TPTP Problem File: SCT126+1.p
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%------------------------------------------------------------------------------
% File : SCT126+1 : TPTP v9.0.0. Released v5.2.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem 433102, 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_433102.500.p [Bla11]
% Status : Theorem
% Rating : 0.24 v9.0.0, 0.28 v8.1.0, 0.33 v7.5.0, 0.41 v7.4.0, 0.37 v7.3.0, 0.31 v7.2.0, 0.28 v7.1.0, 0.35 v7.0.0, 0.37 v6.4.0, 0.42 v6.3.0, 0.38 v6.2.0, 0.44 v6.1.0, 0.50 v6.0.0, 0.43 v5.5.0, 0.63 v5.4.0, 0.68 v5.3.0, 0.67 v5.2.0
% Syntax : Number of formulae : 525 ( 131 unt; 0 def)
% Number of atoms : 1288 ( 349 equ)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 898 ( 135 ~; 22 |; 50 &)
% ( 101 <=>; 590 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of predicates : 32 ( 31 usr; 0 prp; 1-6 aty)
% Number of functors : 65 ( 65 usr; 17 con; 0-6 aty)
% Number of variables : 2091 (2074 !; 17 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-03-01 11:14:28
% : Renamed from SWW128+1
%------------------------------------------------------------------------------
%----Relevant facts (494)
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_assms_I3_J,axiom,
c_Arrow__Order__Mirabelle_OIIA(v_F) ).
fof(fact_u,axiom,
c_Arrow__Order__Mirabelle_Ounanimity(v_F) ).
fof(fact__096Q_A_058_AProf_096,axiom,
hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),v_Q____),c_Arrow__Order__Mirabelle_OProf)) ).
fof(fact_mkbot__Lin,axiom,
! [V_x_2,V_L_2] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_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_L_2,V_x_2)),c_Arrow__Order__Mirabelle_OLin)) ) ).
fof(fact_mktop__Lin,axiom,
! [V_x_2,V_L_2] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_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_L_2,V_x_2)),c_Arrow__Order__Mirabelle_OLin)) ) ).
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_above__Lin,axiom,
! [V_L_2,V_y_2,V_x_2] :
( V_x_2 != V_y_2
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_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)),hAPP(hAPP(hAPP(c_Arrow__Order__Mirabelle_Oabove,V_L_2),V_x_2),V_y_2)),c_Arrow__Order__Mirabelle_OLin)) ) ) ).
fof(fact_below__Lin,axiom,
! [V_L_2,V_y_2,V_x_2] :
( V_x_2 != V_y_2
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_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_L_2,V_x_2,V_y_2)),c_Arrow__Order__Mirabelle_OLin)) ) ) ).
fof(fact_d1_I2_J,axiom,
v_a_H____ != v_b_H____ ).
fof(fact_linear__alt,axiom,
? [B_L] : hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),B_L),c_Arrow__Order__Mirabelle_OLin)) ).
fof(fact_mem__def,axiom,
! [V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
<=> hBOOL(hAPP(V_A_2,V_x_2)) ) ).
fof(fact__096P_H_A_058_AProf_096,axiom,
hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),v_P_H____),c_Arrow__Order__Mirabelle_OProf)) ).
fof(fact_d2_I2_J,axiom,
v_b____ != v_a_H____ ).
fof(fact_d2_I1_J,axiom,
v_a____ != v_b_H____ ).
fof(fact__096P_A_058_AProf_096,axiom,
hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),v_P____),c_Arrow__Order__Mirabelle_OProf)) ).
fof(fact_eq__mem__trans,axiom,
! [V_A_2,T_a,V_ba_2,V_aa_2] :
( V_aa_2 = V_ba_2
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2)) ) ) ).
fof(fact_eqelem__imp__iff,axiom,
! [V_A_2,T_a,V_y_2,V_x_2] :
( V_x_2 = V_y_2
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
<=> hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_A_2)) ) ) ).
fof(fact_eqset__imp__iff,axiom,
! [V_x_2,T_a,V_B_2,V_A_2] :
( V_A_2 = V_B_2
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
<=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_B_2)) ) ) ).
fof(fact_dictator__def,axiom,
! [V_i_2,V_Fa_2] :
( c_Arrow__Order__Mirabelle_Odictator(V_Fa_2,V_i_2)
<=> ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),B_x),c_Arrow__Order__Mirabelle_OProf))
=> hAPP(V_Fa_2,B_x) = hAPP(B_x,V_i_2) ) ) ).
fof(fact_in__below,axiom,
! [V_y_2,V_x_2,V_L_2,V_ba_2,V_aa_2] :
( V_aa_2 != V_ba_2
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_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_x_2),V_y_2)),c_Arrow__Order__Mirabelle_Obelow(V_L_2,V_aa_2,V_ba_2)))
<=> ( V_x_2 != V_y_2
& ( V_y_2 = V_aa_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_x_2),V_ba_2)),V_L_2)) )
& ( V_y_2 != V_aa_2
=> ( ( V_x_2 = V_aa_2
=> ( V_y_2 = V_ba_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_ba_2),V_y_2)),V_L_2)) ) )
& ( V_x_2 != V_aa_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_x_2),V_y_2)),V_L_2)) ) ) ) ) ) ) ) ).
fof(fact_in__above,axiom,
! [V_y_2,V_x_2,V_L_2,V_ba_2,V_aa_2] :
( V_aa_2 != V_ba_2
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_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_x_2),V_y_2)),hAPP(hAPP(hAPP(c_Arrow__Order__Mirabelle_Oabove,V_L_2),V_aa_2),V_ba_2)))
<=> ( V_x_2 != V_y_2
& ( V_x_2 = V_ba_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_aa_2),V_y_2)),V_L_2)) )
& ( V_x_2 != V_ba_2
=> ( ( V_y_2 = V_ba_2
=> ( V_x_2 = V_aa_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_x_2),V_aa_2)),V_L_2)) ) )
& ( V_y_2 != V_ba_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_x_2),V_y_2)),V_L_2)) ) ) ) ) ) ) ) ).
fof(fact_Q__def,axiom,
! [V_i_2] : hAPP(v_Q____,V_i_2) = c_HOL_OLet(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,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(hAPP(c_If(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),hAPP(hAPP(c_fequal,v_a____),v_a_H____)),hAPP(v_P____,V_i_2)),c_Arrow__Order__Mirabelle_Obelow(hAPP(v_P____,V_i_2),v_a_H____,v_a____)),hAPP(hAPP(c_COMBS(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,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),tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_If(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),hAPP(hAPP(c_fequal,v_b____),v_b_H____))),hAPP(hAPP(c_COMBC(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_Arrow__Order__Mirabelle_Oalt,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),c_Arrow__Order__Mirabelle_Oabove),v_b____)),v_b_H____))) ).
fof(fact_assms_I1_J,axiom,
hBOOL(hAPP(hAPP(c_member(tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,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))),v_F),c_FuncSet_OPi(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,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_Arrow__Order__Mirabelle_OProf,c_COMBK(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_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Arrow__Order__Mirabelle_OLin)))) ).
fof(fact_eq__mem,axiom,
! [V_y_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),hAPP(c_fequal,V_y_2)))
<=> V_x_2 = V_y_2 ) ).
fof(fact_converse__in__Lin,axiom,
! [V_L_2] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Relation_Oconverse(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_L_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)),V_L_2),c_Arrow__Order__Mirabelle_OLin)) ) ).
fof(fact_Lin__irrefl,axiom,
! [V_ba_2,V_aa_2,V_L_2] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_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_aa_2),V_ba_2)),V_L_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_ba_2),V_aa_2)),V_L_2)) ) ) ).
fof(fact_notin__Lin__iff,axiom,
! [V_y_2,V_x_2,V_L_2] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_2),c_Arrow__Order__Mirabelle_OLin))
=> ( V_x_2 != V_y_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_x_2),V_y_2)),V_L_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_y_2),V_x_2)),V_L_2)) ) ) ) ).
fof(fact_const__Lin__Prof,axiom,
! [V_L_2] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),V_L_2),c_Arrow__Order__Mirabelle_OLin))
=> hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),c_COMBK(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),tc_Arrow__Order__Mirabelle_Oindi,V_L_2)),c_Arrow__Order__Mirabelle_OProf)) ) ).
fof(fact__C1_C,axiom,
! [B_i] :
( 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____),v_b____)),hAPP(v_P____,B_i)))
<=> 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_H____),v_b_H____)),hAPP(v_P_H____,B_i))) ) ).
fof(fact_d1_I1_J,axiom,
v_a____ != v_b____ ).
fof(fact__096a_A_060_092_060_094bsub_062F_AP_092_060_094esub_062_Ab_096,axiom,
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____),v_b____)),hAPP(v_F,v_P____))) ).
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(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(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,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(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(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,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_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_in__mktop,axiom,
! [V_z_2,V_L_2,V_y_2,V_x_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_x_2),V_y_2)),c_Arrow__Order__Mirabelle_Omktop(V_L_2,V_z_2)))
<=> ( V_x_2 != V_z_2
& ( V_y_2 = V_z_2
=> V_x_2 != V_y_2 )
& ( V_y_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_x_2),V_y_2)),V_L_2)) ) ) ) ).
fof(fact_in__mkbot,axiom,
! [V_z_2,V_L_2,V_y_2,V_x_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_x_2),V_y_2)),c_Arrow__Order__Mirabelle_Omkbot(V_L_2,V_z_2)))
<=> ( V_y_2 != V_z_2
& ( V_x_2 = V_z_2
=> V_x_2 != V_y_2 )
& ( V_x_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_x_2),V_y_2)),V_L_2)) ) ) ) ).
fof(fact_PiE,axiom,
! [V_x_2,V_B_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2)))
=> ( ~ hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_B_2,V_x_2)))
=> ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2)) ) ) ).
fof(fact_dictatorI,axiom,
! [V_i_2,V_Fa_2] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,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))),V_Fa_2),c_FuncSet_OPi(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,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_Arrow__Order__Mirabelle_OProf,c_COMBK(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_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),c_Arrow__Order__Mirabelle_OLin))))
=> ( ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),B_x),c_Arrow__Order__Mirabelle_OProf))
=> ! [B_a,B_b] :
( B_a != B_b
=> ( 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),B_a),B_b)),hAPP(B_x,V_i_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),B_a),B_b)),hAPP(V_Fa_2,B_x))) ) ) )
=> c_Arrow__Order__Mirabelle_Odictator(V_Fa_2,V_i_2) ) ) ).
fof(fact_funcset__mem,axiom,
! [V_x_2,V_B_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_B_2))))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_x_2)),V_B_2)) ) ) ).
fof(fact_Pi__mem,axiom,
! [V_x_2,V_B_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2)))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_B_2,V_x_2))) ) ) ).
fof(fact_converse__iff,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Relation_Oconverse(T_b,T_a,V_r_2)))
<=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_ba_2),V_aa_2)),V_r_2)) ) ).
fof(fact_converseI,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_ba_2),V_aa_2)),c_Relation_Oconverse(T_a,T_b,V_r_2))) ) ).
fof(fact_converseD,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Relation_Oconverse(T_b,T_a,V_r_2)))
=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_ba_2),V_aa_2)),V_r_2)) ) ).
fof(fact_funcset__id,axiom,
! [V_A_2,T_a] : hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_a)),c_COMBI(T_a)),c_FuncSet_OPi(T_a,T_a,V_A_2,c_COMBK(tc_fun(T_a,tc_HOL_Obool),T_a,V_A_2)))) ).
fof(fact_IIA__def,axiom,
! [V_Fa_2] :
( c_Arrow__Order__Mirabelle_OIIA(V_Fa_2)
<=> ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),B_x),c_Arrow__Order__Mirabelle_OProf))
=> ! [B_xa] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),B_xa),c_Arrow__Order__Mirabelle_OProf))
=> ! [B_a,B_b] :
( ! [B_i] :
( 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),B_a),B_b)),hAPP(B_x,B_i)))
<=> 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),B_a),B_b)),hAPP(B_xa,B_i))) )
=> ( 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),B_a),B_b)),hAPP(V_Fa_2,B_x)))
<=> 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),B_a),B_b)),hAPP(V_Fa_2,B_xa))) ) ) ) ) ) ).
fof(fact_unanimity__def,axiom,
! [V_Fa_2] :
( c_Arrow__Order__Mirabelle_Ounanimity(V_Fa_2)
<=> ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool))),B_x),c_Arrow__Order__Mirabelle_OProf))
=> ! [B_a,B_b] :
( ! [B_i] : 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),B_a),B_b)),hAPP(B_x,B_i)))
=> 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),B_a),B_b)),hAPP(V_Fa_2,B_x))) ) ) ) ).
fof(fact_converse__converse,axiom,
! [V_r_2,T_a,T_b] : c_Relation_Oconverse(T_b,T_a,c_Relation_Oconverse(T_a,T_b,V_r_2)) = 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_Pi__I,axiom,
! [V_B_2,V_f_2,T_b,V_A_2,T_a] :
( ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,B_x)),hAPP(V_B_2,B_x))) )
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2))) ) ).
fof(fact_in__rel__def,axiom,
! [V_y_2,V_x_2,V_R_2,T_b,T_a] :
( c_FunDef_Oin__rel(T_a,T_b,V_R_2,V_x_2,V_y_2)
<=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x_2),V_y_2)),V_R_2)) ) ).
fof(fact_converseE,axiom,
! [V_r_2,V_yx_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),V_yx_2),c_Relation_Oconverse(T_b,T_a,V_r_2)))
=> ~ ! [B_x,B_y] :
( V_yx_2 = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_y),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),B_y)),V_r_2)) ) ) ).
fof(fact_same__fstI,axiom,
! [T_a,V_R_2,V_y_2,V_y_H_2,T_b,V_x_2,V_Pa_2] :
( hBOOL(hAPP(V_Pa_2,V_x_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_y_2)),hAPP(V_R_2,V_x_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_x_2),V_y_H_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x_2),V_y_2))),c_Recdef_Osame__fst(T_a,T_b,V_Pa_2,V_R_2))) ) ) ).
fof(fact_RangeI,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),V_ba_2),c_Relation_ORange(T_a,T_b,V_r_2))) ) ).
fof(fact_complete__Lin,axiom,
! [V_ba_2,V_aa_2] :
( V_aa_2 != V_ba_2
=> ? [B_x] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),B_x),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_aa_2),V_ba_2)),B_x)) ) ) ).
fof(fact_Id__on__eqI,axiom,
! [V_A_2,T_a,V_ba_2,V_aa_2] :
( V_aa_2 = V_ba_2
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_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_aa_2),V_ba_2)),c_Relation_OId__on(T_a,V_A_2))) ) ) ).
fof(fact_Id__on__iff,axiom,
! [V_A_2,V_y_2,V_x_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_x_2),V_y_2)),c_Relation_OId__on(T_a,V_A_2)))
<=> ( V_x_2 = V_y_2
& hBOOL(hAPP(hAPP(c_member(T_a),V_x_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_converse__Id__on,axiom,
! [V_A_2,T_a] : c_Relation_Oconverse(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = c_Relation_OId__on(T_a,V_A_2) ).
fof(fact_Range__iff,axiom,
! [V_r_2,T_b,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_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_aa_2)),V_r_2)) ) ).
fof(fact_RangeP__Range__eq,axiom,
! [V_x_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(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(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,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_x_2)
<=> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Relation_ORange(T_a,T_b,V_r_2))) ) ).
fof(fact_in__lex__prod,axiom,
! [V_s_2,V_r_2,V_b_Ha_2,V_a_Ha_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_Ha_2),V_b_Ha_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_aa_2),V_a_Ha_2)),V_r_2))
| ( V_aa_2 = V_a_Ha_2
& hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_ba_2),V_b_Ha_2)),V_s_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_RangeE,axiom,
! [V_r_2,T_b,V_ba_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_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_ba_2)),V_r_2)) ) ).
fof(fact_funcsetI,axiom,
! [V_B_2,V_f_2,T_b,V_A_2,T_a] :
( ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,B_x)),V_B_2)) )
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_B_2)))) ) ).
fof(fact_DomainI,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),c_Relation_ODomain(T_a,T_b,V_r_2))) ) ).
fof(fact_conversep__converse__eq,axiom,
! [V_y_2,V_x_2,V_r_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_Predicate_Oconversep(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(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(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,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_x_2),V_y_2))
<=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_x_2),V_y_2)),c_Relation_Oconverse(T_a,T_b,V_r_2))) ) ).
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_Pi__I_H,axiom,
! [V_B_2,V_f_2,T_b,V_A_2,T_a] :
( ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,B_x)),hAPP(V_B_2,B_x))) )
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2))) ) ).
fof(fact_RangeP_Ointros,axiom,
! [T_b,T_a,V_ba_2,V_aa_2,V_r_2] :
( hBOOL(hAPP(hAPP(V_r_2,V_aa_2),V_ba_2))
=> c_Predicate_ORangeP(T_a,T_b,V_r_2,V_ba_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_conversepD,axiom,
! [V_ba_2,V_aa_2,V_r_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_a,T_b,V_r_2),V_aa_2),V_ba_2))
=> hBOOL(hAPP(hAPP(V_r_2,V_ba_2),V_aa_2)) ) ).
fof(fact_conversep_Ointros,axiom,
! [T_b,T_a,V_ba_2,V_aa_2,V_r_2] :
( hBOOL(hAPP(hAPP(V_r_2,V_aa_2),V_ba_2))
=> hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_a,T_b,V_r_2),V_ba_2),V_aa_2)) ) ).
fof(fact_conversep_Oequations,axiom,
! [V_a1_2,V_a2_2,V_r_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_a,T_b,V_r_2),V_a2_2),V_a1_2))
<=> hBOOL(hAPP(hAPP(V_r_2,V_a1_2),V_a2_2)) ) ).
fof(fact_conversep__iff,axiom,
! [V_ba_2,V_aa_2,V_r_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_a,T_b,V_r_2),V_aa_2),V_ba_2))
<=> hBOOL(hAPP(hAPP(V_r_2,V_ba_2),V_aa_2)) ) ).
fof(fact_conversep__conversep,axiom,
! [V_r_2,T_a,T_b] : c_Predicate_Oconversep(T_b,T_a,c_Predicate_Oconversep(T_a,T_b,V_r_2)) = V_r_2 ).
fof(fact_conversep__eq,axiom,
! [T_a] : c_Predicate_Oconversep(T_a,T_a,c_fequal) = c_fequal ).
fof(fact_conversep__noteq,axiom,
! [V_y_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_a,T_a,hAPP(c_COMBB(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,c_fNot)),c_fequal)),V_x_2),V_y_2))
<=> V_x_2 != V_y_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__def,axiom,
! [V_r_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,V_r_2) = c_Relation_ODomain(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_r_2)) ).
fof(fact_Domain__converse,axiom,
! [V_r_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_r_2)) = c_Relation_ORange(T_b,T_a,V_r_2) ).
fof(fact_Range__converse,axiom,
! [V_r_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Relation_Oconverse(T_a,T_b,V_r_2)) = c_Relation_ODomain(T_a,T_b,V_r_2) ).
fof(fact_Domain__iff,axiom,
! [V_r_2,T_b,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_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_aa_2),B_y)),V_r_2)) ) ).
fof(fact_DomainP__Domain__eq,axiom,
! [V_x_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(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(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,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_x_2)
<=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),c_Relation_ODomain(T_a,T_b,V_r_2))) ) ).
fof(fact_DomainE,axiom,
! [V_r_2,T_b,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_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_aa_2),B_y)),V_r_2)) ) ).
fof(fact_Pi__cong,axiom,
! [V_B_2,T_b,V_g_2,V_f_2,V_A_2,T_a] :
( ! [B_w] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_w),V_A_2))
=> hAPP(V_f_2,B_w) = hAPP(V_g_2,B_w) )
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2)))
<=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_g_2),c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2))) ) ) ).
fof(fact_compose__assoc,axiom,
! [V_D_2,V_h_2,T_d,V_C_2,V_g_2,T_c,V_B_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_B_2))))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,T_c)),V_g_2),c_FuncSet_OPi(T_b,T_c,V_B_2,c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b,V_C_2))))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,T_d)),V_h_2),c_FuncSet_OPi(T_c,T_d,V_C_2,c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c,V_D_2))))
=> c_FuncSet_Ocompose(T_a,T_c,T_d,V_A_2,V_h_2,c_FuncSet_Ocompose(T_a,T_b,T_c,V_A_2,V_g_2,V_f_2)) = c_FuncSet_Ocompose(T_a,T_b,T_d,V_A_2,c_FuncSet_Ocompose(T_b,T_c,T_d,V_B_2,V_h_2,V_g_2),V_f_2) ) ) ) ).
fof(fact_funcset__compose,axiom,
! [V_C_2,V_g_2,T_c,V_B_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_B_2))))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,T_c)),V_g_2),c_FuncSet_OPi(T_b,T_c,V_B_2,c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b,V_C_2))))
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_c)),c_FuncSet_Ocompose(T_a,T_b,T_c,V_A_2,V_g_2,V_f_2)),c_FuncSet_OPi(T_a,T_c,V_A_2,c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_a,V_C_2)))) ) ) ).
fof(fact_acc__downward,axiom,
! [V_aa_2,V_r_2,V_ba_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Wellfounded_Oacc(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_aa_2),V_ba_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),c_Wellfounded_Oacc(T_a,V_r_2))) ) ) ).
fof(fact_acc_Osimps,axiom,
! [V_r_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),c_Wellfounded_Oacc(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),V_aa_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),B_x),c_Wellfounded_Oacc(T_a,V_r_2))) ) ) ).
fof(fact_total__on__def,axiom,
! [V_r_2,V_A_2,T_a] :
( c_Relation_Ototal__on(T_a,V_A_2,V_r_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_A_2))
=> ( B_x != B_xa
=> ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_x),B_xa)),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_xa),B_x)),V_r_2)) ) ) ) ) ) ).
fof(fact_in__inv__image,axiom,
! [V_f_2,V_r_2,T_b,V_y_2,V_x_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_x_2),V_y_2)),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_x_2)),hAPP(V_f_2,V_y_2))),V_r_2)) ) ).
fof(fact_DomainP_Ointros,axiom,
! [T_b,T_a,V_ba_2,V_aa_2,V_r_2] :
( hBOOL(hAPP(hAPP(V_r_2,V_aa_2),V_ba_2))
=> c_Predicate_ODomainP(T_a,T_b,V_r_2,V_aa_2) ) ).
fof(fact_compose__eq,axiom,
! [V_f_2,V_g_2,T_b,T_c,V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(c_FuncSet_Ocompose(T_a,T_c,T_b,V_A_2,V_g_2,V_f_2),V_x_2) = hAPP(V_g_2,hAPP(V_f_2,V_x_2)) ) ).
fof(fact_converse__inv__image,axiom,
! [V_f_2,V_R_2,T_b,T_a] : c_Relation_Oconverse(T_a,T_a,c_Relation_Oinv__image(T_b,T_a,V_R_2,V_f_2)) = c_Relation_Oinv__image(T_b,T_a,c_Relation_Oconverse(T_b,T_b,V_R_2),V_f_2) ).
fof(fact_total__on__converse,axiom,
! [V_r_2,V_A_2,T_a] :
( c_Relation_Ototal__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
<=> c_Relation_Ototal__on(T_a,V_A_2,V_r_2) ) ).
fof(fact_accp__acc__eq,axiom,
! [V_x_2,V_r_2,T_a] :
( hBOOL(hAPP(c_Wellfounded_Oaccp(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(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(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,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_x_2))
<=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),c_Wellfounded_Oacc(T_a,V_r_2))) ) ).
fof(fact_acc_OaccI,axiom,
! [V_r_2,V_x_2,T_a] :
( ! [B_y] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_y),V_x_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),B_y),c_Wellfounded_Oacc(T_a,V_r_2))) )
=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),c_Wellfounded_Oacc(T_a,V_r_2))) ) ).
fof(fact_not__acc__down,axiom,
! [V_R_2,V_x_2,T_a] :
( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),c_Wellfounded_Oacc(T_a,V_R_2)))
=> ~ ! [B_z] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_z),V_x_2)),V_R_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),B_z),c_Wellfounded_Oacc(T_a,V_R_2))) ) ) ).
fof(fact_compose__extensional,axiom,
! [V_g_2,V_f_2,V_A_2,T_c,T_b,T_a] : hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),c_FuncSet_Ocompose(T_a,T_c,T_b,V_A_2,V_f_2,V_g_2)),c_FuncSet_Oextensional(T_a,T_b,V_A_2))) ).
fof(fact_acc__downwards,axiom,
! [V_ba_2,V_r_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),c_Wellfounded_Oacc(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_ba_2),V_aa_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Wellfounded_Oacc(T_a,V_r_2))) ) ) ).
fof(fact_acc__downwards__aux,axiom,
! [V_r_2,V_aa_2,V_ba_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_ba_2),V_aa_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),c_Wellfounded_Oacc(T_a,V_r_2)))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Wellfounded_Oacc(T_a,V_r_2))) ) ) ).
fof(fact_refl__onD,axiom,
! [V_aa_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_aa_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_aa_2),V_aa_2)),V_r_2)) ) ) ).
fof(fact_refl__onD1,axiom,
! [V_y_2,V_x_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_x_2),V_y_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2)) ) ) ).
fof(fact_accp__downward,axiom,
! [V_aa_2,V_ba_2,V_r_2,T_a] :
( hBOOL(hAPP(c_Wellfounded_Oaccp(T_a,V_r_2),V_ba_2))
=> ( hBOOL(hAPP(hAPP(V_r_2,V_aa_2),V_ba_2))
=> hBOOL(hAPP(c_Wellfounded_Oaccp(T_a,V_r_2),V_aa_2)) ) ) ).
fof(fact_accp_Oequations,axiom,
! [V_aa_2,V_r_2,T_a] :
( hBOOL(hAPP(c_Wellfounded_Oaccp(T_a,V_r_2),V_aa_2))
<=> ! [B_x] :
( hBOOL(hAPP(hAPP(V_r_2,B_x),V_aa_2))
=> hBOOL(hAPP(c_Wellfounded_Oaccp(T_a,V_r_2),B_x)) ) ) ).
fof(fact_accp_Osimps,axiom,
! [V_aa_2,V_r_2,T_a] :
( hBOOL(hAPP(c_Wellfounded_Oaccp(T_a,V_r_2),V_aa_2))
<=> ! [B_x] :
( hBOOL(hAPP(hAPP(V_r_2,B_x),V_aa_2))
=> hBOOL(hAPP(c_Wellfounded_Oaccp(T_a,V_r_2),B_x)) ) ) ).
fof(fact_refl__on__converse,axiom,
! [V_r_2,V_A_2,T_a] :
( c_Relation_Orefl__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
<=> c_Relation_Orefl__on(T_a,V_A_2,V_r_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_refl__onD2,axiom,
! [V_y_2,V_x_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_x_2),V_y_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_A_2)) ) ) ).
fof(fact_rtrancl_Ortrancl__refl,axiom,
! [V_r_2,V_aa_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_aa_2),V_aa_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ).
fof(fact_r__into__rtrancl,axiom,
! [V_r_2,V_p_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),V_p_2),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),V_p_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ).
fof(fact_Not__Domain__rtrancl,axiom,
! [V_y_2,V_R_2,V_x_2,T_a] :
( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),c_Relation_ODomain(T_a,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_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_a,V_R_2)))
<=> V_x_2 = V_y_2 ) ) ).
fof(fact_rtrancl__converseD,axiom,
! [V_r_2,V_y_2,V_x_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_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,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_y_2),V_x_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ).
fof(fact_rtrancl__converseI,axiom,
! [V_r_2,V_x_2,V_y_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_y_2),V_x_2)),c_Transitive__Closure_Ortrancl(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_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)))) ) ).
fof(fact_rtrancl__trans,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Ortrancl(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_ba_2),V_c_2)),c_Transitive__Closure_Ortrancl(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_aa_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ) ).
fof(fact_rtrancl_Ortrancl__into__rtrancl,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Ortrancl(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_ba_2),V_c_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_aa_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ) ).
fof(fact_rtrancl__idemp,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Transitive__Closure_Ortrancl(T_a,V_r_2) ).
fof(fact_rtrancl__converse,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) = c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ).
fof(fact_converse__rtrancl__into__rtrancl,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_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_ba_2),V_c_2)),c_Transitive__Closure_Ortrancl(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_aa_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ) ).
fof(fact_compose__Id,axiom,
! [V_B_2,V_A_2,V_g_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_g_2),c_FuncSet_OPi(T_a,T_b,V_A_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_B_2))))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_g_2),c_FuncSet_Oextensional(T_a,T_b,V_A_2)))
=> c_FuncSet_Ocompose(T_a,T_a,T_b,V_A_2,V_g_2,c_FuncSet_Orestrict(T_a,T_a,c_COMBI(T_a),V_A_2)) = V_g_2 ) ) ).
fof(fact_Id__compose,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_B_2))))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional(T_a,T_b,V_A_2)))
=> c_FuncSet_Ocompose(T_a,T_b,T_b,V_A_2,c_FuncSet_Orestrict(T_b,T_b,c_COMBI(T_b),V_B_2),V_f_2) = V_f_2 ) ) ).
fof(fact_single__valued__confluent,axiom,
! [V_z_2,V_y_2,V_x_2,V_r_2,T_a] :
( c_Relation_Osingle__valued(T_a,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_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(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_x_2),V_z_2)),c_Transitive__Closure_Ortrancl(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_y_2),V_z_2)),c_Transitive__Closure_Ortrancl(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_z_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ) ) ) ).
fof(fact_rtrancl__induct2,axiom,
! [V_Pa_2,V_r_2,V_by_2,V_bx_2,V_ay_2,V_ax_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_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)))
=> ( hBOOL(hAPP(hAPP(V_Pa_2,V_ax_2),V_ay_2))
=> ( ! [B_a,B_b] :
( 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_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_a),B_b))),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)))
=> ! [B_aa,B_ba] :
( 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),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_aa),B_ba))),V_r_2))
=> ( hBOOL(hAPP(hAPP(V_Pa_2,B_a),B_b))
=> hBOOL(hAPP(hAPP(V_Pa_2,B_aa),B_ba)) ) ) )
=> hBOOL(hAPP(hAPP(V_Pa_2,V_bx_2),V_by_2)) ) ) ) ).
fof(fact_converse__rtrancl__induct2,axiom,
! [V_Pa_2,V_r_2,V_by_2,V_bx_2,V_ay_2,V_ax_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_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)))
=> ( hBOOL(hAPP(hAPP(V_Pa_2,V_bx_2),V_by_2))
=> ( ! [B_a,B_b,B_aa,B_ba] :
( 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),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_aa),B_ba))),V_r_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),B_aa),B_ba)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)))
=> ( hBOOL(hAPP(hAPP(V_Pa_2,B_aa),B_ba))
=> hBOOL(hAPP(hAPP(V_Pa_2,B_a),B_b)) ) ) )
=> hBOOL(hAPP(hAPP(V_Pa_2,V_ax_2),V_ay_2)) ) ) ) ).
fof(fact_converse__rtranclE2,axiom,
! [V_r_2,V_zb_2,V_za_2,V_xb_2,V_xa_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_xa_2),V_xb_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_za_2),V_zb_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)))
=> ( hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_xa_2),V_xb_2) != hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_za_2),V_zb_2)
=> ~ ! [B_a,B_b] :
( 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_xb_2)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_a),B_b))),V_r_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),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_za_2),V_zb_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))) ) ) ) ).
fof(fact_extensional__arb,axiom,
! [V_x_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional(T_a,T_b,V_A_2)))
=> ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(V_f_2,V_x_2) = c_HOL_Oundefined(T_b) ) ) ).
fof(fact_restrict__apply,axiom,
! [V_f_2,T_b,V_A_2,V_x_2,T_a] :
( ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2),V_x_2) = hAPP(V_f_2,V_x_2) )
& ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2),V_x_2) = c_HOL_Oundefined(T_b) ) ) ).
fof(fact_restrict__def,axiom,
! [V_f_2,T_b,V_A_2,V_x_2,T_a] :
( ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2),V_x_2) = hAPP(V_f_2,V_x_2) )
& ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2),V_x_2) = c_HOL_Oundefined(T_b) ) ) ).
fof(fact_single__valued__Id__on,axiom,
! [V_A_2,T_a] : c_Relation_Osingle__valued(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) ).
fof(fact_compose__def,axiom,
! [V_f_2,V_g_2,V_A_2,T_b,T_c,T_a] : c_FuncSet_Ocompose(T_a,T_c,T_b,V_A_2,V_g_2,V_f_2) = c_FuncSet_Orestrict(T_a,T_b,hAPP(c_COMBB(T_c,T_b,T_a,V_g_2),V_f_2),V_A_2) ).
fof(fact_restrict__extensional,axiom,
! [V_A_2,V_f_2,T_b,T_a] : hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2)),c_FuncSet_Oextensional(T_a,T_b,V_A_2))) ).
fof(fact_extensional__restrict,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional(T_a,T_b,V_A_2)))
=> c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2) = V_f_2 ) ).
fof(fact_single__valuedD,axiom,
! [V_z_2,V_y_2,V_x_2,V_r_2,T_b,T_a] :
( c_Relation_Osingle__valued(T_a,T_b,V_r_2)
=> ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x_2),V_y_2)),V_r_2))
=> ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x_2),V_z_2)),V_r_2))
=> V_y_2 = V_z_2 ) ) ) ).
fof(fact_single__valued__def,axiom,
! [V_r_2,T_b,T_a] :
( c_Relation_Osingle__valued(T_a,T_b,V_r_2)
<=> ! [B_x,B_y] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_x),B_y)),V_r_2))
=> ! [B_z] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_x),B_z)),V_r_2))
=> B_y = B_z ) ) ) ).
fof(fact_restrictI,axiom,
! [V_B_2,V_f_2,T_b,V_A_2,T_a] :
( ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,B_x)),hAPP(V_B_2,B_x))) )
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2)),c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2))) ) ).
fof(fact_restrict__in__funcset,axiom,
! [V_B_2,V_f_2,T_b,V_A_2,T_a] :
( ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,B_x)),V_B_2)) )
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2)),c_FuncSet_OPi(T_a,T_b,V_A_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_B_2)))) ) ).
fof(fact_cut__def,axiom,
! [V_f_2,T_b,V_r_2,V_x_2,V_y_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_y_2),V_x_2)),V_r_2))
=> hAPP(hAPP(hAPP(hAPP(c_Recdef_Ocut(T_a,T_b),V_f_2),V_r_2),V_x_2),V_y_2) = hAPP(V_f_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_y_2),V_x_2)),V_r_2))
=> hAPP(hAPP(hAPP(hAPP(c_Recdef_Ocut(T_a,T_b),V_f_2),V_r_2),V_x_2),V_y_2) = c_HOL_Oundefined(T_b) ) ) ).
fof(fact_extensional__funcset__arb,axiom,
! [V_x_2,V_T_2,V_S_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional__funcset(T_a,T_b,V_S_2,V_T_2)))
=> ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_S_2))
=> hAPP(V_f_2,V_x_2) = c_HOL_Oundefined(T_b) ) ) ).
fof(fact_extensionalityI,axiom,
! [V_g_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional(T_a,T_b,V_A_2)))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_g_2),c_FuncSet_Oextensional(T_a,T_b,V_A_2)))
=> ( ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
=> hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x) )
=> V_f_2 = V_g_2 ) ) ) ).
fof(fact_single__valuedI,axiom,
! [V_r_2,T_b,T_a] :
( ! [B_x,B_y] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_x),B_y)),V_r_2))
=> ! [B_z] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),B_x),B_z)),V_r_2))
=> B_y = B_z ) )
=> c_Relation_Osingle__valued(T_a,T_b,V_r_2) ) ).
fof(fact_rtrancl__induct,axiom,
! [V_Pa_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))
=> ( hBOOL(hAPP(V_Pa_2,V_aa_2))
=> ( ! [B_y] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_aa_2),B_y)),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))
=> ! [B_z] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_y),B_z)),V_r_2))
=> ( hBOOL(hAPP(V_Pa_2,B_y))
=> hBOOL(hAPP(V_Pa_2,B_z)) ) ) )
=> hBOOL(hAPP(V_Pa_2,V_ba_2)) ) ) ) ).
fof(fact_extensional__funcset__mem,axiom,
! [V_x_2,V_T_2,V_S_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional__funcset(T_a,T_b,V_S_2,V_T_2)))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_S_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_x_2)),V_T_2)) ) ) ).
fof(fact_tfl__cut__apply,axiom,
! [T_b,V_aa_2,V_x_2,T_a,B_f,B_R] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_x_2),V_aa_2)),B_R))
=> hAPP(hAPP(hAPP(hAPP(c_Recdef_Ocut(T_a,T_b),B_f),B_R),V_aa_2),V_x_2) = hAPP(B_f,V_x_2) ) ).
fof(fact_cuts__eq,axiom,
! [V_g_2,V_x_2,V_r_2,V_f_2,T_b,T_a] :
( hAPP(hAPP(hAPP(c_Recdef_Ocut(T_a,T_b),V_f_2),V_r_2),V_x_2) = hAPP(hAPP(hAPP(c_Recdef_Ocut(T_a,T_b),V_g_2),V_r_2),V_x_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_y),V_x_2)),V_r_2))
=> hAPP(V_f_2,B_y) = hAPP(V_g_2,B_y) ) ) ).
fof(fact_cut__apply,axiom,
! [V_f_2,T_b,V_r_2,V_aa_2,V_x_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_x_2),V_aa_2)),V_r_2))
=> hAPP(hAPP(hAPP(hAPP(c_Recdef_Ocut(T_a,T_b),V_f_2),V_r_2),V_aa_2),V_x_2) = hAPP(V_f_2,V_x_2) ) ).
fof(fact_adm__lemma,axiom,
! [V_Fa_2,V_R_2,T_b,T_a] : c_Recdef_Oadm__wf(T_a,T_b,V_R_2,hAPP(hAPP(c_COMBC(tc_fun(T_a,T_b),tc_fun(T_a,T_a),tc_fun(T_a,T_b)),hAPP(c_COMBB(tc_fun(T_a,tc_fun(T_a,T_b)),tc_fun(tc_fun(T_a,T_a),tc_fun(T_a,T_b)),tc_fun(T_a,T_b),c_COMBS(T_a,T_a,T_b)),hAPP(c_COMBB(tc_fun(T_a,tc_fun(T_a,T_b)),tc_fun(T_a,tc_fun(T_a,T_b)),tc_fun(T_a,T_b),c_COMBB(tc_fun(T_a,T_b),tc_fun(T_a,T_b),T_a,V_Fa_2)),hAPP(hAPP(c_COMBC(tc_fun(T_a,T_b),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(T_a,tc_fun(T_a,T_b))),c_Recdef_Ocut(T_a,T_b)),V_R_2)))),c_COMBI(T_a))) ).
fof(fact_extensional__funcset__def,axiom,
! [V_T_2,V_S_2,T_b,T_a] : c_FuncSet_Oextensional__funcset(T_a,T_b,V_S_2,V_T_2) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool),c_FuncSet_OPi(T_a,T_b,V_S_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_T_2)),c_FuncSet_Oextensional(T_a,T_b,V_S_2)) ).
fof(fact_rtranclE,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))
=> ( V_aa_2 != V_ba_2
=> ~ ! [B_y] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_aa_2),B_y)),c_Transitive__Closure_Ortrancl(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),B_y),V_ba_2)),V_r_2)) ) ) ) ).
fof(fact_converse__rtranclE,axiom,
! [V_r_2,V_z_2,V_x_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_x_2),V_z_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))
=> ( V_x_2 != V_z_2
=> ~ ! [B_y] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_x_2),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),V_z_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ) ) ).
fof(fact_converse__rtrancl__induct,axiom,
! [V_Pa_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))
=> ( hBOOL(hAPP(V_Pa_2,V_ba_2))
=> ( ! [B_y,B_z] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_y),B_z)),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_z),V_ba_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2)))
=> ( hBOOL(hAPP(V_Pa_2,B_z))
=> hBOOL(hAPP(V_Pa_2,B_y)) ) ) )
=> hBOOL(hAPP(V_Pa_2,V_aa_2)) ) ) ) ).
fof(fact_trancl__into__rtrancl,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Otrancl(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_aa_2),V_ba_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ).
fof(fact_inf1E,axiom,
! [V_x_2,V_B_2,V_A_2,T_a] :
( hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2))
=> ~ ( hBOOL(hAPP(V_A_2,V_x_2))
=> ~ hBOOL(hAPP(V_B_2,V_x_2)) ) ) ).
fof(fact_inf1I,axiom,
! [T_a,V_B_2,V_x_2,V_A_2] :
( hBOOL(hAPP(V_A_2,V_x_2))
=> ( hBOOL(hAPP(V_B_2,V_x_2))
=> hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_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),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),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) ) ) ).
fof(fact_trancl_Or__into__trancl,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_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_aa_2),V_ba_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ).
fof(fact_inf1D2,axiom,
! [V_x_2,V_B_2,V_A_2,T_a] :
( hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2))
=> hBOOL(hAPP(V_B_2,V_x_2)) ) ).
fof(fact_inf1D1,axiom,
! [V_x_2,V_B_2,V_A_2,T_a] :
( hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2))
=> hBOOL(hAPP(V_A_2,V_x_2)) ) ).
fof(fact_Int__assoc,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,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] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,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] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ).
fof(fact_Int__commute,axiom,
! [V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) ).
fof(fact_Int__absorb,axiom,
! [V_A_2,T_a] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) = V_A_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,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),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_IntD2,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),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_IntD1,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),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_Int__iff,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),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_r__into__trancl_H,axiom,
! [V_r_2,V_p_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),V_p_2),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),V_p_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ).
fof(fact_trancl__rtrancl__absorb,axiom,
! [V_R_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Otrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ).
fof(fact_rtrancl__trancl__absorb,axiom,
! [V_R_2,T_a] : c_Transitive__Closure_Otrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ).
fof(fact_trancl__converse,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) = c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) ).
fof(fact_trancl__domain,axiom,
! [V_r_2,T_a] : c_Relation_ODomain(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) = c_Relation_ODomain(T_a,T_a,V_r_2) ).
fof(fact_trancl__range,axiom,
! [V_r_2,T_a] : c_Relation_ORange(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) = c_Relation_ORange(T_a,T_a,V_r_2) ).
fof(fact_inf__Int__eq,axiom,
! [V_x_2,V_S_2,V_R_2,T_a] :
( hBOOL(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_x_2))
<=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_R_2,V_S_2))) ) ).
fof(fact_trancl__trans,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Otrancl(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_ba_2),V_c_2)),c_Transitive__Closure_Otrancl(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_aa_2),V_c_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ) ).
fof(fact_Transitive__Closure_Otrancl__into__trancl,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Otrancl(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_ba_2),V_c_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_aa_2),V_c_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ) ).
fof(fact_trancl__into__trancl2,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_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_ba_2),V_c_2)),c_Transitive__Closure_Otrancl(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_aa_2),V_c_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ) ).
fof(fact_r__r__into__trancl,axiom,
! [V_c_2,V_R_2,V_ba_2,V_aa_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_aa_2),V_ba_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_ba_2),V_c_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_aa_2),V_c_2)),c_Transitive__Closure_Otrancl(T_a,V_R_2))) ) ) ).
fof(fact_rtrancl__eq__or__trancl,axiom,
! [V_R_2,V_y_2,V_x_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_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_a,V_R_2)))
<=> ( V_x_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_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_a,V_R_2))) ) ) ) ).
fof(fact_rtrancl__into__trancl2,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_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_ba_2),V_c_2)),c_Transitive__Closure_Ortrancl(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_aa_2),V_c_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ) ).
fof(fact_rtranclD,axiom,
! [V_R_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Ortrancl(T_a,V_R_2)))
=> ( V_aa_2 = V_ba_2
| ( V_aa_2 != V_ba_2
& hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_aa_2),V_ba_2)),c_Transitive__Closure_Otrancl(T_a,V_R_2))) ) ) ) ).
fof(fact_rtrancl__into__trancl1,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Ortrancl(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_ba_2),V_c_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_aa_2),V_c_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ) ).
fof(fact_trancl__rtrancl__trancl,axiom,
! [V_c_2,V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Otrancl(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_ba_2),V_c_2)),c_Transitive__Closure_Ortrancl(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_aa_2),V_c_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ) ).
fof(fact_rtrancl__trancl__trancl,axiom,
! [V_z_2,V_r_2,V_y_2,V_x_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_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(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_y_2),V_z_2)),c_Transitive__Closure_Otrancl(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_x_2),V_z_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ) ).
fof(fact_trancl__converseD,axiom,
! [V_r_2,V_y_2,V_x_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_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,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_x_2),V_y_2)),c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)))) ) ).
fof(fact_trancl__converseI,axiom,
! [V_r_2,V_y_2,V_x_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_x_2),V_y_2)),c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(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_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)))) ) ).
fof(fact_inf__fun__def,axiom,
! [V_x_2,V_g_2,V_f_2,T_b,T_a] :
( class_Lattices_Olattice(T_a)
=> hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,T_a),V_f_2,V_g_2),V_x_2) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)) ) ).
fof(fact_inf__apply,axiom,
! [V_x_2,V_g_2,V_f_2,T_b,T_a] :
( class_Lattices_Olattice(T_a)
=> hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,T_a),V_f_2,V_g_2),V_x_2) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)) ) ).
fof(fact_tranclD,axiom,
! [V_R_2,V_y_2,V_x_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_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_a,V_R_2)))
=> ? [B_z] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_x_2),B_z)),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_z),V_y_2)),c_Transitive__Closure_Ortrancl(T_a,V_R_2))) ) ) ).
fof(fact_tranclD2,axiom,
! [V_R_2,V_y_2,V_x_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_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_a,V_R_2)))
=> ? [B_z] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_x_2),B_z)),c_Transitive__Closure_Ortrancl(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),B_z),V_y_2)),V_R_2)) ) ) ).
fof(fact_acyclic__def,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Oacyclic(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)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ).
fof(fact_adm__wf__def,axiom,
! [V_Fa_2,V_R_2,T_b,T_a] :
( c_Recdef_Oadm__wf(T_a,T_b,V_R_2,V_Fa_2)
<=> ! [B_f,B_g,B_x] :
( ! [B_z] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),B_z),B_x)),V_R_2))
=> hAPP(B_f,B_z) = hAPP(B_g,B_z) )
=> hAPP(hAPP(V_Fa_2,B_f),B_x) = hAPP(hAPP(V_Fa_2,B_g),B_x) ) ) ).
fof(fact_inf__Int__eq2,axiom,
! [V_y_2,V_x_2,V_S_2,V_R_2,T_b,T_a] :
( hBOOL(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(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(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,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(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(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,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_x_2),V_y_2))
<=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x_2),V_y_2)),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_converse__meet,axiom,
! [V_s_2,V_r_2,T_a,T_b] : c_Predicate_Oconversep(T_b,T_a,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_fun(T_a,tc_HOL_Obool)),V_r_2,V_s_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool)),c_Predicate_Oconversep(T_b,T_a,V_r_2),c_Predicate_Oconversep(T_b,T_a,V_s_2)) ).
fof(fact_converse__Int,axiom,
! [V_s_2,V_r_2,T_a,T_b] : c_Relation_Oconverse(T_b,T_a,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_r_2,V_s_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),c_Relation_Oconverse(T_b,T_a,V_r_2),c_Relation_Oconverse(T_b,T_a,V_s_2)) ).
fof(fact_acyclic__converse,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Oacyclic(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))
<=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) ).
fof(fact_inf__assoc,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Lattices_Osemilattice__inf(T_a)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_z) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_z) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b),V_c) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_b,V_c)) ) ).
fof(fact_inf__left__commute,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Lattices_Osemilattice__inf(T_a)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_b,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_c)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y)) = 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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y)) = 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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b)) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b) ) ).
fof(fact_inf__commute,axiom,
! [V_y,V_x,T_a] :
( class_Lattices_Osemilattice__inf(T_a)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y) = 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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y) = 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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b) = c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_b,V_a) ) ).
fof(fact_inf__idem,axiom,
! [V_x,T_a] :
( class_Lattices_Osemilattice__inf(T_a)
=> 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)
=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_a) = V_a ) ).
fof(fact_irrefl__tranclI,axiom,
! [V_x_2,V_r_2,T_a] :
( c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Oconverse(T_a,T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,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_x_2),V_x_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ).
fof(fact_acyclicI,axiom,
! [V_r_2,T_a] :
( ! [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)),c_Transitive__Closure_Otrancl(T_a,V_r_2)))
=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) ).
fof(fact_acyclic__insert,axiom,
! [V_r_2,V_x_2,V_y_2,T_a] :
( c_Wellfounded_Oacyclic(T_a,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_y_2),V_x_2)),V_r_2))
<=> ( c_Wellfounded_Oacyclic(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_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_a,V_r_2))) ) ) ).
fof(fact_tranclE,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),c_Transitive__Closure_Otrancl(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_aa_2),V_ba_2)),V_r_2))
=> ~ ! [B_c] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_aa_2),B_c)),c_Transitive__Closure_Otrancl(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),B_c),V_ba_2)),V_r_2)) ) ) ) ).
fof(fact_converse__tranclE,axiom,
! [V_r_2,V_z_2,V_x_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_x_2),V_z_2)),c_Transitive__Closure_Otrancl(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_x_2),V_z_2)),V_r_2))
=> ~ ! [B_y] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_x_2),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),V_z_2)),c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ) ) ).
fof(fact_inf2E,axiom,
! [V_y_2,V_x_2,V_B_2,V_A_2,T_b,T_a] :
( hBOOL(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_x_2),V_y_2))
=> ~ ( hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2))
=> ~ hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2)) ) ) ).
fof(fact_inf2I,axiom,
! [T_b,T_a,V_B_2,V_y_2,V_x_2,V_A_2] :
( hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2))
=> ( hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2))
=> hBOOL(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_x_2),V_y_2)) ) ) ).
fof(fact_emptyE,axiom,
! [V_aa_2,T_a] : ~ hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) ).
fof(fact_insertCI,axiom,
! [V_ba_2,V_B_2,V_aa_2,T_a] :
( ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_B_2))
=> V_aa_2 = V_ba_2 )
=> hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_ba_2),V_B_2))) ) ).
fof(fact_insertE,axiom,
! [V_A_2,V_ba_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_ba_2),V_A_2)))
=> ( V_aa_2 != V_ba_2
=> hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2)) ) ) ).
fof(fact_inf2D2,axiom,
! [V_y_2,V_x_2,V_B_2,V_A_2,T_b,T_a] :
( hBOOL(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_x_2),V_y_2))
=> hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2)) ) ).
fof(fact_inf2D1,axiom,
! [V_y_2,V_x_2,V_B_2,V_A_2,T_b,T_a] :
( hBOOL(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_x_2),V_y_2))
=> hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2)) ) ).
fof(fact_Domain__insert,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),V_r_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Relation_ODomain(T_a,T_b,V_r_2)) ).
fof(fact_Range__insert,axiom,
! [V_r_2,V_ba_2,V_aa_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_aa_2),V_ba_2)),V_r_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_ba_2),c_Relation_ORange(T_b,T_a,V_r_2)) ).
fof(fact_insertI1,axiom,
! [V_B_2,V_aa_2,T_a] : hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2))) ).
fof(fact_insert__iff,axiom,
! [V_A_2,V_ba_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_ba_2),V_A_2)))
<=> ( V_aa_2 = V_ba_2
| hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2)) ) ) ).
fof(fact_insert__ident,axiom,
! [V_B_2,V_A_2,V_x_2,T_a] :
( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_B_2))
=> ( hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_B_2)
<=> V_A_2 = V_B_2 ) ) ) ).
fof(fact_insertI2,axiom,
! [V_ba_2,V_B_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_B_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_ba_2),V_B_2))) ) ).
fof(fact_insert__absorb,axiom,
! [V_A_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2))
=> hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_A_2) = V_A_2 ) ).
fof(fact_insert__inter__insert,axiom,
! [V_B_2,V_A_2,V_aa_2,T_a] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ).
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_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_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_equals0D,axiom,
! [V_aa_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_aa_2),V_A_2)) ) ).
fof(fact_extensional__funcset__empty__range,axiom,
! [T_b,T_a,V_S_2] :
( V_S_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
=> c_FuncSet_Oextensional__funcset(T_a,T_b,V_S_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool)) ) ).
fof(fact_Pi__eq__empty,axiom,
! [V_B_2,V_A_2,T_b,T_a] :
( c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool))
<=> ? [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
& hAPP(V_B_2,B_x) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ) ).
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_singletonE,axiom,
! [V_aa_2,V_ba_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
=> V_ba_2 = V_aa_2 ) ).
fof(fact_singleton__iff,axiom,
! [V_aa_2,V_ba_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
<=> V_ba_2 = V_aa_2 ) ).
fof(fact_singleton__inject,axiom,
! [V_ba_2,V_aa_2,T_a] :
( hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(T_a),V_ba_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))
=> V_aa_2 = V_ba_2 ) ).
fof(fact_insert__code,axiom,
! [V_x_2,V_A_2,V_y_2,T_a] :
( hBOOL(hAPP(hAPP(hAPP(c_Set_Oinsert(T_a),V_y_2),V_A_2),V_x_2))
<=> ( V_y_2 = V_x_2
| hBOOL(hAPP(V_A_2,V_x_2)) ) ) ).
fof(fact_doubleton__eq__iff,axiom,
! [V_d_2,V_c_2,V_ba_2,V_aa_2,T_a] :
( hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_ba_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_aa_2 = V_c_2
& V_ba_2 = V_d_2 )
| ( V_aa_2 = V_d_2
& V_ba_2 = V_c_2 ) ) ) ).
fof(fact_insert__commute,axiom,
! [V_A_2,V_y_2,V_x_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_y_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_y_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2)) ).
fof(fact_insert__absorb2,axiom,
! [V_A_2,V_x_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2) ).
fof(fact_insert__not__empty,axiom,
! [V_A_2,V_aa_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_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_aa_2,T_a] : c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) != hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_A_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_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_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_disjoint__iff__not__equal,axiom,
! [V_B_2,V_A_2,T_a] :
( 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__empty__right,axiom,
! [V_A_2,T_a] : 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_Int__empty__left,axiom,
! [V_B_2,T_a] : 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_trancl__empty,axiom,
! [T_a] : c_Transitive__Closure_Otrancl(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(tc_prod(T_a,T_a),tc_HOL_Obool)) ).
fof(fact_bot__empty__eq,axiom,
! [V_x_2,T_a] :
( hBOOL(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_x_2))
<=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) ) ).
fof(fact_total__on__empty,axiom,
! [V_r_2,T_a] : c_Relation_Ototal__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_r_2) ).
fof(fact_inf__bot__right,axiom,
! [V_x,T_a] :
( class_Lattices_Obounded__lattice__bot(T_a)
=> 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_inf__bot__left,axiom,
! [V_x,T_a] :
( class_Lattices_Obounded__lattice__bot(T_a)
=> 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_Int__insert__left__if1,axiom,
! [V_B_2,V_C_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_C_2))
=> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2),V_C_2) = hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),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_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2))
=> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),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_aa_2,T_a] :
( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_C_2))
=> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2),V_C_2) = 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_aa_2,T_a] :
( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2))
=> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2)) = 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_aa_2,T_a] :
( ( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_C_2))
=> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2),V_C_2) = hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),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_aa_2),V_C_2))
=> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2),V_C_2) = 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_aa_2,T_a] :
( ( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2))
=> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),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_aa_2),V_A_2))
=> c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) ).
fof(fact_the__elem__eq,axiom,
! [V_x_2,T_a] : c_Set_Othe__elem(T_a,hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_x_2 ).
fof(fact_bot__fun__def,axiom,
! [V_x_2,T_b,T_a] :
( class_Orderings_Obot(T_a)
=> hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_x_2) = c_Orderings_Obot__class_Obot(T_a) ) ).
fof(fact_bot__apply,axiom,
! [V_x_2,T_b,T_a] :
( class_Orderings_Obot(T_a)
=> hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_x_2) = c_Orderings_Obot__class_Obot(T_a) ) ).
fof(fact_extensional__funcset__restrict__domain,axiom,
! [V_T_2,V_f_2,T_b,V_S_2,V_x_2,T_a] :
( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_S_2))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional__funcset(T_a,T_b,hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_S_2),V_T_2)))
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,c_HOL_Oundefined(T_b))),c_FuncSet_Oextensional__funcset(T_a,T_b,V_S_2,V_T_2))) ) ) ).
fof(fact_extensional__funcset__empty__domain,axiom,
! [V_T_2,T_b,T_a] : c_FuncSet_Oextensional__funcset(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_T_2) = hAPP(hAPP(c_Set_Oinsert(tc_fun(T_a,T_b)),c_COMBK(T_b,T_a,c_HOL_Oundefined(T_b))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool))) ).
fof(fact_extensional__empty,axiom,
! [T_b,T_a] : c_FuncSet_Oextensional(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(tc_fun(T_a,T_b)),c_COMBK(T_b,T_a,c_HOL_Oundefined(T_b))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool))) ).
fof(fact_bot__empty__eq2,axiom,
! [V_y_2,V_x_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_x_2),V_y_2))
<=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)))) ) ).
fof(fact_extensional__funcset__extend__domainI,axiom,
! [V_x_2,V_S_2,V_f_2,T_b,V_T_2,V_y_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_T_2))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,T_a)),V_f_2),c_FuncSet_Oextensional__funcset(T_b,T_a,V_S_2,V_T_2)))
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,T_a)),c_Fun_Ofun__upd(T_b,T_a,V_f_2,V_x_2,V_y_2)),c_FuncSet_Oextensional__funcset(T_b,T_a,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_S_2),V_T_2))) ) ) ).
fof(fact_fun__upd__triv,axiom,
! [V_x_2,V_f_2,T_b,T_a] : c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,hAPP(V_f_2,V_x_2)) = V_f_2 ).
fof(fact_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [V_x_2,V_S_2,V_f_2,T_b,V_T_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_T_2))
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,T_a)),V_f_2),c_FuncSet_Oextensional__funcset(T_b,T_a,V_S_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_T_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))))
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,T_a)),c_Fun_Ofun__upd(T_b,T_a,V_f_2,V_x_2,V_aa_2)),c_FuncSet_Oextensional__funcset(T_b,T_a,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_S_2),V_T_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(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_fun__upd__def,axiom,
! [V_ba_2,V_f_2,T_b,T_a,V_aa_2,V_x_2] :
( ( V_x_2 = V_aa_2
=> hAPP(c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_aa_2,V_ba_2),V_x_2) = V_ba_2 )
& ( V_x_2 != V_aa_2
=> hAPP(c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_aa_2,V_ba_2),V_x_2) = hAPP(V_f_2,V_x_2) ) ) ).
fof(fact_DiffI,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),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))) ) ) ).
fof(fact_DiffE,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),c_Groups_Ominus__class_Ominus(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_bot2E,axiom,
! [V_y_2,V_x_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_x_2),V_y_2)) ).
fof(fact_Diff__cancel,axiom,
! [V_A_2,T_a] : c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).
fof(fact_Diff__empty,axiom,
! [V_A_2,T_a] : c_Groups_Ominus__class_Ominus(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_empty__Diff,axiom,
! [V_A_2,T_a] : c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),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_minus__apply,axiom,
! [V_x_2,V_B_2,V_A_2,T_b,T_a] :
( class_Groups_Ominus(T_a)
=> hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,T_a),V_A_2,V_B_2),V_x_2) = c_Groups_Ominus__class_Ominus(T_a,hAPP(V_A_2,V_x_2),hAPP(V_B_2,V_x_2)) ) ).
fof(fact_fun__diff__def,axiom,
! [V_x_2,V_B_2,V_A_2,T_b,T_a] :
( class_Groups_Ominus(T_a)
=> hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,T_a),V_A_2,V_B_2),V_x_2) = c_Groups_Ominus__class_Ominus(T_a,hAPP(V_A_2,V_x_2),hAPP(V_B_2,V_x_2)) ) ).
fof(fact_Diff__iff,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),c_Groups_Ominus__class_Ominus(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_DiffD1,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),c_Groups_Ominus__class_Ominus(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_DiffD2,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_c_2),c_Groups_Ominus__class_Ominus(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_Diff__idemp,axiom,
! [V_B_2,V_A_2,T_a] : c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ).
fof(fact_Diff__Int2,axiom,
! [V_B_2,V_C_2,V_A_2,T_a] : c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),V_B_2) ).
fof(fact_Diff__Int__distrib2,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) ).
fof(fact_Int__Diff,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] : c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) ).
fof(fact_Diff__Int__distrib,axiom,
! [V_B_2,V_A_2,V_C_2,T_a] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_C_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_B_2)) ).
fof(fact_insert__Diff1,axiom,
! [V_A_2,V_B_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_B_2))
=> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2),V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ).
fof(fact_insert__Diff__if,axiom,
! [V_A_2,V_B_2,V_x_2,T_a] :
( ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_B_2))
=> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2),V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) )
& ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_B_2))
=> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2),V_B_2) = hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ) ).
fof(fact_insert__Diff__single,axiom,
! [V_A_2,V_aa_2,T_a] : hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) = hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_A_2) ).
fof(fact_Diff__insert2,axiom,
! [V_B_2,V_aa_2,V_A_2,T_a] : c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2) ).
fof(fact_Diff__insert,axiom,
! [V_B_2,V_aa_2,V_A_2,T_a] : c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_B_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) ).
fof(fact_Diff__triv,axiom,
! [V_B_2,V_A_2,T_a] :
( 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))
=> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_A_2 ) ).
fof(fact_Diff__disjoint,axiom,
! [V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).
fof(fact_Diff__insert__absorb,axiom,
! [V_A_2,V_x_2,T_a] :
( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_A_2 ) ).
fof(fact_insert__Diff,axiom,
! [V_A_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2))
=> hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) = V_A_2 ) ).
fof(fact_fun__upd__idem__iff,axiom,
! [V_y_2,V_x_2,V_f_2,T_b,T_a] :
( c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_y_2) = V_f_2
<=> hAPP(V_f_2,V_x_2) = V_y_2 ) ).
fof(fact_fun__upd__upd,axiom,
! [V_z_2,V_y_2,V_x_2,V_f_2,T_b,T_a] : c_Fun_Ofun__upd(T_a,T_b,c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_y_2),V_x_2,V_z_2) = c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_z_2) ).
fof(fact_fun__upd__same,axiom,
! [V_y_2,V_x_2,V_f_2,T_a,T_b] : hAPP(c_Fun_Ofun__upd(T_b,T_a,V_f_2,V_x_2,V_y_2),V_x_2) = V_y_2 ).
fof(fact_fun__upd__apply,axiom,
! [V_y_2,V_f_2,T_b,T_a,V_x_2,V_z_2] :
( ( V_z_2 = V_x_2
=> hAPP(c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_y_2),V_z_2) = V_y_2 )
& ( V_z_2 != V_x_2
=> hAPP(c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_y_2),V_z_2) = hAPP(V_f_2,V_z_2) ) ) ).
fof(fact_fun__upd__twist,axiom,
! [V_d_2,V_ba_2,V_m_2,T_b,T_a,V_c_2,V_aa_2] :
( V_aa_2 != V_c_2
=> c_Fun_Ofun__upd(T_a,T_b,c_Fun_Ofun__upd(T_a,T_b,V_m_2,V_aa_2,V_ba_2),V_c_2,V_d_2) = c_Fun_Ofun__upd(T_a,T_b,c_Fun_Ofun__upd(T_a,T_b,V_m_2,V_c_2,V_d_2),V_aa_2,V_ba_2) ) ).
fof(fact_fun__upd__other,axiom,
! [V_y_2,V_f_2,T_b,T_a,V_x_2,V_z_2] :
( V_z_2 != V_x_2
=> hAPP(c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_y_2),V_z_2) = hAPP(V_f_2,V_z_2) ) ).
fof(fact_fun__upd__idem,axiom,
! [T_a,T_b,V_y_2,V_x_2,V_f_2] :
( hAPP(V_f_2,V_x_2) = V_y_2
=> c_Fun_Ofun__upd(T_b,T_a,V_f_2,V_x_2,V_y_2) = V_f_2 ) ).
fof(fact_extensional__funcset__fun__upd__restricts__rangeI,axiom,
! [V_T_2,T_b,V_x_2,V_f_2,V_S_2,T_a] :
( ! [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_S_2))
=> hAPP(V_f_2,V_x_2) != hAPP(V_f_2,B_x) )
=> ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional__funcset(T_a,T_b,hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_S_2),V_T_2)))
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,c_HOL_Oundefined(T_b))),c_FuncSet_Oextensional__funcset(T_a,T_b,V_S_2,c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool),V_T_2,hAPP(hAPP(c_Set_Oinsert(T_b),hAPP(V_f_2,V_x_2)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) ) ) ).
fof(fact_fold__graph_H_Ointros_I2_J,axiom,
! [V_y_2,V_z_2,V_f_2,T_b,V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> ( c_Nitpick_Ofold__graph_H(T_a,T_b,V_f_2,V_z_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_y_2)
=> c_Nitpick_Ofold__graph_H(T_a,T_b,V_f_2,V_z_2,V_A_2,hAPP(hAPP(V_f_2,V_x_2),V_y_2)) ) ) ).
fof(fact_extensional__funcset__fun__upd__inj__onI,axiom,
! [V_x_2,V_aa_2,V_T_2,V_S_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional__funcset(T_a,T_b,V_S_2,c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool),V_T_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))
=> ( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_S_2)
=> c_Fun_Oinj__on(T_a,T_b,c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_aa_2),V_S_2) ) ) ).
fof(fact_inj__on__empty,axiom,
! [V_f_2,T_b,T_a] : c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).
fof(fact_inj__onD,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_A_2))
=> V_x_2 = V_y_2 ) ) ) ) ).
fof(fact_inj__on__iff,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_A_2))
=> ( hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2)
<=> V_x_2 = V_y_2 ) ) ) ) ).
fof(fact_inj__on__contraD,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( V_x_2 != V_y_2
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_A_2))
=> hAPP(V_f_2,V_x_2) != hAPP(V_f_2,V_y_2) ) ) ) ) ).
fof(fact_inj__on__id2,axiom,
! [V_A_2,T_a] : c_Fun_Oinj__on(T_a,T_a,c_COMBI(T_a),V_A_2) ).
fof(fact_inj__on__def,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_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_A_2))
=> ( hAPP(V_f_2,B_x) = hAPP(V_f_2,B_xa)
=> B_x = B_xa ) ) ) ) ).
fof(fact_inj__on__diff,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ).
fof(fact_inj__on__Int,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_B_2)
=> c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ) ).
fof(fact_inj__on__restrict__eq,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,c_FuncSet_Orestrict(T_a,T_b,V_f_2,V_A_2),V_A_2)
<=> c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) ) ).
fof(fact_fold__graph_H_Oequations_I1_J,axiom,
! [V_a2_2,V_a1_2,T_b,T_a] : c_Nitpick_Ofold__graph_H(T_a,T_b,V_a1_2,V_a2_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_a2_2) ).
fof(fact_fold__graph_H_Ointros_I1_J,axiom,
! [V_z_2,V_f_2,T_b,T_a] : c_Nitpick_Ofold__graph_H(T_a,T_b,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_z_2) ).
fof(fact_fold__graph_H_Oequations_I2_J,axiom,
! [V_a5_2,V_a1_2,V_a2_2,V_a4_2,V_a3_2,T_b,T_a] :
( c_Nitpick_Ofold__graph_H(T_a,T_b,V_a3_2,V_a4_2,V_a2_2,hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2))
<=> ( ( V_a2_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
& hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = V_a4_2 )
| ? [B_x,B_y] :
( hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = hAPP(hAPP(V_a3_2,B_x),B_y)
& hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_a2_2))
& c_Nitpick_Ofold__graph_H(T_a,T_b,V_a3_2,V_a4_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_a2_2,hAPP(hAPP(c_Set_Oinsert(T_a),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),B_y) ) ) ) ).
fof(fact_the__sym__eq__trivial,axiom,
! [V_x_2,T_a] : c_HOL_OThe(T_a,hAPP(c_fequal,V_x_2)) = V_x_2 ).
fof(fact_the__eq__trivial,axiom,
! [V_aa_2,T_a] : c_HOL_OThe(T_a,hAPP(hAPP(c_COMBC(T_a,T_a,tc_HOL_Obool),c_fequal),V_aa_2)) = V_aa_2 ).
fof(fact_Image__singleton__iff,axiom,
! [V_aa_2,V_r_2,T_b,V_ba_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),hAPP(c_Relation_OImage(T_b,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_aa_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_aa_2),V_ba_2)),V_r_2)) ) ).
fof(fact_fun__upd__image,axiom,
! [V_y_2,V_f_2,T_b,V_A_2,V_x_2,T_a] :
( ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> c_Set_Oimage(T_a,T_b,c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_y_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),c_Set_Oimage(T_a,T_b,V_f_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) )
& ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> c_Set_Oimage(T_a,T_b,c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_y_2),V_A_2) = c_Set_Oimage(T_a,T_b,V_f_2,V_A_2) ) ) ).
fof(fact_inj__on__insert,axiom,
! [V_A_2,V_aa_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_A_2))
<=> ( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
& ~ hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_aa_2)),c_Set_Oimage(T_a,T_b,V_f_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))))) ) ) ).
fof(fact_image__eqI,axiom,
! [T_a,V_A_2,T_b,V_x_2,V_f_2,V_ba_2] :
( V_ba_2 = hAPP(V_f_2,V_x_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_ba_2),c_Set_Oimage(T_b,T_a,V_f_2,V_A_2))) ) ) ).
fof(fact_image__is__empty,axiom,
! [V_A_2,V_f_2,T_a,T_b] :
( c_Set_Oimage(T_b,T_a,V_f_2,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_b,tc_HOL_Obool)) ) ).
fof(fact_image__insert,axiom,
! [V_B_2,V_aa_2,V_f_2,T_a,T_b] : c_Set_Oimage(T_b,T_a,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_aa_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_a),hAPP(V_f_2,V_aa_2)),c_Set_Oimage(T_b,T_a,V_f_2,V_B_2)) ).
fof(fact_image__empty,axiom,
! [V_f_2,T_a,T_b] : c_Set_Oimage(T_b,T_a,V_f_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_empty__is__image,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)
<=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ).
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_image__ident,axiom,
! [V_Y_2,T_a] : c_Set_Oimage(T_a,T_a,c_COMBI(T_a),V_Y_2) = V_Y_2 ).
fof(fact_image__restrict__eq,axiom,
! [V_A_2,V_f_2,T_a,T_b] : c_Set_Oimage(T_b,T_a,c_FuncSet_Orestrict(T_b,T_a,V_f_2,V_A_2),V_A_2) = c_Set_Oimage(T_b,T_a,V_f_2,V_A_2) ).
fof(fact_image__image,axiom,
! [V_A_2,V_g_2,T_c,V_f_2,T_a,T_b] : c_Set_Oimage(T_b,T_a,V_f_2,c_Set_Oimage(T_c,T_b,V_g_2,V_A_2)) = c_Set_Oimage(T_c,T_a,hAPP(c_COMBB(T_b,T_a,T_c,V_f_2),V_g_2),V_A_2) ).
fof(fact_rev__image__eqI,axiom,
! [T_b,V_f_2,V_ba_2,V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> ( V_ba_2 = hAPP(V_f_2,V_x_2)
=> hBOOL(hAPP(hAPP(c_member(T_b),V_ba_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2))) ) ) ).
fof(fact_imageI,axiom,
! [V_f_2,T_b,V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2))) ) ).
fof(fact_image__iff,axiom,
! [V_A_2,V_f_2,T_b,V_z_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_z_2),c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)))
<=> ? [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
& V_z_2 = hAPP(V_f_2,B_x) ) ) ).
fof(fact_surj__compose,axiom,
! [V_C_2,V_g_2,T_c,V_B_2,V_A_2,V_f_2,T_a,T_b] :
( c_Set_Oimage(T_b,T_a,V_f_2,V_A_2) = V_B_2
=> ( c_Set_Oimage(T_a,T_c,V_g_2,V_B_2) = V_C_2
=> c_Set_Oimage(T_b,T_c,c_FuncSet_Ocompose(T_b,T_a,T_c,V_A_2,V_g_2,V_f_2),V_A_2) = V_C_2 ) ) ).
fof(fact_insert__image,axiom,
! [V_f_2,T_b,V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(hAPP(c_Set_Oinsert(T_b),hAPP(V_f_2,V_x_2)),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = c_Set_Oimage(T_a,T_b,V_f_2,V_A_2) ) ).
fof(fact_image__constant__conv,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))
=> c_Set_Oimage(T_a,T_b,c_COMBK(T_b,T_a,V_c_2),V_A_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))
=> c_Set_Oimage(T_a,T_b,c_COMBK(T_b,T_a,V_c_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,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) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ).
fof(fact_inj__on__fun__updI,axiom,
! [V_x_2,V_y_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)))
=> c_Fun_Oinj__on(T_a,T_b,c_Fun_Ofun__upd(T_a,T_b,V_f_2,V_x_2,V_y_2),V_A_2) ) ) ).
fof(fact_Image__iff,axiom,
! [V_A_2,V_r_2,T_b,V_ba_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_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_ba_2)),V_r_2)) ) ) ).
fof(fact_rev__ImageI,axiom,
! [V_r_2,V_ba_2,T_b,V_A_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_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_aa_2),V_ba_2)),V_r_2))
=> hBOOL(hAPP(hAPP(c_member(T_b),V_ba_2),hAPP(c_Relation_OImage(T_a,T_b,V_r_2),V_A_2))) ) ) ).
fof(fact_image__constant,axiom,
! [V_c_2,T_b,V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> c_Set_Oimage(T_a,T_b,c_COMBK(T_b,T_a,V_c_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) ) ).
fof(fact_Image__Int__eq,axiom,
! [V_B_2,V_A_2,V_R_2,T_b,T_a] :
( c_Relation_Osingle__valued(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_R_2))
=> hAPP(c_Relation_OImage(T_b,T_a,V_R_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)) = 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_ImageE,axiom,
! [V_A_2,V_r_2,T_b,V_ba_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_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_ba_2)),V_r_2))
=> ~ hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) ) ) ).
fof(fact_equiv__class__nondisjoint,axiom,
! [V_ba_2,V_aa_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(T_a),V_x_2),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_aa_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_ba_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_aa_2),V_ba_2)),V_r_2)) ) ) ).
fof(fact_inv__into__funcset,axiom,
! [V_B_2,V_A_2,V_f_2,T_a,T_b] :
( c_Set_Oimage(T_b,T_a,V_f_2,V_A_2) = V_B_2
=> hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),c_FuncSet_Orestrict(T_a,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_a,V_A_2,V_f_2),V_B_2)),c_FuncSet_OPi(T_a,T_b,V_B_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_A_2)))) ) ).
fof(fact_equiv__class__self,axiom,
! [V_aa_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_aa_2),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) ) ) ).
fof(fact_equiv__class__eq,axiom,
! [V_ba_2,V_aa_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_aa_2),V_ba_2)),V_r_2))
=> hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_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_ba_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) ) ) ).
fof(fact_compose__id__inv__into,axiom,
! [V_B_2,V_A_2,V_f_2,T_a,T_b] :
( c_Set_Oimage(T_b,T_a,V_f_2,V_A_2) = V_B_2
=> c_FuncSet_Ocompose(T_a,T_b,T_a,V_B_2,V_f_2,c_FuncSet_Orestrict(T_a,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_a,V_A_2,V_f_2),V_B_2)) = c_FuncSet_Orestrict(T_a,T_a,c_COMBI(T_a),V_B_2) ) ).
fof(fact_eq__equiv__class,axiom,
! [V_A_2,V_ba_2,V_aa_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_aa_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_ba_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_ba_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_aa_2),V_ba_2)),V_r_2)) ) ) ) ).
fof(fact_eq__equiv__class__iff,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(T_a),V_x_2),V_A_2))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_A_2))
=> ( hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_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_y_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_x_2),V_y_2)),V_r_2)) ) ) ) ) ).
fof(fact_equiv__class__eq__iff,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_prod(T_a,T_a)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_a),V_x_2),V_y_2)),V_r_2))
<=> ( hAPP(c_Relation_OImage(T_a,T_a,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_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_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
& hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
& hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_A_2)) ) ) ) ).
fof(fact_inv__into__f__eq,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> ( hAPP(V_f_2,V_x_2) = V_y_2
=> hAPP(c_Hilbert__Choice_Oinv__into(T_a,T_b,V_A_2,V_f_2),V_y_2) = V_x_2 ) ) ) ).
fof(fact_inv__into__f__f,axiom,
! [V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(c_Hilbert__Choice_Oinv__into(T_a,T_b,V_A_2,V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2 ) ) ).
fof(fact_inv__into__injective,axiom,
! [V_y_2,V_x_2,V_f_2,V_A_2,T_b,T_a] :
( hAPP(c_Hilbert__Choice_Oinv__into(T_a,T_b,V_A_2,V_f_2),V_x_2) = hAPP(c_Hilbert__Choice_Oinv__into(T_a,T_b,V_A_2,V_f_2),V_y_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)))
=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)))
=> V_x_2 = V_y_2 ) ) ) ).
fof(fact_f__inv__into__f,axiom,
! [V_A_2,V_f_2,T_b,V_y_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)))
=> hAPP(V_f_2,hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_a,V_A_2,V_f_2),V_y_2)) = V_y_2 ) ).
fof(fact_inv__into__into,axiom,
! [V_A_2,V_f_2,T_b,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)))
=> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_a,V_A_2,V_f_2),V_x_2)),V_A_2)) ) ).
fof(fact_eq__equiv__class__iff2,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(T_a),V_x_2),V_A_2))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_y_2),V_A_2))
=> ( hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_r_2) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),hAPP(hAPP(c_Set_Oinsert(T_a),V_y_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_x_2),V_y_2)),V_r_2)) ) ) ) ) ).
fof(fact_subset__equiv__class,axiom,
! [V_aa_2,V_ba_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_ba_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_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_ba_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_aa_2),V_ba_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_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_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_trancl__mono,axiom,
! [V_s_2,V_r_2,V_p_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),V_p_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)))
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_a,T_a)),V_p_2),c_Transitive__Closure_Otrancl(T_a,V_s_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),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)),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_subset__image__iff,axiom,
! [V_A_2,V_f_2,T_b,V_B_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2))
<=> ? [B_AA] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),B_AA,V_A_2)
& V_B_2 = c_Set_Oimage(T_b,T_a,V_f_2,B_AA) ) ) ).
fof(fact_image__mono,axiom,
! [V_f_2,T_b,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_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_2,V_B_2)) ) ).
fof(fact_subset__inj__on,axiom,
! [V_A_2,V_B_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_B_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) ) ) ).
fof(fact_double__diff,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_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2)) = V_A_2 ) ) ).
fof(fact_Diff__mono,axiom,
! [V_B_2,V_D_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_D_2,V_B_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2)) ) ) ).
fof(fact_Diff__subset,axiom,
! [V_B_2,V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_A_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_le__funE,axiom,
! [V_x_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_x_2),hAPP(V_g_2,V_x_2)) ) ) ).
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_order__antisym__conv,axiom,
! [V_x_2,V_y_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
<=> V_x_2 = V_y_2 ) ) ) ).
fof(fact_le__funD,axiom,
! [V_x_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_x_2),hAPP(V_g_2,V_x_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_order__eq__iff,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_x_2 = V_y_2
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_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_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_acyclic__subset,axiom,
! [V_r_2,V_s_2,T_a] :
( c_Wellfounded_Oacyclic(T_a,V_s_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) ) ).
fof(fact_single__valued__subset,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_Relation_Osingle__valued(T_a,T_b,V_s_2)
=> c_Relation_Osingle__valued(T_a,T_b,V_r_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_in__mono,axiom,
! [V_x_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_x_2),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_B_2)) ) ) ).
fof(fact_set__rev__mp,axiom,
! [V_B_2,V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_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_x_2),V_B_2)) ) ) ).
fof(fact_set__mp,axiom,
! [V_x_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_x_2),V_A_2))
=> hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_B_2)) ) ) ).
fof(fact_acc__subset,axiom,
! [V_R2_2,V_R1_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R1_2,V_R2_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Oacc(T_a,V_R2_2),c_Wellfounded_Oacc(T_a,V_R1_2)) ) ).
fof(fact_Pi__anti__mono,axiom,
! [V_B_2,T_b,V_A_2,V_A_H_2,T_a] :
( 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(tc_fun(T_a,T_b),tc_HOL_Obool),c_FuncSet_OPi(T_a,T_b,V_A_2,V_B_2),c_FuncSet_OPi(T_a,T_b,V_A_H_2,V_B_2)) ) ).
fof(fact_predicate1D,axiom,
! [V_x_2,V_Qa_2,V_Pa_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_Pa_2,V_Qa_2)
=> ( hBOOL(hAPP(V_Pa_2,V_x_2))
=> hBOOL(hAPP(V_Qa_2,V_x_2)) ) ) ).
fof(fact_rev__predicate1D,axiom,
! [V_Qa_2,T_a,V_x_2,V_Pa_2] :
( hBOOL(hAPP(V_Pa_2,V_x_2))
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_Pa_2,V_Qa_2)
=> hBOOL(hAPP(V_Qa_2,V_x_2)) ) ) ).
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_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_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_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_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_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_rtrancl__mono,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_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_s_2)) ) ).
fof(fact_rtrancl__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_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_S_2,c_Transitive__Closure_Ortrancl(T_a,V_R_2))
=> c_Transitive__Closure_Ortrancl(T_a,V_S_2) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ) ) ).
fof(fact_rtrancl__subset__rtrancl,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,c_Transitive__Closure_Ortrancl(T_a,V_s_2))
=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_s_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),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__lower2,axiom,
! [V_B_2,V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),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)
=> 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__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)
=> 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__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,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_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),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_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,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_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,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b),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,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,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b)) ) ) ) ).
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)
=> 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)
=> 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,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,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_a,V_b),V_x) ) ) ).
fof(fact_le__inf__iff,axiom,
! [V_z_2,V_y_2,V_x_2,T_a] :
( class_Lattices_Osemilattice__inf(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y_2,V_z_2))
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_z_2) ) ) ) ).
fof(fact_le__iff__inf,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Lattices_Osemilattice__inf(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
<=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x_2,V_y_2) = V_x_2 ) ) ).
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,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,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,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,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_x) ) ).
fof(fact_subset__insertI,axiom,
! [V_aa_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_aa_2),V_B_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_subset__insertI2,axiom,
! [V_ba_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_ba_2),V_B_2)) ) ).
fof(fact_insert__mono,axiom,
! [V_aa_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_aa_2),V_C_2),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),V_D_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_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_image__diff__subset,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oimage(T_b,T_a,V_f_2,V_A_2),c_Set_Oimage(T_b,T_a,V_f_2,V_B_2)),c_Set_Oimage(T_b,T_a,V_f_2,c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2))) ).
fof(fact_image__Int__subset,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oimage(T_b,T_a,V_f_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Set_Oimage(T_b,T_a,V_f_2,V_A_2),c_Set_Oimage(T_b,T_a,V_f_2,V_B_2))) ).
fof(fact_subset__singletonD,axiom,
! [V_x_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_x_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_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ) ) ).
fof(fact_insert__subset,axiom,
! [V_B_2,V_A_2,V_x_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),V_A_2),V_B_2)
<=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_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__insert,axiom,
! [V_B_2,V_A_2,V_x_2,T_a] :
( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_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_x_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_quotient__empty,axiom,
! [V_r_2,T_a] : hAPP(hAPP(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_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)) = hAPP(hAPP(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__is__empty,axiom,
! [V_r_2,V_A_2,T_a] :
( hAPP(hAPP(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_Image__closed__trancl,axiom,
! [V_X_2,V_r_2,T_a] :
( 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_X_2),V_X_2)
=> hAPP(c_Relation_OImage(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)),V_X_2) = V_X_2 ) ).
fof(fact_image__inv__into__cancel,axiom,
! [V_B_H_2,V_A_H_2,V_A_2,V_f_2,T_a,T_b] :
( c_Set_Oimage(T_b,T_a,V_f_2,V_A_2) = V_A_H_2
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_H_2,V_A_H_2)
=> c_Set_Oimage(T_b,T_a,V_f_2,c_Set_Oimage(T_a,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_a,V_A_2,V_f_2),V_B_H_2)) = V_B_H_2 ) ) ).
fof(fact_Domain__Diff__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_Groups_Ominus__class_Ominus(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)),c_Relation_ODomain(T_a,T_b,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_A_2,V_B_2))) ).
fof(fact_Range__Diff__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_Groups_Ominus__class_Ominus(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)),c_Relation_ORange(T_b,T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_A_2,V_B_2))) ).
fof(fact_extensional__subset,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional(T_a,T_b,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(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_Oextensional(T_a,T_b,V_B_2))) ) ) ).
fof(fact_inj__on__iff__surj,axiom,
! [V_A_H_2,T_b,T_a,V_A_2] :
( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
=> ( ? [B_f] :
( c_Fun_Oinj__on(T_a,T_b,B_f,V_A_2)
& c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,B_f,V_A_2),V_A_H_2) )
<=> ? [B_g] : c_Set_Oimage(T_b,T_a,B_g,V_A_H_2) = V_A_2 ) ) ).
fof(fact_inj__on__image__Int,axiom,
! [V_B_2,V_A_2,V_C_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_C_2)
=> ( 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_Set_Oimage(T_a,T_b,V_f_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_2,V_B_2)) ) ) ) ).
fof(fact_inj__on__image__set__diff,axiom,
! [V_B_2,V_A_2,V_C_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_C_2)
=> ( 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_Set_Oimage(T_a,T_b,V_f_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_2,V_B_2)) ) ) ) ).
fof(fact_inv__into__image__cancel,axiom,
! [V_S_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_S_2,V_A_2)
=> c_Set_Oimage(T_b,T_a,c_Hilbert__Choice_Oinv__into(T_a,T_b,V_A_2,V_f_2),c_Set_Oimage(T_a,T_b,V_f_2,V_S_2)) = V_S_2 ) ) ).
fof(fact_inj__on__inv__into,axiom,
! [V_A_2,V_f_2,T_b,V_B_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2))
=> c_Fun_Oinj__on(T_a,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_a,V_A_2,V_f_2),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,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_A_2,V_B_2)),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_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,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_A_2,V_B_2)),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_funcset__image,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_member(tc_fun(T_a,T_b)),V_f_2),c_FuncSet_OPi(T_a,T_b,V_A_2,c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_a,V_B_2))))
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),V_B_2) ) ).
fof(fact_singleton__quotient,axiom,
! [V_r_2,V_x_2,T_a] : hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),hAPP(hAPP(c_Set_Oinsert(T_a),V_x_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_x_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_diff__single__insert,axiom,
! [V_B_2,V_x_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_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_x_2),V_B_2)) ) ) ).
fof(fact_subset__insert__iff,axiom,
! [V_B_2,V_x_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_x_2),V_B_2))
<=> ( ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2) )
& ( ~ hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_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),hAPP(hAPP(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),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),V_A_2),V_r_2)))
=> ( V_X_2 = V_Y_2
| 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_quotient__diff1,axiom,
! [V_aa_2,V_A_2,V_r_2,T_a] :
( c_Fun_Oinj__on(T_a,tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_a,tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(T_a,tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool)),T_a,c_Equiv__Relations_Oquotient(T_a)),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_r_2),V_A_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2))
=> hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))),V_r_2) = c_Groups_Ominus__class_Ominus(tc_fun(tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),V_A_2),V_r_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),hAPP(hAPP(c_Set_Oinsert(T_a),V_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_r_2)) ) ) ).
fof(fact_quotientI,axiom,
! [V_r_2,V_A_2,V_x_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_x_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_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),V_A_2),V_r_2))) ) ).
fof(fact_equiv__class__subset,axiom,
! [V_ba_2,V_aa_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_aa_2),V_ba_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_aa_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_ba_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))) ) ) ).
fof(fact_quotient__eq__iff,axiom,
! [V_y_2,V_x_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),hAPP(hAPP(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),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),V_A_2),V_r_2)))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_X_2))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_y_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_x_2),V_y_2)),V_r_2)) ) ) ) ) ) ) ).
fof(fact_quotient__eqI,axiom,
! [V_y_2,V_x_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),hAPP(hAPP(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),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),V_A_2),V_r_2)))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_X_2))
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_y_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_x_2),V_y_2)),V_r_2))
=> V_X_2 = V_Y_2 ) ) ) ) ) ) ).
fof(fact_flat__lub__def,axiom,
! [V_ba_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_ba_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
=> c_Partial__Function_Oflat__lub(T_a,V_ba_2,V_A_2) = V_ba_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_ba_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
=> c_Partial__Function_Oflat__lub(T_a,V_ba_2,V_A_2) = c_HOL_OThe(T_a,hAPP(hAPP(c_COMBC(T_a,tc_fun(T_a,tc_HOL_Obool),tc_HOL_Obool),c_member(T_a)),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,hAPP(hAPP(c_Set_Oinsert(T_a),V_ba_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))) ) ) ).
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),hAPP(hAPP(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_accp__subset,axiom,
! [V_R2_2,V_R1_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool)),V_R1_2,V_R2_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Oaccp(T_a,V_R2_2),c_Wellfounded_Oaccp(T_a,V_R1_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(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(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,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(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(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,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_diff__eq__diff__less__eq,axiom,
! [V_d_2,V_c_2,V_ba_2,V_aa_2,T_a] :
( class_Groups_Oordered__ab__group__add(T_a)
=> ( c_Groups_Ominus__class_Ominus(T_a,V_aa_2,V_ba_2) = c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_aa_2,V_ba_2)
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_d_2) ) ) ) ).
fof(fact_the__inv__into__into,axiom,
! [V_B_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Set_Oimage(T_a,T_b,V_f_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),hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_x_2)),V_B_2)) ) ) ) ).
fof(fact_rev__predicate2D,axiom,
! [V_Qa_2,T_b,T_a,V_y_2,V_x_2,V_Pa_2] :
( hBOOL(hAPP(hAPP(V_Pa_2,V_x_2),V_y_2))
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool)),V_Pa_2,V_Qa_2)
=> hBOOL(hAPP(hAPP(V_Qa_2,V_x_2),V_y_2)) ) ) ).
fof(fact_predicate2D,axiom,
! [V_y_2,V_x_2,V_Qa_2,V_Pa_2,T_b,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool)),V_Pa_2,V_Qa_2)
=> ( hBOOL(hAPP(hAPP(V_Pa_2,V_x_2),V_y_2))
=> hBOOL(hAPP(hAPP(V_Qa_2,V_x_2),V_y_2)) ) ) ).
fof(fact_diff__eq__diff__eq,axiom,
! [V_d_2,V_c_2,V_ba_2,V_aa_2,T_a] :
( class_Groups_Oab__group__add(T_a)
=> ( c_Groups_Ominus__class_Ominus(T_a,V_aa_2,V_ba_2) = c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2)
=> ( V_aa_2 = V_ba_2
<=> V_c_2 = V_d_2 ) ) ) ).
fof(fact_the__inv__into__f__eq,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hAPP(V_f_2,V_x_2) = V_y_2
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_y_2) = V_x_2 ) ) ) ).
fof(fact_the__inv__into__f__f,axiom,
! [V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_a),V_x_2),V_A_2))
=> hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2 ) ) ).
fof(fact_inj__on__the__inv__into,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> c_Fun_Oinj__on(T_b,T_a,c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) ) ).
fof(fact_the__inv__into__onto,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> c_Set_Oimage(T_b,T_a,c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = V_A_2 ) ).
fof(fact_the__inv__into__def,axiom,
! [V_x_2,V_f_2,V_A_2,T_b,T_a] : hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_x_2) = c_HOL_OThe(T_a,hAPP(hAPP(c_COMBS(T_a,tc_HOL_Obool,tc_HOL_Obool),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,T_b,tc_HOL_Obool),hAPP(c_COMBB(T_b,tc_fun(T_b,tc_HOL_Obool),T_a,c_fequal),V_f_2)),V_x_2))) ).
fof(fact_f__the__inv__into__f,axiom,
! [V_y_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)))
=> hAPP(V_f_2,hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_y_2)) = V_y_2 ) ) ).
fof(fact_quotient__def,axiom,
! [V_r_2,V_A_2,T_a] : hAPP(hAPP(c_Equiv__Relations_Oquotient(T_a),V_A_2),V_r_2) = 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(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(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_congruentD,axiom,
! [V_z_2,V_y_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_y_2),V_z_2)),V_r_2))
=> hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_z_2) ) ) ).
fof(fact_SUP1__I,axiom,
! [T_b,V_ba_2,V_B_2,V_A_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2))
=> ( hBOOL(hAPP(hAPP(V_B_2,V_aa_2),V_ba_2))
=> hBOOL(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2),V_ba_2)) ) ) ).
fof(fact_SUP1__iff,axiom,
! [V_ba_2,V_B_2,V_A_2,T_b,T_a] :
( hBOOL(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2),V_ba_2))
<=> ? [B_x] :
( hBOOL(hAPP(hAPP(c_member(T_a),B_x),V_A_2))
& hBOOL(hAPP(hAPP(V_B_2,B_x),V_ba_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),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)) = c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_a,tc_HOL_Obool),V_A_2,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_UN__equiv__class,axiom,
! [V_aa_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_aa_2),V_A_2))
=> 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_aa_2),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_f_2) = hAPP(V_f_2,V_aa_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) = c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_a,tc_HOL_Obool),V_B_2,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__I,axiom,
! [V_B_2,V_ba_2,T_b,V_A_2,V_aa_2,T_a] :
( hBOOL(hAPP(hAPP(c_member(T_a),V_aa_2),V_A_2))
=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_ba_2),hAPP(V_B_2,V_aa_2)))
=> hBOOL(hAPP(hAPP(c_member(T_b),V_ba_2),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_a,tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2))) ) ) ).
%----Arity declarations (18)
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__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__inf,axiom,
! [T_2,T_1] :
( class_Lattices_Olattice(T_1)
=> class_Lattices_Osemilattice__inf(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_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_fun__Groups_Ominus,axiom,
! [T_2,T_1] :
( class_Groups_Ominus(T_1)
=> class_Groups_Ominus(tc_fun(T_2,T_1)) ) ).
fof(arity_HOL__Obool__Lattices_Obounded__lattice__bot,axiom,
class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Lattices_Osemilattice__inf,axiom,
class_Lattices_Osemilattice__inf(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_Oord,axiom,
class_Orderings_Oord(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Orderings_Obot,axiom,
class_Orderings_Obot(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Groups_Ominus,axiom,
class_Groups_Ominus(tc_HOL_Obool) ).
%----Helper facts (12)
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_Qa_2,V_Pa_2,T_c,T_a,T_b] : hAPP(hAPP(c_COMBB(T_b,T_a,T_c,V_Pa_2),V_Qa_2),V_R_2) = hAPP(V_Pa_2,hAPP(V_Qa_2,V_R_2)) ).
fof(help_c__COMBC__1,axiom,
! [V_R_2,V_Qa_2,V_Pa_2,T_a,T_c,T_b] : hAPP(hAPP(hAPP(c_COMBC(T_b,T_c,T_a),V_Pa_2),V_Qa_2),V_R_2) = hAPP(hAPP(V_Pa_2,V_R_2),V_Qa_2) ).
fof(help_c__COMBS__1,axiom,
! [V_R_2,V_Qa_2,V_Pa_2,T_a,T_c,T_b] : hAPP(hAPP(hAPP(c_COMBS(T_b,T_c,T_a),V_Pa_2),V_Qa_2),V_R_2) = hAPP(hAPP(V_Pa_2,V_R_2),hAPP(V_Qa_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__fNot__1,axiom,
! [V_Pa_2] :
( ~ hBOOL(hAPP(c_fNot,V_Pa_2))
| ~ hBOOL(V_Pa_2) ) ).
fof(help_c__fNot__2,axiom,
! [V_Pa_2] :
( ~ ~ hBOOL(V_Pa_2)
| hBOOL(hAPP(c_fNot,V_Pa_2)) ) ).
fof(help_c__fconj__1,axiom,
! [V_Qa_2,V_Pa_2] :
( ~ hBOOL(V_Pa_2)
| ~ hBOOL(V_Qa_2)
| hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2)) ) ).
fof(help_c__fconj__2,axiom,
! [V_Qa_2,V_Pa_2] :
( ~ hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2))
| hBOOL(V_Pa_2) ) ).
fof(help_c__fconj__3,axiom,
! [V_Qa_2,V_Pa_2] :
( ~ hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2))
| hBOOL(V_Qa_2) ) ).
%----Conjectures (1)
fof(conj_0,conjecture,
hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool)),hAPP(v_F,v_Q____)),c_Arrow__Order__Mirabelle_OLin)) ).
%------------------------------------------------------------------------------