TPTP Problem File: SWW473_1.p
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%------------------------------------------------------------------------------
% File : SWW473_1 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 383, 100 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_100_tff_l383 [Bla11]
% Status : Theorem
% Rating : 0.00 v5.4.0, 0.33 v5.3.0
% Syntax : Number of formulae : 673 ( 98 unt; 305 typ; 0 def)
% Number of atoms : 786 ( 168 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 485 ( 67 ~; 40 |; 23 &)
% ( 42 <=>; 313 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 91 ( 90 usr)
% Number of type conns : 290 ( 162 >; 128 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 214 ( 214 usr; 53 con; 0-2 aty)
% Number of variables : 822 ( 816 !; 6 ?; 822 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 15:40:48
%------------------------------------------------------------------------------
%----Should-be-implicit typings (90)
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tff(sy_c_Set_Oinsert_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__,type,
insert2003652156l_bool: ( fun_fu425979586l_bool * fun_fu814369080l_bool ) > fun_fu814369080l_bool ).
tff(sy_c_fNot,type,
fNot: fun_bool_bool ).
tff(sy_c_fconj,type,
fconj: fun_bo1549164019l_bool ).
tff(sy_c_fdisj,type,
fdisj: fun_bo1549164019l_bool ).
tff(sy_c_fequal_000t__a,type,
fequal_a: fun_a_fun_a_bool ).
tff(sy_c_fequal_000tc__Com__Opname,type,
fequal_pname: fun_pn800050071e_bool ).
tff(sy_c_fequal_000tc__Nat__Onat,type,
fequal_nat: fun_nat_fun_nat_bool ).
tff(sy_c_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
fequal_fun_a_bool: fun_fu1471507361l_bool ).
tff(sy_c_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
fequal533582459e_bool: fun_fu802393907l_bool ).
tff(sy_c_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
fequal_fun_nat_bool: fun_fu1217155507l_bool ).
tff(sy_c_fimplies,type,
fimplies: fun_bo1549164019l_bool ).
tff(sy_c_hAPP_000t__a_000tc__HOL__Obool,type,
hAPP_a_bool: ( fun_a_bool * x_a ) > bool ).
tff(sy_c_hAPP_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
hAPP_a_fun_a_bool: ( fun_a_fun_a_bool * x_a ) > fun_a_bool ).
tff(sy_c_hAPP_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
hAPP_a93125764e_bool: ( fun_a_fun_pname_bool * x_a ) > fun_pname_bool ).
tff(sy_c_hAPP_000t__a_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_a_fun_bool_bool: ( fun_a_fun_bool_bool * x_a ) > fun_bool_bool ).
tff(sy_c_hAPP_000t__a_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
hAPP_a85458249l_bool: ( fun_a_1255737515l_bool * x_a ) > fun_fun_a_bool_bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000t__a,type,
hAPP_pname_a: ( fun_pname_a * pname ) > x_a ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__HOL__Obool,type,
hAPP_pname_bool: ( fun_pname_bool * pname ) > bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
hAPP_p1534023578a_bool: ( fun_pname_fun_a_bool * pname ) > fun_a_bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
hAPP_p61793385e_bool: ( fun_pn800050071e_bool * pname ) > fun_pname_bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_p393069232l_bool: ( fun_pn250273176l_bool * pname ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,type,
hAPP_p338031245l_bool: ( fun_pn422929397l_bool * pname ) > fun_fu1430349052l_bool ).
tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).
tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
hAPP_nat_nat: ( fun_nat_nat * nat ) > nat ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
hAPP_nat_fun_a_bool: ( fun_nat_fun_a_bool * nat ) > fun_a_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
hAPP_n1025906991e_bool: ( fun_na936072029e_bool * nat ) > fun_pname_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_n1006566506l_bool: ( fun_na1469252690l_bool * nat ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
hAPP_n1699378549t_bool: ( fun_nat_fun_nat_bool * nat ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_,type,
hAPP_n215258509l_bool: ( fun_na1436237685l_bool * nat ) > fun_fu425979586l_bool ).
tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
hAPP_fun_a_bool_bool: ( fun_fun_a_bool_bool * fun_a_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
hAPP_fun_a_bool_nat: ( fun_fun_a_bool_nat * fun_a_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
hAPP_f2050579477a_bool: ( fun_fu1731003005a_bool * fun_a_bool ) > fun_a_bool ).
tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__HOL__Obool_Mtc__HOL,type,
hAPP_f198738859l_bool: ( fun_fu554186387l_bool * fun_a_bool ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL_,type,
hAPP_f1631501043l_bool: ( fun_fu1471507361l_bool * fun_a_bool ) > fun_fun_a_bool_bool ).
tff(sy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__fun_It__a_,type,
hAPP_f285962445l_bool: ( fun_fu386216885l_bool * fun_a_bool ) > fun_fu885608257l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
hAPP_f1664156314l_bool: ( fun_fu1430349052l_bool * fun_pname_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
hAPP_f921600141ol_nat: ( fun_fu1668467777ol_nat * fun_pname_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__Com__Opn,type,
hAPP_f759274231e_bool: ( fun_fu410713561e_bool * fun_pname_bool ) > fun_pname_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__HOL__Obo,type,
hAPP_f1476298914l_bool: ( fun_fu31783638l_bool * fun_pname_bool ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc_,type,
hAPP_f434788991l_bool: ( fun_fu802393907l_bool * fun_pname_bool ) > fun_fu1430349052l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__029,type,
hAPP_f556039215l_bool: ( fun_fu931343505l_bool * fun_pname_bool ) > fun_fu399576434l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
hAPP_f54304608l_bool: ( fun_fu425979586l_bool * fun_nat_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Onat,type,
hAPP_f22106695ol_nat: ( fun_fun_nat_bool_nat * fun_nat_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__HOL__Obool,type,
hAPP_f1748468828l_bool: ( fun_fu1016514960l_bool * fun_nat_bool ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat_,type,
hAPP_f800510211t_bool: ( fun_fu821463397t_bool * fun_nat_bool ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__N,type,
hAPP_f103356543l_bool: ( fun_fu1217155507l_bool * fun_nat_bool ) > fun_fu425979586l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__f,type,
hAPP_f1951378235l_bool: ( fun_fu1436348701l_bool * fun_nat_bool ) > fun_fu814369080l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__H,type,
hAPP_f621171935l_bool: ( fun_fu885608257l_bool * fun_fun_a_bool_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__N,type,
hAPP_f2009550088ol_nat: ( fun_fu2020802748ol_nat * fun_fun_a_bool_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__f,type,
hAPP_f2117159681l_bool: ( fun_fu911136611l_bool * fun_fun_a_bool_bool ) > fun_fun_a_bool_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__f_030,type,
hAPP_f1434722111l_bool: ( fun_fu418465139l_bool * fun_fun_a_bool_bool ) > fun_fu885608257l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool,type,
hAPP_f1935102916l_bool: ( fun_fu399576434l_bool * fun_fu1430349052l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_031,type,
hAPP_f55526627ol_nat: ( fun_fu1730389579ol_nat * fun_fu1430349052l_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_032,type,
hAPP_f559147733l_bool: ( fun_fu2087345469l_bool * fun_fu1430349052l_bool ) > fun_fu1430349052l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_033,type,
hAPP_f510955609l_bool: ( fun_fu1911931399l_bool * fun_fu1430349052l_bool ) > fun_fu399576434l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
hAPP_f1637334154l_bool: ( fun_fu814369080l_bool * fun_fu425979586l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_034,type,
hAPP_f696928925ol_nat: ( fun_fu411113733ol_nat * fun_fu425979586l_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_035,type,
hAPP_f1246832597l_bool: ( fun_fu616551101l_bool * fun_fu425979586l_bool ) > fun_fu425979586l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_036,type,
hAPP_f1772781669l_bool: ( fun_fu140186515l_bool * fun_fu425979586l_bool ) > fun_fu814369080l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
hAPP_f292226953l_bool: ( fun_fu255076663l_bool * fun_fu885608257l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_037,type,
hAPP_f1253658590ol_nat: ( fun_fu48515398ol_nat * fun_fu885608257l_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_038,type,
hAPP_f1050622307l_bool: ( fun_fu821736593l_bool * fun_fu885608257l_bool ) > fun_fu255076663l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__H,type,
hAPP_f389811538l_bool: ( fun_fu1438281908l_bool * fun_fu399576434l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__H_039,type,
hAPP_f98387925ol_nat: ( fun_fu1701008009ol_nat * fun_fu399576434l_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__H_040,type,
hAPP_f1759205631l_bool: ( fun_fu1086940979l_bool * fun_fu399576434l_bool ) > fun_fu1438281908l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL,type,
hAPP_f937997336l_bool: ( fun_fu61768826l_bool * fun_fu814369080l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL_041,type,
hAPP_f1690079119ol_nat: ( fun_fu1297083715ol_nat * fun_fu814369080l_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL_042,type,
hAPP_f760187903l_bool: ( fun_fu1137991347l_bool * fun_fu814369080l_bool ) > fun_fu61768826l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL,type,
hAPP_f1363661463l_bool: ( fun_fu754241017l_bool * fun_fu255076663l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool,type,
hAPP_f595608956l_bool: ( fun_fu2065874474l_bool * fun_fu1438281908l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
hAPP_f1295398978l_bool: ( fun_fu1971389424l_bool * fun_fu61768826l_bool ) > bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_c_member_000t__a,type,
member_a: fun_a_1255737515l_bool ).
tff(sy_c_member_000tc__Com__Opname,type,
member_pname: fun_pn422929397l_bool ).
tff(sy_c_member_000tc__Nat__Onat,type,
member_nat: fun_na1436237685l_bool ).
tff(sy_c_member_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
member_fun_a_bool: fun_fu386216885l_bool ).
tff(sy_c_member_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
member799430823e_bool: fun_fu931343505l_bool ).
tff(sy_c_member_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
member_fun_nat_bool: fun_fu1436348701l_bool ).
tff(sy_v_G,type,
g: fun_a_bool ).
tff(sy_v_P,type,
p: fun_a_bool > fun_fun_a_bool_bool ).
tff(sy_v_U,type,
u: fun_pname_bool ).
tff(sy_v_mgt__call,type,
mgt_call: fun_pname_a ).
tff(sy_v_na,type,
na: nat ).
tff(sy_v_pn,type,
pn: pname ).
%----Relevant facts (300)
tff(fact_0_assms_I1_J,axiom,
! [Ts: fun_a_bool,G: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Ts),G))
=> hBOOL(hAPP_fun_a_bool_bool(p(G),Ts)) ) ).
tff(fact_1_finite__Collect__subsets,axiom,
! [A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(ord_le1568362934t_bool),A)))) ) ).
tff(fact_2_finite__Collect__subsets,axiom,
! [A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(ord_le313189616e_bool),A)))) ) ).
tff(fact_3_finite__Collect__subsets,axiom,
! [A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(ord_le1311769555a_bool),A)))) ) ).
tff(fact_4_finite__Collect__subsets,axiom,
! [A: fun_fu814369080l_bool] :
( hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,A))
=> hBOOL(hAPP_f1295398978l_bool(finite719726885l_bool,collec1874991203l_bool(hAPP_f760187903l_bool(cOMBC_1269652216l_bool(ord_le65145710l_bool),A)))) ) ).
tff(fact_5_finite__Collect__subsets,axiom,
! [A: fun_fu399576434l_bool] :
( hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,A))
=> hBOOL(hAPP_f595608956l_bool(finite1491191519l_bool,collec792590109l_bool(hAPP_f1759205631l_bool(cOMBC_336095980l_bool(ord_le1375671464l_bool),A)))) ) ).
tff(fact_6_finite__Collect__subsets,axiom,
! [A: fun_fu885608257l_bool] :
( hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,A))
=> hBOOL(hAPP_f1363661463l_bool(finite1343359508l_bool,collec1635217238l_bool(hAPP_f1050622307l_bool(cOMBC_636888218l_bool(ord_le967226251l_bool),A)))) ) ).
tff(fact_7_finite__Collect__subsets,axiom,
! [A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
=> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,collec707592106l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(ord_le1375614389l_bool),A)))) ) ).
tff(fact_8_finite__Collect__subsets,axiom,
! [A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
=> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,collec1613912337l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(ord_le675606854l_bool),A)))) ) ).
tff(fact_9_finite__Collect__subsets,axiom,
! [A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
=> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,collec1015864663l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(ord_le1454342156l_bool),A)))) ) ).
tff(fact_10_finite__imageI,axiom,
! [H: fun_pname_a,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(H,F_1))) ) ).
tff(fact_11_finite__imageI,axiom,
! [H: fun_fu411113733ol_nat,F_1: fun_fu814369080l_bool] :
( hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_2089570637ol_nat(H,F_1))) ) ).
tff(fact_12_finite__imageI,axiom,
! [H: fun_fu1730389579ol_nat,F_1: fun_fu399576434l_bool] :
( hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1079571347ol_nat(H,F_1))) ) ).
tff(fact_13_finite__imageI,axiom,
! [H: fun_fu2020802748ol_nat,F_1: fun_fu885608257l_bool] :
( hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1802975832ol_nat(H,F_1))) ) ).
tff(fact_14_finite__imageI,axiom,
! [H: fun_fun_a_bool_nat,F_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_fun_a_bool_nat(H,F_1))) ) ).
tff(fact_15_finite__imageI,axiom,
! [H: fun_fu1668467777ol_nat,F_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1551609309ol_nat(H,F_1))) ) ).
tff(fact_16_finite__imageI,axiom,
! [H: fun_fun_nat_bool_nat,F_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_496248727ol_nat(H,F_1))) ) ).
tff(fact_17_finite__imageI,axiom,
! [H: fun_a_nat,F_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_a_nat(H,F_1))) ) ).
tff(fact_18_finite__imageI,axiom,
! [H: fun_fu881587263_pname,F_1: fun_fu814369080l_bool] :
( hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1604018183_pname(H,F_1))) ) ).
tff(fact_19_finite__imageI,axiom,
! [H: fun_fu1664106117_pname,F_1: fun_fu399576434l_bool] :
( hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1705983821_pname(H,F_1))) ) ).
tff(fact_20_finite__imageI,axiom,
! [H: fun_fu1175941238_pname,F_1: fun_fu885608257l_bool] :
( hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_990671762_pname(H,F_1))) ) ).
tff(fact_21_finite__imageI,axiom,
! [H: fun_fun_a_bool_pname,F_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1854862208_pname(H,F_1))) ) ).
tff(fact_22_finite__imageI,axiom,
! [H: fun_fu1499449723_pname,F_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1283814551_pname(H,F_1))) ) ).
tff(fact_23_finite__imageI,axiom,
! [H: fun_fu665170229_pname,F_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1921560913_pname(H,F_1))) ) ).
tff(fact_24_finite__imageI,axiom,
! [H: fun_a_pname,F_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_a_pname(H,F_1))) ) ).
tff(fact_25_finite__imageI,axiom,
! [H: fun_na1436237685l_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,image_1607900221l_bool(H,F_1))) ) ).
tff(fact_26_finite__imageI,axiom,
! [H: fun_na2122364079l_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,image_1874789623l_bool(H,F_1))) ) ).
tff(fact_27_finite__imageI,axiom,
! [H: fun_na1632405922l_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,image_1208015684l_bool(H,F_1))) ) ).
tff(fact_28_finite__imageI,axiom,
! [H: fun_nat_fun_a_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_nat_fun_a_bool(H,F_1))) ) ).
tff(fact_29_finite__imageI,axiom,
! [H: fun_na936072029e_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_1655916159e_bool(H,F_1))) ) ).
tff(fact_30_finite__imageI,axiom,
! [H: fun_nat_fun_nat_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_26036933t_bool(H,F_1))) ) ).
tff(fact_31_finite__imageI,axiom,
! [H: fun_nat_a,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_nat_a(H,F_1))) ) ).
tff(fact_32_finite__imageI,axiom,
! [H: fun_pn1165013435l_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,image_1154884483l_bool(H,F_1))) ) ).
tff(fact_33_finite__imageI,axiom,
! [H: fun_pn422929397l_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,image_1642285373l_bool(H,F_1))) ) ).
tff(fact_34_finite__imageI,axiom,
! [H: fun_pn1038293468l_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,image_1420695166l_bool(H,F_1))) ) ).
tff(fact_35_finite__imageI,axiom,
! [H: fun_pname_fun_a_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_112932426a_bool(H,F_1))) ) ).
tff(fact_36_finite__imageI,axiom,
! [H: fun_pn800050071e_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_47868345e_bool(H,F_1))) ) ).
tff(fact_37_finite__imageI,axiom,
! [H: fun_pn406123357t_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_2129980159t_bool(H,F_1))) ) ).
tff(fact_38_finite__imageI,axiom,
! [H: fun_pname_pname,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_pname_pname(H,F_1))) ) ).
tff(fact_39_finite__imageI,axiom,
! [H: fun_a_a,F_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_a_a(H,F_1))) ) ).
tff(fact_40_finite__imageI,axiom,
! [H: fun_fun_nat_bool_a,F_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_fun_nat_bool_a(H,F_1))) ) ).
tff(fact_41_finite__imageI,axiom,
! [H: fun_fun_pname_bool_a,F_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_876012084bool_a(H,F_1))) ) ).
tff(fact_42_finite__imageI,axiom,
! [H: fun_fun_a_bool_a,F_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_fun_a_bool_a(H,F_1))) ) ).
tff(fact_43_finite__imageI,axiom,
! [H: fun_pname_nat,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_pname_nat(H,F_1))) ) ).
tff(fact_44_finite__imageI,axiom,
! [H: fun_nat_pname,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_nat_pname(H,F_1))) ) ).
tff(fact_45_finite_OinsertI,axiom,
! [A_1: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,insert_a(A_1,A))) ) ).
tff(fact_46_finite_OinsertI,axiom,
! [A_1: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,insert_nat(A_1,A))) ) ).
tff(fact_47_finite_OinsertI,axiom,
! [A_1: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,insert_pname(A_1,A))) ) ).
tff(fact_48_finite_OinsertI,axiom,
! [A_1: fun_fu425979586l_bool,A: fun_fu814369080l_bool] :
( hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,A))
=> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,insert2003652156l_bool(A_1,A))) ) ).
tff(fact_49_finite_OinsertI,axiom,
! [A_1: fun_fu1430349052l_bool,A: fun_fu399576434l_bool] :
( hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,A))
=> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,insert1117693814l_bool(A_1,A))) ) ).
tff(fact_50_finite_OinsertI,axiom,
! [A_1: fun_fun_a_bool_bool,A: fun_fu885608257l_bool] :
( hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,A))
=> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,insert1457093509l_bool(A_1,A))) ) ).
tff(fact_51_finite_OinsertI,axiom,
! [A_1: fun_a_bool,A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,insert_fun_a_bool(A_1,A))) ) ).
tff(fact_52_finite_OinsertI,axiom,
! [A_1: fun_pname_bool,A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,insert1325755072e_bool(A_1,A))) ) ).
tff(fact_53_finite_OinsertI,axiom,
! [A_1: fun_nat_bool,A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,insert_fun_nat_bool(A_1,A))) ) ).
tff(fact_54_card__image__le,axiom,
! [F: fun_pname_nat,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_pname_nat(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).
tff(fact_55_card__image__le,axiom,
! [F: fun_a_nat,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_a_nat(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).
tff(fact_56_card__image__le,axiom,
! [F: fun_fun_nat_bool_nat,A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_496248727ol_nat(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).
tff(fact_57_card__image__le,axiom,
! [F: fun_fu1668467777ol_nat,A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_1551609309ol_nat(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).
tff(fact_58_card__image__le,axiom,
! [F: fun_fun_a_bool_nat,A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_fun_a_bool_nat(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).
tff(fact_59_card__image__le,axiom,
! [F: fun_a_pname,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_a_pname(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).
tff(fact_60_card__image__le,axiom,
! [F: fun_nat_pname,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_nat_pname(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).
tff(fact_61_card__image__le,axiom,
! [F: fun_pname_a,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).
tff(fact_62_card__image__le,axiom,
! [F: fun_fu897950882bool_a,A: fun_fu814369080l_bool] :
( hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_526090948bool_a(F,A))),hAPP_f1690079119ol_nat(finite1352710292l_bool,A))) ) ).
tff(fact_63_card__image__le,axiom,
! [F: fun_fu2061654492bool_a,A: fun_fu399576434l_bool] :
( hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_349102846bool_a(F,A))),hAPP_f98387925ol_nat(finite269641166l_bool,A))) ) ).
tff(fact_64_card__image__le,axiom,
! [F: fun_fu1373417771bool_a,A: fun_fu885608257l_bool] :
( hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_573985017bool_a(F,A))),hAPP_f1253658590ol_nat(finite1659325229l_bool,A))) ) ).
tff(fact_65_card__image__le,axiom,
! [F: fun_fun_a_bool_a,A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_a_bool_a(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).
tff(fact_66_card__image__le,axiom,
! [F: fun_fun_pname_bool_a,A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_876012084bool_a(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).
tff(fact_67_card__image__le,axiom,
! [F: fun_fun_nat_bool_a,A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_nat_bool_a(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).
tff(fact_68_card__image__le,axiom,
! [F: fun_a_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_a_a(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))) ) ).
tff(fact_69_card__image__le,axiom,
! [F: fun_pname_pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_pname_pname(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).
tff(fact_70_card__image__le,axiom,
! [F: fun_pn406123357t_bool,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,image_2129980159t_bool(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).
tff(fact_71_card__image__le,axiom,
! [F: fun_pn800050071e_bool,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,image_47868345e_bool(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).
tff(fact_72_card__image__le,axiom,
! [F: fun_pname_fun_a_bool,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,image_112932426a_bool(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))) ) ).
tff(fact_73_card__image__le,axiom,
! [F: fun_nat_a,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_nat_a(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).
tff(fact_74_card__image__le,axiom,
! [F: fun_nat_fun_nat_bool,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,image_26036933t_bool(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).
tff(fact_75_card__image__le,axiom,
! [F: fun_na936072029e_bool,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,image_1655916159e_bool(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).
tff(fact_76_card__image__le,axiom,
! [F: fun_nat_fun_a_bool,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,image_nat_fun_a_bool(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).
tff(fact_77_card__image__le,axiom,
! [F: fun_fu665170229_pname,A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_1921560913_pname(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))) ) ).
tff(fact_78_card__image__le,axiom,
! [F: fun_fu1499449723_pname,A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_1283814551_pname(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) ) ).
tff(fact_79_card__image__le,axiom,
! [F: fun_fun_a_bool_pname,A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_1854862208_pname(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) ) ).
tff(fact_80_card__mono,axiom,
! [A: fun_fu425979586l_bool,B: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A)),hAPP_f696928925ol_nat(finite346522414t_bool,B))) ) ) ).
tff(fact_81_card__mono,axiom,
! [A: fun_fu1430349052l_bool,B: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A)),hAPP_f55526627ol_nat(finite1340463720e_bool,B))) ) ) ).
tff(fact_82_card__mono,axiom,
! [A: fun_fun_a_bool_bool,B: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)),hAPP_f2009550088ol_nat(finite1306199131a_bool,B))) ) ) ).
tff(fact_83_card__mono,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,B))) ) ) ).
tff(fact_84_card__mono,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,B))) ) ) ).
tff(fact_85_card__mono,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,B))) ) ) ).
tff(fact_86_card__seteq,axiom,
! [A: fun_fu425979586l_bool,B: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,B)),hAPP_f696928925ol_nat(finite346522414t_bool,A)))
=> ( A = B ) ) ) ) ).
tff(fact_87_card__seteq,axiom,
! [A: fun_fu1430349052l_bool,B: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,B)),hAPP_f55526627ol_nat(finite1340463720e_bool,A)))
=> ( A = B ) ) ) ) ).
tff(fact_88_card__seteq,axiom,
! [A: fun_fun_a_bool_bool,B: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,B)),hAPP_f2009550088ol_nat(finite1306199131a_bool,A)))
=> ( A = B ) ) ) ) ).
tff(fact_89_card__seteq,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,B)),hAPP_f921600141ol_nat(finite_card_pname,A)))
=> ( A = B ) ) ) ) ).
tff(fact_90_card__seteq,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,B)),hAPP_fun_a_bool_nat(finite_card_a,A)))
=> ( A = B ) ) ) ) ).
tff(fact_91_card__seteq,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,B)),hAPP_f22106695ol_nat(finite_card_nat,A)))
=> ( A = B ) ) ) ) ).
tff(fact_92_card__insert__le,axiom,
! [X_2: fun_nat_bool,A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A)),hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_2,A)))) ) ).
tff(fact_93_card__insert__le,axiom,
! [X_2: fun_pname_bool,A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A)),hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_2,A)))) ) ).
tff(fact_94_card__insert__le,axiom,
! [X_2: fun_a_bool,A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)),hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_2,A)))) ) ).
tff(fact_95_card__insert__le,axiom,
! [X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_2,A)))) ) ).
tff(fact_96_card__insert__le,axiom,
! [X_2: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_2,A)))) ) ).
tff(fact_97_card__insert__le,axiom,
! [X_2: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_2,A)))) ) ).
tff(fact_98_card__insert__if,axiom,
! [X_2: fun_nat_bool,A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
=> ( ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_2),A))
=> ( hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_2,A)) = hAPP_f696928925ol_nat(finite346522414t_bool,A) ) )
& ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_2),A))
=> ( hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A)) ) ) ) ) ).
tff(fact_99_card__insert__if,axiom,
! [X_2: fun_pname_bool,A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
=> ( ( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A))
=> ( hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_2,A)) = hAPP_f55526627ol_nat(finite1340463720e_bool,A) ) )
& ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A))
=> ( hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A)) ) ) ) ) ).
tff(fact_100_card__insert__if,axiom,
! [X_2: fun_a_bool,A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
=> ( ( hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_2),A))
=> ( hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_2,A)) = hAPP_f2009550088ol_nat(finite1306199131a_bool,A) ) )
& ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_2),A))
=> ( hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)) ) ) ) ) ).
tff(fact_101_card__insert__if,axiom,
! [X_2: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> ( hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_2,A)) = hAPP_f22106695ol_nat(finite_card_nat,A) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> ( hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_2,A)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A)) ) ) ) ) ).
tff(fact_102_card__insert__if,axiom,
! [X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_2,A)) = hAPP_f921600141ol_nat(finite_card_pname,A) ) )
& ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_2,A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A)) ) ) ) ) ).
tff(fact_103_card__insert__if,axiom,
! [X_2: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
=> ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
=> ( hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_2,A)) = hAPP_fun_a_bool_nat(finite_card_a,A) ) )
& ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
=> ( hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_2,A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A)) ) ) ) ) ).
tff(fact_104_card__insert__disjoint,axiom,
! [X_2: fun_nat_bool,A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))
=> ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_2),A))
=> ( hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A)) ) ) ) ).
tff(fact_105_card__insert__disjoint,axiom,
! [X_2: fun_pname_bool,A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))
=> ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A))
=> ( hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A)) ) ) ) ).
tff(fact_106_card__insert__disjoint,axiom,
! [X_2: fun_a_bool,A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))
=> ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_2),A))
=> ( hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)) ) ) ) ).
tff(fact_107_card__insert__disjoint,axiom,
! [X_2: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> ( hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_2,A)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A)) ) ) ) ).
tff(fact_108_card__insert__disjoint,axiom,
! [X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_2,A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A)) ) ) ) ).
tff(fact_109_card__insert__disjoint,axiom,
! [X_2: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))
=> ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
=> ( hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_2,A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A)) ) ) ) ).
tff(fact_110_finite__Collect__conjI,axiom,
! [Q_1: fun_fu425979586l_bool,Pa: fun_fu425979586l_bool] :
( ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa)))
| hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_1))) )
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,Pa),Q_1)))) ) ).
tff(fact_111_finite__Collect__conjI,axiom,
! [Q_1: fun_fu1430349052l_bool,Pa: fun_fu1430349052l_bool] :
( ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa)))
| hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_1))) )
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,Pa),Q_1)))) ) ).
tff(fact_112_finite__Collect__conjI,axiom,
! [Q_1: fun_fun_a_bool_bool,Pa: fun_fun_a_bool_bool] :
( ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa)))
| hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_1))) )
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,Pa),Q_1)))) ) ).
tff(fact_113_finite__Collect__conjI,axiom,
! [Q_1: fun_a_bool,Pa: fun_a_bool] :
( ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa)))
| hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_1))) )
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,Pa),Q_1)))) ) ).
tff(fact_114_finite__Collect__conjI,axiom,
! [Q_1: fun_pname_bool,Pa: fun_pname_bool] :
( ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa)))
| hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_1))) )
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,Pa),Q_1)))) ) ).
tff(fact_115_finite__Collect__conjI,axiom,
! [Q_1: fun_nat_bool,Pa: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
| hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) )
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,Pa),Q_1)))) ) ).
tff(fact_116_Suc__diff__le,axiom,
! [N_1: nat,M_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_2))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M_2)),N_1) = hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(M_2),N_1)) ) ) ).
tff(fact_117_finite__Collect__le__nat,axiom,
! [K: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),K)))) ).
tff(fact_118_card__Collect__le__nat,axiom,
! [Na: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),Na))) = hAPP_nat_nat(suc,Na) ).
tff(fact_119_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
=> ( X = Y ) ) ).
tff(fact_120_nat_Oinject,axiom,
! [Nat_1: nat,Nat: nat] :
( ( hAPP_nat_nat(suc,Nat_1) = hAPP_nat_nat(suc,Nat) )
<=> ( Nat_1 = Nat ) ) ).
tff(fact_121_Suc__n__not__n,axiom,
! [N_1: nat] : hAPP_nat_nat(suc,N_1) != N_1 ).
tff(fact_122_n__not__Suc__n,axiom,
! [N_1: nat] : N_1 != hAPP_nat_nat(suc,N_1) ).
tff(fact_123_le__antisym,axiom,
! [M_2: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_2))
=> ( M_2 = N_1 ) ) ) ).
tff(fact_124_le__trans,axiom,
! [K_1: nat,I_1: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1)) ) ) ).
tff(fact_125_eq__imp__le,axiom,
! [M_2: nat,N_1: nat] :
( ( M_2 = N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) ) ).
tff(fact_126_nat__le__linear,axiom,
! [M_2: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_2)) ) ).
tff(fact_127_le__refl,axiom,
! [N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1)) ).
tff(fact_128_diff__commute,axiom,
! [I_1: nat,J: nat,K_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),J)),K_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),K_1)),J) ).
tff(fact_129_finite__Collect__disjI,axiom,
! [Pa: fun_pname_bool,Q_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa)))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_1))) ) ) ).
tff(fact_130_finite__Collect__disjI,axiom,
! [Pa: fun_fu425979586l_bool,Q_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa)))
& hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_1))) ) ) ).
tff(fact_131_finite__Collect__disjI,axiom,
! [Pa: fun_fu1430349052l_bool,Q_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa)))
& hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_1))) ) ) ).
tff(fact_132_finite__Collect__disjI,axiom,
! [Pa: fun_fun_a_bool_bool,Q_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa)))
& hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_1))) ) ) ).
tff(fact_133_finite__Collect__disjI,axiom,
! [Pa: fun_nat_bool,Q_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) ) ) ).
tff(fact_134_finite__Collect__disjI,axiom,
! [Pa: fun_a_bool,Q_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa)))
& hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_1))) ) ) ).
tff(fact_135_finite__insert,axiom,
! [A_1: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,insert_nat(A_1,A)))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ).
tff(fact_136_finite__insert,axiom,
! [A_1: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,insert_pname(A_1,A)))
<=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ).
tff(fact_137_finite__insert,axiom,
! [A_1: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,insert_a(A_1,A)))
<=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ).
tff(fact_138_finite__insert,axiom,
! [A_1: fun_nat_bool,A: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,insert_fun_nat_bool(A_1,A)))
<=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ).
tff(fact_139_finite__insert,axiom,
! [A_1: fun_pname_bool,A: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,insert1325755072e_bool(A_1,A)))
<=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ).
tff(fact_140_finite__insert,axiom,
! [A_1: fun_a_bool,A: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,insert_fun_a_bool(A_1,A)))
<=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ).
tff(fact_141_finite__subset,axiom,
! [A: fun_fu425979586l_bool,B: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ) ).
tff(fact_142_finite__subset,axiom,
! [A: fun_fu1430349052l_bool,B: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ) ).
tff(fact_143_finite__subset,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ) ).
tff(fact_144_finite__subset,axiom,
! [A: fun_fun_a_bool_bool,B: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ) ).
tff(fact_145_finite__subset,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).
tff(fact_146_finite__subset,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ) ).
tff(fact_147_rev__finite__subset,axiom,
! [A: fun_fu425979586l_bool,B: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) ) ) ).
tff(fact_148_rev__finite__subset,axiom,
! [A: fun_fu1430349052l_bool,B: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) ) ) ).
tff(fact_149_rev__finite__subset,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) ) ) ).
tff(fact_150_rev__finite__subset,axiom,
! [A: fun_fun_a_bool_bool,B: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) ) ) ).
tff(fact_151_rev__finite__subset,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) ) ) ).
tff(fact_152_rev__finite__subset,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) ) ) ).
tff(fact_153_Suc__leD,axiom,
! [M_2: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_2)),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) ) ).
tff(fact_154_le__SucE,axiom,
! [M_2: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(suc,N_1)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
=> ( M_2 = hAPP_nat_nat(suc,N_1) ) ) ) ).
tff(fact_155_le__SucI,axiom,
! [M_2: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(suc,N_1))) ) ).
tff(fact_156_Suc__le__mono,axiom,
! [Na: nat,M_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,M_3)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),M_3)) ) ).
tff(fact_157_le__Suc__eq,axiom,
! [M_3: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),hAPP_nat_nat(suc,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),Na))
| ( M_3 = hAPP_nat_nat(suc,Na) ) ) ) ).
tff(fact_158_not__less__eq__eq,axiom,
! [M_3: nat,Na: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),Na))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),M_3)) ) ).
tff(fact_159_Suc__n__not__le__n,axiom,
! [N_1: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),N_1)) ).
tff(fact_160_Suc__diff__diff,axiom,
! [M_2: nat,N_1: nat,K_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M_2)),N_1)),hAPP_nat_nat(suc,K_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M_2),N_1)),K_1) ).
tff(fact_161_diff__Suc__Suc,axiom,
! [M_2: nat,N_1: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M_2)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(minus_minus_nat(M_2),N_1) ).
tff(fact_162_le__diff__iff,axiom,
! [Na: nat,K: nat,M_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_3),K)),hAPP_nat_nat(minus_minus_nat(Na),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),Na)) ) ) ) ).
tff(fact_163_Nat_Odiff__diff__eq,axiom,
! [N_1: nat,K_1: nat,M_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M_2),K_1)),hAPP_nat_nat(minus_minus_nat(N_1),K_1)) = hAPP_nat_nat(minus_minus_nat(M_2),N_1) ) ) ) ).
tff(fact_164_eq__diff__iff,axiom,
! [Na: nat,K: nat,M_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
=> ( ( hAPP_nat_nat(minus_minus_nat(M_3),K) = hAPP_nat_nat(minus_minus_nat(Na),K) )
<=> ( M_3 = Na ) ) ) ) ).
tff(fact_165_diff__diff__cancel,axiom,
! [I_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),N_1))
=> ( hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(minus_minus_nat(N_1),I_1)) = I_1 ) ) ).
tff(fact_166_diff__le__mono,axiom,
! [L: nat,M_2: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_2),L)),hAPP_nat_nat(minus_minus_nat(N_1),L))) ) ).
tff(fact_167_diff__le__mono2,axiom,
! [L: nat,M_2: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(L),N_1)),hAPP_nat_nat(minus_minus_nat(L),M_2))) ) ).
tff(fact_168_diff__le__self,axiom,
! [M_2: nat,N_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_2),N_1)),M_2)) ).
tff(fact_169_finite__surj,axiom,
! [B: fun_a_bool,F: fun_pname_a,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) ) ) ).
tff(fact_170_finite__subset__image,axiom,
! [F: fun_pname_a,A: fun_pname_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
=> ? [C_2: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_2),A))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_2))
& ( B = image_pname_a(F,C_2) ) ) ) ) ).
tff(fact_171_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_nat_fun_nat_bool] :
( ! [N_2: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,N_2)),hAPP_n1699378549t_bool(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,Na)),hAPP_n1699378549t_bool(F,N_3))) ) ) ).
tff(fact_172_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_na936072029e_bool] :
( ! [N_2: nat] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,N_2)),hAPP_n1025906991e_bool(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,Na)),hAPP_n1025906991e_bool(F,N_3))) ) ) ).
tff(fact_173_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_nat_nat] :
( ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_3))) ) ) ).
tff(fact_174_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_nat_fun_a_bool] :
( ! [N_2: nat] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,N_2)),hAPP_nat_fun_a_bool(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,Na)),hAPP_nat_fun_a_bool(F,N_3))) ) ) ).
tff(fact_175_pigeonhole__infinite,axiom,
! [F: fun_pname_a,A: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))
=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(F,A)))
=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
& ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A)),hAPP_a93125764e_bool(cOMBC_pname_a_bool(cOMBB_1897541054_pname(fequal_a,F)),hAPP_pname_a(F,X_1)))))) ) ) ) ).
tff(fact_176_image__eqI,axiom,
! [A: fun_pname_bool,B_1: x_a,F: fun_pname_a,X_2: pname] :
( ( B_1 = hAPP_pname_a(F,X_2) )
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A))) ) ) ).
tff(fact_177_equalityI,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))
=> ( A = B ) ) ) ).
tff(fact_178_equalityI,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A))
=> ( A = B ) ) ) ).
tff(fact_179_equalityI,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A))
=> ( A = B ) ) ) ).
tff(fact_180_subsetD,axiom,
! [C_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),B)) ) ) ).
tff(fact_181_subsetD,axiom,
! [C_1: x_a,A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_1),A))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_1),B)) ) ) ).
tff(fact_182_subsetD,axiom,
! [C_1: pname,A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_1),A))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_1),B)) ) ) ).
tff(fact_183_insertCI,axiom,
! [B_1: nat,A_1: nat,B: fun_nat_bool] :
( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B))
=> ( A_1 = B_1 ) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(B_1,B))) ) ).
tff(fact_184_insertCI,axiom,
! [B_1: pname,A_1: pname,B: fun_pname_bool] :
( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B))
=> ( A_1 = B_1 ) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B_1,B))) ) ).
tff(fact_185_insertCI,axiom,
! [B_1: x_a,A_1: x_a,B: fun_a_bool] :
( ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),B))
=> ( A_1 = B_1 ) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(B_1,B))) ) ).
tff(fact_186_insertE,axiom,
! [A_1: nat,B_1: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(B_1,A)))
=> ( ( A_1 != B_1 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) ) ) ).
tff(fact_187_insertE,axiom,
! [A_1: pname,B_1: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B_1,A)))
=> ( ( A_1 != B_1 )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) ) ) ).
tff(fact_188_insertE,axiom,
! [A_1: x_a,B_1: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(B_1,A)))
=> ( ( A_1 != B_1 )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),A)) ) ) ).
tff(fact_189_insertI1,axiom,
! [A_1: nat,B: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(A_1,B))) ).
tff(fact_190_insertI1,axiom,
! [A_1: pname,B: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(A_1,B))) ).
tff(fact_191_insertI1,axiom,
! [A_1: x_a,B: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(A_1,B))) ).
tff(fact_192_insert__compr,axiom,
! [A_1: nat,B: fun_nat_bool] : insert_nat(A_1,B) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_1)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B))) ).
tff(fact_193_insert__compr,axiom,
! [A_1: pname,B: fun_pname_bool] : insert_pname(A_1,B) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_1)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),B))) ).
tff(fact_194_insert__compr,axiom,
! [A_1: x_a,B: fun_a_bool] : insert_a(A_1,B) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_1)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B))) ).
tff(fact_195_insert__compr,axiom,
! [A_1: fun_nat_bool,B: fun_fu425979586l_bool] : insert_fun_nat_bool(A_1,B) = collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_1)),hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(member_fun_nat_bool),B))) ).
tff(fact_196_insert__compr,axiom,
! [A_1: fun_pname_bool,B: fun_fu1430349052l_bool] : insert1325755072e_bool(A_1,B) = collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_1)),hAPP_f559147733l_bool(cOMBC_1988546018l_bool(member799430823e_bool),B))) ).
tff(fact_197_insert__compr,axiom,
! [A_1: fun_a_bool,B: fun_fun_a_bool_bool] : insert_fun_a_bool(A_1,B) = collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_1)),hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(member_fun_a_bool),B))) ).
tff(fact_198_insert__Collect,axiom,
! [A_1: nat,Pa: fun_nat_bool] : insert_nat(A_1,collect_nat(Pa)) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fimplies,cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_1))),Pa)) ).
tff(fact_199_insert__Collect,axiom,
! [A_1: pname,Pa: fun_pname_bool] : insert_pname(A_1,collect_pname(Pa)) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fimplies,cOMBB_647938656_pname(fNot,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_1))),Pa)) ).
tff(fact_200_insert__Collect,axiom,
! [A_1: x_a,Pa: fun_a_bool] : insert_a(A_1,collect_a(Pa)) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fimplies,cOMBB_bool_bool_a(fNot,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_1))),Pa)) ).
tff(fact_201_insert__Collect,axiom,
! [A_1: fun_nat_bool,Pa: fun_fu425979586l_bool] : insert_fun_nat_bool(A_1,collect_fun_nat_bool(Pa)) = collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fimplies,cOMBB_238756964t_bool(fNot,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_1))),Pa)) ).
tff(fact_202_insert__Collect,axiom,
! [A_1: fun_pname_bool,Pa: fun_fu1430349052l_bool] : insert1325755072e_bool(A_1,collec1974731493e_bool(Pa)) = collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fimplies,cOMBB_307249310e_bool(fNot,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_1))),Pa)) ).
tff(fact_203_insert__Collect,axiom,
! [A_1: fun_a_bool,Pa: fun_fun_a_bool_bool] : insert_fun_a_bool(A_1,collect_fun_a_bool(Pa)) = collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fimplies,cOMBB_2140588453a_bool(fNot,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_1))),Pa)) ).
tff(fact_204_insert__absorb2,axiom,
! [X_2: nat,A: fun_nat_bool] : insert_nat(X_2,insert_nat(X_2,A)) = insert_nat(X_2,A) ).
tff(fact_205_insert__absorb2,axiom,
! [X_2: pname,A: fun_pname_bool] : insert_pname(X_2,insert_pname(X_2,A)) = insert_pname(X_2,A) ).
tff(fact_206_insert__absorb2,axiom,
! [X_2: x_a,A: fun_a_bool] : insert_a(X_2,insert_a(X_2,A)) = insert_a(X_2,A) ).
tff(fact_207_insert__commute,axiom,
! [X_2: nat,Y_1: nat,A: fun_nat_bool] : insert_nat(X_2,insert_nat(Y_1,A)) = insert_nat(Y_1,insert_nat(X_2,A)) ).
tff(fact_208_insert__commute,axiom,
! [X_2: pname,Y_1: pname,A: fun_pname_bool] : insert_pname(X_2,insert_pname(Y_1,A)) = insert_pname(Y_1,insert_pname(X_2,A)) ).
tff(fact_209_insert__commute,axiom,
! [X_2: x_a,Y_1: x_a,A: fun_a_bool] : insert_a(X_2,insert_a(Y_1,A)) = insert_a(Y_1,insert_a(X_2,A)) ).
tff(fact_210_insert__iff,axiom,
! [A_1: nat,B_1: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(B_1,A)))
<=> ( ( A_1 = B_1 )
| hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) ) ) ).
tff(fact_211_insert__iff,axiom,
! [A_1: pname,B_1: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B_1,A)))
<=> ( ( A_1 = B_1 )
| hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) ) ) ).
tff(fact_212_insert__iff,axiom,
! [A_1: x_a,B_1: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(B_1,A)))
<=> ( ( A_1 = B_1 )
| hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),A)) ) ) ).
tff(fact_213_insert__code,axiom,
! [Y_1: nat,A: fun_nat_bool,X_2: nat] :
( hBOOL(hAPP_nat_bool(insert_nat(Y_1,A),X_2))
<=> ( ( Y_1 = X_2 )
| hBOOL(hAPP_nat_bool(A,X_2)) ) ) ).
tff(fact_214_insert__code,axiom,
! [Y_1: pname,A: fun_pname_bool,X_2: pname] :
( hBOOL(hAPP_pname_bool(insert_pname(Y_1,A),X_2))
<=> ( ( Y_1 = X_2 )
| hBOOL(hAPP_pname_bool(A,X_2)) ) ) ).
tff(fact_215_insert__code,axiom,
! [Y_1: x_a,A: fun_a_bool,X_2: x_a] :
( hBOOL(hAPP_a_bool(insert_a(Y_1,A),X_2))
<=> ( ( Y_1 = X_2 )
| hBOOL(hAPP_a_bool(A,X_2)) ) ) ).
tff(fact_216_insert__ident,axiom,
! [B: fun_nat_bool,X_2: nat,A: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))
=> ( ( insert_nat(X_2,A) = insert_nat(X_2,B) )
<=> ( A = B ) ) ) ) ).
tff(fact_217_insert__ident,axiom,
! [B: fun_pname_bool,X_2: pname,A: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B))
=> ( ( insert_pname(X_2,A) = insert_pname(X_2,B) )
<=> ( A = B ) ) ) ) ).
tff(fact_218_insert__ident,axiom,
! [B: fun_a_bool,X_2: x_a,A: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
=> ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B))
=> ( ( insert_a(X_2,A) = insert_a(X_2,B) )
<=> ( A = B ) ) ) ) ).
tff(fact_219_insertI2,axiom,
! [B_1: nat,A_1: nat,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(B_1,B))) ) ).
tff(fact_220_insertI2,axiom,
! [B_1: pname,A_1: pname,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B_1,B))) ) ).
tff(fact_221_insertI2,axiom,
! [B_1: x_a,A_1: x_a,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),B))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(B_1,B))) ) ).
tff(fact_222_insert__absorb,axiom,
! [A_1: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))
=> ( insert_nat(A_1,A) = A ) ) ).
tff(fact_223_insert__absorb,axiom,
! [A_1: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A))
=> ( insert_pname(A_1,A) = A ) ) ).
tff(fact_224_insert__absorb,axiom,
! [A_1: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),A))
=> ( insert_a(A_1,A) = A ) ) ).
tff(fact_225_subset__refl,axiom,
! [A: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),A)) ).
tff(fact_226_subset__refl,axiom,
! [A: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),A)) ).
tff(fact_227_subset__refl,axiom,
! [A: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),A)) ).
tff(fact_228_set__eq__subset,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( ( A = B )
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).
tff(fact_229_set__eq__subset,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( ( A = B )
<=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
& hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A)) ) ) ).
tff(fact_230_set__eq__subset,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( ( A = B )
<=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
& hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ) ).
tff(fact_231_equalityD1,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( ( A = B )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).
tff(fact_232_equalityD1,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( ( A = B )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ).
tff(fact_233_equalityD1,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( ( A = B )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ).
tff(fact_234_equalityD2,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( ( A = B )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ).
tff(fact_235_equalityD2,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( ( A = B )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A)) ) ).
tff(fact_236_equalityD2,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( ( A = B )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ).
tff(fact_237_in__mono,axiom,
! [X_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) ) ) ).
tff(fact_238_in__mono,axiom,
! [X_2: x_a,A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B)) ) ) ).
tff(fact_239_in__mono,axiom,
! [X_2: pname,A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B)) ) ) ).
tff(fact_240_set__rev__mp,axiom,
! [B: fun_nat_bool,X_2: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) ) ) ).
tff(fact_241_set__rev__mp,axiom,
! [B: fun_a_bool,X_2: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B)) ) ) ).
tff(fact_242_set__rev__mp,axiom,
! [B: fun_pname_bool,X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B)) ) ) ).
tff(fact_243_set__mp,axiom,
! [X_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) ) ) ).
tff(fact_244_set__mp,axiom,
! [X_2: x_a,A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B)) ) ) ).
tff(fact_245_set__mp,axiom,
! [X_2: pname,A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B)) ) ) ).
tff(fact_246_subset__trans,axiom,
! [C: fun_nat_bool,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C)) ) ) ).
tff(fact_247_subset__trans,axiom,
! [C: fun_pname_bool,A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),C))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),C)) ) ) ).
tff(fact_248_subset__trans,axiom,
! [C: fun_a_bool,A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),C))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),C)) ) ) ).
tff(fact_249_equalityE,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( ( A = B )
=> ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) ) ) ).
tff(fact_250_equalityE,axiom,
! [A: fun_pname_bool,B: fun_pname_bool] :
( ( A = B )
=> ~ ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> ~ hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A)) ) ) ).
tff(fact_251_equalityE,axiom,
! [A: fun_a_bool,B: fun_a_bool] :
( ( A = B )
=> ~ ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) ) ) ).
tff(fact_252_mem__def,axiom,
! [X_2: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
<=> hBOOL(hAPP_nat_bool(A,X_2)) ) ).
tff(fact_253_mem__def,axiom,
! [X_2: x_a,A: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
<=> hBOOL(hAPP_a_bool(A,X_2)) ) ).
tff(fact_254_mem__def,axiom,
! [X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
<=> hBOOL(hAPP_pname_bool(A,X_2)) ) ).
tff(fact_255_Collect__def,axiom,
! [Pa: fun_pname_bool] : collect_pname(Pa) = Pa ).
tff(fact_256_Collect__def,axiom,
! [Pa: fun_fu425979586l_bool] : collect_fun_nat_bool(Pa) = Pa ).
tff(fact_257_Collect__def,axiom,
! [Pa: fun_fu1430349052l_bool] : collec1974731493e_bool(Pa) = Pa ).
tff(fact_258_Collect__def,axiom,
! [Pa: fun_fun_a_bool_bool] : collect_fun_a_bool(Pa) = Pa ).
tff(fact_259_Collect__def,axiom,
! [Pa: fun_nat_bool] : collect_nat(Pa) = Pa ).
tff(fact_260_image__iff,axiom,
! [Z: x_a,F: fun_pname_a,A: fun_pname_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,Z),image_pname_a(F,A)))
<=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
& ( Z = hAPP_pname_a(F,X_1) ) ) ) ).
tff(fact_261_imageI,axiom,
! [F: fun_pname_a,X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_2)),image_pname_a(F,A))) ) ).
tff(fact_262_rev__image__eqI,axiom,
! [B_1: x_a,F: fun_pname_a,X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( ( B_1 = hAPP_pname_a(F,X_2) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A))) ) ) ).
tff(fact_263_insert__compr__raw,axiom,
! [X_1: nat,Xa: fun_nat_bool] : insert_nat(X_1,Xa) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_1)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),Xa))) ).
tff(fact_264_insert__compr__raw,axiom,
! [X_1: pname,Xa: fun_pname_bool] : insert_pname(X_1,Xa) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),X_1)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),Xa))) ).
tff(fact_265_insert__compr__raw,axiom,
! [X_1: x_a,Xa: fun_a_bool] : insert_a(X_1,Xa) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),X_1)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),Xa))) ).
tff(fact_266_insert__compr__raw,axiom,
! [X_1: fun_nat_bool,Xa: fun_fu425979586l_bool] : insert_fun_nat_bool(X_1,Xa) = collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),X_1)),hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(member_fun_nat_bool),Xa))) ).
tff(fact_267_insert__compr__raw,axiom,
! [X_1: fun_pname_bool,Xa: fun_fu1430349052l_bool] : insert1325755072e_bool(X_1,Xa) = collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),X_1)),hAPP_f559147733l_bool(cOMBC_1988546018l_bool(member799430823e_bool),Xa))) ).
tff(fact_268_insert__compr__raw,axiom,
! [X_1: fun_a_bool,Xa: fun_fun_a_bool_bool] : insert_fun_a_bool(X_1,Xa) = collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),X_1)),hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(member_fun_a_bool),Xa))) ).
tff(fact_269_subset__insertI,axiom,
! [B: fun_nat_bool,A_1: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),insert_nat(A_1,B))) ).
tff(fact_270_subset__insertI,axiom,
! [B: fun_pname_bool,A_1: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),insert_pname(A_1,B))) ).
tff(fact_271_subset__insertI,axiom,
! [B: fun_a_bool,A_1: x_a] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),insert_a(A_1,B))) ).
tff(fact_272_insert__subset,axiom,
! [X_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,insert_nat(X_2,A)),B))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).
tff(fact_273_insert__subset,axiom,
! [X_2: pname,A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,insert_pname(X_2,A)),B))
<=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B))
& hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ) ).
tff(fact_274_insert__subset,axiom,
! [X_2: x_a,A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(X_2,A)),B))
<=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B))
& hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ) ).
tff(fact_275_subset__insert,axiom,
! [B: fun_nat_bool,X_2: nat,A: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),insert_nat(X_2,B)))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ) ).
tff(fact_276_subset__insert,axiom,
! [B: fun_pname_bool,X_2: pname,A: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),insert_pname(X_2,B)))
<=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ) ).
tff(fact_277_subset__insert,axiom,
! [B: fun_a_bool,X_2: x_a,A: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),insert_a(X_2,B)))
<=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ) ).
tff(fact_278_subset__insertI2,axiom,
! [B_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),insert_nat(B_1,B))) ) ).
tff(fact_279_subset__insertI2,axiom,
! [B_1: pname,A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),insert_pname(B_1,B))) ) ).
tff(fact_280_subset__insertI2,axiom,
! [B_1: x_a,A: fun_a_bool,B: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),insert_a(B_1,B))) ) ).
tff(fact_281_insert__mono,axiom,
! [A_1: nat,C: fun_nat_bool,D: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),D))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,insert_nat(A_1,C)),insert_nat(A_1,D))) ) ).
tff(fact_282_insert__mono,axiom,
! [A_1: pname,C: fun_pname_bool,D: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C),D))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,insert_pname(A_1,C)),insert_pname(A_1,D))) ) ).
tff(fact_283_insert__mono,axiom,
! [A_1: x_a,C: fun_a_bool,D: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C),D))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(A_1,C)),insert_a(A_1,D))) ) ).
tff(fact_284_image__insert,axiom,
! [F: fun_pname_a,A_1: pname,B: fun_pname_bool] : image_pname_a(F,insert_pname(A_1,B)) = insert_a(hAPP_pname_a(F,A_1),image_pname_a(F,B)) ).
tff(fact_285_insert__image,axiom,
! [F: fun_pname_a,X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( insert_a(hAPP_pname_a(F,X_2),image_pname_a(F,A)) = image_pname_a(F,A) ) ) ).
tff(fact_286_subset__image__iff,axiom,
! [B: fun_a_bool,F: fun_pname_a,A: fun_pname_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))
<=> ? [AA: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,AA),A))
& ( B = image_pname_a(F,AA) ) ) ) ).
tff(fact_287_image__mono,axiom,
! [F: fun_pname_a,A: fun_pname_bool,B: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A)),image_pname_a(F,B))) ) ).
tff(fact_288_imageE,axiom,
! [B_1: x_a,F: fun_pname_a,A: fun_pname_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A)))
=> ~ ! [X_1: pname] :
( ( B_1 = hAPP_pname_a(F,X_1) )
=> ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A)) ) ) ).
tff(fact_289_subsetI,axiom,
! [B: fun_nat_bool,A: fun_nat_bool] :
( ! [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).
tff(fact_290_subsetI,axiom,
! [B: fun_a_bool,A: fun_a_bool] :
( ! [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B)) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) ) ).
tff(fact_291_subsetI,axiom,
! [B: fun_pname_bool,A: fun_pname_bool] :
( ! [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B)) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) ) ).
tff(fact_292_zero__induct__lemma,axiom,
! [I: nat,Pa: fun_nat_bool,K: nat] :
( hBOOL(hAPP_nat_bool(Pa,K))
=> ( ! [N_2: nat] :
( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_2)))
=> hBOOL(hAPP_nat_bool(Pa,N_2)) )
=> hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(minus_minus_nat(K),I))) ) ) ).
tff(fact_293_Suc__le__D,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_1))
=> ? [M: nat] : M_1 = hAPP_nat_nat(suc,M) ) ).
tff(fact_294_image__subsetI,axiom,
! [F: fun_pname_a,B: fun_a_bool,A: fun_pname_bool] :
( ! [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_1)),B)) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A)),B)) ) ).
tff(fact_295_order__refl,axiom,
! [X_2: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_2),X_2)) ).
tff(fact_296_order__refl,axiom,
! [X_2: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_2),X_2)) ).
tff(fact_297_order__refl,axiom,
! [X_3: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),X_3)) ).
tff(fact_298_order__refl,axiom,
! [X_2: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_2),X_2)) ).
tff(fact_299_finite__nat__set__iff__bounded__le,axiom,
! [N: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N))
<=> ? [M: nat] :
! [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),M)) ) ) ).
%----Helper facts (61)
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ hBOOL(hAPP_bool_bool(fNot,P))
| ~ hBOOL(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(fNot,P)) ) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q))
| hBOOL(P)
| hBOOL(Q) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q))
| ~ hBOOL(P)
| hBOOL(Q) ) ).
tff(help_fequal_1_1_fequal_000t__a_T,axiom,
! [X: x_a,Y: x_a] :
( ~ hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000t__a_T,axiom,
! [X: x_a,Y: x_a] :
( ( X != Y )
| hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y)) ) ).
tff(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).
tff(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
! [X: pname,Y: pname] :
( ~ hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
! [X: pname,Y: pname] :
( ( X != Y )
| hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y)) ) ).
tff(help_COMBC_1_1_COMBC_000t__a_000t__a_000tc__HOL__Obool_U,axiom,
! [P: fun_a_fun_a_bool,Q: x_a,R: x_a] : hAPP_a_bool(hAPP_a_fun_a_bool(cOMBC_a_a_bool(P),Q),R) = hAPP_a_bool(hAPP_a_fun_a_bool(P,R),Q) ).
tff(help_fequal_1_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_a_bool,Y: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_a_bool,Y: fun_a_bool] :
( ( X != Y )
| hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y)) ) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000t__a_U,axiom,
! [P: fun_bool_bool,Q: fun_a_bool,R: x_a] : hAPP_a_bool(cOMBB_bool_bool_a(P,Q),R) = hAPP_bool_bool(P,hAPP_a_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_a_fun_bool_bool,Q: fun_a_bool,R: x_a] : hAPP_a_bool(cOMBS_a_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_a_fun_bool_bool(P,R),hAPP_a_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000t__a_000tc__HOL__Obool_U,axiom,
! [P: fun_pname_fun_a_bool,Q: x_a,R: pname] : hAPP_pname_bool(hAPP_a93125764e_bool(cOMBC_pname_a_bool(P),Q),R) = hAPP_a_bool(hAPP_p1534023578a_bool(P,R),Q) ).
tff(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_nat_bool,Y: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_nat_bool,Y: fun_nat_bool] :
( ( X != Y )
| hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)) ) ).
tff(help_fequal_1_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_pname_bool,Y: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_pname_bool,Y: fun_pname_bool] :
( ( X != Y )
| hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y)) ) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
! [P: fun_nat_fun_nat_bool,Q: nat,R: nat] : hAPP_nat_bool(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P: fun_bool_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(cOMBB_bool_bool_nat(P,Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_na1469252690l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(cOMBS_nat_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname_U,axiom,
! [P: fun_bool_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(cOMBB_647938656_pname(P,Q),R) = hAPP_bool_bool(P,hAPP_pname_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_pn250273176l_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(cOMBS_568398431l_bool(P,Q),R) = hAPP_bool_bool(hAPP_p393069232l_bool(P,R),hAPP_pname_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
! [P: fun_pn800050071e_bool,Q: pname,R: pname] : hAPP_pname_bool(hAPP_p61793385e_bool(cOMBC_1149511130e_bool(P),Q),R) = hAPP_pname_bool(hAPP_p61793385e_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Oboo,axiom,
! [P: fun_a_1255737515l_bool,Q: fun_a_bool,R: x_a] : hAPP_a_bool(hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opna,axiom,
! [P: fun_a_fun_a_bool,Q: fun_pname_a,R: pname] : hAPP_p1534023578a_bool(cOMBB_1897541054_pname(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_pname_a(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_It__a_Mtc__H,axiom,
! [P: fun_bool_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : hAPP_fun_a_bool_bool(cOMBB_2140588453a_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_fun_a_bool_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_a_bool,R: x_a] : hAPP_a_fun_bool_bool(cOMBB_1972296269bool_a(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_a_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc_,axiom,
! [P: fun_fu554186387l_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : hAPP_fun_a_bool_bool(cOMBS_1035972772l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f198738859l_bool(P,R),hAPP_fun_a_bool_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P: fun_na1436237685l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(cOMBC_226598744l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Nat__On,axiom,
! [P: fun_bool_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(cOMBB_238756964t_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_f54304608l_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_043,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : hAPP_n1006566506l_bool(cOMBB_1015721476ol_nat(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
! [P: fun_fu1016514960l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(cOMBS_1187019125l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1748468828l_bool(P,R),hAPP_f54304608l_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Com__Op,axiom,
! [P: fun_bool_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : hAPP_f1664156314l_bool(cOMBB_307249310e_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_f1664156314l_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_044,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_pname_bool,R: pname] : hAPP_p393069232l_bool(cOMBB_675860798_pname(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_pname_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__O,axiom,
! [P: fun_fu31783638l_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : hAPP_f1664156314l_bool(cOMBS_350070575l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1476298914l_bool(P,R),hAPP_f1664156314l_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob,axiom,
! [P: fun_pn422929397l_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(hAPP_f759274231e_bool(cOMBC_1058051404l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_p338031245l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HO,axiom,
! [P: fun_fu1471507361l_bool,Q: fun_a_bool,R: fun_a_bool] : hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_045,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : hAPP_f198738859l_bool(cOMBB_338059395a_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_fun_a_bool_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P: fun_fu1217155507l_bool,Q: fun_nat_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_046,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f1748468828l_bool(cOMBB_444170502t_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f54304608l_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_047,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : hAPP_f1476298914l_bool(cOMBB_2095475776e_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f1664156314l_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
! [P: fun_fu802393907l_bool,Q: fun_pname_bool,R: fun_pname_bool] : hAPP_f1664156314l_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_f434788991l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__,axiom,
! [P: fun_fu386216885l_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : hAPP_fun_a_bool_bool(hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(P),Q),R) = hAPP_f621171935l_bool(hAPP_f285962445l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__048,axiom,
! [P: fun_fu1436348701l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(P),Q),R) = hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It_049,axiom,
! [P: fun_fu931343505l_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : hAPP_f1664156314l_bool(hAPP_f559147733l_bool(cOMBC_1988546018l_bool(P),Q),R) = hAPP_f1935102916l_bool(hAPP_f556039215l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool,axiom,
! [P: fun_fu418465139l_bool,Q: fun_fun_a_bool_bool,R: fun_fun_a_bool_bool] : hAPP_f621171935l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(P),Q),R) = hAPP_f621171935l_bool(hAPP_f1434722111l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__H,axiom,
! [P: fun_fu140186515l_bool,Q: fun_fu425979586l_bool,R: fun_fu425979586l_bool] : hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(P),Q),R) = hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc_,axiom,
! [P: fun_fu1911931399l_bool,Q: fun_fu1430349052l_bool,R: fun_fu1430349052l_bool] : hAPP_f1935102916l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(P),Q),R) = hAPP_f1935102916l_bool(hAPP_f510955609l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__H,axiom,
! [P: fun_fu821736593l_bool,Q: fun_fu885608257l_bool,R: fun_fu885608257l_bool] : hAPP_f292226953l_bool(hAPP_f1050622307l_bool(cOMBC_636888218l_bool(P),Q),R) = hAPP_f292226953l_bool(hAPP_f1050622307l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool,axiom,
! [P: fun_fu1137991347l_bool,Q: fun_fu814369080l_bool,R: fun_fu814369080l_bool] : hAPP_f937997336l_bool(hAPP_f760187903l_bool(cOMBC_1269652216l_bool(P),Q),R) = hAPP_f937997336l_bool(hAPP_f760187903l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obo,axiom,
! [P: fun_fu1086940979l_bool,Q: fun_fu399576434l_bool,R: fun_fu399576434l_bool] : hAPP_f389811538l_bool(hAPP_f1759205631l_bool(cOMBC_336095980l_bool(P),Q),R) = hAPP_f389811538l_bool(hAPP_f1759205631l_bool(P,R),Q) ).
%----Conjectures (7)
tff(conj_0,hypothesis,
hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,u)) ).
tff(conj_1,hypothesis,
hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,g),image_pname_a(mgt_call,u))) ).
tff(conj_2,hypothesis,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,na)),hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(mgt_call,u)))) ).
tff(conj_3,hypothesis,
hAPP_fun_a_bool_nat(finite_card_a,g) = hAPP_nat_nat(minus_minus_nat(hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(mgt_call,u))),hAPP_nat_nat(suc,na)) ).
tff(conj_4,hypothesis,
hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,pn),u)) ).
tff(conj_5,hypothesis,
~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(mgt_call,pn)),g)) ).
tff(conj_6,conjecture,
hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(hAPP_pname_a(mgt_call,pn),g)),image_pname_a(mgt_call,u))) ).
%------------------------------------------------------------------------------