_r_2)) ) ).

fof(fact_DomainP_Ointros,axiom,
    ! [T_b,T_a,V_b_2,V_a_2,V_r_2] :
      ( hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2))
     => c_Predicate_ODomainP(T_a,T_b,V_r_2,V_a_2) ) ).

fof(fact_RangeP_Ointros,axiom,
    ! [T_b,T_a,V_b_2,V_a_2,V_r_2] :
      ( hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2))
     => c_Predicate_ORangeP(T_a,T_b,V_r_2,V_b_2) ) ).

fof(fact_RangeP_Oequations,axiom,
    ! [V_a2_2,V_r_2,T_b,T_a] :
      ( c_Predicate_ORangeP(T_a,T_b,V_r_2,V_a2_2)
    <=> ? [B_a] : hBOOL(hAPP(hAPP(V_r_2,B_a),V_a2_2)) ) ).

fof(fact_DomainP_Oequations,axiom,
    ! [V_a1_2,V_r_2,T_b,T_a] :
      ( c_Predicate_ODomainP(T_a,T_b,V_r_2,V_a1_2)
    <=> ? [B_x1] : hBOOL(hAPP(hAPP(V_r_2,V_a1_2),B_x1)) ) ).

fof(fact_in__rel__def,axiom,
    ! [V_ya_2,V_xa_2,V_R_2,T_b,T_a] :
      ( c_FunDef_Oin__rel(T_a,T_b,V_R_2,V_xa_2,V_ya_2)
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_xa_2),V_ya_2)),V_R_2)) ) ).

fof(fact_rp__inv__image__def,axiom,
    ! [T_b,T_a] : c_FunDef_Orp__inv__image(T_a,T_b) = c_Product__Type_Oprod_Oprod__case(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_fun(T_b,T_a),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_fun(T_b,T_a),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_fun(T_b,T_a),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_fun(T_b,T_a),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),tc_fun(tc_fun(T_b,T_a),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))),tc_fun(tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_fun(T_b,T_a),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_fun(T_b,T_a),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))))),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),c_COMBB(tc_fun(tc_fun(T_b,T_a),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),tc_fun(tc_fun(T_b,T_a),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,T_a),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))),tc_fun(tc_fun(tc_fun(T_b,T_a),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),tc_fun(tc_fun(T_b,T_a),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),c_COMBS(tc_fun(T_b,T_a),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,T_a),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),tc_fun(tc_fun(T_b,T_a),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))),tc_fun(T_b,T_a)),c_Product__Type_OPair(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))),c_Relation_Oinv__image(T_a,T_b))))),c_Relation_Oinv__image(T_a,T_b))) ).

fof(fact_pred__comp__rel__comp__eq,axiom,
    ! [V_ya_2,V_xa_2,V_s_2,V_r_2,T_c,T_b,T_a] :
      ( c_Predicate_Opred__comp(T_a,T_b,T_c,hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_r_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_s_2),V_xa_2,V_ya_2)
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_c),V_xa_2),V_ya_2)),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_b,T_c),V_r_2),V_s_2))) ) ).

fof(fact_refl__on__def_H,axiom,
    ! [V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
    <=> ( ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),B_x),V_r_2))
           => hBOOL(hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_a,tc_HOL_Obool,hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_a)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_A_2)))),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_A_2))),B_x)) )
        & ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
           => hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_x),B_x)),V_r_2)) ) ) ) ).

fof(fact_the__sym__eq__trivial,axiom,
    ! [V_xa_2,T_a] : c_HOL_OThe(T_a,hAPP(c_fequal,V_xa_2)) = V_xa_2 ).

fof(fact_pred__comp_Ointros,axiom,
    ! [T_c,T_b,T_a,V_c_2,V_s_2,V_b_2,V_a_2,V_r_2] :
      ( hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2))
     => ( hBOOL(hAPP(hAPP(V_s_2,V_b_2),V_c_2))
       => c_Predicate_Opred__comp(T_a,T_b,T_c,V_r_2,V_s_2,V_a_2,V_c_2) ) ) ).

fof(fact_pred__comp_Oequations,axiom,
    ! [V_a3_2,V_a1_2,V_s_2,V_r_2,T_c,T_b,T_a] :
      ( c_Predicate_Opred__comp(T_a,T_b,T_c,V_r_2,V_s_2,V_a1_2,V_a3_2)
    <=> ? [B_b] :
          ( hBOOL(hAPP(hAPP(V_r_2,V_a1_2),B_b))
          & hBOOL(hAPP(hAPP(V_s_2,B_b),V_a3_2)) ) ) ).

fof(fact_refl__on__empty,axiom,
    ! [T_a] : c_Relation_Orefl__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).

fof(fact_refl__on__Un,axiom,
    ! [V_s_2,V_B_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( c_Relation_Orefl__on(T_a,V_B_2,V_s_2)
       => c_Relation_Orefl__on(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),V_r_2),V_s_2)) ) ) ).

fof(fact_refl__onD,axiom,
    ! [V_a_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_a_2)),V_r_2)) ) ) ).

fof(fact_refl__onD1,axiom,
    ! [V_ya_2,V_xa_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),V_r_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2)) ) ) ).

fof(fact_refl__onD2,axiom,
    ! [V_ya_2,V_xa_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),V_r_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_ya_2),V_A_2)) ) ) ).

fof(fact_the__eq__trivial,axiom,
    ! [V_a_2,T_a] : c_HOL_OThe(T_a,hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_a_2)) = V_a_2 ).

fof(fact_reflp__def,axiom,
    ! [V_r_2,T_a] :
      ( c_Predicate_Oreflp(T_a,V_r_2)
    <=> c_Relation_Orefl__on(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),c_Set_OCollect(tc_prod(T_a,T_a),c_Product__Type_Oprod_Oprod__case(T_a,T_a,tc_HOL_Obool,V_r_2))) ) ).

fof(fact_rp__inv__image__rp,axiom,
    ! [V_f_2,T_b,V_P_2,T_a] :
      ( c_FunDef_Oreduction__pair(T_a,V_P_2)
     => c_FunDef_Oreduction__pair(T_b,hAPP(hAPP(c_FunDef_Orp__inv__image(T_a,T_b),V_P_2),V_f_2)) ) ).

fof(fact_Image__singleton,axiom,
    ! [V_a_2,V_r_2,T_a,T_b] : hAPP(c_Relation_OImage(T_b,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Set_OCollect(T_a,hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_prod(T_b,T_a),tc_fun(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),tc_HOL_Obool),T_a),c_member(tc_prod(T_b,T_a))),hAPP(c_Product__Type_OPair(T_b,T_a),V_a_2))),V_r_2)) ).

fof(fact_mem__splitI2,axiom,
    ! [V_c_2,V_z_2,T_c,T_b,T_a,V_p_2] :
      ( ! [B_a,B_b] :
          ( V_p_2 = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_a),B_b)
         => hBOOL(hAPP(hAPP(c_member(T_c),V_z_2),hAPP(hAPP(V_c_2,B_a),B_b))) )
     => hBOOL(hAPP(hAPP(c_member(T_c),V_z_2),hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_b,tc_fun(T_c,tc_HOL_Obool),V_c_2),V_p_2))) ) ).

fof(fact_mem__splitE,axiom,
    ! [V_p_2,V_c_2,T_c,T_b,V_z_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_z_2),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_fun(T_a,tc_HOL_Obool),V_c_2),V_p_2)))
     => ~ ! [B_x,B_y] :
            ( V_p_2 = hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_x),B_y)
           => ~ hBOOL(hAPP(hAPP(c_member(T_a),V_z_2),hAPP(hAPP(V_c_2,B_x),B_y))) ) ) ).

fof(fact_Image__Un,axiom,
    ! [V_B_2,V_A_2,V_R_2,T_a,T_b] : hAPP(c_Relation_OImage(T_b,T_a,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_a,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_b,T_a,V_R_2),V_B_2)) ).

fof(fact_reflpE,axiom,
    ! [V_xa_2,V_r_2,T_a] :
      ( c_Predicate_Oreflp(T_a,V_r_2)
     => hBOOL(hAPP(hAPP(V_r_2,V_xa_2),V_xa_2)) ) ).

fof(fact_Image__empty,axiom,
    ! [V_R_2,T_a,T_b] : hAPP(c_Relation_OImage(T_b,T_a,V_R_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_Un__Image,axiom,
    ! [V_A_2,V_S_2,V_R_2,T_a,T_b] : hAPP(c_Relation_OImage(T_b,T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)),V_R_2),V_S_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_a,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_b,T_a,V_S_2),V_A_2)) ).

fof(fact_rev__ImageI,axiom,
    ! [V_r_2,V_b_2,T_b,V_A_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2)),V_r_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Relation_OImage(T_a,T_b,V_r_2),V_A_2))) ) ) ).

fof(fact_Image__iff,axiom,
    ! [V_A_2,V_r_2,T_b,V_b_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),hAPP(c_Relation_OImage(T_b,T_a,V_r_2),V_A_2)))
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
          & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),B_x),V_b_2)),V_r_2)) ) ) ).

fof(fact_Image__singleton__iff,axiom,
    ! [V_a_2,V_r_2,T_b,V_b_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),hAPP(c_Relation_OImage(T_b,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_a_2),V_b_2)),V_r_2)) ) ).

fof(fact_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Relation_Orefl__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2)
     => ( c_Relation_Oantisym(T_a,V_r_2)
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Relation_OField(T_a,V_r_2)))
         => ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),c_Relation_OField(T_a,V_r_2)))
           => ( hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
            <=> V_a_2 = V_b_2 ) ) ) ) ) ).

fof(fact_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Order__Relation_Opartial__order__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Relation_OField(T_a,V_r_2)))
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),c_Relation_OField(T_a,V_r_2)))
         => ( hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
          <=> V_a_2 = V_b_2 ) ) ) ) ).

fof(fact_ImageE,axiom,
    ! [V_A_2,V_r_2,T_b,V_b_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),hAPP(c_Relation_OImage(T_b,T_a,V_r_2),V_A_2)))
     => ~ ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),B_x),V_b_2)),V_r_2))
           => ~ hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) ) ) ).

fof(fact_equiv__class__eq__iff,axiom,
    ! [V_ya_2,V_xa_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),V_r_2))
      <=> ( hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_ya_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
          & hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
          & hBOOL(hAPP(hAPP(c_member(T_a),V_ya_2),V_A_2)) ) ) ) ).

fof(fact_eq__equiv__class,axiom,
    ! [V_A_2,V_b_2,V_a_2,V_r_2,T_a] :
      ( hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
     => ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_b_2)),V_r_2)) ) ) ) ).

fof(fact_antisym__empty,axiom,
    ! [T_a] : c_Relation_Oantisym(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).

fof(fact_partial__order__on__empty,axiom,
    ! [T_a] : c_Order__Relation_Opartial__order__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).

fof(fact_antisym__def,axiom,
    ! [V_r_2,T_a] :
      ( c_Relation_Oantisym(T_a,V_r_2)
    <=> ! [B_x,B_y] :
          ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_x),B_y)),V_r_2))
         => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_y),B_x)),V_r_2))
           => B_x = B_y ) ) ) ).

fof(fact_antisymD,axiom,
    ! [V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Relation_Oantisym(T_a,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_b_2)),V_r_2))
       => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_b_2),V_a_2)),V_r_2))
         => V_a_2 = V_b_2 ) ) ) ).

fof(fact_equiv__class__self,axiom,
    ! [V_a_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) ) ) ).

fof(fact_equiv__class__eq,axiom,
    ! [V_b_2,V_a_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_b_2)),V_r_2))
       => hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) ) ) ).

fof(fact_eq__equiv__class__iff,axiom,
    ! [V_ya_2,V_xa_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_ya_2),V_A_2))
         => ( hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_ya_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
          <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),V_r_2)) ) ) ) ) ).

fof(fact_congruent__def,axiom,
    ! [V_f_2,V_r_2,T_b,T_a] :
      ( c_Equiv__Relations_Ocongruent(T_a,T_b,V_r_2,V_f_2)
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),B_x),V_r_2))
         => hBOOL(hAPP(c_Product__Type_Oprod_Oprod__case(T_a,T_a,tc_HOL_Obool,hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,T_b),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_a,T_b),tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBB(T_b,tc_HOL_Obool,T_a)),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,tc_HOL_Obool),T_a),c_fequal),V_f_2))),V_f_2)),B_x)) ) ) ).

fof(fact_eq__equiv__class__iff2,axiom,
    ! [V_ya_2,V_xa_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_ya_2),V_A_2))
         => ( c_Equiv__Relations_Oquotient(T_a,hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_r_2) = c_Equiv__Relations_Oquotient(T_a,hAPP(hAPP(c_Set_Oinsert(T_a),V_ya_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_r_2)
          <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),V_r_2)) ) ) ) ) ).

fof(fact_subset__equiv__class,axiom,
    ! [V_a_2,V_b_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_b_2)),V_r_2)) ) ) ) ).

fof(fact_equiv__class__nondisjoint,axiom,
    ! [V_b_2,V_a_2,V_xa_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_b_2)),V_r_2)) ) ) ).

fof(fact_order__refl,axiom,
    ! [V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_x) ) ).

fof(fact_equalityI,axiom,
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)
       => V_A_2 = V_B_2 ) ) ).

fof(fact_inf1E,axiom,
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_xa_2))
     => ~ ( hBOOL(hAPP(V_A_2,V_xa_2))
         => ~ hBOOL(hAPP(V_B_2,V_xa_2)) ) ) ).

fof(fact_inf1I,axiom,
    ! [T_a,V_B_2,V_xa_2,V_A_2] :
      ( hBOOL(hAPP(V_A_2,V_xa_2))
     => ( hBOOL(hAPP(V_B_2,V_xa_2))
       => hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_xa_2)) ) ) ).

fof(fact_subsetD,axiom,
    ! [V_c_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2)) ) ) ).

fof(fact_IntE,axiom,
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
     => ~ ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
         => ~ hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2)) ) ) ).

fof(fact_IntI,axiom,
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2))) ) ) ).

fof(fact_empty__subsetI,axiom,
    ! [V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_A_2) ).

fof(fact_less__by__empty,axiom,
    ! [V_B_2,T_a,V_A_2] :
      ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
     => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_A_2,V_B_2) ) ).

fof(fact_rel__comp__mono,axiom,
    ! [V_s_2,V_s_H_2,T_c,V_r_2,V_r_H_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_H_2,V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),V_s_H_2,V_s_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_b,T_c),V_r_H_2),V_s_H_2),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_b,T_c),V_r_2),V_s_2)) ) ) ).

fof(fact_refl__on__Int,axiom,
    ! [V_s_2,V_B_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( c_Relation_Orefl__on(T_a,V_B_2,V_s_2)
       => c_Relation_Orefl__on(T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),V_r_2),V_s_2)) ) ) ).

fof(fact_inf__fun__def,axiom,
    ! [V_xa_2,V_g_2,V_f_2,T_b,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,T_a)),V_f_2),V_g_2),V_xa_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(V_f_2,V_xa_2)),hAPP(V_g_2,V_xa_2)) ) ).

fof(fact_insert__inter__insert,axiom,
    ! [V_B_2,V_A_2,V_a_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_A_2)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) ).

fof(fact_in__mono,axiom,
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_B_2)) ) ) ).

fof(fact_set__rev__mp,axiom,
    ! [V_B_2,V_A_2,V_xa_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
       => hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_B_2)) ) ) ).

fof(fact_set__mp,axiom,
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_B_2)) ) ) ).

fof(fact_Int__UNIV__left,axiom,
    ! [V_B_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))),V_B_2) = V_B_2 ).

fof(fact_Int__UNIV__right,axiom,
    ! [V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = V_A_2 ).

fof(fact_bot__least,axiom,
    ! [V_x,T_a] :
      ( class_Orderings_Obot(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,c_Orderings_Obot__class_Obot(T_a),V_x) ) ).

fof(fact_Int__Un__distrib,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_C_2)) ).

fof(fact_Un__Int__distrib,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_C_2)) ).

fof(fact_Int__Un__distrib2,axiom,
    ! [V_A_2,V_C_2,V_B_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_A_2)) ).

fof(fact_Un__Int__distrib2,axiom,
    ! [V_A_2,V_C_2,V_B_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_A_2)) ).

fof(fact_Un__Int__crazy,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_A_2)) ).

fof(fact_subset__empty,axiom,
    ! [V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ).

fof(fact_inf__sup__ord_I3_J,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_x,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) ) ).

fof(fact_sup__ge1,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_x,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) ) ).

fof(fact_inf__sup__ord_I4_J,axiom,
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_y,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) ) ).

fof(fact_sup__ge2,axiom,
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_y,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) ) ).

fof(fact_le__iff__sup,axiom,
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
      <=> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_xa_2),V_ya_2) = V_ya_2 ) ) ).

fof(fact_le__sup__iff,axiom,
    ! [V_z_2,V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_xa_2),V_ya_2),V_z_2)
      <=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_z_2)
          & c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_z_2) ) ) ) ).

fof(fact_le__supI1,axiom,
    ! [V_b,V_a,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a)
       => c_Orderings_Oord__class_Oless__eq(T_a,V_x,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)) ) ) ).

fof(fact_le__supI2,axiom,
    ! [V_a,V_b,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_b)
       => c_Orderings_Oord__class_Oless__eq(T_a,V_x,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)) ) ) ).

fof(fact_sup__absorb2,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = V_y ) ) ).

fof(fact_sup__absorb1,axiom,
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = V_x ) ) ).

fof(fact_le__supI,axiom,
    ! [V_b,V_x,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x)
         => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b),V_x) ) ) ) ).

fof(fact_sup__least,axiom,
    ! [V_z,V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x)
         => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z),V_x) ) ) ) ).

fof(fact_sup__mono,axiom,
    ! [V_d,V_b,V_c,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_d)
         => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_c),V_d)) ) ) ) ).

fof(fact_le__supE,axiom,
    ! [V_x,V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b),V_x)
       => ~ ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x)
           => ~ c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x) ) ) ) ).

fof(fact_top__greatest,axiom,
    ! [V_x,T_a] :
      ( class_Orderings_Otop(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Orderings_Otop__class_Otop(T_a)) ) ).

fof(fact_Range__Int__subset,axiom,
    ! [V_B_2,V_A_2,T_b,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_ORange(T_b,T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),c_Relation_ORange(T_b,T_a,V_A_2)),c_Relation_ORange(T_b,T_a,V_B_2))) ).

fof(fact_Domain__Int__subset,axiom,
    ! [V_B_2,V_A_2,T_b,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),c_Relation_ODomain(T_a,T_b,V_A_2)),c_Relation_ODomain(T_a,T_b,V_B_2))) ).

fof(fact_mono__Field,axiom,
    ! [V_s_2,V_r_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OField(T_a,V_r_2),c_Relation_OField(T_a,V_s_2)) ) ).

fof(fact_Domain__mono,axiom,
    ! [V_s_2,V_r_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_2,V_s_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_r_2),c_Relation_ODomain(T_a,T_b,V_s_2)) ) ).

fof(fact_linorder__le__cases,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) ) ) ).

fof(fact_inf1D2,axiom,
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_xa_2))
     => hBOOL(hAPP(V_B_2,V_xa_2)) ) ).

fof(fact_inf1D1,axiom,
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_xa_2))
     => hBOOL(hAPP(V_A_2,V_xa_2)) ) ).

fof(fact_le__infE,axiom,
    ! [V_b,V_a,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b))
       => ~ ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a)
           => ~ c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_b) ) ) ) ).

fof(fact_le__funE,axiom,
    ! [V_xa_2,V_g_2,V_f_2,T_a,T_b] :
      ( class_Orderings_Oord(T_b)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
       => c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2)) ) ) ).

fof(fact_inf__mono,axiom,
    ! [V_d,V_b,V_c,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_d)
         => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_c),V_d)) ) ) ) ).

fof(fact_inf__greatest,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z)
         => c_Orderings_Oord__class_Oless__eq(T_a,V_x,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) ) ) ) ).

fof(fact_le__infI,axiom,
    ! [V_b,V_a,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_b)
         => c_Orderings_Oord__class_Oless__eq(T_a,V_x,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)) ) ) ) ).

fof(fact_xt1_I6_J,axiom,
    ! [V_z,V_x,V_y,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y)
         => c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x) ) ) ) ).

fof(fact_xt1_I5_J,axiom,
    ! [V_x,V_y,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
         => V_x = V_y ) ) ) ).

fof(fact_order__trans,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z)
         => c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z) ) ) ) ).

fof(fact_order__antisym,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
         => V_x = V_y ) ) ) ).

fof(fact_xt1_I4_J,axiom,
    ! [V_c,V_a,V_b,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
       => ( V_b = V_c
         => c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a) ) ) ) ).

fof(fact_ord__le__eq__trans,axiom,
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oord(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
       => ( V_b = V_c
         => c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c) ) ) ) ).

fof(fact_xt1_I3_J,axiom,
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( V_a = V_b
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b)
         => c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a) ) ) ) ).

fof(fact_ord__eq__le__trans,axiom,
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oord(T_a)
     => ( V_a = V_b
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c)
         => c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c) ) ) ) ).

fof(fact_predicate1D,axiom,
    ! [V_xa_2,V_Q_2,V_P_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2)
     => ( hBOOL(hAPP(V_P_2,V_xa_2))
       => hBOOL(hAPP(V_Q_2,V_xa_2)) ) ) ).

fof(fact_order__antisym__conv,axiom,
    ! [V_xa_2,V_ya_2,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
        <=> V_xa_2 = V_ya_2 ) ) ) ).

fof(fact_inf__absorb2,axiom,
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = V_y ) ) ).

fof(fact_inf__absorb1,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = V_x ) ) ).

fof(fact_le__infI2,axiom,
    ! [V_a,V_x,V_b,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x)
       => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b),V_x) ) ) ).

fof(fact_le__infI1,axiom,
    ! [V_b,V_x,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x)
       => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b),V_x) ) ) ).

fof(fact_le__funD,axiom,
    ! [V_xa_2,V_g_2,V_f_2,T_a,T_b] :
      ( class_Orderings_Oord(T_b)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
       => c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2)) ) ) ).

fof(fact_order__eq__refl,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( V_x = V_y
       => c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) ) ) ).

fof(fact_rev__predicate1D,axiom,
    ! [V_Q_2,T_a,V_xa_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,V_xa_2))
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2)
       => hBOOL(hAPP(V_Q_2,V_xa_2)) ) ) ).

fof(fact_inf__apply,axiom,
    ! [V_xa_2,V_g_2,V_f_2,T_b,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,T_a)),V_f_2),V_g_2),V_xa_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(V_f_2,V_xa_2)),hAPP(V_g_2,V_xa_2)) ) ).

fof(fact_inf__assoc,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) ) ).

fof(fact_inf__sup__aci_I2_J,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) ) ).

fof(fact_inf_Oassoc,axiom,
    ! [V_c,V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)) ) ).

fof(fact_le__inf__iff,axiom,
    ! [V_z_2,V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_ya_2),V_z_2))
      <=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
          & c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_z_2) ) ) ) ).

fof(fact_inf__left__commute,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)) ) ).

fof(fact_inf__sup__aci_I3_J,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)) ) ).

fof(fact_inf_Oleft__commute,axiom,
    ! [V_c,V_a,V_b,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)) ) ).

fof(fact_inf__left__idem,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) ) ).

fof(fact_inf__sup__aci_I4_J,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) ) ).

fof(fact_inf_Oleft__idem,axiom,
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b) ) ).

fof(fact_order__eq__iff,axiom,
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( V_xa_2 = V_ya_2
      <=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
          & c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) ) ) ) ).

fof(fact_le__iff__inf,axiom,
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
      <=> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_xa_2),V_ya_2) = V_xa_2 ) ) ).

fof(fact_linorder__linear,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
        | c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) ) ) ).

fof(fact_inf__commute,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x) ) ).

fof(fact_inf__sup__aci_I1_J,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x) ) ).

fof(fact_inf_Ocommute,axiom,
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_a) ) ).

fof(fact_le__fun__def,axiom,
    ! [V_g_2,V_f_2,T_a,T_b] :
      ( class_Orderings_Oord(T_b)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
      <=> ! [B_x] : c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x)) ) ) ).

fof(fact_inf__le2,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y),V_y) ) ).

fof(fact_inf__sup__ord_I2_J,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y),V_y) ) ).

fof(fact_inf__le1,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y),V_x) ) ).

fof(fact_inf__sup__ord_I1_J,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y),V_x) ) ).

fof(fact_inf__idem,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_x) = V_x ) ).

fof(fact_inf_Oidem,axiom,
    ! [V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_a) = V_a ) ).

fof(fact_equalityE,axiom,
    ! [T_a,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => ~ ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
         => ~ c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) ) ) ).

fof(fact_Int__mono,axiom,
    ! [V_D_2,V_B_2,V_C_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_D_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_D_2)) ) ) ).

fof(fact_Int__greatest,axiom,
    ! [V_B_2,V_A_2,V_C_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_B_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) ).

fof(fact_subset__trans,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2) ) ) ).

fof(fact_Int__absorb1,axiom,
    ! [V_A_2,V_B_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2 ) ).

fof(fact_Int__absorb2,axiom,
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2 ) ).

fof(fact_equalityD2,axiom,
    ! [T_a,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) ) ).

fof(fact_equalityD1,axiom,
    ! [T_a,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ).

fof(fact_Int__assoc,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) ).

fof(fact_Int__left__commute,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_C_2)) ).

fof(fact_Int__left__absorb,axiom,
    ! [V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) ).

fof(fact_set__eq__subset,axiom,
    ! [T_a,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
    <=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
        & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) ) ) ).

fof(fact_Int__commute,axiom,
    ! [V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_A_2) ).

fof(fact_Int__lower2,axiom,
    ! [V_B_2,V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_B_2) ).

fof(fact_Int__lower1,axiom,
    ! [V_B_2,V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_A_2) ).

fof(fact_Int__absorb,axiom,
    ! [V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2 ).

fof(fact_subset__refl,axiom,
    ! [V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) ).

fof(fact_mono__Chain,axiom,
    ! [V_s_2,V_r_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_Zorn_OChain(T_a,V_r_2),c_Zorn_OChain(T_a,V_s_2)) ) ).

fof(fact_Un__Int__assoc__eq,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_a] :
      ( hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2))
    <=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2) ) ).

fof(fact_distrib__inf__le,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z))) ) ).

fof(fact_distrib__sup__le,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z))) ) ).

fof(fact_Int__empty__left,axiom,
    ! [V_B_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_Int__empty__right,axiom,
    ! [V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_disjoint__iff__not__equal,axiom,
    ! [V_B_2,V_A_2,T_a] :
      ( hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
         => ! [B_xa] :
              ( hBOOL(hAPP(hAPP(c_member(T_a),B_xa),V_B_2))
             => B_x != B_xa ) ) ) ).

fof(fact_Int__iff,axiom,
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
    <=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2))
        & hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2)) ) ) ).

fof(fact_IntD1,axiom,
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
     => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_A_2)) ) ).

fof(fact_IntD2,axiom,
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
     => hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),V_B_2)) ) ).

fof(fact_inf__top__left,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__top(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),c_Orderings_Otop__class_Otop(T_a)),V_x) = V_x ) ).

fof(fact_inf__top__right,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__top(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Otop__class_Otop(T_a)) = V_x ) ).

fof(fact_inf__eq__top__iff,axiom,
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Obounded__lattice__top(T_a)
     => ( hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_xa_2),V_ya_2) = c_Orderings_Otop__class_Otop(T_a)
      <=> ( V_xa_2 = c_Orderings_Otop__class_Otop(T_a)
          & V_ya_2 = c_Orderings_Otop__class_Otop(T_a) ) ) ) ).

fof(fact_inf__sup__absorb,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = V_x ) ).

fof(fact_sup__inf__absorb,axiom,
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = V_x ) ).

fof(fact_inf__sup__distrib1,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Odistrib__lattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)) ) ).

fof(fact_sup__inf__distrib1,axiom,
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Odistrib__lattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)) ) ).

fof(fact_inf__sup__distrib2,axiom,
    ! [V_x,V_z,V_y,T_a] :
      ( class_Lattices_Odistrib__lattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_z),V_x)) ) ).

fof(fact_sup__inf__distrib2,axiom,
    ! [V_x,V_z,V_y,T_a] :
      ( class_Lattices_Odistrib__lattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_z),V_x)) ) ).

fof(fact_inf__bot__left,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x) = c_Orderings_Obot__class_Obot(T_a) ) ).

fof(fact_inf__bot__right,axiom,
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)) = c_Orderings_Obot__class_Obot(T_a) ) ).

fof(fact_Un__upper1,axiom,
    ! [V_B_2,V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) ).

fof(fact_Un__upper2,axiom,
    ! [V_A_2,V_B_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) ).

fof(fact_subset__Un__eq,axiom,
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
    <=> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2 ) ).

fof(fact_Un__absorb1,axiom,
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2 ) ).

fof(fact_Un__absorb2,axiom,
    ! [V_A_2,V_B_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2 ) ).

fof(fact_Un__least,axiom,
    ! [V_B_2,V_C_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),V_C_2) ) ) ).

fof(fact_Un__mono,axiom,
    ! [V_D_2,V_B_2,V_C_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_D_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_D_2)) ) ) ).

fof(fact_subset__UNIV,axiom,
    ! [V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) ).

fof(fact_subset__insertI,axiom,
    ! [V_a_2,V_B_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)) ).

fof(fact_subset__insertI2,axiom,
    ! [V_b_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),V_B_2)) ) ).

fof(fact_insert__mono,axiom,
    ! [V_a_2,V_D_2,V_C_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_C_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_D_2)) ) ).

fof(fact_Image__Int__subset,axiom,
    ! [V_B_2,V_A_2,V_R_2,T_b,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(c_Relation_OImage(T_b,T_a,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_a,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_b,T_a,V_R_2),V_B_2))) ).

fof(fact_Image__mono,axiom,
    ! [V_A_2,V_A_H_2,V_r_2,V_r_H_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_H_2,V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_H_2,V_A_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Relation_OImage(T_a,T_b,V_r_H_2),V_A_H_2),hAPP(c_Relation_OImage(T_a,T_b,V_r_2),V_A_2)) ) ) ).

fof(fact_quotient__is__empty,axiom,
    ! [V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool))
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ).

fof(fact_quotient__is__empty2,axiom,
    ! [V_r_2,V_A_2,T_a] :
      ( c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)) = c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ).

fof(fact_quotient__empty,axiom,
    ! [V_r_2,T_a] : c_Equiv__Relations_Oquotient(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)) ).

fof(fact_antisym__subset,axiom,
    ! [V_s_2,V_r_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
     => ( c_Relation_Oantisym(T_a,V_s_2)
       => c_Relation_Oantisym(T_a,V_r_2) ) ) ).

fof(fact_inf__Int__eq,axiom,
    ! [V_xa_2,V_S_2,V_R_2,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_R_2)),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_S_2)),V_xa_2))
    <=> hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_R_2),V_S_2))) ) ).

fof(fact_Collect__conj__eq,axiom,
    ! [V_Q_2,V_P_2,T_a] : c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),V_P_2)),V_Q_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),c_Set_OCollect(T_a,V_P_2)),c_Set_OCollect(T_a,V_Q_2)) ).

fof(fact_pred__subset__eq,axiom,
    ! [V_S_2,V_R_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_R_2),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_S_2))
    <=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_R_2,V_S_2) ) ).

fof(fact_quotient__disj,axiom,
    ! [V_Y_2,V_X_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_X_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
       => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_Y_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
         => ( V_X_2 = V_Y_2
            | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_X_2),V_Y_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ) ) ).

fof(fact_Int__insert__left__if1,axiom,
    ! [V_B_2,V_C_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_C_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) ) ).

fof(fact_Int__insert__right__if1,axiom,
    ! [V_B_2,V_A_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) ) ).

fof(fact_Int__insert__left__if0,axiom,
    ! [V_B_2,V_C_2,V_a_2,T_a] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_C_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2) ) ).

fof(fact_Int__insert__right__if0,axiom,
    ! [V_B_2,V_A_2,V_a_2,T_a] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) ) ).

fof(fact_Int__insert__left,axiom,
    ! [V_B_2,V_C_2,V_a_2,T_a] :
      ( ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_C_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2)) )
      & ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_C_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_C_2) ) ) ).

fof(fact_Int__insert__right,axiom,
    ! [V_B_2,V_A_2,V_a_2,T_a] :
      ( ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) )
      & ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) ) ) ).

fof(fact_subset__insert,axiom,
    ! [V_B_2,V_A_2,V_xa_2,T_a] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),V_B_2))
      <=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) ).

fof(fact_insert__subset,axiom,
    ! [V_B_2,V_A_2,V_xa_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),V_A_2),V_B_2)
    <=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_B_2))
        & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) ).

fof(fact_subset__singletonD,axiom,
    ! [V_xa_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
     => ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
        | V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ) ) ).

fof(fact_Int__Collect,axiom,
    ! [V_P_2,V_A_2,V_xa_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),c_Set_OCollect(T_a,V_P_2))))
    <=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
        & hBOOL(hAPP(V_P_2,V_xa_2)) ) ) ).

fof(fact_Int__def,axiom,
    ! [V_B_2,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = c_Set_OCollect(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_A_2))),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),V_B_2))) ).

fof(fact_singleton__quotient,axiom,
    ! [V_r_2,V_xa_2,T_a] : c_Equiv__Relations_Oquotient(T_a,hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_r_2) = hAPP(hAPP(c_Set_Oinsert(tc_fun(T_a,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool))) ).

fof(fact_congruentD,axiom,
    ! [V_z_2,V_ya_2,V_f_2,V_r_2,T_b,T_a] :
      ( c_Equiv__Relations_Ocongruent(T_a,T_b,V_r_2,V_f_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_ya_2),V_z_2)),V_r_2))
       => hAPP(V_f_2,V_ya_2) = hAPP(V_f_2,V_z_2) ) ) ).

fof(fact_quotientI,axiom,
    ! [V_r_2,V_A_2,V_xa_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2))
     => hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_xa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2))) ) ).

fof(fact_equiv__class__subset,axiom,
    ! [V_b_2,V_a_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_b_2)),V_r_2))
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) ) ) ).

fof(fact_quotient__eqI,axiom,
    ! [V_ya_2,V_xa_2,V_Y_2,V_X_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_X_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
       => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_Y_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
         => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_X_2))
           => ( hBOOL(hAPP(hAPP(c_member(T_a),V_ya_2),V_Y_2))
             => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),V_r_2))
               => V_X_2 = V_Y_2 ) ) ) ) ) ) ).

fof(fact_quotient__eq__iff,axiom,
    ! [V_ya_2,V_xa_2,V_Y_2,V_X_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_X_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
       => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_Y_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
         => ( hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_X_2))
           => ( hBOOL(hAPP(hAPP(c_member(T_a),V_ya_2),V_Y_2))
             => ( V_X_2 = V_Y_2
              <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),V_r_2)) ) ) ) ) ) ) ).

fof(fact_subset__Image1__Image1__iff,axiom,
    ! [V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Order__Relation_Opreorder__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),c_Relation_OField(T_a,V_r_2)))
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),c_Relation_OField(T_a,V_r_2)))
         => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
          <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_b_2),V_a_2)),V_r_2)) ) ) ) ) ).

fof(fact_min__ext__compat,axiom,
    ! [V_S_2,V_R_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),V_R_2),V_S_2),V_R_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool),hAPP(hAPP(c_Relation_Orel__comp(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Wellfounded_Omin__ext(T_a,V_R_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool)),c_Wellfounded_Omin__ext(T_a,V_S_2)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool))))),c_Wellfounded_Omin__ext(T_a,V_R_2)) ) ).

fof(fact_congruent2__implies__congruent,axiom,
    ! [V_a_2,V_f_2,V_r2_2,T_c,T_b,V_r1_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r1_2)
     => ( c_Equiv__Relations_Ocongruent2(T_a,T_b,T_c,V_r1_2,V_r2_2,V_f_2)
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
         => c_Equiv__Relations_Ocongruent(T_b,T_c,V_r2_2,hAPP(V_f_2,V_a_2)) ) ) ) ).

fof(fact_quotientE,axiom,
    ! [V_r_2,V_A_2,V_X_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_X_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
     => ~ ! [B_x] :
            ( V_X_2 = hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
           => ~ hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2)) ) ) ).

fof(fact_inf__Int__eq2,axiom,
    ! [V_ya_2,V_xa_2,V_S_2,V_R_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_R_2)),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_S_2)),V_xa_2),V_ya_2))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_xa_2),V_ya_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),V_R_2),V_S_2))) ) ).

fof(fact_pred__subset__eq2,axiom,
    ! [V_S_2,V_R_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_R_2),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_COMBC(T_b,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_a,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_b),tc_fun(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_a,T_b)))),c_Product__Type_OPair(T_a,T_b)))),V_S_2))
    <=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_R_2,V_S_2) ) ).

fof(fact_preorder__on__empty,axiom,
    ! [T_a] : c_Order__Relation_Opreorder__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).

fof(fact_congruent2D,axiom,
    ! [V_z2_2,V_y2_2,V_z1_2,V_y1_2,V_f_2,V_r2_2,V_r1_2,T_c,T_b,T_a] :
      ( c_Equiv__Relations_Ocongruent2(T_a,T_b,T_c,V_r1_2,V_r2_2,V_f_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_y1_2),V_z1_2)),V_r1_2))
       => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y2_2),V_z2_2)),V_r2_2))
         => hAPP(hAPP(V_f_2,V_y1_2),V_y2_2) = hAPP(hAPP(V_f_2,V_z1_2),V_z2_2) ) ) ) ).

fof(fact_partial__order__on__def,axiom,
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Opartial__order__on(T_a,V_A_2,V_r_2)
    <=> ( c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2)
        & c_Relation_Oantisym(T_a,V_r_2) ) ) ).

fof(fact_max__ext__compat,axiom,
    ! [V_S_2,V_R_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),V_R_2),V_S_2),V_R_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool),hAPP(hAPP(c_Relation_Orel__comp(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Wellfounded_Omax__ext(T_a,V_R_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool)),c_Wellfounded_Omax__ext(T_a,V_S_2)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool))))),c_Wellfounded_Omax__ext(T_a,V_R_2)) ) ).

fof(fact_subset__Image__Image__iff,axiom,
    ! [V_B_2,V_A_2,V_r_2,T_a] :
      ( c_Order__Relation_Opreorder__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Relation_OField(T_a,V_r_2))
       => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Relation_OField(T_a,V_r_2))
         => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),V_A_2),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),V_B_2))
          <=> ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
               => ? [B_xa] :
                    ( hBOOL(hAPP(hAPP(c_member(T_a),B_xa),V_B_2))
                    & hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_xa),B_x)),V_r_2)) ) ) ) ) ) ) ).

fof(fact_inf2E,axiom,
    ! [V_ya_2,V_xa_2,V_B_2,V_A_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_A_2),V_B_2),V_xa_2),V_ya_2))
     => ~ ( hBOOL(hAPP(hAPP(V_A_2,V_xa_2),V_ya_2))
         => ~ hBOOL(hAPP(hAPP(V_B_2,V_xa_2),V_ya_2)) ) ) ).

fof(fact_inf2I,axiom,
    ! [T_b,T_a,V_B_2,V_ya_2,V_xa_2,V_A_2] :
      ( hBOOL(hAPP(hAPP(V_A_2,V_xa_2),V_ya_2))
     => ( hBOOL(hAPP(hAPP(V_B_2,V_xa_2),V_ya_2))
       => hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_A_2),V_B_2),V_xa_2),V_ya_2)) ) ) ).

fof(fact_inf2D2,axiom,
    ! [V_ya_2,V_xa_2,V_B_2,V_A_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_A_2),V_B_2),V_xa_2),V_ya_2))
     => hBOOL(hAPP(hAPP(V_B_2,V_xa_2),V_ya_2)) ) ).

fof(fact_inf2D1,axiom,
    ! [V_ya_2,V_xa_2,V_B_2,V_A_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool))),V_A_2),V_B_2),V_xa_2),V_ya_2))
     => hBOOL(hAPP(hAPP(V_A_2,V_xa_2),V_ya_2)) ) ).

fof(fact_rev__predicate2D,axiom,
    ! [V_Q_2,T_b,T_a,V_ya_2,V_xa_2,V_P_2] :
      ( hBOOL(hAPP(hAPP(V_P_2,V_xa_2),V_ya_2))
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool)),V_P_2,V_Q_2)
       => hBOOL(hAPP(hAPP(V_Q_2,V_xa_2),V_ya_2)) ) ) ).

fof(fact_predicate2D,axiom,
    ! [V_ya_2,V_xa_2,V_Q_2,V_P_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool)),V_P_2,V_Q_2)
     => ( hBOOL(hAPP(hAPP(V_P_2,V_xa_2),V_ya_2))
       => hBOOL(hAPP(hAPP(V_Q_2,V_xa_2),V_ya_2)) ) ) ).

fof(fact_max__ext__additive,axiom,
    ! [V_D_2,V_C_2,V_R_2,V_B_2,V_A_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)),c_Wellfounded_Omax__ext(T_a,V_R_2)))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_D_2)),c_Wellfounded_Omax__ext(T_a,V_R_2)))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_D_2))),c_Wellfounded_Omax__ext(T_a,V_R_2))) ) ) ).

fof(fact_max__extp__max__ext__eq,axiom,
    ! [V_xb_2,V_xa_2,V_R_2,T_a] :
      ( c_Wellfounded_Omax__extp(T_a,hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_prod(T_a,T_a)),tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_a),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool),T_a),c_member(tc_prod(T_a,T_a)))),c_Product__Type_OPair(T_a,T_a)))),V_R_2),V_xa_2,V_xb_2)
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),V_xa_2),V_xb_2)),c_Wellfounded_Omax__ext(T_a,V_R_2))) ) ).

fof(fact_in__lex__prod,axiom,
    ! [V_s_2,V_r_2,V_b_H_2,V_a_H_2,V_b_2,V_a_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_H_2),V_b_H_2))),c_Wellfounded_Olex__prod(T_a,T_b,V_r_2,V_s_2)))
    <=> ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_a_H_2)),V_r_2))
        | ( V_a_2 = V_a_H_2
          & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_b_H_2)),V_s_2)) ) ) ) ).

fof(fact_equivp__equiv,axiom,
    ! [V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),V_A_2)
    <=> c_Equiv__Relations_Oequivp(T_a,hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_prod(T_a,T_a)),tc_fun(T_a,tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_prod(T_a,T_a),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_HOL_Obool),T_a),c_member(tc_prod(T_a,T_a)))),c_Product__Type_OPair(T_a,T_a)))),V_A_2)) ) ).

fof(fact_equivp__transp,axiom,
    ! [V_z_2,V_ya_2,V_xa_2,V_R_2,T_a] :
      ( c_Equiv__Relations_Oequivp(T_a,V_R_2)
     => ( hBOOL(hAPP(hAPP(V_R_2,V_xa_2),V_ya_2))
       => ( hBOOL(hAPP(hAPP(V_R_2,V_ya_2),V_z_2))
         => hBOOL(hAPP(hAPP(V_R_2,V_xa_2),V_z_2)) ) ) ) ).

fof(fact_equivp__symp,axiom,
    ! [V_ya_2,V_xa_2,V_R_2,T_a] :
      ( c_Equiv__Relations_Oequivp(T_a,V_R_2)
     => ( hBOOL(hAPP(hAPP(V_R_2,V_xa_2),V_ya_2))
       => hBOOL(hAPP(hAPP(V_R_2,V_ya_2),V_xa_2)) ) ) ).

fof(fact_equivp__reflp,axiom,
    ! [V_xa_2,V_R_2,T_a] :
      ( c_Equiv__Relations_Oequivp(T_a,V_R_2)
     => hBOOL(hAPP(hAPP(V_R_2,V_xa_2),V_xa_2)) ) ).

fof(fact_equivp__def,axiom,
    ! [V_R_2,T_a] :
      ( c_Equiv__Relations_Oequivp(T_a,V_R_2)
    <=> ! [B_x,B_y] :
          ( hBOOL(hAPP(hAPP(V_R_2,B_x),B_y))
        <=> hAPP(V_R_2,B_x) = hAPP(V_R_2,B_y) ) ) ).

fof(fact_identity__equivp,axiom,
    ! [T_a] : c_Equiv__Relations_Oequivp(T_a,c_fequal) ).

fof(fact_same__fstI,axiom,
    ! [T_a,V_R_2,V_ya_2,V_y_H_2,T_b,V_xa_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,V_xa_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_H_2),V_ya_2)),hAPP(V_R_2,V_xa_2)))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_xa_2),V_y_H_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_xa_2),V_ya_2))),c_Recdef_Osame__fst(T_a,T_b,V_P_2,V_R_2))) ) ) ).

fof(fact_quotient__def,axiom,
    ! [V_r_2,V_A_2,T_a] : c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)),T_a),c_Set_Oinsert(tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),T_a),c_Relation_OImage(T_a,T_a,V_r_2)),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Set_Oinsert(T_a)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)))) ).

fof(fact_SUP1__I,axiom,
    ! [T_b,V_b_2,V_B_2,V_A_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(V_B_2,V_a_2),V_b_2))
       => hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)) ) ) ).

fof(fact_SUP__UN__eq,axiom,
    ! [V_xa_2,V_r_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b))),V_r_2)),V_xa_2))
    <=> hBOOL(hAPP(hAPP(c_member(T_b),V_xa_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))),V_r_2))) ) ).

fof(fact_SUP1__iff,axiom,
    ! [V_b_2,V_B_2,V_A_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2))
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
          & hBOOL(hAPP(hAPP(V_B_2,B_x),V_b_2)) ) ) ).

fof(fact_Image__UN,axiom,
    ! [V_B_2,V_A_2,T_c,V_r_2,T_a,T_b] : hAPP(c_Relation_OImage(T_b,T_a,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),T_c),c_Relation_OImage(T_b,T_a,V_r_2)),V_B_2)) ).

fof(fact_Image__eq__UN,axiom,
    ! [V_B_2,V_r_2,T_a,T_b] : hAPP(c_Relation_OImage(T_b,T_a,V_r_2),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),T_b),c_Relation_OImage(T_b,T_a,V_r_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) ).

fof(fact_UN__equiv__class2,axiom,
    ! [V_a2_2,V_a1_2,V_f_2,T_c,V_r2_2,V_A2_2,T_b,V_r1_2,V_A1_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A1_2,V_r1_2)
     => ( c_Equiv__Relations_Oequiv(T_b,V_A2_2,V_r2_2)
       => ( c_Equiv__Relations_Ocongruent2(T_a,T_b,tc_fun(T_c,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2)
         => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a1_2),V_A1_2))
           => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a2_2),V_A2_2))
             => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_a,T_a,V_r1_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a1_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_a),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r2_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a2_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),V_f_2)) = hAPP(hAPP(V_f_2,V_a1_2),V_a2_2) ) ) ) ) ) ).

fof(fact_congruent2__implies__congruent__UN,axiom,
    ! [V_a_2,V_f_2,T_c,V_r2_2,V_A2_2,T_b,V_r1_2,V_A1_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A1_2,V_r1_2)
     => ( c_Equiv__Relations_Oequiv(T_b,V_A2_2,V_r2_2)
       => ( c_Equiv__Relations_Ocongruent2(T_a,T_b,tc_fun(T_c,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2)
         => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A2_2))
           => c_Equiv__Relations_Ocongruent(T_a,tc_fun(T_c,tc_HOL_Obool),V_r1_2,hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_a),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r2_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),V_f_2)) ) ) ) ) ).

fof(fact_UN__equiv__class,axiom,
    ! [V_a_2,V_f_2,T_b,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( c_Equiv__Relations_Ocongruent(T_a,tc_fun(T_b,tc_HOL_Obool),V_r_2,V_f_2)
       => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
         => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))),V_f_2) = hAPP(V_f_2,V_a_2) ) ) ) ).

fof(fact_UN__I,axiom,
    ! [V_B_2,V_b_2,T_b,V_A_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(V_B_2,V_a_2)))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) ) ).

fof(fact_UN__insert,axiom,
    ! [V_B_2,V_A_2,V_a_2,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(V_B_2,V_a_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)) ).

fof(fact_rel__comp__UNION__distrib2,axiom,
    ! [V_s_2,V_r_2,V_I_2,T_d,T_b,T_c,T_a] : hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool)),V_I_2),V_r_2)),V_s_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),T_d),c_Relation_Orel__comp(T_a,T_c,T_b)),V_r_2)),V_s_2)) ).

fof(fact_rel__comp__UNION__distrib,axiom,
    ! [V_r_2,V_I_2,T_d,V_s_2,T_b,T_c,T_a] : hAPP(hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_s_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_I_2),V_r_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),T_d),hAPP(c_Relation_Orel__comp(T_a,T_c,T_b),V_s_2)),V_r_2)) ).

fof(fact_ball__UN,axiom,
    ! [V_P_2,V_B_2,V_A_2,T_b,T_a] :
      ( ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
         => hBOOL(hAPP(V_P_2,B_x)) )
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
         => ! [B_xa] :
              ( hBOOL(hAPP(hAPP(c_member(T_a),B_xa),hAPP(V_B_2,B_x)))
             => hBOOL(hAPP(V_P_2,B_xa)) ) ) ) ).

fof(fact_SUP__commute,axiom,
    ! [V_f_2,V_B_2,T_c,V_A_2,T_b,T_a] :
      ( class_Complete__Lattice_Ocomplete__lattice(T_a)
     => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_a),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_a),T_a,T_b),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_a),V_B_2)),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_a),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,T_a),T_a,T_c),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_a),V_A_2)),hAPP(c_COMBC(T_b,T_c,T_a),V_f_2))) ) ).

fof(fact_UN__UN__flatten,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_c,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),T_c),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool))),V_B_2)),V_C_2)) ).

fof(fact_UN__simps_I9_J,axiom,
    ! [V_C_2,V_B_2,V_A_2,T_c,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),T_c),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool))),V_B_2)),V_C_2)) ).

fof(fact_UN__extend__simps_I9_J,axiom,
    ! [V_C_2,V_B_2,T_c,V_A_2,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool)),T_b),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_a,tc_HOL_Obool))),V_B_2)),V_C_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) ).

fof(fact_SUP__le__iff,axiom,
    ! [V_u_2,V_M_2,V_A_2,T_b,T_a] :
      ( class_Complete__Lattice_Ocomplete__lattice(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_a),V_A_2),V_M_2),V_u_2)
      <=> ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
           => c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_M_2,B_x),V_u_2) ) ) ) ).

fof(fact_UN__iff,axiom,
    ! [V_B_2,V_A_2,T_b,V_b_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)))
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
          & hBOOL(hAPP(hAPP(c_member(T_a),V_b_2),hAPP(V_B_2,B_x))) ) ) ).

fof(fact_SUP__const,axiom,
    ! [V_M_2,T_a,V_A_2,T_b] :
      ( class_Complete__Lattice_Ocomplete__lattice(T_b)
     => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,T_b),V_A_2),c_COMBK(T_b,T_a,V_M_2)) = V_M_2 ) ) ).

fof(fact_UNION__empty__conv_I1_J,axiom,
    ! [V_B_2,V_A_2,T_b,T_a] :
      ( c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2)
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
         => hAPP(V_B_2,B_x) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ).

fof(fact_UN__empty2,axiom,
    ! [V_A_2,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_A_2),c_COMBK(tc_fun(T_a,tc_HOL_Obool),T_b,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).

fof(fact_UN__constant,axiom,
    ! [V_c_2,T_b,T_a,V_A_2] :
      ( ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_c_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )
      & ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_c_2)) = V_c_2 ) ) ).

fof(fact_UNION__empty__conv_I2_J,axiom,
    ! [V_B_2,V_A_2,T_a,T_b] :
      ( hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
         => hAPP(V_B_2,B_x) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ).

fof(fact_UN__subset__iff,axiom,
    ! [V_B_2,V_A_2,V_I_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_I_2),V_A_2),V_B_2)
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_I_2))
         => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(V_A_2,B_x),V_B_2) ) ) ).

fof(fact_UN__Un__distrib,axiom,
    ! [V_B_2,V_A_2,V_I_2,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_I_2),V_B_2)) ).

fof(fact_UN__Un,axiom,
    ! [V_M_2,V_B_2,V_A_2,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_M_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_M_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_B_2),V_M_2)) ).

fof(fact_UN__simps_I5_J,axiom,
    ! [V_B_2,V_A_2,V_C_2,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_B_2)) ).

fof(fact_UN__simps_I4_J,axiom,
    ! [V_B_2,V_A_2,V_C_2,T_a,T_b] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) ).

fof(fact_UN__extend__simps_I5_J,axiom,
    ! [V_B_2,V_C_2,T_b,V_A_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2)),V_B_2)) ).

fof(fact_Int__UN__distrib,axiom,
    ! [V_A_2,V_I_2,T_b,V_B_2,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_B_2)),V_A_2)) ).

fof(fact_UN__extend__simps_I4_J,axiom,
    ! [V_B_2,V_A_2,V_C_2,T_b,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool))),V_A_2)),V_B_2)) ).

fof(fact_Int__UN__distrib2,axiom,
    ! [V_B_2,V_J_2,T_c,V_A_2,V_I_2,T_b,T_a] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_a,tc_HOL_Obool)),V_J_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_a,tc_HOL_Obool),T_b),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_a,tc_HOL_Obool)),V_J_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_a,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_a,tc_HOL_Obool))),T_b),c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool))),V_A_2))),V_B_2))) ).

fof(fact_SUPR__apply,axiom,
    ! [V_xa_2,V_f_2,V_A_2,T_c,T_b,T_a] :
      ( class_Complete__Lattice_Ocomplete__lattice(T_a)
     => hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,T_a)),V_A_2),V_f_2),V_xa_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_a),V_A_2),hAPP(hAPP(c_COMBC(T_b,T_c,T_a),V_f_2),V_xa_2)) ) ).

fof(fact_SUP__subset,axiom,
    ! [V_f_2,V_B_2,V_A_2,T_a,T_b] :
      ( class_Complete__Lattice_Ocomplete__lattice(T_b)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
       => c_Orderings_Oord__class_Oless__eq(T_b,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,T_b),V_A_2),V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,T_b),V_B_2),V_f_2)) ) ) ).

fof(fact_le__SUPI,axiom,
    ! [V_M_2,V_A_2,V_i_2,T_a,T_b] :
      ( class_Complete__Lattice_Ocomplete__lattice(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_i_2),V_A_2))
       => c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_M_2,V_i_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,T_b),V_A_2),V_M_2)) ) ) ).

fof(fact_UN__insert__distrib,axiom,
    ! [V_B_2,V_a_2,T_b,V_A_2,V_u_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_u_2),V_A_2))
     => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_Set_Oinsert(T_b),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) ).

fof(fact_UN__upper,axiom,
    ! [V_B_2,T_b,V_A_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),hAPP(V_B_2,V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) ).

fof(fact_UN__absorb,axiom,
    ! [V_A_2,T_b,V_I_2,V_k_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_k_2),V_I_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(V_A_2,V_k_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2) ) ).

fof(fact_UN__simps_I1_J,axiom,
    ! [V_B_2,V_a_2,T_b,T_a,V_C_2] :
      ( ( V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_Set_Oinsert(T_b),V_a_2)),V_B_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )
      & ( V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_Set_Oinsert(T_b),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) ) ) ).

fof(fact_UN__singleton,axiom,
    ! [V_A_2,T_a] : hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_a,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Set_Oinsert(T_a)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_A_2 ).

fof(fact_UN__extend__simps_I1_J,axiom,
    ! [V_B_2,V_a_2,T_b,T_a,V_C_2] :
      ( ( V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) )
      & ( V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_Set_Oinsert(T_b),V_a_2)),V_B_2)) ) ) ).

fof(fact_UN__simps_I3_J,axiom,
    ! [V_B_2,V_A_2,T_b,T_a,V_C_2] :
      ( ( V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )
      & ( V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) ) ) ).

fof(fact_UN__simps_I2_J,axiom,
    ! [V_B_2,V_A_2,T_b,T_a,V_C_2] :
      ( ( V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )
      & ( V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) ) ) ).

fof(fact_UN__extend__simps_I3_J,axiom,
    ! [V_B_2,V_A_2,T_b,T_a,V_C_2] :
      ( ( V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) = V_A_2 )
      & ( V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_a),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2)) ) ) ).

fof(fact_UN__extend__simps_I2_J,axiom,
    ! [V_B_2,V_A_2,T_b,T_a,V_C_2] :
      ( ( V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = V_B_2 )
      & ( V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_a,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_a),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) ) ) ).

fof(fact_UN__equiv__class__inject,axiom,
    ! [V_Y_2,V_X_2,V_f_2,T_b,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( c_Equiv__Relations_Ocongruent(T_a,tc_fun(T_b,tc_HOL_Obool),V_r_2,V_f_2)
       => ( hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_f_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_Y_2),V_f_2)
         => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_X_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
           => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_Y_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
             => ( ! [B_x] :
                    ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
                   => ! [B_y] :
                        ( hBOOL(hAPP(hAPP(c_member(T_a),B_y),V_A_2))
                       => ( hAPP(V_f_2,B_x) = hAPP(V_f_2,B_y)
                         => hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_x),B_y)),V_r_2)) ) ) )
               => V_X_2 = V_Y_2 ) ) ) ) ) ) ).

fof(fact_UN__equiv__class__type2,axiom,
    ! [V_B_2,V_X2_2,V_X1_2,V_f_2,T_c,V_r2_2,V_A2_2,T_b,V_r1_2,V_A1_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A1_2,V_r1_2)
     => ( c_Equiv__Relations_Oequiv(T_b,V_A2_2,V_r2_2)
       => ( c_Equiv__Relations_Ocongruent2(T_a,T_b,tc_fun(T_c,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2)
         => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_X1_2),c_Equiv__Relations_Oquotient(T_a,V_A1_2,V_r1_2)))
           => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X2_2),c_Equiv__Relations_Oquotient(T_b,V_A2_2,V_r2_2)))
             => ( ! [B_x1] :
                    ( hBOOL(hAPP(hAPP(c_member(T_a),B_x1),V_A1_2))
                   => ! [B_x2] :
                        ( hBOOL(hAPP(hAPP(c_member(T_b),B_x2),V_A2_2))
                       => hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(V_f_2,B_x1),B_x2)),V_B_2)) ) )
               => hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_c,tc_HOL_Obool)),V_X1_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_a),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_X2_2)),V_f_2))),V_B_2)) ) ) ) ) ) ) ).

fof(fact_SUP2__I,axiom,
    ! [T_c,T_b,V_c_2,V_b_2,V_B_2,V_A_2,V_a_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(hAPP(V_B_2,V_a_2),V_b_2),V_c_2))
       => hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)) ) ) ).

fof(fact_SUP__UN__eq2,axiom,
    ! [V_ya_2,V_xa_2,V_r_2,T_c,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),T_a),hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c))))),V_r_2)),V_xa_2),V_ya_2))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_xa_2),V_ya_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))),V_r_2))) ) ).

fof(fact_UN__equiv__class__type,axiom,
    ! [V_B_2,V_X_2,V_f_2,T_b,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( c_Equiv__Relations_Ocongruent(T_a,tc_fun(T_b,tc_HOL_Obool),V_r_2,V_f_2)
       => ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,tc_HOL_Obool)),V_X_2),c_Equiv__Relations_Oquotient(T_a,V_A_2,V_r_2)))
         => ( ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
               => hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),hAPP(V_f_2,B_x)),V_B_2)) )
           => hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_f_2)),V_B_2)) ) ) ) ) ).

fof(fact_Id__on__def,axiom,
    ! [V_A_2,T_a] : c_Relation_OId__on(T_a,V_A_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_prod(T_a,T_a),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),T_a),c_Set_Oinsert(tc_prod(T_a,T_a))),hAPP(hAPP(c_COMBS(T_a,T_a,tc_prod(T_a,T_a)),c_Product__Type_OPair(T_a,T_a)),c_COMBI(T_a)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)))) ).

fof(fact_SUP2__iff,axiom,
    ! [V_c_2,V_b_2,V_B_2,V_A_2,T_c,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2))
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
          & hBOOL(hAPP(hAPP(hAPP(V_B_2,B_x),V_b_2),V_c_2)) ) ) ).

fof(fact_Id__on__empty,axiom,
    ! [T_a] : c_Relation_OId__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) ).

fof(fact_Image__Id__on,axiom,
    ! [V_B_2,V_A_2,T_a] : hAPP(c_Relation_OImage(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool)),V_A_2),V_B_2) ).

fof(fact_Domain__Id__on,axiom,
    ! [V_A_2,T_a] : c_Relation_ODomain(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2 ).

fof(fact_Range__Id__on,axiom,
    ! [V_A_2,T_a] : c_Relation_ORange(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2 ).

fof(fact_refl__on__Id__on,axiom,
    ! [V_A_2,T_a] : c_Relation_Orefl__on(T_a,V_A_2,c_Relation_OId__on(T_a,V_A_2)) ).

fof(fact_antisym__Id__on,axiom,
    ! [V_A_2,T_a] : c_Relation_Oantisym(T_a,c_Relation_OId__on(T_a,V_A_2)) ).

fof(fact_Id__on__def_H,axiom,
    ! [V_A_2,T_a] : c_Relation_OId__on(T_a,V_A_2) = c_Product__Type_Oprod_Oprod__case(T_a,T_a,tc_HOL_Obool,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_fun(T_a,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_HOL_Obool,tc_fun(T_a,tc_HOL_Obool)),T_a),c_COMBC(T_a,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_a),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_a),c_fconj)),c_fequal))),V_A_2)) ).

fof(fact_Id__on__eqI,axiom,
    ! [V_A_2,T_a,V_b_2,V_a_2] :
      ( V_a_2 = V_b_2
     => ( hBOOL(hAPP(hAPP(c_member(T_a),V_a_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_a_2),V_b_2)),c_Relation_OId__on(T_a,V_A_2))) ) ) ).

fof(fact_Id__on__iff,axiom,
    ! [V_A_2,V_ya_2,V_xa_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_xa_2),V_ya_2)),c_Relation_OId__on(T_a,V_A_2)))
    <=> ( V_xa_2 = V_ya_2
        & hBOOL(hAPP(hAPP(c_member(T_a),V_xa_2),V_A_2)) ) ) ).

fof(fact_Id__onE,axiom,
    ! [V_A_2,V_c_2,T_a] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),V_c_2),c_Relation_OId__on(T_a,V_A_2)))
     => ~ ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
           => V_c_2 != hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_x),B_x) ) ) ).

fof(fact_Powp__mono,axiom,
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_Predicate_OPowp(T_a,V_A_2),c_Predicate_OPowp(T_a,V_B_2)) ) ).

fof(fact_Powp__def,axiom,
    ! [V_B_2,V_A_2,T_a] :
      ( hBOOL(hAPP(c_Predicate_OPowp(T_a,V_A_2),V_B_2))
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_B_2))
         => hBOOL(hAPP(V_A_2,B_x)) ) ) ).

fof(fact_well__order__on__empty,axiom,
    ! [T_a] : c_Order__Relation_Owell__order__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).

fof(fact_reduction__pairI,axiom,
    ! [V_S_2,V_R_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_R_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),hAPP(hAPP(c_Relation_Orel__comp(T_a,T_a,T_a),V_R_2),V_S_2),V_R_2)
       => c_FunDef_Oreduction__pair(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)),V_R_2),V_S_2)) ) ) ).

fof(fact_wf__empty,axiom,
    ! [T_a] : c_Wellfounded_Owf(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).

fof(fact_wf__inv__image,axiom,
    ! [V_f_2,T_b,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Relation_Oinv__image(T_a,T_b),V_r_2),V_f_2)) ) ).

%----Arity declarations (26)
fof(arity_HOL__Obool__Lattices_Obounded__lattice,axiom,
    class_Lattices_Obounded__lattice(tc_HOL_Obool) ).

fof(arity_fun__Lattices_Obounded__lattice,axiom,
    ! [T_2,T_1] :
      ( class_Lattices_Obounded__lattice(T_1)
     => class_Lattices_Obounded__lattice(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Complete__Lattice_Ocomplete__lattice,axiom,
    ! [T_2,T_1] :
      ( class_Complete__Lattice_Ocomplete__lattice(T_1)
     => class_Complete__Lattice_Ocomplete__lattice(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Lattices_Obounded__lattice__top,axiom,
    ! [T_2,T_1] :
      ( class_Lattices_Obounded__lattice(T_1)
     => class_Lattices_Obounded__lattice__top(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Lattices_Obounded__lattice__bot,axiom,
    ! [T_2,T_1] :
      ( class_Lattices_Obounded__lattice(T_1)
     => class_Lattices_Obounded__lattice__bot(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Lattices_Osemilattice__sup,axiom,
    ! [T_2,T_1] :
      ( class_Lattices_Olattice(T_1)
     => class_Lattices_Osemilattice__sup(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Lattices_Osemilattice__inf,axiom,
    ! [T_2,T_1] :
      ( class_Lattices_Olattice(T_1)
     => class_Lattices_Osemilattice__inf(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Lattices_Odistrib__lattice,axiom,
    ! [T_2,T_1] :
      ( class_Lattices_Odistrib__lattice(T_1)
     => class_Lattices_Odistrib__lattice(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Orderings_Opreorder,axiom,
    ! [T_2,T_1] :
      ( class_Orderings_Opreorder(T_1)
     => class_Orderings_Opreorder(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Lattices_Olattice,axiom,
    ! [T_2,T_1] :
      ( class_Lattices_Olattice(T_1)
     => class_Lattices_Olattice(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Orderings_Oorder,axiom,
    ! [T_2,T_1] :
      ( class_Orderings_Oorder(T_1)
     => class_Orderings_Oorder(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Orderings_Otop,axiom,
    ! [T_2,T_1] :
      ( class_Orderings_Otop(T_1)
     => class_Orderings_Otop(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Orderings_Oord,axiom,
    ! [T_2,T_1] :
      ( class_Orderings_Oord(T_1)
     => class_Orderings_Oord(tc_fun(T_2,T_1)) ) ).

fof(arity_fun__Orderings_Obot,axiom,
    ! [T_2,T_1] :
      ( class_Orderings_Obot(T_1)
     => class_Orderings_Obot(tc_fun(T_2,T_1)) ) ).

fof(arity_HOL__Obool__Complete__Lattice_Ocomplete__lattice,axiom,
    class_Complete__Lattice_Ocomplete__lattice(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Lattices_Obounded__lattice__top,axiom,
    class_Lattices_Obounded__lattice__top(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Lattices_Obounded__lattice__bot,axiom,
    class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Lattices_Osemilattice__sup,axiom,
    class_Lattices_Osemilattice__sup(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Lattices_Osemilattice__inf,axiom,
    class_Lattices_Osemilattice__inf(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Lattices_Odistrib__lattice,axiom,
    class_Lattices_Odistrib__lattice(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Orderings_Opreorder,axiom,
    class_Orderings_Opreorder(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Lattices_Olattice,axiom,
    class_Lattices_Olattice(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Orderings_Oorder,axiom,
    class_Orderings_Oorder(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Orderings_Otop,axiom,
    class_Orderings_Otop(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Orderings_Oord,axiom,
    class_Orderings_Oord(tc_HOL_Obool) ).

fof(arity_HOL__Obool__Orderings_Obot,axiom,
    class_Orderings_Obot(tc_HOL_Obool) ).

%----Helper facts (20)
fof(help_c__COMBI__1,axiom,
    ! [V_P,T_a] : hAPP(c_COMBI(T_a),V_P) = V_P ).

fof(help_c__COMBK__1,axiom,
    ! [V_Q,V_P,T_b,T_a] : hAPP(c_COMBK(T_a,T_b,V_P),V_Q) = V_P ).

fof(help_c__COMBB__1,axiom,
    ! [V_R_2,V_Q_2,V_P_2,T_c,T_a,T_b] : hAPP(hAPP(hAPP(c_COMBB(T_b,T_a,T_c),V_P_2),V_Q_2),V_R_2) = hAPP(V_P_2,hAPP(V_Q_2,V_R_2)) ).

fof(help_c__COMBC__1,axiom,
    ! [V_R_2,V_Q_2,V_P_2,T_a,T_c,T_b] : hAPP(hAPP(hAPP(c_COMBC(T_b,T_c,T_a),V_P_2),V_Q_2),V_R_2) = hAPP(hAPP(V_P_2,V_R_2),V_Q_2) ).

fof(help_c__COMBS__1,axiom,
    ! [V_R_2,V_Q_2,V_P_2,T_a,T_c,T_b] : hAPP(hAPP(hAPP(c_COMBS(T_b,T_c,T_a),V_P_2),V_Q_2),V_R_2) = hAPP(hAPP(V_P_2,V_R_2),hAPP(V_Q_2,V_R_2)) ).

fof(help_c__fequal__1,axiom,
    ! [V_y_2,V_x_2] :
      ( ~ hBOOL(hAPP(hAPP(c_fequal,V_x_2),V_y_2))
      | V_x_2 = V_y_2 ) ).

fof(help_c__fequal__2,axiom,
    ! [V_y_2,V_x_2] :
      ( V_x_2 != V_y_2
      | hBOOL(hAPP(hAPP(c_fequal,V_x_2),V_y_2)) ) ).

fof(help_c__fFalse__1,axiom,
    ~ hBOOL(c_fFalse) ).

fof(help_c__fTrue__1,axiom,
    hBOOL(c_fTrue) ).

fof(help_c__fNot__1,axiom,
    ! [V_P_2] :
      ( ~ hBOOL(hAPP(c_fNot,V_P_2))
      | ~ hBOOL(V_P_2) ) ).

fof(help_c__fNot__2,axiom,
    ! [V_P_2] :
      ( ~ ~ hBOOL(V_P_2)
      | hBOOL(hAPP(c_fNot,V_P_2)) ) ).

fof(help_c__fconj__1,axiom,
    ! [V_Q_2,V_P_2] :
      ( ~ hBOOL(V_P_2)
      | ~ hBOOL(V_Q_2)
      | hBOOL(hAPP(hAPP(c_fconj,V_P_2),V_Q_2)) ) ).

fof(help_c__fconj__2,axiom,
    ! [V_Q_2,V_P_2] :
      ( ~ hBOOL(hAPP(hAPP(c_fconj,V_P_2),V_Q_2))
      | hBOOL(V_P_2) ) ).

fof(help_c__fconj__3,axiom,
    ! [V_Q_2,V_P_2] :
      ( ~ hBOOL(hAPP(hAPP(c_fconj,V_P_2),V_Q_2))
      | hBOOL(V_Q_2) ) ).

fof(help_c__fdisj__1,axiom,
    ! [V_Q_2,V_P_2] :
      ( ~ hBOOL(V_P_2)
      | hBOOL(hAPP(hAPP(c_fdisj,V_P_2),V_Q_2)) ) ).

fof(help_c__fdisj__2,axiom,
    ! [V_P_2,V_Q_2] :
      ( ~ hBOOL(V_Q_2)
      | hBOOL(hAPP(hAPP(c_fdisj,V_P_2),V_Q_2)) ) ).

fof(help_c__fdisj__3,axiom,
    ! [V_Q_2,V_P_2] :
      ( ~ hBOOL(hAPP(hAPP(c_fdisj,V_P_2),V_Q_2))
      | hBOOL(V_P_2)
      | hBOOL(V_Q_2) ) ).

fof(help_c__fimplies__1,axiom,
    ! [V_Q_2,V_P_2] :
      ( hBOOL(V_P_2)
      | hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2)) ) ).

fof(help_c__fimplies__2,axiom,
    ! [V_P_2,V_Q_2] :
      ( ~ hBOOL(V_Q_2)
      | hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2)) ) ).

fof(help_c__fimplies__3,axiom,
    ! [V_Q_2,V_P_2] :
      ( ~ hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2))
      | ~ hBOOL(V_P_2)
      | hBOOL(V_Q_2) ) ).

%----Conjectures (3)
fof(conj_0,hypothesis,
    v_x != v_y ).

fof(conj_1,hypothesis,
    hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),v_L),c_Set_OCollect(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),hAPP(hAPP(c_COMBS(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_fconj),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_HOL_Obool,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_HOL_OAll(tc_Arrow__Order__Mirabelle_Oalt)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_HOL_Obool,tc_Arrow__Order__Mirabelle_Oalt),c_HOL_OAll(tc_Arrow__Order__Mirabelle_Oalt))),hAPP(hAPP(c_COMBS(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool))),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool))),tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool))),tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBS(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_fimplies))),hAPP(c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),c_COMBC(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool)),tc_Arrow__Order__Mirabelle_Oalt),hAPP(c_COMBB(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_fun(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt),c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt)))))))),hAPP(hAPP(c_COMBB(tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_HOL_Obool))),tc_fun(tc_Arrow__Order__Mirabelle_Oalt,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).

fof(conj_2,conjecture,
    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).

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