TPTP Problem File: SWW473^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW473^2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 383, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_500_thf_l383 [Bla11]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.46 v8.1.0, 0.45 v7.5.0, 0.57 v7.4.0, 0.56 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.80 v6.2.0, 0.57 v6.1.0, 0.71 v5.5.0, 0.67 v5.4.0, 1.00 v5.3.0
% Syntax : Number of formulae : 826 ( 137 unt; 107 typ; 0 def)
% Number of atoms : 2653 ( 494 equ; 85 cnn)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 5606 ( 202 ~; 57 |; 185 &;4242 @)
% ( 137 <=>; 783 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 1635 (1635 >; 0 *; 0 +; 0 <<)
% Number of symbols : 108 ( 102 usr; 11 con; 0-3 aty)
% Number of variables : 1787 ( 105 ^;1617 !; 65 ?;1787 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:38:20
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
thf(ty_ty_t__a,type,
x_a: $tType ).
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Opname,type,
pname: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,
option_com: $tType ).
%----Explicit typings (102)
thf(sy_c_Com_Obody,type,
body: pname > option_com ).
thf(sy_c_Finite__Set_Ocard_000_062_It__a_M_Eo_J,type,
finite_card_a_o: ( ( x_a > $o ) > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000_062_Itc__Com__Opname_M_Eo_J,type,
finite_card_pname_o: ( ( pname > $o ) > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000_062_Itc__Nat__Onat_M_Eo_J,type,
finite_card_nat_o: ( ( nat > $o ) > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000t__a,type,
finite_card_a: ( x_a > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Com__Opname,type,
finite_card_pname: ( pname > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
finite_card_nat: ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ofinite_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
finite_finite_a_o_o: ( ( ( x_a > $o ) > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,
finite1066544169me_o_o: ( ( ( pname > $o ) > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
finite1676163439at_o_o: ( ( ( nat > $o ) > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_It__a_M_Eo_J,type,
finite_finite_a_o: ( ( x_a > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Com__Opname_M_Eo_J,type,
finite297249702name_o: ( ( pname > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Nat__Onat_M_Eo_J,type,
finite_finite_nat_o: ( ( nat > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000t__a,type,
finite_finite_a: ( x_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
finite_finite_pname: ( pname > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
finite_finite_nat: ( nat > $o ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Onat__case_000_Eo,type,
nat_case_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Option_Othe_000tc__Com__Ocom,type,
the_com: option_com > com ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
bot_bot_a_o_o: ( x_a > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,
bot_bot_pname_o_o: ( pname > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
bot_bot_nat_o_o: ( nat > $o ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_It__a_M_Eo_J,type,
bot_bot_a_o: x_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Opname_M_Eo_J,type,
bot_bot_pname_o: pname > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
ord_less_eq_a_o_o: ( ( x_a > $o ) > $o ) > ( ( x_a > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_,type,
ord_le1205211808me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
ord_less_eq_nat_o_o: ( ( nat > $o ) > $o ) > ( ( nat > $o ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_It__a_M_Eo_J,type,
ord_less_eq_a_o: ( x_a > $o ) > ( x_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Com__Opname_M_Eo_J,type,
ord_less_eq_pname_o: ( pname > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
collect_a_o_o: ( ( ( x_a > $o ) > $o ) > $o ) > ( ( x_a > $o ) > $o ) > $o ).
thf(sy_c_Set_OCollect_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,
collect_pname_o_o: ( ( ( pname > $o ) > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o ).
thf(sy_c_Set_OCollect_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
collect_nat_o_o: ( ( ( nat > $o ) > $o ) > $o ) > ( ( nat > $o ) > $o ) > $o ).
thf(sy_c_Set_OCollect_000_062_It__a_M_Eo_J,type,
collect_a_o: ( ( x_a > $o ) > $o ) > ( x_a > $o ) > $o ).
thf(sy_c_Set_OCollect_000_062_Itc__Com__Opname_M_Eo_J,type,
collect_pname_o: ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_OCollect_000_062_Itc__Nat__Onat_M_Eo_J,type,
collect_nat_o: ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Set_OCollect_000t__a,type,
collect_a: ( x_a > $o ) > x_a > $o ).
thf(sy_c_Set_OCollect_000tc__Com__Opname,type,
collect_pname: ( pname > $o ) > pname > $o ).
thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000t__a,type,
image_a_o_a: ( ( x_a > $o ) > x_a ) > ( ( x_a > $o ) > $o ) > x_a > $o ).
thf(sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000tc__Com__Opname,type,
image_a_o_pname: ( ( x_a > $o ) > pname ) > ( ( x_a > $o ) > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000tc__Nat__Onat,type,
image_a_o_nat: ( ( x_a > $o ) > nat ) > ( ( x_a > $o ) > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000t__a,type,
image_pname_o_a: ( ( pname > $o ) > x_a ) > ( ( pname > $o ) > $o ) > x_a > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Com__Opname,type,
image_pname_o_pname: ( ( pname > $o ) > pname ) > ( ( pname > $o ) > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Nat__Onat,type,
image_pname_o_nat: ( ( pname > $o ) > nat ) > ( ( pname > $o ) > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000t__a,type,
image_nat_o_a: ( ( nat > $o ) > x_a ) > ( ( nat > $o ) > $o ) > x_a > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Com__Opname,type,
image_nat_o_pname: ( ( nat > $o ) > pname ) > ( ( nat > $o ) > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Nat__Onat,type,
image_nat_o_nat: ( ( nat > $o ) > nat ) > ( ( nat > $o ) > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000t__a_000_062_It__a_M_Eo_J,type,
image_a_a_o: ( x_a > x_a > $o ) > ( x_a > $o ) > ( x_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000t__a_000_062_Itc__Com__Opname_M_Eo_J,type,
image_a_pname_o: ( x_a > pname > $o ) > ( x_a > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_Oimage_000t__a_000_062_Itc__Nat__Onat_M_Eo_J,type,
image_a_nat_o: ( x_a > nat > $o ) > ( x_a > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Set_Oimage_000t__a_000t__a,type,
image_a_a: ( x_a > x_a ) > ( x_a > $o ) > x_a > $o ).
thf(sy_c_Set_Oimage_000t__a_000tc__Com__Opname,type,
image_a_pname: ( x_a > pname ) > ( x_a > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000t__a_000tc__Nat__Onat,type,
image_a_nat: ( x_a > nat ) > ( x_a > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_It__a_M_Eo_J,type,
image_pname_a_o: ( pname > x_a > $o ) > ( pname > $o ) > ( x_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Com__Opname_M_Eo_J,type,
image_pname_pname_o: ( pname > pname > $o ) > ( pname > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Nat__Onat_M_Eo_J,type,
image_pname_nat_o: ( pname > nat > $o ) > ( pname > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000t__a,type,
image_pname_a: ( pname > x_a ) > ( pname > $o ) > x_a > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,
image_pname_pname: ( pname > pname ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Nat__Onat,type,
image_pname_nat: ( pname > nat ) > ( pname > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000_062_It__a_M_Eo_J,type,
image_nat_a_o: ( nat > x_a > $o ) > ( nat > $o ) > ( x_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000_062_Itc__Com__Opname_M_Eo_J,type,
image_nat_pname_o: ( nat > pname > $o ) > ( nat > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
image_nat_nat_o: ( nat > nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000t__a,type,
image_nat_a: ( nat > x_a ) > ( nat > $o ) > x_a > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Com__Opname,type,
image_nat_pname: ( nat > pname ) > ( nat > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Oinsert_000_062_It__a_M_Eo_J,type,
insert_a_o: ( x_a > $o ) > ( ( x_a > $o ) > $o ) > ( x_a > $o ) > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Com__Opname_M_Eo_J,type,
insert_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_Oinsert_000_062_Itc__Nat__Onat_M_Eo_J,type,
insert_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Set_Oinsert_000t__a,type,
insert_a: x_a > ( x_a > $o ) > x_a > $o ).
thf(sy_c_Set_Oinsert_000tc__Com__Opname,type,
insert_pname: pname > ( pname > $o ) > pname > $o ).
thf(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
insert_nat: nat > ( nat > $o ) > nat > $o ).
thf(sy_c_fequal_000_062_It__a_M_Eo_J,type,
fequal_a_o: ( x_a > $o ) > ( x_a > $o ) > $o ).
thf(sy_c_fequal_000_062_Itc__Com__Opname_M_Eo_J,type,
fequal_pname_o: ( pname > $o ) > ( pname > $o ) > $o ).
thf(sy_c_fequal_000_062_Itc__Nat__Onat_M_Eo_J,type,
fequal_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_fequal_000t__a,type,
fequal_a: x_a > x_a > $o ).
thf(sy_c_fequal_000tc__Com__Opname,type,
fequal_pname: pname > pname > $o ).
thf(sy_c_fequal_000tc__Nat__Onat,type,
fequal_nat: nat > nat > $o ).
thf(sy_c_member_000_062_It__a_M_Eo_J,type,
member_a_o: ( x_a > $o ) > ( ( x_a > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Com__Opname_M_Eo_J,type,
member_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > $o ).
thf(sy_c_member_000_062_Itc__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > $o ).
thf(sy_c_member_000t__a,type,
member_a: x_a > ( x_a > $o ) > $o ).
thf(sy_c_member_000tc__Com__Opname,type,
member_pname: pname > ( pname > $o ) > $o ).
thf(sy_c_member_000tc__Nat__Onat,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(sy_v_G,type,
g: x_a > $o ).
thf(sy_v_P,type,
p: ( x_a > $o ) > ( x_a > $o ) > $o ).
thf(sy_v_U,type,
u: pname > $o ).
thf(sy_v_mgt,type,
mgt: com > x_a ).
thf(sy_v_mgt__call,type,
mgt_call: pname > x_a ).
thf(sy_v_na,type,
na: nat ).
thf(sy_v_pn,type,
pn: pname ).
thf(sy_v_wt,type,
wt: com > $o ).
%----Relevant facts (700)
thf(fact_0_assms_I1_J,axiom,
! [Ts: x_a > $o,G: x_a > $o] :
( ( ord_less_eq_a_o @ Ts @ G )
=> ( p @ G @ Ts ) ) ).
thf(fact_1_finite__Collect__subsets,axiom,
! [A_92: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_92 )
=> ( finite1676163439at_o_o
@ ( collect_nat_o_o
@ ^ [B_47: ( nat > $o ) > $o] : ( ord_less_eq_nat_o_o @ B_47 @ A_92 ) ) ) ) ).
thf(fact_2_finite__Collect__subsets,axiom,
! [A_92: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_92 )
=> ( finite1066544169me_o_o
@ ( collect_pname_o_o
@ ^ [B_47: ( pname > $o ) > $o] : ( ord_le1205211808me_o_o @ B_47 @ A_92 ) ) ) ) ).
thf(fact_3_finite__Collect__subsets,axiom,
! [A_92: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_92 )
=> ( finite_finite_a_o_o
@ ( collect_a_o_o
@ ^ [B_47: ( x_a > $o ) > $o] : ( ord_less_eq_a_o_o @ B_47 @ A_92 ) ) ) ) ).
thf(fact_4_finite__Collect__subsets,axiom,
! [A_92: x_a > $o] :
( ( finite_finite_a @ A_92 )
=> ( finite_finite_a_o
@ ( collect_a_o
@ ^ [B_47: x_a > $o] : ( ord_less_eq_a_o @ B_47 @ A_92 ) ) ) ) ).
thf(fact_5_finite__Collect__subsets,axiom,
! [A_92: pname > $o] :
( ( finite_finite_pname @ A_92 )
=> ( finite297249702name_o
@ ( collect_pname_o
@ ^ [B_47: pname > $o] : ( ord_less_eq_pname_o @ B_47 @ A_92 ) ) ) ) ).
thf(fact_6_finite__Collect__subsets,axiom,
! [A_92: nat > $o] :
( ( finite_finite_nat @ A_92 )
=> ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [B_47: nat > $o] : ( ord_less_eq_nat_o @ B_47 @ A_92 ) ) ) ) ).
thf(fact_7_finite__imageI,axiom,
! [H: pname > nat > $o,F_25: pname > $o] :
( ( finite_finite_pname @ F_25 )
=> ( finite_finite_nat_o @ ( image_pname_nat_o @ H @ F_25 ) ) ) ).
thf(fact_8_finite__imageI,axiom,
! [H: pname > pname > $o,F_25: pname > $o] :
( ( finite_finite_pname @ F_25 )
=> ( finite297249702name_o @ ( image_pname_pname_o @ H @ F_25 ) ) ) ).
thf(fact_9_finite__imageI,axiom,
! [H: pname > x_a > $o,F_25: pname > $o] :
( ( finite_finite_pname @ F_25 )
=> ( finite_finite_a_o @ ( image_pname_a_o @ H @ F_25 ) ) ) ).
thf(fact_10_finite__imageI,axiom,
! [H: nat > x_a,F_25: nat > $o] :
( ( finite_finite_nat @ F_25 )
=> ( finite_finite_a @ ( image_nat_a @ H @ F_25 ) ) ) ).
thf(fact_11_finite__imageI,axiom,
! [H: nat > nat > $o,F_25: nat > $o] :
( ( finite_finite_nat @ F_25 )
=> ( finite_finite_nat_o @ ( image_nat_nat_o @ H @ F_25 ) ) ) ).
thf(fact_12_finite__imageI,axiom,
! [H: nat > pname > $o,F_25: nat > $o] :
( ( finite_finite_nat @ F_25 )
=> ( finite297249702name_o @ ( image_nat_pname_o @ H @ F_25 ) ) ) ).
thf(fact_13_finite__imageI,axiom,
! [H: nat > x_a > $o,F_25: nat > $o] :
( ( finite_finite_nat @ F_25 )
=> ( finite_finite_a_o @ ( image_nat_a_o @ H @ F_25 ) ) ) ).
thf(fact_14_finite__imageI,axiom,
! [H: x_a > pname,F_25: x_a > $o] :
( ( finite_finite_a @ F_25 )
=> ( finite_finite_pname @ ( image_a_pname @ H @ F_25 ) ) ) ).
thf(fact_15_finite__imageI,axiom,
! [H: ( nat > $o ) > pname,F_25: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_25 )
=> ( finite_finite_pname @ ( image_nat_o_pname @ H @ F_25 ) ) ) ).
thf(fact_16_finite__imageI,axiom,
! [H: ( pname > $o ) > pname,F_25: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_25 )
=> ( finite_finite_pname @ ( image_pname_o_pname @ H @ F_25 ) ) ) ).
thf(fact_17_finite__imageI,axiom,
! [H: ( x_a > $o ) > pname,F_25: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_25 )
=> ( finite_finite_pname @ ( image_a_o_pname @ H @ F_25 ) ) ) ).
thf(fact_18_finite__imageI,axiom,
! [H: x_a > nat,F_25: x_a > $o] :
( ( finite_finite_a @ F_25 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F_25 ) ) ) ).
thf(fact_19_finite__imageI,axiom,
! [H: ( nat > $o ) > nat,F_25: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_25 )
=> ( finite_finite_nat @ ( image_nat_o_nat @ H @ F_25 ) ) ) ).
thf(fact_20_finite__imageI,axiom,
! [H: ( pname > $o ) > nat,F_25: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_25 )
=> ( finite_finite_nat @ ( image_pname_o_nat @ H @ F_25 ) ) ) ).
thf(fact_21_finite__imageI,axiom,
! [H: ( x_a > $o ) > nat,F_25: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_25 )
=> ( finite_finite_nat @ ( image_a_o_nat @ H @ F_25 ) ) ) ).
thf(fact_22_finite__imageI,axiom,
! [H: pname > x_a,F_25: pname > $o] :
( ( finite_finite_pname @ F_25 )
=> ( finite_finite_a @ ( image_pname_a @ H @ F_25 ) ) ) ).
thf(fact_23_finite_OinsertI,axiom,
! [A_91: nat > $o,A_90: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_90 )
=> ( finite_finite_nat_o @ ( insert_nat_o @ A_91 @ A_90 ) ) ) ).
thf(fact_24_finite_OinsertI,axiom,
! [A_91: pname > $o,A_90: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_90 )
=> ( finite297249702name_o @ ( insert_pname_o @ A_91 @ A_90 ) ) ) ).
thf(fact_25_finite_OinsertI,axiom,
! [A_91: x_a > $o,A_90: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_90 )
=> ( finite_finite_a_o @ ( insert_a_o @ A_91 @ A_90 ) ) ) ).
thf(fact_26_finite_OinsertI,axiom,
! [A_91: pname,A_90: pname > $o] :
( ( finite_finite_pname @ A_90 )
=> ( finite_finite_pname @ ( insert_pname @ A_91 @ A_90 ) ) ) ).
thf(fact_27_finite_OinsertI,axiom,
! [A_91: nat,A_90: nat > $o] :
( ( finite_finite_nat @ A_90 )
=> ( finite_finite_nat @ ( insert_nat @ A_91 @ A_90 ) ) ) ).
thf(fact_28_finite_OinsertI,axiom,
! [A_91: x_a,A_90: x_a > $o] :
( ( finite_finite_a @ A_90 )
=> ( finite_finite_a @ ( insert_a @ A_91 @ A_90 ) ) ) ).
thf(fact_29_card__image__le,axiom,
! [F_24: pname > pname,A_89: pname > $o] :
( ( finite_finite_pname @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ ( image_pname_pname @ F_24 @ A_89 ) ) @ ( finite_card_pname @ A_89 ) ) ) ).
thf(fact_30_card__image__le,axiom,
! [F_24: x_a > x_a,A_89: x_a > $o] :
( ( finite_finite_a @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F_24 @ A_89 ) ) @ ( finite_card_a @ A_89 ) ) ) ).
thf(fact_31_card__image__le,axiom,
! [F_24: ( nat > $o ) > x_a,A_89: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_o_a @ F_24 @ A_89 ) ) @ ( finite_card_nat_o @ A_89 ) ) ) ).
thf(fact_32_card__image__le,axiom,
! [F_24: ( pname > $o ) > x_a,A_89: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_pname_o_a @ F_24 @ A_89 ) ) @ ( finite_card_pname_o @ A_89 ) ) ) ).
thf(fact_33_card__image__le,axiom,
! [F_24: ( x_a > $o ) > x_a,A_89: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_o_a @ F_24 @ A_89 ) ) @ ( finite_card_a_o @ A_89 ) ) ) ).
thf(fact_34_card__image__le,axiom,
! [F_24: pname > nat,A_89: pname > $o] :
( ( finite_finite_pname @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_pname_nat @ F_24 @ A_89 ) ) @ ( finite_card_pname @ A_89 ) ) ) ).
thf(fact_35_card__image__le,axiom,
! [F_24: x_a > nat,A_89: x_a > $o] :
( ( finite_finite_a @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_nat @ F_24 @ A_89 ) ) @ ( finite_card_a @ A_89 ) ) ) ).
thf(fact_36_card__image__le,axiom,
! [F_24: ( nat > $o ) > nat,A_89: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_o_nat @ F_24 @ A_89 ) ) @ ( finite_card_nat_o @ A_89 ) ) ) ).
thf(fact_37_card__image__le,axiom,
! [F_24: ( pname > $o ) > nat,A_89: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_pname_o_nat @ F_24 @ A_89 ) ) @ ( finite_card_pname_o @ A_89 ) ) ) ).
thf(fact_38_card__image__le,axiom,
! [F_24: ( x_a > $o ) > nat,A_89: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_o_nat @ F_24 @ A_89 ) ) @ ( finite_card_a_o @ A_89 ) ) ) ).
thf(fact_39_card__image__le,axiom,
! [F_24: x_a > pname,A_89: x_a > $o] :
( ( finite_finite_a @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ ( image_a_pname @ F_24 @ A_89 ) ) @ ( finite_card_a @ A_89 ) ) ) ).
thf(fact_40_card__image__le,axiom,
! [F_24: nat > pname,A_89: nat > $o] :
( ( finite_finite_nat @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ ( image_nat_pname @ F_24 @ A_89 ) ) @ ( finite_card_nat @ A_89 ) ) ) ).
thf(fact_41_card__image__le,axiom,
! [F_24: pname > x_a,A_89: pname > $o] :
( ( finite_finite_pname @ A_89 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_pname_a @ F_24 @ A_89 ) ) @ ( finite_card_pname @ A_89 ) ) ) ).
thf(fact_42_card__mono,axiom,
! [A_88: ( nat > $o ) > $o,B_46: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_46 )
=> ( ( ord_less_eq_nat_o_o @ A_88 @ B_46 )
=> ( ord_less_eq_nat @ ( finite_card_nat_o @ A_88 ) @ ( finite_card_nat_o @ B_46 ) ) ) ) ).
thf(fact_43_card__mono,axiom,
! [A_88: ( pname > $o ) > $o,B_46: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_46 )
=> ( ( ord_le1205211808me_o_o @ A_88 @ B_46 )
=> ( ord_less_eq_nat @ ( finite_card_pname_o @ A_88 ) @ ( finite_card_pname_o @ B_46 ) ) ) ) ).
thf(fact_44_card__mono,axiom,
! [A_88: ( x_a > $o ) > $o,B_46: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_46 )
=> ( ( ord_less_eq_a_o_o @ A_88 @ B_46 )
=> ( ord_less_eq_nat @ ( finite_card_a_o @ A_88 ) @ ( finite_card_a_o @ B_46 ) ) ) ) ).
thf(fact_45_card__mono,axiom,
! [A_88: pname > $o,B_46: pname > $o] :
( ( finite_finite_pname @ B_46 )
=> ( ( ord_less_eq_pname_o @ A_88 @ B_46 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ A_88 ) @ ( finite_card_pname @ B_46 ) ) ) ) ).
thf(fact_46_card__mono,axiom,
! [A_88: x_a > $o,B_46: x_a > $o] :
( ( finite_finite_a @ B_46 )
=> ( ( ord_less_eq_a_o @ A_88 @ B_46 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A_88 ) @ ( finite_card_a @ B_46 ) ) ) ) ).
thf(fact_47_card__mono,axiom,
! [A_88: nat > $o,B_46: nat > $o] :
( ( finite_finite_nat @ B_46 )
=> ( ( ord_less_eq_nat_o @ A_88 @ B_46 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A_88 ) @ ( finite_card_nat @ B_46 ) ) ) ) ).
thf(fact_48_card__seteq,axiom,
! [A_87: ( nat > $o ) > $o,B_45: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_45 )
=> ( ( ord_less_eq_nat_o_o @ A_87 @ B_45 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_o @ B_45 ) @ ( finite_card_nat_o @ A_87 ) )
=> ( A_87 = B_45 ) ) ) ) ).
thf(fact_49_card__seteq,axiom,
! [A_87: ( pname > $o ) > $o,B_45: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_45 )
=> ( ( ord_le1205211808me_o_o @ A_87 @ B_45 )
=> ( ( ord_less_eq_nat @ ( finite_card_pname_o @ B_45 ) @ ( finite_card_pname_o @ A_87 ) )
=> ( A_87 = B_45 ) ) ) ) ).
thf(fact_50_card__seteq,axiom,
! [A_87: ( x_a > $o ) > $o,B_45: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_45 )
=> ( ( ord_less_eq_a_o_o @ A_87 @ B_45 )
=> ( ( ord_less_eq_nat @ ( finite_card_a_o @ B_45 ) @ ( finite_card_a_o @ A_87 ) )
=> ( A_87 = B_45 ) ) ) ) ).
thf(fact_51_card__seteq,axiom,
! [A_87: pname > $o,B_45: pname > $o] :
( ( finite_finite_pname @ B_45 )
=> ( ( ord_less_eq_pname_o @ A_87 @ B_45 )
=> ( ( ord_less_eq_nat @ ( finite_card_pname @ B_45 ) @ ( finite_card_pname @ A_87 ) )
=> ( A_87 = B_45 ) ) ) ) ).
thf(fact_52_card__seteq,axiom,
! [A_87: x_a > $o,B_45: x_a > $o] :
( ( finite_finite_a @ B_45 )
=> ( ( ord_less_eq_a_o @ A_87 @ B_45 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B_45 ) @ ( finite_card_a @ A_87 ) )
=> ( A_87 = B_45 ) ) ) ) ).
thf(fact_53_card__seteq,axiom,
! [A_87: nat > $o,B_45: nat > $o] :
( ( finite_finite_nat @ B_45 )
=> ( ( ord_less_eq_nat_o @ A_87 @ B_45 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B_45 ) @ ( finite_card_nat @ A_87 ) )
=> ( A_87 = B_45 ) ) ) ) ).
thf(fact_54_card__insert__le,axiom,
! [X_33: nat > $o,A_86: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_86 )
=> ( ord_less_eq_nat @ ( finite_card_nat_o @ A_86 ) @ ( finite_card_nat_o @ ( insert_nat_o @ X_33 @ A_86 ) ) ) ) ).
thf(fact_55_card__insert__le,axiom,
! [X_33: pname > $o,A_86: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_86 )
=> ( ord_less_eq_nat @ ( finite_card_pname_o @ A_86 ) @ ( finite_card_pname_o @ ( insert_pname_o @ X_33 @ A_86 ) ) ) ) ).
thf(fact_56_card__insert__le,axiom,
! [X_33: x_a > $o,A_86: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_86 )
=> ( ord_less_eq_nat @ ( finite_card_a_o @ A_86 ) @ ( finite_card_a_o @ ( insert_a_o @ X_33 @ A_86 ) ) ) ) ).
thf(fact_57_card__insert__le,axiom,
! [X_33: pname,A_86: pname > $o] :
( ( finite_finite_pname @ A_86 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ A_86 ) @ ( finite_card_pname @ ( insert_pname @ X_33 @ A_86 ) ) ) ) ).
thf(fact_58_card__insert__le,axiom,
! [X_33: nat,A_86: nat > $o] :
( ( finite_finite_nat @ A_86 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A_86 ) @ ( finite_card_nat @ ( insert_nat @ X_33 @ A_86 ) ) ) ) ).
thf(fact_59_card__insert__le,axiom,
! [X_33: x_a,A_86: x_a > $o] :
( ( finite_finite_a @ A_86 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A_86 ) @ ( finite_card_a @ ( insert_a @ X_33 @ A_86 ) ) ) ) ).
thf(fact_60_card__insert__if,axiom,
! [X_32: nat > $o,A_85: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_85 )
=> ( ( ( member_nat_o @ X_32 @ A_85 )
=> ( ( finite_card_nat_o @ ( insert_nat_o @ X_32 @ A_85 ) )
= ( finite_card_nat_o @ A_85 ) ) )
& ( ~ ( member_nat_o @ X_32 @ A_85 )
=> ( ( finite_card_nat_o @ ( insert_nat_o @ X_32 @ A_85 ) )
= ( suc @ ( finite_card_nat_o @ A_85 ) ) ) ) ) ) ).
thf(fact_61_card__insert__if,axiom,
! [X_32: pname > $o,A_85: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_85 )
=> ( ( ( member_pname_o @ X_32 @ A_85 )
=> ( ( finite_card_pname_o @ ( insert_pname_o @ X_32 @ A_85 ) )
= ( finite_card_pname_o @ A_85 ) ) )
& ( ~ ( member_pname_o @ X_32 @ A_85 )
=> ( ( finite_card_pname_o @ ( insert_pname_o @ X_32 @ A_85 ) )
= ( suc @ ( finite_card_pname_o @ A_85 ) ) ) ) ) ) ).
thf(fact_62_card__insert__if,axiom,
! [X_32: x_a > $o,A_85: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_85 )
=> ( ( ( member_a_o @ X_32 @ A_85 )
=> ( ( finite_card_a_o @ ( insert_a_o @ X_32 @ A_85 ) )
= ( finite_card_a_o @ A_85 ) ) )
& ( ~ ( member_a_o @ X_32 @ A_85 )
=> ( ( finite_card_a_o @ ( insert_a_o @ X_32 @ A_85 ) )
= ( suc @ ( finite_card_a_o @ A_85 ) ) ) ) ) ) ).
thf(fact_63_card__insert__if,axiom,
! [X_32: nat,A_85: nat > $o] :
( ( finite_finite_nat @ A_85 )
=> ( ( ( member_nat @ X_32 @ A_85 )
=> ( ( finite_card_nat @ ( insert_nat @ X_32 @ A_85 ) )
= ( finite_card_nat @ A_85 ) ) )
& ( ~ ( member_nat @ X_32 @ A_85 )
=> ( ( finite_card_nat @ ( insert_nat @ X_32 @ A_85 ) )
= ( suc @ ( finite_card_nat @ A_85 ) ) ) ) ) ) ).
thf(fact_64_card__insert__if,axiom,
! [X_32: pname,A_85: pname > $o] :
( ( finite_finite_pname @ A_85 )
=> ( ( ( member_pname @ X_32 @ A_85 )
=> ( ( finite_card_pname @ ( insert_pname @ X_32 @ A_85 ) )
= ( finite_card_pname @ A_85 ) ) )
& ( ~ ( member_pname @ X_32 @ A_85 )
=> ( ( finite_card_pname @ ( insert_pname @ X_32 @ A_85 ) )
= ( suc @ ( finite_card_pname @ A_85 ) ) ) ) ) ) ).
thf(fact_65_card__insert__if,axiom,
! [X_32: x_a,A_85: x_a > $o] :
( ( finite_finite_a @ A_85 )
=> ( ( ( member_a @ X_32 @ A_85 )
=> ( ( finite_card_a @ ( insert_a @ X_32 @ A_85 ) )
= ( finite_card_a @ A_85 ) ) )
& ( ~ ( member_a @ X_32 @ A_85 )
=> ( ( finite_card_a @ ( insert_a @ X_32 @ A_85 ) )
= ( suc @ ( finite_card_a @ A_85 ) ) ) ) ) ) ).
thf(fact_66_card__insert__disjoint,axiom,
! [X_31: nat > $o,A_84: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_84 )
=> ( ~ ( member_nat_o @ X_31 @ A_84 )
=> ( ( finite_card_nat_o @ ( insert_nat_o @ X_31 @ A_84 ) )
= ( suc @ ( finite_card_nat_o @ A_84 ) ) ) ) ) ).
thf(fact_67_card__insert__disjoint,axiom,
! [X_31: pname > $o,A_84: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_84 )
=> ( ~ ( member_pname_o @ X_31 @ A_84 )
=> ( ( finite_card_pname_o @ ( insert_pname_o @ X_31 @ A_84 ) )
= ( suc @ ( finite_card_pname_o @ A_84 ) ) ) ) ) ).
thf(fact_68_card__insert__disjoint,axiom,
! [X_31: x_a > $o,A_84: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_84 )
=> ( ~ ( member_a_o @ X_31 @ A_84 )
=> ( ( finite_card_a_o @ ( insert_a_o @ X_31 @ A_84 ) )
= ( suc @ ( finite_card_a_o @ A_84 ) ) ) ) ) ).
thf(fact_69_card__insert__disjoint,axiom,
! [X_31: nat,A_84: nat > $o] :
( ( finite_finite_nat @ A_84 )
=> ( ~ ( member_nat @ X_31 @ A_84 )
=> ( ( finite_card_nat @ ( insert_nat @ X_31 @ A_84 ) )
= ( suc @ ( finite_card_nat @ A_84 ) ) ) ) ) ).
thf(fact_70_card__insert__disjoint,axiom,
! [X_31: pname,A_84: pname > $o] :
( ( finite_finite_pname @ A_84 )
=> ( ~ ( member_pname @ X_31 @ A_84 )
=> ( ( finite_card_pname @ ( insert_pname @ X_31 @ A_84 ) )
= ( suc @ ( finite_card_pname @ A_84 ) ) ) ) ) ).
thf(fact_71_card__insert__disjoint,axiom,
! [X_31: x_a,A_84: x_a > $o] :
( ( finite_finite_a @ A_84 )
=> ( ~ ( member_a @ X_31 @ A_84 )
=> ( ( finite_card_a @ ( insert_a @ X_31 @ A_84 ) )
= ( suc @ ( finite_card_a @ A_84 ) ) ) ) ) ).
thf(fact_72_finite__Collect__conjI,axiom,
! [Q_1: x_a > $o,P_10: x_a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P_10 ) )
| ( finite_finite_a @ ( collect_a @ Q_1 ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( P_10 @ X_1 ) @ ( Q_1 @ X_1 ) ) ) ) ) ).
thf(fact_73_finite__Collect__conjI,axiom,
! [Q_1: ( nat > $o ) > $o,P_10: ( nat > $o ) > $o] :
( ( ( finite_finite_nat_o @ ( collect_nat_o @ P_10 ) )
| ( finite_finite_nat_o @ ( collect_nat_o @ Q_1 ) ) )
=> ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (&) @ ( P_10 @ X_1 ) @ ( Q_1 @ X_1 ) ) ) ) ) ).
thf(fact_74_finite__Collect__conjI,axiom,
! [Q_1: ( pname > $o ) > $o,P_10: ( pname > $o ) > $o] :
( ( ( finite297249702name_o @ ( collect_pname_o @ P_10 ) )
| ( finite297249702name_o @ ( collect_pname_o @ Q_1 ) ) )
=> ( finite297249702name_o
@ ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (&) @ ( P_10 @ X_1 ) @ ( Q_1 @ X_1 ) ) ) ) ) ).
thf(fact_75_finite__Collect__conjI,axiom,
! [Q_1: ( x_a > $o ) > $o,P_10: ( x_a > $o ) > $o] :
( ( ( finite_finite_a_o @ ( collect_a_o @ P_10 ) )
| ( finite_finite_a_o @ ( collect_a_o @ Q_1 ) ) )
=> ( finite_finite_a_o
@ ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (&) @ ( P_10 @ X_1 ) @ ( Q_1 @ X_1 ) ) ) ) ) ).
thf(fact_76_finite__Collect__conjI,axiom,
! [Q_1: pname > $o,P_10: pname > $o] :
( ( ( finite_finite_pname @ ( collect_pname @ P_10 ) )
| ( finite_finite_pname @ ( collect_pname @ Q_1 ) ) )
=> ( finite_finite_pname
@ ( collect_pname
@ ^ [X_1: pname] : ( (&) @ ( P_10 @ X_1 ) @ ( Q_1 @ X_1 ) ) ) ) ) ).
thf(fact_77_finite__Collect__conjI,axiom,
! [Q_1: nat > $o,P_10: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P_10 ) )
| ( finite_finite_nat @ ( collect_nat @ Q_1 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( P_10 @ X_1 ) @ ( Q_1 @ X_1 ) ) ) ) ) ).
thf(fact_78_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_79_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_2: nat] : ( ord_less_eq_nat @ N_2 @ K ) ) ) ).
thf(fact_80_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_eq_nat @ I_1 @ N ) ) )
= ( suc @ N ) ) ).
thf(fact_81_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_82_nat_Oinject,axiom,
! [Nat_2: nat,Nat: nat] :
( ( ( suc @ Nat_2 )
= ( suc @ Nat ) )
<=> ( Nat_2 = Nat ) ) ).
thf(fact_83_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
thf(fact_84_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
thf(fact_85_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_86_le__trans,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ( ord_less_eq_nat @ J_1 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
thf(fact_87_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_88_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_89_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_90_diff__commute,axiom,
! [I: nat,J_1: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J_1 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J_1 ) ) ).
thf(fact_91_finite__Collect__disjI,axiom,
! [P_9: x_a > $o,Q: x_a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X_1: x_a] : ( (|) @ ( P_9 @ X_1 ) @ ( Q @ X_1 ) ) ) )
<=> ( ( finite_finite_a @ ( collect_a @ P_9 ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
thf(fact_92_finite__Collect__disjI,axiom,
! [P_9: ( nat > $o ) > $o,Q: ( nat > $o ) > $o] :
( ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (|) @ ( P_9 @ X_1 ) @ ( Q @ X_1 ) ) ) )
<=> ( ( finite_finite_nat_o @ ( collect_nat_o @ P_9 ) )
& ( finite_finite_nat_o @ ( collect_nat_o @ Q ) ) ) ) ).
thf(fact_93_finite__Collect__disjI,axiom,
! [P_9: ( pname > $o ) > $o,Q: ( pname > $o ) > $o] :
( ( finite297249702name_o
@ ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (|) @ ( P_9 @ X_1 ) @ ( Q @ X_1 ) ) ) )
<=> ( ( finite297249702name_o @ ( collect_pname_o @ P_9 ) )
& ( finite297249702name_o @ ( collect_pname_o @ Q ) ) ) ) ).
thf(fact_94_finite__Collect__disjI,axiom,
! [P_9: ( x_a > $o ) > $o,Q: ( x_a > $o ) > $o] :
( ( finite_finite_a_o
@ ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (|) @ ( P_9 @ X_1 ) @ ( Q @ X_1 ) ) ) )
<=> ( ( finite_finite_a_o @ ( collect_a_o @ P_9 ) )
& ( finite_finite_a_o @ ( collect_a_o @ Q ) ) ) ) ).
thf(fact_95_finite__Collect__disjI,axiom,
! [P_9: pname > $o,Q: pname > $o] :
( ( finite_finite_pname
@ ( collect_pname
@ ^ [X_1: pname] : ( (|) @ ( P_9 @ X_1 ) @ ( Q @ X_1 ) ) ) )
<=> ( ( finite_finite_pname @ ( collect_pname @ P_9 ) )
& ( finite_finite_pname @ ( collect_pname @ Q ) ) ) ) ).
thf(fact_96_finite__Collect__disjI,axiom,
! [P_9: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X_1: nat] : ( (|) @ ( P_9 @ X_1 ) @ ( Q @ X_1 ) ) ) )
<=> ( ( finite_finite_nat @ ( collect_nat @ P_9 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
thf(fact_97_finite__insert,axiom,
! [A_83: nat > $o,A_82: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ ( insert_nat_o @ A_83 @ A_82 ) )
<=> ( finite_finite_nat_o @ A_82 ) ) ).
thf(fact_98_finite__insert,axiom,
! [A_83: pname > $o,A_82: ( pname > $o ) > $o] :
( ( finite297249702name_o @ ( insert_pname_o @ A_83 @ A_82 ) )
<=> ( finite297249702name_o @ A_82 ) ) ).
thf(fact_99_finite__insert,axiom,
! [A_83: x_a > $o,A_82: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ ( insert_a_o @ A_83 @ A_82 ) )
<=> ( finite_finite_a_o @ A_82 ) ) ).
thf(fact_100_finite__insert,axiom,
! [A_83: pname,A_82: pname > $o] :
( ( finite_finite_pname @ ( insert_pname @ A_83 @ A_82 ) )
<=> ( finite_finite_pname @ A_82 ) ) ).
thf(fact_101_finite__insert,axiom,
! [A_83: nat,A_82: nat > $o] :
( ( finite_finite_nat @ ( insert_nat @ A_83 @ A_82 ) )
<=> ( finite_finite_nat @ A_82 ) ) ).
thf(fact_102_finite__insert,axiom,
! [A_83: x_a,A_82: x_a > $o] :
( ( finite_finite_a @ ( insert_a @ A_83 @ A_82 ) )
<=> ( finite_finite_a @ A_82 ) ) ).
thf(fact_103_finite__subset,axiom,
! [A_81: ( nat > $o ) > $o,B_44: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ A_81 @ B_44 )
=> ( ( finite_finite_nat_o @ B_44 )
=> ( finite_finite_nat_o @ A_81 ) ) ) ).
thf(fact_104_finite__subset,axiom,
! [A_81: ( pname > $o ) > $o,B_44: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ A_81 @ B_44 )
=> ( ( finite297249702name_o @ B_44 )
=> ( finite297249702name_o @ A_81 ) ) ) ).
thf(fact_105_finite__subset,axiom,
! [A_81: ( x_a > $o ) > $o,B_44: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ A_81 @ B_44 )
=> ( ( finite_finite_a_o @ B_44 )
=> ( finite_finite_a_o @ A_81 ) ) ) ).
thf(fact_106_finite__subset,axiom,
! [A_81: x_a > $o,B_44: x_a > $o] :
( ( ord_less_eq_a_o @ A_81 @ B_44 )
=> ( ( finite_finite_a @ B_44 )
=> ( finite_finite_a @ A_81 ) ) ) ).
thf(fact_107_finite__subset,axiom,
! [A_81: pname > $o,B_44: pname > $o] :
( ( ord_less_eq_pname_o @ A_81 @ B_44 )
=> ( ( finite_finite_pname @ B_44 )
=> ( finite_finite_pname @ A_81 ) ) ) ).
thf(fact_108_finite__subset,axiom,
! [A_81: nat > $o,B_44: nat > $o] :
( ( ord_less_eq_nat_o @ A_81 @ B_44 )
=> ( ( finite_finite_nat @ B_44 )
=> ( finite_finite_nat @ A_81 ) ) ) ).
thf(fact_109_rev__finite__subset,axiom,
! [A_80: ( nat > $o ) > $o,B_43: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_43 )
=> ( ( ord_less_eq_nat_o_o @ A_80 @ B_43 )
=> ( finite_finite_nat_o @ A_80 ) ) ) ).
thf(fact_110_rev__finite__subset,axiom,
! [A_80: ( pname > $o ) > $o,B_43: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_43 )
=> ( ( ord_le1205211808me_o_o @ A_80 @ B_43 )
=> ( finite297249702name_o @ A_80 ) ) ) ).
thf(fact_111_rev__finite__subset,axiom,
! [A_80: ( x_a > $o ) > $o,B_43: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_43 )
=> ( ( ord_less_eq_a_o_o @ A_80 @ B_43 )
=> ( finite_finite_a_o @ A_80 ) ) ) ).
thf(fact_112_rev__finite__subset,axiom,
! [A_80: x_a > $o,B_43: x_a > $o] :
( ( finite_finite_a @ B_43 )
=> ( ( ord_less_eq_a_o @ A_80 @ B_43 )
=> ( finite_finite_a @ A_80 ) ) ) ).
thf(fact_113_rev__finite__subset,axiom,
! [A_80: pname > $o,B_43: pname > $o] :
( ( finite_finite_pname @ B_43 )
=> ( ( ord_less_eq_pname_o @ A_80 @ B_43 )
=> ( finite_finite_pname @ A_80 ) ) ) ).
thf(fact_114_rev__finite__subset,axiom,
! [A_80: nat > $o,B_43: nat > $o] :
( ( finite_finite_nat @ B_43 )
=> ( ( ord_less_eq_nat_o @ A_80 @ B_43 )
=> ( finite_finite_nat @ A_80 ) ) ) ).
thf(fact_115_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_116_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
thf(fact_117_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_118_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_119_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
thf(fact_120_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_eq_nat @ M @ N )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_121_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
thf(fact_122_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
thf(fact_123_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_124_le__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
thf(fact_125_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_126_eq__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
<=> ( M = N ) ) ) ) ).
thf(fact_127_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
thf(fact_128_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
thf(fact_129_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
thf(fact_130_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_131_finite__surj,axiom,
! [B_42: x_a > $o,F_23: x_a > x_a,A_79: x_a > $o] :
( ( finite_finite_a @ A_79 )
=> ( ( ord_less_eq_a_o @ B_42 @ ( image_a_a @ F_23 @ A_79 ) )
=> ( finite_finite_a @ B_42 ) ) ) ).
thf(fact_132_finite__surj,axiom,
! [B_42: x_a > $o,F_23: ( nat > $o ) > x_a,A_79: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_79 )
=> ( ( ord_less_eq_a_o @ B_42 @ ( image_nat_o_a @ F_23 @ A_79 ) )
=> ( finite_finite_a @ B_42 ) ) ) ).
thf(fact_133_finite__surj,axiom,
! [B_42: x_a > $o,F_23: ( pname > $o ) > x_a,A_79: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_79 )
=> ( ( ord_less_eq_a_o @ B_42 @ ( image_pname_o_a @ F_23 @ A_79 ) )
=> ( finite_finite_a @ B_42 ) ) ) ).
thf(fact_134_finite__surj,axiom,
! [B_42: x_a > $o,F_23: ( x_a > $o ) > x_a,A_79: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_79 )
=> ( ( ord_less_eq_a_o @ B_42 @ ( image_a_o_a @ F_23 @ A_79 ) )
=> ( finite_finite_a @ B_42 ) ) ) ).
thf(fact_135_finite__surj,axiom,
! [B_42: ( nat > $o ) > $o,F_23: pname > nat > $o,A_79: pname > $o] :
( ( finite_finite_pname @ A_79 )
=> ( ( ord_less_eq_nat_o_o @ B_42 @ ( image_pname_nat_o @ F_23 @ A_79 ) )
=> ( finite_finite_nat_o @ B_42 ) ) ) ).
thf(fact_136_finite__surj,axiom,
! [B_42: ( pname > $o ) > $o,F_23: pname > pname > $o,A_79: pname > $o] :
( ( finite_finite_pname @ A_79 )
=> ( ( ord_le1205211808me_o_o @ B_42 @ ( image_pname_pname_o @ F_23 @ A_79 ) )
=> ( finite297249702name_o @ B_42 ) ) ) ).
thf(fact_137_finite__surj,axiom,
! [B_42: ( x_a > $o ) > $o,F_23: pname > x_a > $o,A_79: pname > $o] :
( ( finite_finite_pname @ A_79 )
=> ( ( ord_less_eq_a_o_o @ B_42 @ ( image_pname_a_o @ F_23 @ A_79 ) )
=> ( finite_finite_a_o @ B_42 ) ) ) ).
thf(fact_138_finite__surj,axiom,
! [B_42: pname > $o,F_23: pname > pname,A_79: pname > $o] :
( ( finite_finite_pname @ A_79 )
=> ( ( ord_less_eq_pname_o @ B_42 @ ( image_pname_pname @ F_23 @ A_79 ) )
=> ( finite_finite_pname @ B_42 ) ) ) ).
thf(fact_139_finite__surj,axiom,
! [B_42: nat > $o,F_23: pname > nat,A_79: pname > $o] :
( ( finite_finite_pname @ A_79 )
=> ( ( ord_less_eq_nat_o @ B_42 @ ( image_pname_nat @ F_23 @ A_79 ) )
=> ( finite_finite_nat @ B_42 ) ) ) ).
thf(fact_140_finite__surj,axiom,
! [B_42: x_a > $o,F_23: nat > x_a,A_79: nat > $o] :
( ( finite_finite_nat @ A_79 )
=> ( ( ord_less_eq_a_o @ B_42 @ ( image_nat_a @ F_23 @ A_79 ) )
=> ( finite_finite_a @ B_42 ) ) ) ).
thf(fact_141_finite__surj,axiom,
! [B_42: ( nat > $o ) > $o,F_23: nat > nat > $o,A_79: nat > $o] :
( ( finite_finite_nat @ A_79 )
=> ( ( ord_less_eq_nat_o_o @ B_42 @ ( image_nat_nat_o @ F_23 @ A_79 ) )
=> ( finite_finite_nat_o @ B_42 ) ) ) ).
thf(fact_142_finite__surj,axiom,
! [B_42: ( pname > $o ) > $o,F_23: nat > pname > $o,A_79: nat > $o] :
( ( finite_finite_nat @ A_79 )
=> ( ( ord_le1205211808me_o_o @ B_42 @ ( image_nat_pname_o @ F_23 @ A_79 ) )
=> ( finite297249702name_o @ B_42 ) ) ) ).
thf(fact_143_finite__surj,axiom,
! [B_42: ( x_a > $o ) > $o,F_23: nat > x_a > $o,A_79: nat > $o] :
( ( finite_finite_nat @ A_79 )
=> ( ( ord_less_eq_a_o_o @ B_42 @ ( image_nat_a_o @ F_23 @ A_79 ) )
=> ( finite_finite_a_o @ B_42 ) ) ) ).
thf(fact_144_finite__surj,axiom,
! [B_42: pname > $o,F_23: nat > pname,A_79: nat > $o] :
( ( finite_finite_nat @ A_79 )
=> ( ( ord_less_eq_pname_o @ B_42 @ ( image_nat_pname @ F_23 @ A_79 ) )
=> ( finite_finite_pname @ B_42 ) ) ) ).
thf(fact_145_finite__surj,axiom,
! [B_42: nat > $o,F_23: nat > nat,A_79: nat > $o] :
( ( finite_finite_nat @ A_79 )
=> ( ( ord_less_eq_nat_o @ B_42 @ ( image_nat_nat @ F_23 @ A_79 ) )
=> ( finite_finite_nat @ B_42 ) ) ) ).
thf(fact_146_finite__surj,axiom,
! [B_42: pname > $o,F_23: x_a > pname,A_79: x_a > $o] :
( ( finite_finite_a @ A_79 )
=> ( ( ord_less_eq_pname_o @ B_42 @ ( image_a_pname @ F_23 @ A_79 ) )
=> ( finite_finite_pname @ B_42 ) ) ) ).
thf(fact_147_finite__surj,axiom,
! [B_42: pname > $o,F_23: ( nat > $o ) > pname,A_79: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_79 )
=> ( ( ord_less_eq_pname_o @ B_42 @ ( image_nat_o_pname @ F_23 @ A_79 ) )
=> ( finite_finite_pname @ B_42 ) ) ) ).
thf(fact_148_finite__surj,axiom,
! [B_42: pname > $o,F_23: ( pname > $o ) > pname,A_79: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_79 )
=> ( ( ord_less_eq_pname_o @ B_42 @ ( image_pname_o_pname @ F_23 @ A_79 ) )
=> ( finite_finite_pname @ B_42 ) ) ) ).
thf(fact_149_finite__surj,axiom,
! [B_42: pname > $o,F_23: ( x_a > $o ) > pname,A_79: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_79 )
=> ( ( ord_less_eq_pname_o @ B_42 @ ( image_a_o_pname @ F_23 @ A_79 ) )
=> ( finite_finite_pname @ B_42 ) ) ) ).
thf(fact_150_finite__surj,axiom,
! [B_42: nat > $o,F_23: x_a > nat,A_79: x_a > $o] :
( ( finite_finite_a @ A_79 )
=> ( ( ord_less_eq_nat_o @ B_42 @ ( image_a_nat @ F_23 @ A_79 ) )
=> ( finite_finite_nat @ B_42 ) ) ) ).
thf(fact_151_finite__surj,axiom,
! [B_42: nat > $o,F_23: ( nat > $o ) > nat,A_79: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_79 )
=> ( ( ord_less_eq_nat_o @ B_42 @ ( image_nat_o_nat @ F_23 @ A_79 ) )
=> ( finite_finite_nat @ B_42 ) ) ) ).
thf(fact_152_finite__surj,axiom,
! [B_42: nat > $o,F_23: ( pname > $o ) > nat,A_79: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_79 )
=> ( ( ord_less_eq_nat_o @ B_42 @ ( image_pname_o_nat @ F_23 @ A_79 ) )
=> ( finite_finite_nat @ B_42 ) ) ) ).
thf(fact_153_finite__surj,axiom,
! [B_42: nat > $o,F_23: ( x_a > $o ) > nat,A_79: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_79 )
=> ( ( ord_less_eq_nat_o @ B_42 @ ( image_a_o_nat @ F_23 @ A_79 ) )
=> ( finite_finite_nat @ B_42 ) ) ) ).
thf(fact_154_finite__surj,axiom,
! [B_42: x_a > $o,F_23: pname > x_a,A_79: pname > $o] :
( ( finite_finite_pname @ A_79 )
=> ( ( ord_less_eq_a_o @ B_42 @ ( image_pname_a @ F_23 @ A_79 ) )
=> ( finite_finite_a @ B_42 ) ) ) ).
thf(fact_155_finite__subset__image,axiom,
! [F_22: ( nat > $o ) > x_a,A_78: ( nat > $o ) > $o,B_41: x_a > $o] :
( ( finite_finite_a @ B_41 )
=> ( ( ord_less_eq_a_o @ B_41 @ ( image_nat_o_a @ F_22 @ A_78 ) )
=> ? [C_12: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ C_12 @ A_78 )
& ( finite_finite_nat_o @ C_12 )
& ( B_41
= ( image_nat_o_a @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_156_finite__subset__image,axiom,
! [F_22: ( pname > $o ) > x_a,A_78: ( pname > $o ) > $o,B_41: x_a > $o] :
( ( finite_finite_a @ B_41 )
=> ( ( ord_less_eq_a_o @ B_41 @ ( image_pname_o_a @ F_22 @ A_78 ) )
=> ? [C_12: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ C_12 @ A_78 )
& ( finite297249702name_o @ C_12 )
& ( B_41
= ( image_pname_o_a @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_157_finite__subset__image,axiom,
! [F_22: ( x_a > $o ) > x_a,A_78: ( x_a > $o ) > $o,B_41: x_a > $o] :
( ( finite_finite_a @ B_41 )
=> ( ( ord_less_eq_a_o @ B_41 @ ( image_a_o_a @ F_22 @ A_78 ) )
=> ? [C_12: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ C_12 @ A_78 )
& ( finite_finite_a_o @ C_12 )
& ( B_41
= ( image_a_o_a @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_158_finite__subset__image,axiom,
! [F_22: x_a > x_a,A_78: x_a > $o,B_41: x_a > $o] :
( ( finite_finite_a @ B_41 )
=> ( ( ord_less_eq_a_o @ B_41 @ ( image_a_a @ F_22 @ A_78 ) )
=> ? [C_12: x_a > $o] :
( ( ord_less_eq_a_o @ C_12 @ A_78 )
& ( finite_finite_a @ C_12 )
& ( B_41
= ( image_a_a @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_159_finite__subset__image,axiom,
! [F_22: x_a > nat > $o,A_78: x_a > $o,B_41: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_41 )
=> ( ( ord_less_eq_nat_o_o @ B_41 @ ( image_a_nat_o @ F_22 @ A_78 ) )
=> ? [C_12: x_a > $o] :
( ( ord_less_eq_a_o @ C_12 @ A_78 )
& ( finite_finite_a @ C_12 )
& ( B_41
= ( image_a_nat_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_160_finite__subset__image,axiom,
! [F_22: x_a > pname > $o,A_78: x_a > $o,B_41: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_41 )
=> ( ( ord_le1205211808me_o_o @ B_41 @ ( image_a_pname_o @ F_22 @ A_78 ) )
=> ? [C_12: x_a > $o] :
( ( ord_less_eq_a_o @ C_12 @ A_78 )
& ( finite_finite_a @ C_12 )
& ( B_41
= ( image_a_pname_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_161_finite__subset__image,axiom,
! [F_22: x_a > x_a > $o,A_78: x_a > $o,B_41: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_41 )
=> ( ( ord_less_eq_a_o_o @ B_41 @ ( image_a_a_o @ F_22 @ A_78 ) )
=> ? [C_12: x_a > $o] :
( ( ord_less_eq_a_o @ C_12 @ A_78 )
& ( finite_finite_a @ C_12 )
& ( B_41
= ( image_a_a_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_162_finite__subset__image,axiom,
! [F_22: x_a > pname,A_78: x_a > $o,B_41: pname > $o] :
( ( finite_finite_pname @ B_41 )
=> ( ( ord_less_eq_pname_o @ B_41 @ ( image_a_pname @ F_22 @ A_78 ) )
=> ? [C_12: x_a > $o] :
( ( ord_less_eq_a_o @ C_12 @ A_78 )
& ( finite_finite_a @ C_12 )
& ( B_41
= ( image_a_pname @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_163_finite__subset__image,axiom,
! [F_22: ( nat > $o ) > pname,A_78: ( nat > $o ) > $o,B_41: pname > $o] :
( ( finite_finite_pname @ B_41 )
=> ( ( ord_less_eq_pname_o @ B_41 @ ( image_nat_o_pname @ F_22 @ A_78 ) )
=> ? [C_12: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ C_12 @ A_78 )
& ( finite_finite_nat_o @ C_12 )
& ( B_41
= ( image_nat_o_pname @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_164_finite__subset__image,axiom,
! [F_22: ( pname > $o ) > pname,A_78: ( pname > $o ) > $o,B_41: pname > $o] :
( ( finite_finite_pname @ B_41 )
=> ( ( ord_less_eq_pname_o @ B_41 @ ( image_pname_o_pname @ F_22 @ A_78 ) )
=> ? [C_12: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ C_12 @ A_78 )
& ( finite297249702name_o @ C_12 )
& ( B_41
= ( image_pname_o_pname @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_165_finite__subset__image,axiom,
! [F_22: ( x_a > $o ) > pname,A_78: ( x_a > $o ) > $o,B_41: pname > $o] :
( ( finite_finite_pname @ B_41 )
=> ( ( ord_less_eq_pname_o @ B_41 @ ( image_a_o_pname @ F_22 @ A_78 ) )
=> ? [C_12: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ C_12 @ A_78 )
& ( finite_finite_a_o @ C_12 )
& ( B_41
= ( image_a_o_pname @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_166_finite__subset__image,axiom,
! [F_22: x_a > nat,A_78: x_a > $o,B_41: nat > $o] :
( ( finite_finite_nat @ B_41 )
=> ( ( ord_less_eq_nat_o @ B_41 @ ( image_a_nat @ F_22 @ A_78 ) )
=> ? [C_12: x_a > $o] :
( ( ord_less_eq_a_o @ C_12 @ A_78 )
& ( finite_finite_a @ C_12 )
& ( B_41
= ( image_a_nat @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_167_finite__subset__image,axiom,
! [F_22: ( nat > $o ) > nat,A_78: ( nat > $o ) > $o,B_41: nat > $o] :
( ( finite_finite_nat @ B_41 )
=> ( ( ord_less_eq_nat_o @ B_41 @ ( image_nat_o_nat @ F_22 @ A_78 ) )
=> ? [C_12: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ C_12 @ A_78 )
& ( finite_finite_nat_o @ C_12 )
& ( B_41
= ( image_nat_o_nat @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_168_finite__subset__image,axiom,
! [F_22: ( pname > $o ) > nat,A_78: ( pname > $o ) > $o,B_41: nat > $o] :
( ( finite_finite_nat @ B_41 )
=> ( ( ord_less_eq_nat_o @ B_41 @ ( image_pname_o_nat @ F_22 @ A_78 ) )
=> ? [C_12: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ C_12 @ A_78 )
& ( finite297249702name_o @ C_12 )
& ( B_41
= ( image_pname_o_nat @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_169_finite__subset__image,axiom,
! [F_22: ( x_a > $o ) > nat,A_78: ( x_a > $o ) > $o,B_41: nat > $o] :
( ( finite_finite_nat @ B_41 )
=> ( ( ord_less_eq_nat_o @ B_41 @ ( image_a_o_nat @ F_22 @ A_78 ) )
=> ? [C_12: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ C_12 @ A_78 )
& ( finite_finite_a_o @ C_12 )
& ( B_41
= ( image_a_o_nat @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_170_finite__subset__image,axiom,
! [F_22: pname > nat > $o,A_78: pname > $o,B_41: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_41 )
=> ( ( ord_less_eq_nat_o_o @ B_41 @ ( image_pname_nat_o @ F_22 @ A_78 ) )
=> ? [C_12: pname > $o] :
( ( ord_less_eq_pname_o @ C_12 @ A_78 )
& ( finite_finite_pname @ C_12 )
& ( B_41
= ( image_pname_nat_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_171_finite__subset__image,axiom,
! [F_22: pname > pname > $o,A_78: pname > $o,B_41: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_41 )
=> ( ( ord_le1205211808me_o_o @ B_41 @ ( image_pname_pname_o @ F_22 @ A_78 ) )
=> ? [C_12: pname > $o] :
( ( ord_less_eq_pname_o @ C_12 @ A_78 )
& ( finite_finite_pname @ C_12 )
& ( B_41
= ( image_pname_pname_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_172_finite__subset__image,axiom,
! [F_22: pname > x_a > $o,A_78: pname > $o,B_41: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_41 )
=> ( ( ord_less_eq_a_o_o @ B_41 @ ( image_pname_a_o @ F_22 @ A_78 ) )
=> ? [C_12: pname > $o] :
( ( ord_less_eq_pname_o @ C_12 @ A_78 )
& ( finite_finite_pname @ C_12 )
& ( B_41
= ( image_pname_a_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_173_finite__subset__image,axiom,
! [F_22: pname > pname,A_78: pname > $o,B_41: pname > $o] :
( ( finite_finite_pname @ B_41 )
=> ( ( ord_less_eq_pname_o @ B_41 @ ( image_pname_pname @ F_22 @ A_78 ) )
=> ? [C_12: pname > $o] :
( ( ord_less_eq_pname_o @ C_12 @ A_78 )
& ( finite_finite_pname @ C_12 )
& ( B_41
= ( image_pname_pname @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_174_finite__subset__image,axiom,
! [F_22: pname > nat,A_78: pname > $o,B_41: nat > $o] :
( ( finite_finite_nat @ B_41 )
=> ( ( ord_less_eq_nat_o @ B_41 @ ( image_pname_nat @ F_22 @ A_78 ) )
=> ? [C_12: pname > $o] :
( ( ord_less_eq_pname_o @ C_12 @ A_78 )
& ( finite_finite_pname @ C_12 )
& ( B_41
= ( image_pname_nat @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_175_finite__subset__image,axiom,
! [F_22: nat > x_a,A_78: nat > $o,B_41: x_a > $o] :
( ( finite_finite_a @ B_41 )
=> ( ( ord_less_eq_a_o @ B_41 @ ( image_nat_a @ F_22 @ A_78 ) )
=> ? [C_12: nat > $o] :
( ( ord_less_eq_nat_o @ C_12 @ A_78 )
& ( finite_finite_nat @ C_12 )
& ( B_41
= ( image_nat_a @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_176_finite__subset__image,axiom,
! [F_22: nat > nat > $o,A_78: nat > $o,B_41: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_41 )
=> ( ( ord_less_eq_nat_o_o @ B_41 @ ( image_nat_nat_o @ F_22 @ A_78 ) )
=> ? [C_12: nat > $o] :
( ( ord_less_eq_nat_o @ C_12 @ A_78 )
& ( finite_finite_nat @ C_12 )
& ( B_41
= ( image_nat_nat_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_177_finite__subset__image,axiom,
! [F_22: nat > pname > $o,A_78: nat > $o,B_41: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_41 )
=> ( ( ord_le1205211808me_o_o @ B_41 @ ( image_nat_pname_o @ F_22 @ A_78 ) )
=> ? [C_12: nat > $o] :
( ( ord_less_eq_nat_o @ C_12 @ A_78 )
& ( finite_finite_nat @ C_12 )
& ( B_41
= ( image_nat_pname_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_178_finite__subset__image,axiom,
! [F_22: nat > x_a > $o,A_78: nat > $o,B_41: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_41 )
=> ( ( ord_less_eq_a_o_o @ B_41 @ ( image_nat_a_o @ F_22 @ A_78 ) )
=> ? [C_12: nat > $o] :
( ( ord_less_eq_nat_o @ C_12 @ A_78 )
& ( finite_finite_nat @ C_12 )
& ( B_41
= ( image_nat_a_o @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_179_finite__subset__image,axiom,
! [F_22: nat > pname,A_78: nat > $o,B_41: pname > $o] :
( ( finite_finite_pname @ B_41 )
=> ( ( ord_less_eq_pname_o @ B_41 @ ( image_nat_pname @ F_22 @ A_78 ) )
=> ? [C_12: nat > $o] :
( ( ord_less_eq_nat_o @ C_12 @ A_78 )
& ( finite_finite_nat @ C_12 )
& ( B_41
= ( image_nat_pname @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_180_finite__subset__image,axiom,
! [F_22: nat > nat,A_78: nat > $o,B_41: nat > $o] :
( ( finite_finite_nat @ B_41 )
=> ( ( ord_less_eq_nat_o @ B_41 @ ( image_nat_nat @ F_22 @ A_78 ) )
=> ? [C_12: nat > $o] :
( ( ord_less_eq_nat_o @ C_12 @ A_78 )
& ( finite_finite_nat @ C_12 )
& ( B_41
= ( image_nat_nat @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_181_finite__subset__image,axiom,
! [F_22: pname > x_a,A_78: pname > $o,B_41: x_a > $o] :
( ( finite_finite_a @ B_41 )
=> ( ( ord_less_eq_a_o @ B_41 @ ( image_pname_a @ F_22 @ A_78 ) )
=> ? [C_12: pname > $o] :
( ( ord_less_eq_pname_o @ C_12 @ A_78 )
& ( finite_finite_pname @ C_12 )
& ( B_41
= ( image_pname_a @ F_22 @ C_12 ) ) ) ) ) ).
thf(fact_182_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_21: nat > $o] :
( ! [N_2: nat] : ( ord_less_eq_o @ ( F_21 @ N_2 ) @ ( F_21 @ ( suc @ N_2 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_o @ ( F_21 @ N_4 ) @ ( F_21 @ N_3 ) ) ) ) ).
thf(fact_183_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_21: nat > pname > $o] :
( ! [N_2: nat] : ( ord_less_eq_pname_o @ ( F_21 @ N_2 ) @ ( F_21 @ ( suc @ N_2 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_pname_o @ ( F_21 @ N_4 ) @ ( F_21 @ N_3 ) ) ) ) ).
thf(fact_184_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_21: nat > nat > $o] :
( ! [N_2: nat] : ( ord_less_eq_nat_o @ ( F_21 @ N_2 ) @ ( F_21 @ ( suc @ N_2 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_nat_o @ ( F_21 @ N_4 ) @ ( F_21 @ N_3 ) ) ) ) ).
thf(fact_185_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_21: nat > x_a > $o] :
( ! [N_2: nat] : ( ord_less_eq_a_o @ ( F_21 @ N_2 ) @ ( F_21 @ ( suc @ N_2 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_a_o @ ( F_21 @ N_4 ) @ ( F_21 @ N_3 ) ) ) ) ).
thf(fact_186_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_21: nat > nat] :
( ! [N_2: nat] : ( ord_less_eq_nat @ ( F_21 @ N_2 ) @ ( F_21 @ ( suc @ N_2 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_nat @ ( F_21 @ N_4 ) @ ( F_21 @ N_3 ) ) ) ) ).
thf(fact_187_pigeonhole__infinite,axiom,
! [F_20: nat > pname,A_77: nat > $o] :
( ~ ( finite_finite_nat @ A_77 )
=> ( ( finite_finite_pname @ ( image_nat_pname @ F_20 @ A_77 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_77 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_2: nat] :
( (&) @ ( member_nat @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_188_pigeonhole__infinite,axiom,
! [F_20: x_a > pname,A_77: x_a > $o] :
( ~ ( finite_finite_a @ A_77 )
=> ( ( finite_finite_pname @ ( image_a_pname @ F_20 @ A_77 ) )
=> ? [X_1: x_a] :
( ( member_a @ X_1 @ A_77 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A_2: x_a] :
( (&) @ ( member_a @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_189_pigeonhole__infinite,axiom,
! [F_20: pname > pname,A_77: pname > $o] :
( ~ ( finite_finite_pname @ A_77 )
=> ( ( finite_finite_pname @ ( image_pname_pname @ F_20 @ A_77 ) )
=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_77 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_2: pname] :
( (&) @ ( member_pname @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_190_pigeonhole__infinite,axiom,
! [F_20: ( nat > $o ) > pname,A_77: ( nat > $o ) > $o] :
( ~ ( finite_finite_nat_o @ A_77 )
=> ( ( finite_finite_pname @ ( image_nat_o_pname @ F_20 @ A_77 ) )
=> ? [X_1: nat > $o] :
( ( member_nat_o @ X_1 @ A_77 )
& ~ ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [A_2: nat > $o] :
( (&) @ ( member_nat_o @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_191_pigeonhole__infinite,axiom,
! [F_20: ( pname > $o ) > pname,A_77: ( pname > $o ) > $o] :
( ~ ( finite297249702name_o @ A_77 )
=> ( ( finite_finite_pname @ ( image_pname_o_pname @ F_20 @ A_77 ) )
=> ? [X_1: pname > $o] :
( ( member_pname_o @ X_1 @ A_77 )
& ~ ( finite297249702name_o
@ ( collect_pname_o
@ ^ [A_2: pname > $o] :
( (&) @ ( member_pname_o @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_192_pigeonhole__infinite,axiom,
! [F_20: ( x_a > $o ) > pname,A_77: ( x_a > $o ) > $o] :
( ~ ( finite_finite_a_o @ A_77 )
=> ( ( finite_finite_pname @ ( image_a_o_pname @ F_20 @ A_77 ) )
=> ? [X_1: x_a > $o] :
( ( member_a_o @ X_1 @ A_77 )
& ~ ( finite_finite_a_o
@ ( collect_a_o
@ ^ [A_2: x_a > $o] :
( (&) @ ( member_a_o @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_193_pigeonhole__infinite,axiom,
! [F_20: nat > nat,A_77: nat > $o] :
( ~ ( finite_finite_nat @ A_77 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F_20 @ A_77 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_77 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_2: nat] :
( (&) @ ( member_nat @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_194_pigeonhole__infinite,axiom,
! [F_20: x_a > nat,A_77: x_a > $o] :
( ~ ( finite_finite_a @ A_77 )
=> ( ( finite_finite_nat @ ( image_a_nat @ F_20 @ A_77 ) )
=> ? [X_1: x_a] :
( ( member_a @ X_1 @ A_77 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A_2: x_a] :
( (&) @ ( member_a @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_195_pigeonhole__infinite,axiom,
! [F_20: pname > nat,A_77: pname > $o] :
( ~ ( finite_finite_pname @ A_77 )
=> ( ( finite_finite_nat @ ( image_pname_nat @ F_20 @ A_77 ) )
=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_77 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_2: pname] :
( (&) @ ( member_pname @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_196_pigeonhole__infinite,axiom,
! [F_20: ( nat > $o ) > nat,A_77: ( nat > $o ) > $o] :
( ~ ( finite_finite_nat_o @ A_77 )
=> ( ( finite_finite_nat @ ( image_nat_o_nat @ F_20 @ A_77 ) )
=> ? [X_1: nat > $o] :
( ( member_nat_o @ X_1 @ A_77 )
& ~ ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [A_2: nat > $o] :
( (&) @ ( member_nat_o @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_197_pigeonhole__infinite,axiom,
! [F_20: ( pname > $o ) > nat,A_77: ( pname > $o ) > $o] :
( ~ ( finite297249702name_o @ A_77 )
=> ( ( finite_finite_nat @ ( image_pname_o_nat @ F_20 @ A_77 ) )
=> ? [X_1: pname > $o] :
( ( member_pname_o @ X_1 @ A_77 )
& ~ ( finite297249702name_o
@ ( collect_pname_o
@ ^ [A_2: pname > $o] :
( (&) @ ( member_pname_o @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_198_pigeonhole__infinite,axiom,
! [F_20: ( x_a > $o ) > nat,A_77: ( x_a > $o ) > $o] :
( ~ ( finite_finite_a_o @ A_77 )
=> ( ( finite_finite_nat @ ( image_a_o_nat @ F_20 @ A_77 ) )
=> ? [X_1: x_a > $o] :
( ( member_a_o @ X_1 @ A_77 )
& ~ ( finite_finite_a_o
@ ( collect_a_o
@ ^ [A_2: x_a > $o] :
( (&) @ ( member_a_o @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_199_pigeonhole__infinite,axiom,
! [F_20: nat > x_a,A_77: nat > $o] :
( ~ ( finite_finite_nat @ A_77 )
=> ( ( finite_finite_a @ ( image_nat_a @ F_20 @ A_77 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_77 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_2: nat] :
( (&) @ ( member_nat @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_200_pigeonhole__infinite,axiom,
! [F_20: nat > nat > $o,A_77: nat > $o] :
( ~ ( finite_finite_nat @ A_77 )
=> ( ( finite_finite_nat_o @ ( image_nat_nat_o @ F_20 @ A_77 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_77 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_2: nat] :
( (&) @ ( member_nat @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_201_pigeonhole__infinite,axiom,
! [F_20: nat > pname > $o,A_77: nat > $o] :
( ~ ( finite_finite_nat @ A_77 )
=> ( ( finite297249702name_o @ ( image_nat_pname_o @ F_20 @ A_77 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_77 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_2: nat] :
( (&) @ ( member_nat @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_202_pigeonhole__infinite,axiom,
! [F_20: nat > x_a > $o,A_77: nat > $o] :
( ~ ( finite_finite_nat @ A_77 )
=> ( ( finite_finite_a_o @ ( image_nat_a_o @ F_20 @ A_77 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_77 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_2: nat] :
( (&) @ ( member_nat @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_203_pigeonhole__infinite,axiom,
! [F_20: pname > nat > $o,A_77: pname > $o] :
( ~ ( finite_finite_pname @ A_77 )
=> ( ( finite_finite_nat_o @ ( image_pname_nat_o @ F_20 @ A_77 ) )
=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_77 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_2: pname] :
( (&) @ ( member_pname @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_204_pigeonhole__infinite,axiom,
! [F_20: pname > pname > $o,A_77: pname > $o] :
( ~ ( finite_finite_pname @ A_77 )
=> ( ( finite297249702name_o @ ( image_pname_pname_o @ F_20 @ A_77 ) )
=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_77 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_2: pname] :
( (&) @ ( member_pname @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_205_pigeonhole__infinite,axiom,
! [F_20: pname > x_a > $o,A_77: pname > $o] :
( ~ ( finite_finite_pname @ A_77 )
=> ( ( finite_finite_a_o @ ( image_pname_a_o @ F_20 @ A_77 ) )
=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_77 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_2: pname] :
( (&) @ ( member_pname @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_206_pigeonhole__infinite,axiom,
! [F_20: x_a > x_a,A_77: x_a > $o] :
( ~ ( finite_finite_a @ A_77 )
=> ( ( finite_finite_a @ ( image_a_a @ F_20 @ A_77 ) )
=> ? [X_1: x_a] :
( ( member_a @ X_1 @ A_77 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A_2: x_a] :
( (&) @ ( member_a @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_207_pigeonhole__infinite,axiom,
! [F_20: x_a > nat > $o,A_77: x_a > $o] :
( ~ ( finite_finite_a @ A_77 )
=> ( ( finite_finite_nat_o @ ( image_a_nat_o @ F_20 @ A_77 ) )
=> ? [X_1: x_a] :
( ( member_a @ X_1 @ A_77 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A_2: x_a] :
( (&) @ ( member_a @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_208_pigeonhole__infinite,axiom,
! [F_20: x_a > pname > $o,A_77: x_a > $o] :
( ~ ( finite_finite_a @ A_77 )
=> ( ( finite297249702name_o @ ( image_a_pname_o @ F_20 @ A_77 ) )
=> ? [X_1: x_a] :
( ( member_a @ X_1 @ A_77 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A_2: x_a] :
( (&) @ ( member_a @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_209_pigeonhole__infinite,axiom,
! [F_20: x_a > x_a > $o,A_77: x_a > $o] :
( ~ ( finite_finite_a @ A_77 )
=> ( ( finite_finite_a_o @ ( image_a_a_o @ F_20 @ A_77 ) )
=> ? [X_1: x_a] :
( ( member_a @ X_1 @ A_77 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A_2: x_a] :
( (&) @ ( member_a @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_210_pigeonhole__infinite,axiom,
! [F_20: pname > x_a,A_77: pname > $o] :
( ~ ( finite_finite_pname @ A_77 )
=> ( ( finite_finite_a @ ( image_pname_a @ F_20 @ A_77 ) )
=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_77 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_2: pname] :
( (&) @ ( member_pname @ A_2 @ A_77 )
@ ( ( F_20 @ A_2 )
= ( F_20 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_211_image__eqI,axiom,
! [A_76: pname > $o,B_40: nat,F_19: pname > nat,X_30: pname] :
( ( B_40
= ( F_19 @ X_30 ) )
=> ( ( member_pname @ X_30 @ A_76 )
=> ( member_nat @ B_40 @ ( image_pname_nat @ F_19 @ A_76 ) ) ) ) ).
thf(fact_212_image__eqI,axiom,
! [A_76: x_a > $o,B_40: nat,F_19: x_a > nat,X_30: x_a] :
( ( B_40
= ( F_19 @ X_30 ) )
=> ( ( member_a @ X_30 @ A_76 )
=> ( member_nat @ B_40 @ ( image_a_nat @ F_19 @ A_76 ) ) ) ) ).
thf(fact_213_image__eqI,axiom,
! [A_76: nat > $o,B_40: pname,F_19: nat > pname,X_30: nat] :
( ( B_40
= ( F_19 @ X_30 ) )
=> ( ( member_nat @ X_30 @ A_76 )
=> ( member_pname @ B_40 @ ( image_nat_pname @ F_19 @ A_76 ) ) ) ) ).
thf(fact_214_image__eqI,axiom,
! [A_76: nat > $o,B_40: x_a,F_19: nat > x_a,X_30: nat] :
( ( B_40
= ( F_19 @ X_30 ) )
=> ( ( member_nat @ X_30 @ A_76 )
=> ( member_a @ B_40 @ ( image_nat_a @ F_19 @ A_76 ) ) ) ) ).
thf(fact_215_image__eqI,axiom,
! [A_76: pname > $o,B_40: x_a,F_19: pname > x_a,X_30: pname] :
( ( B_40
= ( F_19 @ X_30 ) )
=> ( ( member_pname @ X_30 @ A_76 )
=> ( member_a @ B_40 @ ( image_pname_a @ F_19 @ A_76 ) ) ) ) ).
thf(fact_216_equalityI,axiom,
! [A_75: pname > $o,B_39: pname > $o] :
( ( ord_less_eq_pname_o @ A_75 @ B_39 )
=> ( ( ord_less_eq_pname_o @ B_39 @ A_75 )
=> ( A_75 = B_39 ) ) ) ).
thf(fact_217_equalityI,axiom,
! [A_75: nat > $o,B_39: nat > $o] :
( ( ord_less_eq_nat_o @ A_75 @ B_39 )
=> ( ( ord_less_eq_nat_o @ B_39 @ A_75 )
=> ( A_75 = B_39 ) ) ) ).
thf(fact_218_equalityI,axiom,
! [A_75: x_a > $o,B_39: x_a > $o] :
( ( ord_less_eq_a_o @ A_75 @ B_39 )
=> ( ( ord_less_eq_a_o @ B_39 @ A_75 )
=> ( A_75 = B_39 ) ) ) ).
thf(fact_219_subsetD,axiom,
! [C_11: nat,A_74: nat > $o,B_38: nat > $o] :
( ( ord_less_eq_nat_o @ A_74 @ B_38 )
=> ( ( member_nat @ C_11 @ A_74 )
=> ( member_nat @ C_11 @ B_38 ) ) ) ).
thf(fact_220_subsetD,axiom,
! [C_11: pname,A_74: pname > $o,B_38: pname > $o] :
( ( ord_less_eq_pname_o @ A_74 @ B_38 )
=> ( ( member_pname @ C_11 @ A_74 )
=> ( member_pname @ C_11 @ B_38 ) ) ) ).
thf(fact_221_subsetD,axiom,
! [C_11: x_a,A_74: x_a > $o,B_38: x_a > $o] :
( ( ord_less_eq_a_o @ A_74 @ B_38 )
=> ( ( member_a @ C_11 @ A_74 )
=> ( member_a @ C_11 @ B_38 ) ) ) ).
thf(fact_222_insertCI,axiom,
! [B_37: nat,A_73: nat,B_36: nat > $o] :
( ( ~ ( member_nat @ A_73 @ B_36 )
=> ( A_73 = B_37 ) )
=> ( member_nat @ A_73 @ ( insert_nat @ B_37 @ B_36 ) ) ) ).
thf(fact_223_insertCI,axiom,
! [B_37: pname,A_73: pname,B_36: pname > $o] :
( ( ~ ( member_pname @ A_73 @ B_36 )
=> ( A_73 = B_37 ) )
=> ( member_pname @ A_73 @ ( insert_pname @ B_37 @ B_36 ) ) ) ).
thf(fact_224_insertCI,axiom,
! [B_37: x_a,A_73: x_a,B_36: x_a > $o] :
( ( ~ ( member_a @ A_73 @ B_36 )
=> ( A_73 = B_37 ) )
=> ( member_a @ A_73 @ ( insert_a @ B_37 @ B_36 ) ) ) ).
thf(fact_225_insertE,axiom,
! [A_72: nat,B_35: nat,A_71: nat > $o] :
( ( member_nat @ A_72 @ ( insert_nat @ B_35 @ A_71 ) )
=> ( ( A_72 != B_35 )
=> ( member_nat @ A_72 @ A_71 ) ) ) ).
thf(fact_226_insertE,axiom,
! [A_72: pname,B_35: pname,A_71: pname > $o] :
( ( member_pname @ A_72 @ ( insert_pname @ B_35 @ A_71 ) )
=> ( ( A_72 != B_35 )
=> ( member_pname @ A_72 @ A_71 ) ) ) ).
thf(fact_227_insertE,axiom,
! [A_72: x_a,B_35: x_a,A_71: x_a > $o] :
( ( member_a @ A_72 @ ( insert_a @ B_35 @ A_71 ) )
=> ( ( A_72 != B_35 )
=> ( member_a @ A_72 @ A_71 ) ) ) ).
thf(fact_228_zero__induct__lemma,axiom,
! [I: nat,P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_2: nat] :
( ( P @ ( suc @ N_2 ) )
=> ( P @ N_2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
thf(fact_229_Suc__le__D,axiom,
! [N: nat,M_2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M_2 )
=> ? [M_1: nat] :
( M_2
= ( suc @ M_1 ) ) ) ).
thf(fact_230_insertI1,axiom,
! [A_70: nat,B_34: nat > $o] : ( member_nat @ A_70 @ ( insert_nat @ A_70 @ B_34 ) ) ).
thf(fact_231_insertI1,axiom,
! [A_70: pname,B_34: pname > $o] : ( member_pname @ A_70 @ ( insert_pname @ A_70 @ B_34 ) ) ).
thf(fact_232_insertI1,axiom,
! [A_70: x_a,B_34: x_a > $o] : ( member_a @ A_70 @ ( insert_a @ A_70 @ B_34 ) ) ).
thf(fact_233_insert__compr,axiom,
! [A_69: nat > $o,B_33: ( nat > $o ) > $o] :
( ( insert_nat_o @ A_69 @ B_33 )
= ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (|) @ ( X_1 = A_69 ) @ ( member_nat_o @ X_1 @ B_33 ) ) ) ) ).
thf(fact_234_insert__compr,axiom,
! [A_69: pname > $o,B_33: ( pname > $o ) > $o] :
( ( insert_pname_o @ A_69 @ B_33 )
= ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (|) @ ( X_1 = A_69 ) @ ( member_pname_o @ X_1 @ B_33 ) ) ) ) ).
thf(fact_235_insert__compr,axiom,
! [A_69: x_a > $o,B_33: ( x_a > $o ) > $o] :
( ( insert_a_o @ A_69 @ B_33 )
= ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (|) @ ( X_1 = A_69 ) @ ( member_a_o @ X_1 @ B_33 ) ) ) ) ).
thf(fact_236_insert__compr,axiom,
! [A_69: nat,B_33: nat > $o] :
( ( insert_nat @ A_69 @ B_33 )
= ( collect_nat
@ ^ [X_1: nat] : ( (|) @ ( X_1 = A_69 ) @ ( member_nat @ X_1 @ B_33 ) ) ) ) ).
thf(fact_237_insert__compr,axiom,
! [A_69: pname,B_33: pname > $o] :
( ( insert_pname @ A_69 @ B_33 )
= ( collect_pname
@ ^ [X_1: pname] : ( (|) @ ( X_1 = A_69 ) @ ( member_pname @ X_1 @ B_33 ) ) ) ) ).
thf(fact_238_insert__compr,axiom,
! [A_69: x_a,B_33: x_a > $o] :
( ( insert_a @ A_69 @ B_33 )
= ( collect_a
@ ^ [X_1: x_a] : ( (|) @ ( X_1 = A_69 ) @ ( member_a @ X_1 @ B_33 ) ) ) ) ).
thf(fact_239_insert__Collect,axiom,
! [A_68: pname,P_8: pname > $o] :
( ( insert_pname @ A_68 @ ( collect_pname @ P_8 ) )
= ( collect_pname
@ ^ [U_1: pname] : ( (=>) @ ( (~) @ ( U_1 = A_68 ) ) @ ( P_8 @ U_1 ) ) ) ) ).
thf(fact_240_insert__Collect,axiom,
! [A_68: nat > $o,P_8: ( nat > $o ) > $o] :
( ( insert_nat_o @ A_68 @ ( collect_nat_o @ P_8 ) )
= ( collect_nat_o
@ ^ [U_1: nat > $o] : ( (=>) @ ( (~) @ ( U_1 = A_68 ) ) @ ( P_8 @ U_1 ) ) ) ) ).
thf(fact_241_insert__Collect,axiom,
! [A_68: pname > $o,P_8: ( pname > $o ) > $o] :
( ( insert_pname_o @ A_68 @ ( collect_pname_o @ P_8 ) )
= ( collect_pname_o
@ ^ [U_1: pname > $o] : ( (=>) @ ( (~) @ ( U_1 = A_68 ) ) @ ( P_8 @ U_1 ) ) ) ) ).
thf(fact_242_insert__Collect,axiom,
! [A_68: x_a > $o,P_8: ( x_a > $o ) > $o] :
( ( insert_a_o @ A_68 @ ( collect_a_o @ P_8 ) )
= ( collect_a_o
@ ^ [U_1: x_a > $o] : ( (=>) @ ( (~) @ ( U_1 = A_68 ) ) @ ( P_8 @ U_1 ) ) ) ) ).
thf(fact_243_insert__Collect,axiom,
! [A_68: nat,P_8: nat > $o] :
( ( insert_nat @ A_68 @ ( collect_nat @ P_8 ) )
= ( collect_nat
@ ^ [U_1: nat] : ( (=>) @ ( (~) @ ( U_1 = A_68 ) ) @ ( P_8 @ U_1 ) ) ) ) ).
thf(fact_244_insert__Collect,axiom,
! [A_68: x_a,P_8: x_a > $o] :
( ( insert_a @ A_68 @ ( collect_a @ P_8 ) )
= ( collect_a
@ ^ [U_1: x_a] : ( (=>) @ ( (~) @ ( U_1 = A_68 ) ) @ ( P_8 @ U_1 ) ) ) ) ).
thf(fact_245_insert__absorb2,axiom,
! [X_29: pname,A_67: pname > $o] :
( ( insert_pname @ X_29 @ ( insert_pname @ X_29 @ A_67 ) )
= ( insert_pname @ X_29 @ A_67 ) ) ).
thf(fact_246_insert__absorb2,axiom,
! [X_29: nat,A_67: nat > $o] :
( ( insert_nat @ X_29 @ ( insert_nat @ X_29 @ A_67 ) )
= ( insert_nat @ X_29 @ A_67 ) ) ).
thf(fact_247_insert__absorb2,axiom,
! [X_29: x_a,A_67: x_a > $o] :
( ( insert_a @ X_29 @ ( insert_a @ X_29 @ A_67 ) )
= ( insert_a @ X_29 @ A_67 ) ) ).
thf(fact_248_insert__commute,axiom,
! [X_28: pname,Y_12: pname,A_66: pname > $o] :
( ( insert_pname @ X_28 @ ( insert_pname @ Y_12 @ A_66 ) )
= ( insert_pname @ Y_12 @ ( insert_pname @ X_28 @ A_66 ) ) ) ).
thf(fact_249_insert__commute,axiom,
! [X_28: nat,Y_12: nat,A_66: nat > $o] :
( ( insert_nat @ X_28 @ ( insert_nat @ Y_12 @ A_66 ) )
= ( insert_nat @ Y_12 @ ( insert_nat @ X_28 @ A_66 ) ) ) ).
thf(fact_250_insert__commute,axiom,
! [X_28: x_a,Y_12: x_a,A_66: x_a > $o] :
( ( insert_a @ X_28 @ ( insert_a @ Y_12 @ A_66 ) )
= ( insert_a @ Y_12 @ ( insert_a @ X_28 @ A_66 ) ) ) ).
thf(fact_251_insert__iff,axiom,
! [A_65: nat,B_32: nat,A_64: nat > $o] :
( ( member_nat @ A_65 @ ( insert_nat @ B_32 @ A_64 ) )
<=> ( ( A_65 = B_32 )
| ( member_nat @ A_65 @ A_64 ) ) ) ).
thf(fact_252_insert__iff,axiom,
! [A_65: pname,B_32: pname,A_64: pname > $o] :
( ( member_pname @ A_65 @ ( insert_pname @ B_32 @ A_64 ) )
<=> ( ( A_65 = B_32 )
| ( member_pname @ A_65 @ A_64 ) ) ) ).
thf(fact_253_insert__iff,axiom,
! [A_65: x_a,B_32: x_a,A_64: x_a > $o] :
( ( member_a @ A_65 @ ( insert_a @ B_32 @ A_64 ) )
<=> ( ( A_65 = B_32 )
| ( member_a @ A_65 @ A_64 ) ) ) ).
thf(fact_254_insert__code,axiom,
! [Y_11: pname,A_63: pname > $o,X_27: pname] :
( ( insert_pname @ Y_11 @ A_63 @ X_27 )
<=> ( ( Y_11 = X_27 )
| ( A_63 @ X_27 ) ) ) ).
thf(fact_255_insert__code,axiom,
! [Y_11: nat,A_63: nat > $o,X_27: nat] :
( ( insert_nat @ Y_11 @ A_63 @ X_27 )
<=> ( ( Y_11 = X_27 )
| ( A_63 @ X_27 ) ) ) ).
thf(fact_256_insert__code,axiom,
! [Y_11: x_a,A_63: x_a > $o,X_27: x_a] :
( ( insert_a @ Y_11 @ A_63 @ X_27 )
<=> ( ( Y_11 = X_27 )
| ( A_63 @ X_27 ) ) ) ).
thf(fact_257_insert__ident,axiom,
! [B_31: nat > $o,X_26: nat,A_62: nat > $o] :
( ~ ( member_nat @ X_26 @ A_62 )
=> ( ~ ( member_nat @ X_26 @ B_31 )
=> ( ( ( insert_nat @ X_26 @ A_62 )
= ( insert_nat @ X_26 @ B_31 ) )
<=> ( A_62 = B_31 ) ) ) ) ).
thf(fact_258_insert__ident,axiom,
! [B_31: pname > $o,X_26: pname,A_62: pname > $o] :
( ~ ( member_pname @ X_26 @ A_62 )
=> ( ~ ( member_pname @ X_26 @ B_31 )
=> ( ( ( insert_pname @ X_26 @ A_62 )
= ( insert_pname @ X_26 @ B_31 ) )
<=> ( A_62 = B_31 ) ) ) ) ).
thf(fact_259_insert__ident,axiom,
! [B_31: x_a > $o,X_26: x_a,A_62: x_a > $o] :
( ~ ( member_a @ X_26 @ A_62 )
=> ( ~ ( member_a @ X_26 @ B_31 )
=> ( ( ( insert_a @ X_26 @ A_62 )
= ( insert_a @ X_26 @ B_31 ) )
<=> ( A_62 = B_31 ) ) ) ) ).
thf(fact_260_insertI2,axiom,
! [B_30: nat,A_61: nat,B_29: nat > $o] :
( ( member_nat @ A_61 @ B_29 )
=> ( member_nat @ A_61 @ ( insert_nat @ B_30 @ B_29 ) ) ) ).
thf(fact_261_insertI2,axiom,
! [B_30: pname,A_61: pname,B_29: pname > $o] :
( ( member_pname @ A_61 @ B_29 )
=> ( member_pname @ A_61 @ ( insert_pname @ B_30 @ B_29 ) ) ) ).
thf(fact_262_insertI2,axiom,
! [B_30: x_a,A_61: x_a,B_29: x_a > $o] :
( ( member_a @ A_61 @ B_29 )
=> ( member_a @ A_61 @ ( insert_a @ B_30 @ B_29 ) ) ) ).
thf(fact_263_insert__absorb,axiom,
! [A_60: nat,A_59: nat > $o] :
( ( member_nat @ A_60 @ A_59 )
=> ( ( insert_nat @ A_60 @ A_59 )
= A_59 ) ) ).
thf(fact_264_insert__absorb,axiom,
! [A_60: pname,A_59: pname > $o] :
( ( member_pname @ A_60 @ A_59 )
=> ( ( insert_pname @ A_60 @ A_59 )
= A_59 ) ) ).
thf(fact_265_insert__absorb,axiom,
! [A_60: x_a,A_59: x_a > $o] :
( ( member_a @ A_60 @ A_59 )
=> ( ( insert_a @ A_60 @ A_59 )
= A_59 ) ) ).
thf(fact_266_subset__refl,axiom,
! [A_58: pname > $o] : ( ord_less_eq_pname_o @ A_58 @ A_58 ) ).
thf(fact_267_subset__refl,axiom,
! [A_58: nat > $o] : ( ord_less_eq_nat_o @ A_58 @ A_58 ) ).
thf(fact_268_subset__refl,axiom,
! [A_58: x_a > $o] : ( ord_less_eq_a_o @ A_58 @ A_58 ) ).
thf(fact_269_set__eq__subset,axiom,
! [A_57: pname > $o,B_28: pname > $o] :
( ( A_57 = B_28 )
<=> ( ( ord_less_eq_pname_o @ A_57 @ B_28 )
& ( ord_less_eq_pname_o @ B_28 @ A_57 ) ) ) ).
thf(fact_270_set__eq__subset,axiom,
! [A_57: nat > $o,B_28: nat > $o] :
( ( A_57 = B_28 )
<=> ( ( ord_less_eq_nat_o @ A_57 @ B_28 )
& ( ord_less_eq_nat_o @ B_28 @ A_57 ) ) ) ).
thf(fact_271_set__eq__subset,axiom,
! [A_57: x_a > $o,B_28: x_a > $o] :
( ( A_57 = B_28 )
<=> ( ( ord_less_eq_a_o @ A_57 @ B_28 )
& ( ord_less_eq_a_o @ B_28 @ A_57 ) ) ) ).
thf(fact_272_equalityD1,axiom,
! [A_56: pname > $o,B_27: pname > $o] :
( ( A_56 = B_27 )
=> ( ord_less_eq_pname_o @ A_56 @ B_27 ) ) ).
thf(fact_273_equalityD1,axiom,
! [A_56: nat > $o,B_27: nat > $o] :
( ( A_56 = B_27 )
=> ( ord_less_eq_nat_o @ A_56 @ B_27 ) ) ).
thf(fact_274_equalityD1,axiom,
! [A_56: x_a > $o,B_27: x_a > $o] :
( ( A_56 = B_27 )
=> ( ord_less_eq_a_o @ A_56 @ B_27 ) ) ).
thf(fact_275_equalityD2,axiom,
! [A_55: pname > $o,B_26: pname > $o] :
( ( A_55 = B_26 )
=> ( ord_less_eq_pname_o @ B_26 @ A_55 ) ) ).
thf(fact_276_equalityD2,axiom,
! [A_55: nat > $o,B_26: nat > $o] :
( ( A_55 = B_26 )
=> ( ord_less_eq_nat_o @ B_26 @ A_55 ) ) ).
thf(fact_277_equalityD2,axiom,
! [A_55: x_a > $o,B_26: x_a > $o] :
( ( A_55 = B_26 )
=> ( ord_less_eq_a_o @ B_26 @ A_55 ) ) ).
thf(fact_278_in__mono,axiom,
! [X_25: nat,A_54: nat > $o,B_25: nat > $o] :
( ( ord_less_eq_nat_o @ A_54 @ B_25 )
=> ( ( member_nat @ X_25 @ A_54 )
=> ( member_nat @ X_25 @ B_25 ) ) ) ).
thf(fact_279_in__mono,axiom,
! [X_25: pname,A_54: pname > $o,B_25: pname > $o] :
( ( ord_less_eq_pname_o @ A_54 @ B_25 )
=> ( ( member_pname @ X_25 @ A_54 )
=> ( member_pname @ X_25 @ B_25 ) ) ) ).
thf(fact_280_in__mono,axiom,
! [X_25: x_a,A_54: x_a > $o,B_25: x_a > $o] :
( ( ord_less_eq_a_o @ A_54 @ B_25 )
=> ( ( member_a @ X_25 @ A_54 )
=> ( member_a @ X_25 @ B_25 ) ) ) ).
thf(fact_281_set__rev__mp,axiom,
! [B_24: nat > $o,X_24: nat,A_53: nat > $o] :
( ( member_nat @ X_24 @ A_53 )
=> ( ( ord_less_eq_nat_o @ A_53 @ B_24 )
=> ( member_nat @ X_24 @ B_24 ) ) ) ).
thf(fact_282_set__rev__mp,axiom,
! [B_24: pname > $o,X_24: pname,A_53: pname > $o] :
( ( member_pname @ X_24 @ A_53 )
=> ( ( ord_less_eq_pname_o @ A_53 @ B_24 )
=> ( member_pname @ X_24 @ B_24 ) ) ) ).
thf(fact_283_set__rev__mp,axiom,
! [B_24: x_a > $o,X_24: x_a,A_53: x_a > $o] :
( ( member_a @ X_24 @ A_53 )
=> ( ( ord_less_eq_a_o @ A_53 @ B_24 )
=> ( member_a @ X_24 @ B_24 ) ) ) ).
thf(fact_284_set__mp,axiom,
! [X_23: nat,A_52: nat > $o,B_23: nat > $o] :
( ( ord_less_eq_nat_o @ A_52 @ B_23 )
=> ( ( member_nat @ X_23 @ A_52 )
=> ( member_nat @ X_23 @ B_23 ) ) ) ).
thf(fact_285_set__mp,axiom,
! [X_23: pname,A_52: pname > $o,B_23: pname > $o] :
( ( ord_less_eq_pname_o @ A_52 @ B_23 )
=> ( ( member_pname @ X_23 @ A_52 )
=> ( member_pname @ X_23 @ B_23 ) ) ) ).
thf(fact_286_set__mp,axiom,
! [X_23: x_a,A_52: x_a > $o,B_23: x_a > $o] :
( ( ord_less_eq_a_o @ A_52 @ B_23 )
=> ( ( member_a @ X_23 @ A_52 )
=> ( member_a @ X_23 @ B_23 ) ) ) ).
thf(fact_287_mem__def,axiom,
! [X_22: nat,A_51: nat > $o] :
( ( member_nat @ X_22 @ A_51 )
<=> ( A_51 @ X_22 ) ) ).
thf(fact_288_mem__def,axiom,
! [X_22: pname,A_51: pname > $o] :
( ( member_pname @ X_22 @ A_51 )
<=> ( A_51 @ X_22 ) ) ).
thf(fact_289_mem__def,axiom,
! [X_22: x_a,A_51: x_a > $o] :
( ( member_a @ X_22 @ A_51 )
<=> ( A_51 @ X_22 ) ) ).
thf(fact_290_Collect__def,axiom,
! [P_7: x_a > $o] :
( ( collect_a @ P_7 )
= P_7 ) ).
thf(fact_291_Collect__def,axiom,
! [P_7: pname > $o] :
( ( collect_pname @ P_7 )
= P_7 ) ).
thf(fact_292_Collect__def,axiom,
! [P_7: ( nat > $o ) > $o] :
( ( collect_nat_o @ P_7 )
= P_7 ) ).
thf(fact_293_Collect__def,axiom,
! [P_7: ( pname > $o ) > $o] :
( ( collect_pname_o @ P_7 )
= P_7 ) ).
thf(fact_294_Collect__def,axiom,
! [P_7: ( x_a > $o ) > $o] :
( ( collect_a_o @ P_7 )
= P_7 ) ).
thf(fact_295_Collect__def,axiom,
! [P_7: nat > $o] :
( ( collect_nat @ P_7 )
= P_7 ) ).
thf(fact_296_subset__trans,axiom,
! [C_10: pname > $o,A_50: pname > $o,B_22: pname > $o] :
( ( ord_less_eq_pname_o @ A_50 @ B_22 )
=> ( ( ord_less_eq_pname_o @ B_22 @ C_10 )
=> ( ord_less_eq_pname_o @ A_50 @ C_10 ) ) ) ).
thf(fact_297_subset__trans,axiom,
! [C_10: nat > $o,A_50: nat > $o,B_22: nat > $o] :
( ( ord_less_eq_nat_o @ A_50 @ B_22 )
=> ( ( ord_less_eq_nat_o @ B_22 @ C_10 )
=> ( ord_less_eq_nat_o @ A_50 @ C_10 ) ) ) ).
thf(fact_298_subset__trans,axiom,
! [C_10: x_a > $o,A_50: x_a > $o,B_22: x_a > $o] :
( ( ord_less_eq_a_o @ A_50 @ B_22 )
=> ( ( ord_less_eq_a_o @ B_22 @ C_10 )
=> ( ord_less_eq_a_o @ A_50 @ C_10 ) ) ) ).
thf(fact_299_equalityE,axiom,
! [A_49: pname > $o,B_21: pname > $o] :
( ( A_49 = B_21 )
=> ~ ( ( ord_less_eq_pname_o @ A_49 @ B_21 )
=> ~ ( ord_less_eq_pname_o @ B_21 @ A_49 ) ) ) ).
thf(fact_300_equalityE,axiom,
! [A_49: nat > $o,B_21: nat > $o] :
( ( A_49 = B_21 )
=> ~ ( ( ord_less_eq_nat_o @ A_49 @ B_21 )
=> ~ ( ord_less_eq_nat_o @ B_21 @ A_49 ) ) ) ).
thf(fact_301_equalityE,axiom,
! [A_49: x_a > $o,B_21: x_a > $o] :
( ( A_49 = B_21 )
=> ~ ( ( ord_less_eq_a_o @ A_49 @ B_21 )
=> ~ ( ord_less_eq_a_o @ B_21 @ A_49 ) ) ) ).
thf(fact_302_image__iff,axiom,
! [Z_3: x_a,F_18: pname > x_a,A_48: pname > $o] :
( ( member_a @ Z_3 @ ( image_pname_a @ F_18 @ A_48 ) )
<=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_48 )
& ( Z_3
= ( F_18 @ X_1 ) ) ) ) ).
thf(fact_303_imageI,axiom,
! [F_17: pname > nat,X_21: pname,A_47: pname > $o] :
( ( member_pname @ X_21 @ A_47 )
=> ( member_nat @ ( F_17 @ X_21 ) @ ( image_pname_nat @ F_17 @ A_47 ) ) ) ).
thf(fact_304_imageI,axiom,
! [F_17: x_a > nat,X_21: x_a,A_47: x_a > $o] :
( ( member_a @ X_21 @ A_47 )
=> ( member_nat @ ( F_17 @ X_21 ) @ ( image_a_nat @ F_17 @ A_47 ) ) ) ).
thf(fact_305_imageI,axiom,
! [F_17: nat > pname,X_21: nat,A_47: nat > $o] :
( ( member_nat @ X_21 @ A_47 )
=> ( member_pname @ ( F_17 @ X_21 ) @ ( image_nat_pname @ F_17 @ A_47 ) ) ) ).
thf(fact_306_imageI,axiom,
! [F_17: nat > x_a,X_21: nat,A_47: nat > $o] :
( ( member_nat @ X_21 @ A_47 )
=> ( member_a @ ( F_17 @ X_21 ) @ ( image_nat_a @ F_17 @ A_47 ) ) ) ).
thf(fact_307_imageI,axiom,
! [F_17: pname > x_a,X_21: pname,A_47: pname > $o] :
( ( member_pname @ X_21 @ A_47 )
=> ( member_a @ ( F_17 @ X_21 ) @ ( image_pname_a @ F_17 @ A_47 ) ) ) ).
thf(fact_308_rev__image__eqI,axiom,
! [B_20: nat,F_16: pname > nat,X_20: pname,A_46: pname > $o] :
( ( member_pname @ X_20 @ A_46 )
=> ( ( B_20
= ( F_16 @ X_20 ) )
=> ( member_nat @ B_20 @ ( image_pname_nat @ F_16 @ A_46 ) ) ) ) ).
thf(fact_309_rev__image__eqI,axiom,
! [B_20: nat,F_16: x_a > nat,X_20: x_a,A_46: x_a > $o] :
( ( member_a @ X_20 @ A_46 )
=> ( ( B_20
= ( F_16 @ X_20 ) )
=> ( member_nat @ B_20 @ ( image_a_nat @ F_16 @ A_46 ) ) ) ) ).
thf(fact_310_rev__image__eqI,axiom,
! [B_20: pname,F_16: nat > pname,X_20: nat,A_46: nat > $o] :
( ( member_nat @ X_20 @ A_46 )
=> ( ( B_20
= ( F_16 @ X_20 ) )
=> ( member_pname @ B_20 @ ( image_nat_pname @ F_16 @ A_46 ) ) ) ) ).
thf(fact_311_rev__image__eqI,axiom,
! [B_20: x_a,F_16: nat > x_a,X_20: nat,A_46: nat > $o] :
( ( member_nat @ X_20 @ A_46 )
=> ( ( B_20
= ( F_16 @ X_20 ) )
=> ( member_a @ B_20 @ ( image_nat_a @ F_16 @ A_46 ) ) ) ) ).
thf(fact_312_rev__image__eqI,axiom,
! [B_20: x_a,F_16: pname > x_a,X_20: pname,A_46: pname > $o] :
( ( member_pname @ X_20 @ A_46 )
=> ( ( B_20
= ( F_16 @ X_20 ) )
=> ( member_a @ B_20 @ ( image_pname_a @ F_16 @ A_46 ) ) ) ) ).
thf(fact_313_insert__compr__raw,axiom,
! [X_1: nat > $o,Xa: ( nat > $o ) > $o] :
( ( insert_nat_o @ X_1 @ Xa )
= ( collect_nat_o
@ ^ [Y_10: nat > $o] : ( (|) @ ( Y_10 = X_1 ) @ ( member_nat_o @ Y_10 @ Xa ) ) ) ) ).
thf(fact_314_insert__compr__raw,axiom,
! [X_1: pname > $o,Xa: ( pname > $o ) > $o] :
( ( insert_pname_o @ X_1 @ Xa )
= ( collect_pname_o
@ ^ [Y_10: pname > $o] : ( (|) @ ( Y_10 = X_1 ) @ ( member_pname_o @ Y_10 @ Xa ) ) ) ) ).
thf(fact_315_insert__compr__raw,axiom,
! [X_1: x_a > $o,Xa: ( x_a > $o ) > $o] :
( ( insert_a_o @ X_1 @ Xa )
= ( collect_a_o
@ ^ [Y_10: x_a > $o] : ( (|) @ ( Y_10 = X_1 ) @ ( member_a_o @ Y_10 @ Xa ) ) ) ) ).
thf(fact_316_insert__compr__raw,axiom,
! [X_1: nat,Xa: nat > $o] :
( ( insert_nat @ X_1 @ Xa )
= ( collect_nat
@ ^ [Y_10: nat] : ( (|) @ ( Y_10 = X_1 ) @ ( member_nat @ Y_10 @ Xa ) ) ) ) ).
thf(fact_317_insert__compr__raw,axiom,
! [X_1: pname,Xa: pname > $o] :
( ( insert_pname @ X_1 @ Xa )
= ( collect_pname
@ ^ [Y_10: pname] : ( (|) @ ( Y_10 = X_1 ) @ ( member_pname @ Y_10 @ Xa ) ) ) ) ).
thf(fact_318_insert__compr__raw,axiom,
! [X_1: x_a,Xa: x_a > $o] :
( ( insert_a @ X_1 @ Xa )
= ( collect_a
@ ^ [Y_10: x_a] : ( (|) @ ( Y_10 = X_1 ) @ ( member_a @ Y_10 @ Xa ) ) ) ) ).
thf(fact_319_subset__insertI,axiom,
! [B_19: pname > $o,A_45: pname] : ( ord_less_eq_pname_o @ B_19 @ ( insert_pname @ A_45 @ B_19 ) ) ).
thf(fact_320_subset__insertI,axiom,
! [B_19: nat > $o,A_45: nat] : ( ord_less_eq_nat_o @ B_19 @ ( insert_nat @ A_45 @ B_19 ) ) ).
thf(fact_321_subset__insertI,axiom,
! [B_19: x_a > $o,A_45: x_a] : ( ord_less_eq_a_o @ B_19 @ ( insert_a @ A_45 @ B_19 ) ) ).
thf(fact_322_insert__subset,axiom,
! [X_19: nat,A_44: nat > $o,B_18: nat > $o] :
( ( ord_less_eq_nat_o @ ( insert_nat @ X_19 @ A_44 ) @ B_18 )
<=> ( ( member_nat @ X_19 @ B_18 )
& ( ord_less_eq_nat_o @ A_44 @ B_18 ) ) ) ).
thf(fact_323_insert__subset,axiom,
! [X_19: pname,A_44: pname > $o,B_18: pname > $o] :
( ( ord_less_eq_pname_o @ ( insert_pname @ X_19 @ A_44 ) @ B_18 )
<=> ( ( member_pname @ X_19 @ B_18 )
& ( ord_less_eq_pname_o @ A_44 @ B_18 ) ) ) ).
thf(fact_324_insert__subset,axiom,
! [X_19: x_a,A_44: x_a > $o,B_18: x_a > $o] :
( ( ord_less_eq_a_o @ ( insert_a @ X_19 @ A_44 ) @ B_18 )
<=> ( ( member_a @ X_19 @ B_18 )
& ( ord_less_eq_a_o @ A_44 @ B_18 ) ) ) ).
thf(fact_325_subset__insert,axiom,
! [B_17: nat > $o,X_18: nat,A_43: nat > $o] :
( ~ ( member_nat @ X_18 @ A_43 )
=> ( ( ord_less_eq_nat_o @ A_43 @ ( insert_nat @ X_18 @ B_17 ) )
<=> ( ord_less_eq_nat_o @ A_43 @ B_17 ) ) ) ).
thf(fact_326_subset__insert,axiom,
! [B_17: pname > $o,X_18: pname,A_43: pname > $o] :
( ~ ( member_pname @ X_18 @ A_43 )
=> ( ( ord_less_eq_pname_o @ A_43 @ ( insert_pname @ X_18 @ B_17 ) )
<=> ( ord_less_eq_pname_o @ A_43 @ B_17 ) ) ) ).
thf(fact_327_subset__insert,axiom,
! [B_17: x_a > $o,X_18: x_a,A_43: x_a > $o] :
( ~ ( member_a @ X_18 @ A_43 )
=> ( ( ord_less_eq_a_o @ A_43 @ ( insert_a @ X_18 @ B_17 ) )
<=> ( ord_less_eq_a_o @ A_43 @ B_17 ) ) ) ).
thf(fact_328_subset__insertI2,axiom,
! [B_16: pname,A_42: pname > $o,B_15: pname > $o] :
( ( ord_less_eq_pname_o @ A_42 @ B_15 )
=> ( ord_less_eq_pname_o @ A_42 @ ( insert_pname @ B_16 @ B_15 ) ) ) ).
thf(fact_329_subset__insertI2,axiom,
! [B_16: nat,A_42: nat > $o,B_15: nat > $o] :
( ( ord_less_eq_nat_o @ A_42 @ B_15 )
=> ( ord_less_eq_nat_o @ A_42 @ ( insert_nat @ B_16 @ B_15 ) ) ) ).
thf(fact_330_subset__insertI2,axiom,
! [B_16: x_a,A_42: x_a > $o,B_15: x_a > $o] :
( ( ord_less_eq_a_o @ A_42 @ B_15 )
=> ( ord_less_eq_a_o @ A_42 @ ( insert_a @ B_16 @ B_15 ) ) ) ).
thf(fact_331_insert__mono,axiom,
! [A_41: pname,C_9: pname > $o,D_1: pname > $o] :
( ( ord_less_eq_pname_o @ C_9 @ D_1 )
=> ( ord_less_eq_pname_o @ ( insert_pname @ A_41 @ C_9 ) @ ( insert_pname @ A_41 @ D_1 ) ) ) ).
thf(fact_332_insert__mono,axiom,
! [A_41: nat,C_9: nat > $o,D_1: nat > $o] :
( ( ord_less_eq_nat_o @ C_9 @ D_1 )
=> ( ord_less_eq_nat_o @ ( insert_nat @ A_41 @ C_9 ) @ ( insert_nat @ A_41 @ D_1 ) ) ) ).
thf(fact_333_insert__mono,axiom,
! [A_41: x_a,C_9: x_a > $o,D_1: x_a > $o] :
( ( ord_less_eq_a_o @ C_9 @ D_1 )
=> ( ord_less_eq_a_o @ ( insert_a @ A_41 @ C_9 ) @ ( insert_a @ A_41 @ D_1 ) ) ) ).
thf(fact_334_image__insert,axiom,
! [F_15: x_a > pname,A_40: x_a,B_14: x_a > $o] :
( ( image_a_pname @ F_15 @ ( insert_a @ A_40 @ B_14 ) )
= ( insert_pname @ ( F_15 @ A_40 ) @ ( image_a_pname @ F_15 @ B_14 ) ) ) ).
thf(fact_335_image__insert,axiom,
! [F_15: x_a > nat,A_40: x_a,B_14: x_a > $o] :
( ( image_a_nat @ F_15 @ ( insert_a @ A_40 @ B_14 ) )
= ( insert_nat @ ( F_15 @ A_40 ) @ ( image_a_nat @ F_15 @ B_14 ) ) ) ).
thf(fact_336_image__insert,axiom,
! [F_15: nat > x_a,A_40: nat,B_14: nat > $o] :
( ( image_nat_a @ F_15 @ ( insert_nat @ A_40 @ B_14 ) )
= ( insert_a @ ( F_15 @ A_40 ) @ ( image_nat_a @ F_15 @ B_14 ) ) ) ).
thf(fact_337_image__insert,axiom,
! [F_15: pname > x_a,A_40: pname,B_14: pname > $o] :
( ( image_pname_a @ F_15 @ ( insert_pname @ A_40 @ B_14 ) )
= ( insert_a @ ( F_15 @ A_40 ) @ ( image_pname_a @ F_15 @ B_14 ) ) ) ).
thf(fact_338_insert__image,axiom,
! [F_14: pname > pname,X_17: pname,A_39: pname > $o] :
( ( member_pname @ X_17 @ A_39 )
=> ( ( insert_pname @ ( F_14 @ X_17 ) @ ( image_pname_pname @ F_14 @ A_39 ) )
= ( image_pname_pname @ F_14 @ A_39 ) ) ) ).
thf(fact_339_insert__image,axiom,
! [F_14: pname > nat,X_17: pname,A_39: pname > $o] :
( ( member_pname @ X_17 @ A_39 )
=> ( ( insert_nat @ ( F_14 @ X_17 ) @ ( image_pname_nat @ F_14 @ A_39 ) )
= ( image_pname_nat @ F_14 @ A_39 ) ) ) ).
thf(fact_340_insert__image,axiom,
! [F_14: x_a > pname,X_17: x_a,A_39: x_a > $o] :
( ( member_a @ X_17 @ A_39 )
=> ( ( insert_pname @ ( F_14 @ X_17 ) @ ( image_a_pname @ F_14 @ A_39 ) )
= ( image_a_pname @ F_14 @ A_39 ) ) ) ).
thf(fact_341_insert__image,axiom,
! [F_14: x_a > nat,X_17: x_a,A_39: x_a > $o] :
( ( member_a @ X_17 @ A_39 )
=> ( ( insert_nat @ ( F_14 @ X_17 ) @ ( image_a_nat @ F_14 @ A_39 ) )
= ( image_a_nat @ F_14 @ A_39 ) ) ) ).
thf(fact_342_insert__image,axiom,
! [F_14: nat > x_a,X_17: nat,A_39: nat > $o] :
( ( member_nat @ X_17 @ A_39 )
=> ( ( insert_a @ ( F_14 @ X_17 ) @ ( image_nat_a @ F_14 @ A_39 ) )
= ( image_nat_a @ F_14 @ A_39 ) ) ) ).
thf(fact_343_insert__image,axiom,
! [F_14: pname > x_a,X_17: pname,A_39: pname > $o] :
( ( member_pname @ X_17 @ A_39 )
=> ( ( insert_a @ ( F_14 @ X_17 ) @ ( image_pname_a @ F_14 @ A_39 ) )
= ( image_pname_a @ F_14 @ A_39 ) ) ) ).
thf(fact_344_subset__image__iff,axiom,
! [B_13: x_a > $o,F_13: nat > x_a,A_38: nat > $o] :
( ( ord_less_eq_a_o @ B_13 @ ( image_nat_a @ F_13 @ A_38 ) )
<=> ? [AA: nat > $o] :
( ( ord_less_eq_nat_o @ AA @ A_38 )
& ( B_13
= ( image_nat_a @ F_13 @ AA ) ) ) ) ).
thf(fact_345_subset__image__iff,axiom,
! [B_13: pname > $o,F_13: x_a > pname,A_38: x_a > $o] :
( ( ord_less_eq_pname_o @ B_13 @ ( image_a_pname @ F_13 @ A_38 ) )
<=> ? [AA: x_a > $o] :
( ( ord_less_eq_a_o @ AA @ A_38 )
& ( B_13
= ( image_a_pname @ F_13 @ AA ) ) ) ) ).
thf(fact_346_subset__image__iff,axiom,
! [B_13: nat > $o,F_13: x_a > nat,A_38: x_a > $o] :
( ( ord_less_eq_nat_o @ B_13 @ ( image_a_nat @ F_13 @ A_38 ) )
<=> ? [AA: x_a > $o] :
( ( ord_less_eq_a_o @ AA @ A_38 )
& ( B_13
= ( image_a_nat @ F_13 @ AA ) ) ) ) ).
thf(fact_347_subset__image__iff,axiom,
! [B_13: x_a > $o,F_13: pname > x_a,A_38: pname > $o] :
( ( ord_less_eq_a_o @ B_13 @ ( image_pname_a @ F_13 @ A_38 ) )
<=> ? [AA: pname > $o] :
( ( ord_less_eq_pname_o @ AA @ A_38 )
& ( B_13
= ( image_pname_a @ F_13 @ AA ) ) ) ) ).
thf(fact_348_image__mono,axiom,
! [F_12: x_a > pname,A_37: x_a > $o,B_12: x_a > $o] :
( ( ord_less_eq_a_o @ A_37 @ B_12 )
=> ( ord_less_eq_pname_o @ ( image_a_pname @ F_12 @ A_37 ) @ ( image_a_pname @ F_12 @ B_12 ) ) ) ).
thf(fact_349_image__mono,axiom,
! [F_12: x_a > nat,A_37: x_a > $o,B_12: x_a > $o] :
( ( ord_less_eq_a_o @ A_37 @ B_12 )
=> ( ord_less_eq_nat_o @ ( image_a_nat @ F_12 @ A_37 ) @ ( image_a_nat @ F_12 @ B_12 ) ) ) ).
thf(fact_350_image__mono,axiom,
! [F_12: nat > x_a,A_37: nat > $o,B_12: nat > $o] :
( ( ord_less_eq_nat_o @ A_37 @ B_12 )
=> ( ord_less_eq_a_o @ ( image_nat_a @ F_12 @ A_37 ) @ ( image_nat_a @ F_12 @ B_12 ) ) ) ).
thf(fact_351_image__mono,axiom,
! [F_12: pname > x_a,A_37: pname > $o,B_12: pname > $o] :
( ( ord_less_eq_pname_o @ A_37 @ B_12 )
=> ( ord_less_eq_a_o @ ( image_pname_a @ F_12 @ A_37 ) @ ( image_pname_a @ F_12 @ B_12 ) ) ) ).
thf(fact_352_imageE,axiom,
! [B_11: pname,F_11: nat > pname,A_36: nat > $o] :
( ( member_pname @ B_11 @ ( image_nat_pname @ F_11 @ A_36 ) )
=> ~ ! [X_1: nat] :
( ( B_11
= ( F_11 @ X_1 ) )
=> ~ ( member_nat @ X_1 @ A_36 ) ) ) ).
thf(fact_353_imageE,axiom,
! [B_11: x_a,F_11: nat > x_a,A_36: nat > $o] :
( ( member_a @ B_11 @ ( image_nat_a @ F_11 @ A_36 ) )
=> ~ ! [X_1: nat] :
( ( B_11
= ( F_11 @ X_1 ) )
=> ~ ( member_nat @ X_1 @ A_36 ) ) ) ).
thf(fact_354_imageE,axiom,
! [B_11: nat,F_11: pname > nat,A_36: pname > $o] :
( ( member_nat @ B_11 @ ( image_pname_nat @ F_11 @ A_36 ) )
=> ~ ! [X_1: pname] :
( ( B_11
= ( F_11 @ X_1 ) )
=> ~ ( member_pname @ X_1 @ A_36 ) ) ) ).
thf(fact_355_imageE,axiom,
! [B_11: nat,F_11: x_a > nat,A_36: x_a > $o] :
( ( member_nat @ B_11 @ ( image_a_nat @ F_11 @ A_36 ) )
=> ~ ! [X_1: x_a] :
( ( B_11
= ( F_11 @ X_1 ) )
=> ~ ( member_a @ X_1 @ A_36 ) ) ) ).
thf(fact_356_imageE,axiom,
! [B_11: x_a,F_11: pname > x_a,A_36: pname > $o] :
( ( member_a @ B_11 @ ( image_pname_a @ F_11 @ A_36 ) )
=> ~ ! [X_1: pname] :
( ( B_11
= ( F_11 @ X_1 ) )
=> ~ ( member_pname @ X_1 @ A_36 ) ) ) ).
thf(fact_357_subsetI,axiom,
! [B_10: nat > $o,A_35: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_35 )
=> ( member_nat @ X_1 @ B_10 ) )
=> ( ord_less_eq_nat_o @ A_35 @ B_10 ) ) ).
thf(fact_358_subsetI,axiom,
! [B_10: pname > $o,A_35: pname > $o] :
( ! [X_1: pname] :
( ( member_pname @ X_1 @ A_35 )
=> ( member_pname @ X_1 @ B_10 ) )
=> ( ord_less_eq_pname_o @ A_35 @ B_10 ) ) ).
thf(fact_359_subsetI,axiom,
! [B_10: x_a > $o,A_35: x_a > $o] :
( ! [X_1: x_a] :
( ( member_a @ X_1 @ A_35 )
=> ( member_a @ X_1 @ B_10 ) )
=> ( ord_less_eq_a_o @ A_35 @ B_10 ) ) ).
thf(fact_360_image__subsetI,axiom,
! [F_10: pname > nat,B_9: nat > $o,A_34: pname > $o] :
( ! [X_1: pname] :
( ( member_pname @ X_1 @ A_34 )
=> ( member_nat @ ( F_10 @ X_1 ) @ B_9 ) )
=> ( ord_less_eq_nat_o @ ( image_pname_nat @ F_10 @ A_34 ) @ B_9 ) ) ).
thf(fact_361_image__subsetI,axiom,
! [F_10: pname > pname,B_9: pname > $o,A_34: pname > $o] :
( ! [X_1: pname] :
( ( member_pname @ X_1 @ A_34 )
=> ( member_pname @ ( F_10 @ X_1 ) @ B_9 ) )
=> ( ord_less_eq_pname_o @ ( image_pname_pname @ F_10 @ A_34 ) @ B_9 ) ) ).
thf(fact_362_image__subsetI,axiom,
! [F_10: x_a > nat,B_9: nat > $o,A_34: x_a > $o] :
( ! [X_1: x_a] :
( ( member_a @ X_1 @ A_34 )
=> ( member_nat @ ( F_10 @ X_1 ) @ B_9 ) )
=> ( ord_less_eq_nat_o @ ( image_a_nat @ F_10 @ A_34 ) @ B_9 ) ) ).
thf(fact_363_image__subsetI,axiom,
! [F_10: x_a > pname,B_9: pname > $o,A_34: x_a > $o] :
( ! [X_1: x_a] :
( ( member_a @ X_1 @ A_34 )
=> ( member_pname @ ( F_10 @ X_1 ) @ B_9 ) )
=> ( ord_less_eq_pname_o @ ( image_a_pname @ F_10 @ A_34 ) @ B_9 ) ) ).
thf(fact_364_image__subsetI,axiom,
! [F_10: nat > pname,B_9: pname > $o,A_34: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_34 )
=> ( member_pname @ ( F_10 @ X_1 ) @ B_9 ) )
=> ( ord_less_eq_pname_o @ ( image_nat_pname @ F_10 @ A_34 ) @ B_9 ) ) ).
thf(fact_365_image__subsetI,axiom,
! [F_10: nat > x_a,B_9: x_a > $o,A_34: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_34 )
=> ( member_a @ ( F_10 @ X_1 ) @ B_9 ) )
=> ( ord_less_eq_a_o @ ( image_nat_a @ F_10 @ A_34 ) @ B_9 ) ) ).
thf(fact_366_image__subsetI,axiom,
! [F_10: pname > x_a,B_9: x_a > $o,A_34: pname > $o] :
( ! [X_1: pname] :
( ( member_pname @ X_1 @ A_34 )
=> ( member_a @ ( F_10 @ X_1 ) @ B_9 ) )
=> ( ord_less_eq_a_o @ ( image_pname_a @ F_10 @ A_34 ) @ B_9 ) ) ).
thf(fact_367_order__refl,axiom,
! [X_16: $o] : ( ord_less_eq_o @ X_16 @ X_16 ) ).
thf(fact_368_order__refl,axiom,
! [X_16: pname > $o] : ( ord_less_eq_pname_o @ X_16 @ X_16 ) ).
thf(fact_369_order__refl,axiom,
! [X_16: nat > $o] : ( ord_less_eq_nat_o @ X_16 @ X_16 ) ).
thf(fact_370_order__refl,axiom,
! [X_16: x_a > $o] : ( ord_less_eq_a_o @ X_16 @ X_16 ) ).
thf(fact_371_order__refl,axiom,
! [X_16: nat] : ( ord_less_eq_nat @ X_16 @ X_16 ) ).
thf(fact_372_finite__nat__set__iff__bounded__le,axiom,
! [N_1: nat > $o] :
( ( finite_finite_nat @ N_1 )
<=> ? [M_1: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_1 )
=> ( ord_less_eq_nat @ X_1 @ M_1 ) ) ) ).
thf(fact_373_assms_I3_J,axiom,
! [G: x_a > $o,C: com] :
( ( wt @ C )
=> ( ! [X_1: pname] :
( ( member_pname @ X_1 @ u )
=> ( p @ G @ ( insert_a @ ( mgt_call @ X_1 ) @ bot_bot_a_o ) ) )
=> ( p @ G @ ( insert_a @ ( mgt @ C ) @ bot_bot_a_o ) ) ) ) ).
thf(fact_374_le__fun__def,axiom,
! [F_9: pname > $o,G_3: pname > $o] :
( ( ord_less_eq_pname_o @ F_9 @ G_3 )
<=> ! [X_1: pname] : ( ord_less_eq_o @ ( F_9 @ X_1 ) @ ( G_3 @ X_1 ) ) ) ).
thf(fact_375_le__fun__def,axiom,
! [F_9: nat > $o,G_3: nat > $o] :
( ( ord_less_eq_nat_o @ F_9 @ G_3 )
<=> ! [X_1: nat] : ( ord_less_eq_o @ ( F_9 @ X_1 ) @ ( G_3 @ X_1 ) ) ) ).
thf(fact_376_le__fun__def,axiom,
! [F_9: x_a > $o,G_3: x_a > $o] :
( ( ord_less_eq_a_o @ F_9 @ G_3 )
<=> ! [X_1: x_a] : ( ord_less_eq_o @ ( F_9 @ X_1 ) @ ( G_3 @ X_1 ) ) ) ).
thf(fact_377_le__funD,axiom,
! [X_15: pname,F_8: pname > $o,G_2: pname > $o] :
( ( ord_less_eq_pname_o @ F_8 @ G_2 )
=> ( ord_less_eq_o @ ( F_8 @ X_15 ) @ ( G_2 @ X_15 ) ) ) ).
thf(fact_378_le__funD,axiom,
! [X_15: nat,F_8: nat > $o,G_2: nat > $o] :
( ( ord_less_eq_nat_o @ F_8 @ G_2 )
=> ( ord_less_eq_o @ ( F_8 @ X_15 ) @ ( G_2 @ X_15 ) ) ) ).
thf(fact_379_le__funD,axiom,
! [X_15: x_a,F_8: x_a > $o,G_2: x_a > $o] :
( ( ord_less_eq_a_o @ F_8 @ G_2 )
=> ( ord_less_eq_o @ ( F_8 @ X_15 ) @ ( G_2 @ X_15 ) ) ) ).
thf(fact_380_le__funE,axiom,
! [X_14: pname,F_7: pname > $o,G_1: pname > $o] :
( ( ord_less_eq_pname_o @ F_7 @ G_1 )
=> ( ord_less_eq_o @ ( F_7 @ X_14 ) @ ( G_1 @ X_14 ) ) ) ).
thf(fact_381_le__funE,axiom,
! [X_14: nat,F_7: nat > $o,G_1: nat > $o] :
( ( ord_less_eq_nat_o @ F_7 @ G_1 )
=> ( ord_less_eq_o @ ( F_7 @ X_14 ) @ ( G_1 @ X_14 ) ) ) ).
thf(fact_382_le__funE,axiom,
! [X_14: x_a,F_7: x_a > $o,G_1: x_a > $o] :
( ( ord_less_eq_a_o @ F_7 @ G_1 )
=> ( ord_less_eq_o @ ( F_7 @ X_14 ) @ ( G_1 @ X_14 ) ) ) ).
thf(fact_383_emptyE,axiom,
! [A_33: nat] :
~ ( member_nat @ A_33 @ bot_bot_nat_o ) ).
thf(fact_384_emptyE,axiom,
! [A_33: pname] :
~ ( member_pname @ A_33 @ bot_bot_pname_o ) ).
thf(fact_385_emptyE,axiom,
! [A_33: x_a] :
~ ( member_a @ A_33 @ bot_bot_a_o ) ).
thf(fact_386_finite_OemptyI,axiom,
finite_finite_nat_o @ bot_bot_nat_o_o ).
thf(fact_387_finite_OemptyI,axiom,
finite297249702name_o @ bot_bot_pname_o_o ).
thf(fact_388_finite_OemptyI,axiom,
finite_finite_a_o @ bot_bot_a_o_o ).
thf(fact_389_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_a_o ).
thf(fact_390_finite_OemptyI,axiom,
finite_finite_pname @ bot_bot_pname_o ).
thf(fact_391_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_nat_o ).
thf(fact_392_empty__subsetI,axiom,
! [A_32: pname > $o] : ( ord_less_eq_pname_o @ bot_bot_pname_o @ A_32 ) ).
thf(fact_393_empty__subsetI,axiom,
! [A_32: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_32 ) ).
thf(fact_394_empty__subsetI,axiom,
! [A_32: x_a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A_32 ) ).
thf(fact_395_equals0D,axiom,
! [A_31: nat,A_30: nat > $o] :
( ( A_30 = bot_bot_nat_o )
=> ~ ( member_nat @ A_31 @ A_30 ) ) ).
thf(fact_396_equals0D,axiom,
! [A_31: pname,A_30: pname > $o] :
( ( A_30 = bot_bot_pname_o )
=> ~ ( member_pname @ A_31 @ A_30 ) ) ).
thf(fact_397_equals0D,axiom,
! [A_31: x_a,A_30: x_a > $o] :
( ( A_30 = bot_bot_a_o )
=> ~ ( member_a @ A_31 @ A_30 ) ) ).
thf(fact_398_Collect__empty__eq,axiom,
! [P_6: pname > $o] :
( ( ( collect_pname @ P_6 )
= bot_bot_pname_o )
<=> ! [X_1: pname] :
~ ( P_6 @ X_1 ) ) ).
thf(fact_399_Collect__empty__eq,axiom,
! [P_6: ( nat > $o ) > $o] :
( ( ( collect_nat_o @ P_6 )
= bot_bot_nat_o_o )
<=> ! [X_1: nat > $o] :
~ ( P_6 @ X_1 ) ) ).
thf(fact_400_Collect__empty__eq,axiom,
! [P_6: ( pname > $o ) > $o] :
( ( ( collect_pname_o @ P_6 )
= bot_bot_pname_o_o )
<=> ! [X_1: pname > $o] :
~ ( P_6 @ X_1 ) ) ).
thf(fact_401_Collect__empty__eq,axiom,
! [P_6: ( x_a > $o ) > $o] :
( ( ( collect_a_o @ P_6 )
= bot_bot_a_o_o )
<=> ! [X_1: x_a > $o] :
~ ( P_6 @ X_1 ) ) ).
thf(fact_402_Collect__empty__eq,axiom,
! [P_6: x_a > $o] :
( ( ( collect_a @ P_6 )
= bot_bot_a_o )
<=> ! [X_1: x_a] :
~ ( P_6 @ X_1 ) ) ).
thf(fact_403_Collect__empty__eq,axiom,
! [P_6: nat > $o] :
( ( ( collect_nat @ P_6 )
= bot_bot_nat_o )
<=> ! [X_1: nat] :
~ ( P_6 @ X_1 ) ) ).
thf(fact_404_empty__iff,axiom,
! [C_8: nat] :
~ ( member_nat @ C_8 @ bot_bot_nat_o ) ).
thf(fact_405_empty__iff,axiom,
! [C_8: pname] :
~ ( member_pname @ C_8 @ bot_bot_pname_o ) ).
thf(fact_406_empty__iff,axiom,
! [C_8: x_a] :
~ ( member_a @ C_8 @ bot_bot_a_o ) ).
thf(fact_407_empty__Collect__eq,axiom,
! [P_5: pname > $o] :
( ( bot_bot_pname_o
= ( collect_pname @ P_5 ) )
<=> ! [X_1: pname] :
~ ( P_5 @ X_1 ) ) ).
thf(fact_408_empty__Collect__eq,axiom,
! [P_5: ( nat > $o ) > $o] :
( ( bot_bot_nat_o_o
= ( collect_nat_o @ P_5 ) )
<=> ! [X_1: nat > $o] :
~ ( P_5 @ X_1 ) ) ).
thf(fact_409_empty__Collect__eq,axiom,
! [P_5: ( pname > $o ) > $o] :
( ( bot_bot_pname_o_o
= ( collect_pname_o @ P_5 ) )
<=> ! [X_1: pname > $o] :
~ ( P_5 @ X_1 ) ) ).
thf(fact_410_empty__Collect__eq,axiom,
! [P_5: ( x_a > $o ) > $o] :
( ( bot_bot_a_o_o
= ( collect_a_o @ P_5 ) )
<=> ! [X_1: x_a > $o] :
~ ( P_5 @ X_1 ) ) ).
thf(fact_411_empty__Collect__eq,axiom,
! [P_5: x_a > $o] :
( ( bot_bot_a_o
= ( collect_a @ P_5 ) )
<=> ! [X_1: x_a] :
~ ( P_5 @ X_1 ) ) ).
thf(fact_412_empty__Collect__eq,axiom,
! [P_5: nat > $o] :
( ( bot_bot_nat_o
= ( collect_nat @ P_5 ) )
<=> ! [X_1: nat] :
~ ( P_5 @ X_1 ) ) ).
thf(fact_413_ex__in__conv,axiom,
! [A_29: nat > $o] :
( ? [X_1: nat] : ( member_nat @ X_1 @ A_29 )
<=> ( A_29 != bot_bot_nat_o ) ) ).
thf(fact_414_ex__in__conv,axiom,
! [A_29: pname > $o] :
( ? [X_1: pname] : ( member_pname @ X_1 @ A_29 )
<=> ( A_29 != bot_bot_pname_o ) ) ).
thf(fact_415_ex__in__conv,axiom,
! [A_29: x_a > $o] :
( ? [X_1: x_a] : ( member_a @ X_1 @ A_29 )
<=> ( A_29 != bot_bot_a_o ) ) ).
thf(fact_416_all__not__in__conv,axiom,
! [A_28: nat > $o] :
( ! [X_1: nat] :
~ ( member_nat @ X_1 @ A_28 )
<=> ( A_28 = bot_bot_nat_o ) ) ).
thf(fact_417_all__not__in__conv,axiom,
! [A_28: pname > $o] :
( ! [X_1: pname] :
~ ( member_pname @ X_1 @ A_28 )
<=> ( A_28 = bot_bot_pname_o ) ) ).
thf(fact_418_all__not__in__conv,axiom,
! [A_28: x_a > $o] :
( ! [X_1: x_a] :
~ ( member_a @ X_1 @ A_28 )
<=> ( A_28 = bot_bot_a_o ) ) ).
thf(fact_419_empty__def,axiom,
( bot_bot_pname_o
= ( collect_pname
@ ^ [X_1: pname] : $false ) ) ).
thf(fact_420_empty__def,axiom,
( bot_bot_nat_o_o
= ( collect_nat_o
@ ^ [X_1: nat > $o] : $false ) ) ).
thf(fact_421_empty__def,axiom,
( bot_bot_pname_o_o
= ( collect_pname_o
@ ^ [X_1: pname > $o] : $false ) ) ).
thf(fact_422_empty__def,axiom,
( bot_bot_a_o_o
= ( collect_a_o
@ ^ [X_1: x_a > $o] : $false ) ) ).
thf(fact_423_empty__def,axiom,
( bot_bot_a_o
= ( collect_a
@ ^ [X_1: x_a] : $false ) ) ).
thf(fact_424_empty__def,axiom,
( bot_bot_nat_o
= ( collect_nat
@ ^ [X_1: nat] : $false ) ) ).
thf(fact_425_bot__fun__def,axiom,
! [X_1: pname] :
( ( bot_bot_pname_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_426_bot__fun__def,axiom,
! [X_1: nat] :
( ( bot_bot_nat_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_427_bot__fun__def,axiom,
! [X_1: x_a] :
( ( bot_bot_a_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_428_bot__apply,axiom,
! [X_13: pname] :
( ( bot_bot_pname_o @ X_13 )
<=> bot_bot_o ) ).
thf(fact_429_bot__apply,axiom,
! [X_13: nat] :
( ( bot_bot_nat_o @ X_13 )
<=> bot_bot_o ) ).
thf(fact_430_bot__apply,axiom,
! [X_13: x_a] :
( ( bot_bot_a_o @ X_13 )
<=> bot_bot_o ) ).
thf(fact_431_le__bot,axiom,
! [A_27: pname > $o] :
( ( ord_less_eq_pname_o @ A_27 @ bot_bot_pname_o )
=> ( A_27 = bot_bot_pname_o ) ) ).
thf(fact_432_le__bot,axiom,
! [A_27: $o] :
( ( ord_less_eq_o @ A_27 @ bot_bot_o )
=> ( A_27
<=> bot_bot_o ) ) ).
thf(fact_433_le__bot,axiom,
! [A_27: nat > $o] :
( ( ord_less_eq_nat_o @ A_27 @ bot_bot_nat_o )
=> ( A_27 = bot_bot_nat_o ) ) ).
thf(fact_434_le__bot,axiom,
! [A_27: x_a > $o] :
( ( ord_less_eq_a_o @ A_27 @ bot_bot_a_o )
=> ( A_27 = bot_bot_a_o ) ) ).
thf(fact_435_le__bot,axiom,
! [A_27: nat] :
( ( ord_less_eq_nat @ A_27 @ bot_bot_nat )
=> ( A_27 = bot_bot_nat ) ) ).
thf(fact_436_bot__unique,axiom,
! [A_26: pname > $o] :
( ( ord_less_eq_pname_o @ A_26 @ bot_bot_pname_o )
<=> ( A_26 = bot_bot_pname_o ) ) ).
thf(fact_437_bot__unique,axiom,
! [A_26: $o] :
( ( ord_less_eq_o @ A_26 @ bot_bot_o )
<=> ( A_26
<=> bot_bot_o ) ) ).
thf(fact_438_bot__unique,axiom,
! [A_26: nat > $o] :
( ( ord_less_eq_nat_o @ A_26 @ bot_bot_nat_o )
<=> ( A_26 = bot_bot_nat_o ) ) ).
thf(fact_439_bot__unique,axiom,
! [A_26: x_a > $o] :
( ( ord_less_eq_a_o @ A_26 @ bot_bot_a_o )
<=> ( A_26 = bot_bot_a_o ) ) ).
thf(fact_440_bot__unique,axiom,
! [A_26: nat] :
( ( ord_less_eq_nat @ A_26 @ bot_bot_nat )
<=> ( A_26 = bot_bot_nat ) ) ).
thf(fact_441_bot__least,axiom,
! [A_25: pname > $o] : ( ord_less_eq_pname_o @ bot_bot_pname_o @ A_25 ) ).
thf(fact_442_bot__least,axiom,
! [A_25: $o] : ( ord_less_eq_o @ bot_bot_o @ A_25 ) ).
thf(fact_443_bot__least,axiom,
! [A_25: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_25 ) ).
thf(fact_444_bot__least,axiom,
! [A_25: x_a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A_25 ) ).
thf(fact_445_bot__least,axiom,
! [A_25: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_25 ) ).
thf(fact_446_singleton__inject,axiom,
! [A_24: pname,B_8: pname] :
( ( ( insert_pname @ A_24 @ bot_bot_pname_o )
= ( insert_pname @ B_8 @ bot_bot_pname_o ) )
=> ( A_24 = B_8 ) ) ).
thf(fact_447_singleton__inject,axiom,
! [A_24: nat,B_8: nat] :
( ( ( insert_nat @ A_24 @ bot_bot_nat_o )
= ( insert_nat @ B_8 @ bot_bot_nat_o ) )
=> ( A_24 = B_8 ) ) ).
thf(fact_448_singleton__inject,axiom,
! [A_24: x_a,B_8: x_a] :
( ( ( insert_a @ A_24 @ bot_bot_a_o )
= ( insert_a @ B_8 @ bot_bot_a_o ) )
=> ( A_24 = B_8 ) ) ).
thf(fact_449_singletonE,axiom,
! [B_7: nat,A_23: nat] :
( ( member_nat @ B_7 @ ( insert_nat @ A_23 @ bot_bot_nat_o ) )
=> ( B_7 = A_23 ) ) ).
thf(fact_450_singletonE,axiom,
! [B_7: pname,A_23: pname] :
( ( member_pname @ B_7 @ ( insert_pname @ A_23 @ bot_bot_pname_o ) )
=> ( B_7 = A_23 ) ) ).
thf(fact_451_singletonE,axiom,
! [B_7: x_a,A_23: x_a] :
( ( member_a @ B_7 @ ( insert_a @ A_23 @ bot_bot_a_o ) )
=> ( B_7 = A_23 ) ) ).
thf(fact_452_doubleton__eq__iff,axiom,
! [A_22: pname,B_6: pname,C_7: pname,D: pname] :
( ( ( insert_pname @ A_22 @ ( insert_pname @ B_6 @ bot_bot_pname_o ) )
= ( insert_pname @ C_7 @ ( insert_pname @ D @ bot_bot_pname_o ) ) )
<=> ( ( ( A_22 = C_7 )
& ( B_6 = D ) )
| ( ( A_22 = D )
& ( B_6 = C_7 ) ) ) ) ).
thf(fact_453_doubleton__eq__iff,axiom,
! [A_22: nat,B_6: nat,C_7: nat,D: nat] :
( ( ( insert_nat @ A_22 @ ( insert_nat @ B_6 @ bot_bot_nat_o ) )
= ( insert_nat @ C_7 @ ( insert_nat @ D @ bot_bot_nat_o ) ) )
<=> ( ( ( A_22 = C_7 )
& ( B_6 = D ) )
| ( ( A_22 = D )
& ( B_6 = C_7 ) ) ) ) ).
thf(fact_454_doubleton__eq__iff,axiom,
! [A_22: x_a,B_6: x_a,C_7: x_a,D: x_a] :
( ( ( insert_a @ A_22 @ ( insert_a @ B_6 @ bot_bot_a_o ) )
= ( insert_a @ C_7 @ ( insert_a @ D @ bot_bot_a_o ) ) )
<=> ( ( ( A_22 = C_7 )
& ( B_6 = D ) )
| ( ( A_22 = D )
& ( B_6 = C_7 ) ) ) ) ).
thf(fact_455_singleton__iff,axiom,
! [B_5: nat,A_21: nat] :
( ( member_nat @ B_5 @ ( insert_nat @ A_21 @ bot_bot_nat_o ) )
<=> ( B_5 = A_21 ) ) ).
thf(fact_456_singleton__iff,axiom,
! [B_5: pname,A_21: pname] :
( ( member_pname @ B_5 @ ( insert_pname @ A_21 @ bot_bot_pname_o ) )
<=> ( B_5 = A_21 ) ) ).
thf(fact_457_singleton__iff,axiom,
! [B_5: x_a,A_21: x_a] :
( ( member_a @ B_5 @ ( insert_a @ A_21 @ bot_bot_a_o ) )
<=> ( B_5 = A_21 ) ) ).
thf(fact_458_insert__not__empty,axiom,
! [A_20: pname,A_19: pname > $o] :
( ( insert_pname @ A_20 @ A_19 )
!= bot_bot_pname_o ) ).
thf(fact_459_insert__not__empty,axiom,
! [A_20: nat,A_19: nat > $o] :
( ( insert_nat @ A_20 @ A_19 )
!= bot_bot_nat_o ) ).
thf(fact_460_insert__not__empty,axiom,
! [A_20: x_a,A_19: x_a > $o] :
( ( insert_a @ A_20 @ A_19 )
!= bot_bot_a_o ) ).
thf(fact_461_empty__not__insert,axiom,
! [A_18: pname,A_17: pname > $o] :
( bot_bot_pname_o
!= ( insert_pname @ A_18 @ A_17 ) ) ).
thf(fact_462_empty__not__insert,axiom,
! [A_18: nat,A_17: nat > $o] :
( bot_bot_nat_o
!= ( insert_nat @ A_18 @ A_17 ) ) ).
thf(fact_463_empty__not__insert,axiom,
! [A_18: x_a,A_17: x_a > $o] :
( bot_bot_a_o
!= ( insert_a @ A_18 @ A_17 ) ) ).
thf(fact_464_subset__empty,axiom,
! [A_16: nat > $o] :
( ( ord_less_eq_nat_o @ A_16 @ bot_bot_nat_o )
<=> ( A_16 = bot_bot_nat_o ) ) ).
thf(fact_465_subset__empty,axiom,
! [A_16: pname > $o] :
( ( ord_less_eq_pname_o @ A_16 @ bot_bot_pname_o )
<=> ( A_16 = bot_bot_pname_o ) ) ).
thf(fact_466_subset__empty,axiom,
! [A_16: x_a > $o] :
( ( ord_less_eq_a_o @ A_16 @ bot_bot_a_o )
<=> ( A_16 = bot_bot_a_o ) ) ).
thf(fact_467_image__is__empty,axiom,
! [F_6: pname > x_a,A_15: pname > $o] :
( ( ( image_pname_a @ F_6 @ A_15 )
= bot_bot_a_o )
<=> ( A_15 = bot_bot_pname_o ) ) ).
thf(fact_468_image__empty,axiom,
! [F_5: pname > x_a] :
( ( image_pname_a @ F_5 @ bot_bot_pname_o )
= bot_bot_a_o ) ).
thf(fact_469_empty__is__image,axiom,
! [F_4: pname > x_a,A_14: pname > $o] :
( ( bot_bot_a_o
= ( image_pname_a @ F_4 @ A_14 ) )
<=> ( A_14 = bot_bot_pname_o ) ) ).
thf(fact_470_Collect__conv__if,axiom,
! [P_4: nat > $o,A_13: nat] :
( ( ( P_4 @ A_13 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= ( insert_nat @ A_13 @ bot_bot_nat_o ) ) )
& ( ~ ( P_4 @ A_13 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_471_Collect__conv__if,axiom,
! [P_4: pname > $o,A_13: pname] :
( ( ( P_4 @ A_13 )
=> ( ( collect_pname
@ ^ [X_1: pname] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= ( insert_pname @ A_13 @ bot_bot_pname_o ) ) )
& ( ~ ( P_4 @ A_13 )
=> ( ( collect_pname
@ ^ [X_1: pname] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= bot_bot_pname_o ) ) ) ).
thf(fact_472_Collect__conv__if,axiom,
! [P_4: x_a > $o,A_13: x_a] :
( ( ( P_4 @ A_13 )
=> ( ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= ( insert_a @ A_13 @ bot_bot_a_o ) ) )
& ( ~ ( P_4 @ A_13 )
=> ( ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= bot_bot_a_o ) ) ) ).
thf(fact_473_Collect__conv__if,axiom,
! [P_4: ( nat > $o ) > $o,A_13: nat > $o] :
( ( ( P_4 @ A_13 )
=> ( ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= ( insert_nat_o @ A_13 @ bot_bot_nat_o_o ) ) )
& ( ~ ( P_4 @ A_13 )
=> ( ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= bot_bot_nat_o_o ) ) ) ).
thf(fact_474_Collect__conv__if,axiom,
! [P_4: ( pname > $o ) > $o,A_13: pname > $o] :
( ( ( P_4 @ A_13 )
=> ( ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= ( insert_pname_o @ A_13 @ bot_bot_pname_o_o ) ) )
& ( ~ ( P_4 @ A_13 )
=> ( ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= bot_bot_pname_o_o ) ) ) ).
thf(fact_475_Collect__conv__if,axiom,
! [P_4: ( x_a > $o ) > $o,A_13: x_a > $o] :
( ( ( P_4 @ A_13 )
=> ( ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= ( insert_a_o @ A_13 @ bot_bot_a_o_o ) ) )
& ( ~ ( P_4 @ A_13 )
=> ( ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (&) @ ( X_1 = A_13 ) @ ( P_4 @ X_1 ) ) )
= bot_bot_a_o_o ) ) ) ).
thf(fact_476_Collect__conv__if2,axiom,
! [P_3: nat > $o,A_12: nat] :
( ( ( P_3 @ A_12 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= ( insert_nat @ A_12 @ bot_bot_nat_o ) ) )
& ( ~ ( P_3 @ A_12 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_477_Collect__conv__if2,axiom,
! [P_3: pname > $o,A_12: pname] :
( ( ( P_3 @ A_12 )
=> ( ( collect_pname
@ ^ [X_1: pname] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= ( insert_pname @ A_12 @ bot_bot_pname_o ) ) )
& ( ~ ( P_3 @ A_12 )
=> ( ( collect_pname
@ ^ [X_1: pname] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= bot_bot_pname_o ) ) ) ).
thf(fact_478_Collect__conv__if2,axiom,
! [P_3: x_a > $o,A_12: x_a] :
( ( ( P_3 @ A_12 )
=> ( ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= ( insert_a @ A_12 @ bot_bot_a_o ) ) )
& ( ~ ( P_3 @ A_12 )
=> ( ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= bot_bot_a_o ) ) ) ).
thf(fact_479_Collect__conv__if2,axiom,
! [P_3: ( nat > $o ) > $o,A_12: nat > $o] :
( ( ( P_3 @ A_12 )
=> ( ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= ( insert_nat_o @ A_12 @ bot_bot_nat_o_o ) ) )
& ( ~ ( P_3 @ A_12 )
=> ( ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= bot_bot_nat_o_o ) ) ) ).
thf(fact_480_Collect__conv__if2,axiom,
! [P_3: ( pname > $o ) > $o,A_12: pname > $o] :
( ( ( P_3 @ A_12 )
=> ( ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= ( insert_pname_o @ A_12 @ bot_bot_pname_o_o ) ) )
& ( ~ ( P_3 @ A_12 )
=> ( ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= bot_bot_pname_o_o ) ) ) ).
thf(fact_481_Collect__conv__if2,axiom,
! [P_3: ( x_a > $o ) > $o,A_12: x_a > $o] :
( ( ( P_3 @ A_12 )
=> ( ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= ( insert_a_o @ A_12 @ bot_bot_a_o_o ) ) )
& ( ~ ( P_3 @ A_12 )
=> ( ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (&) @ ( A_12 = X_1 ) @ ( P_3 @ X_1 ) ) )
= bot_bot_a_o_o ) ) ) ).
thf(fact_482_singleton__conv,axiom,
! [A_11: nat] :
( ( collect_nat
@ ^ [X_1: nat] : X_1 = A_11 )
= ( insert_nat @ A_11 @ bot_bot_nat_o ) ) ).
thf(fact_483_singleton__conv,axiom,
! [A_11: pname] :
( ( collect_pname
@ ^ [X_1: pname] : X_1 = A_11 )
= ( insert_pname @ A_11 @ bot_bot_pname_o ) ) ).
thf(fact_484_singleton__conv,axiom,
! [A_11: x_a] :
( ( collect_a
@ ^ [X_1: x_a] : X_1 = A_11 )
= ( insert_a @ A_11 @ bot_bot_a_o ) ) ).
thf(fact_485_singleton__conv,axiom,
! [A_11: nat > $o] :
( ( collect_nat_o
@ ^ [X_1: nat > $o] : X_1 = A_11 )
= ( insert_nat_o @ A_11 @ bot_bot_nat_o_o ) ) ).
thf(fact_486_singleton__conv,axiom,
! [A_11: pname > $o] :
( ( collect_pname_o
@ ^ [X_1: pname > $o] : X_1 = A_11 )
= ( insert_pname_o @ A_11 @ bot_bot_pname_o_o ) ) ).
thf(fact_487_singleton__conv,axiom,
! [A_11: x_a > $o] :
( ( collect_a_o
@ ^ [X_1: x_a > $o] : X_1 = A_11 )
= ( insert_a_o @ A_11 @ bot_bot_a_o_o ) ) ).
thf(fact_488_singleton__conv2,axiom,
! [A_10: nat] :
( ( collect_nat @ ( fequal_nat @ A_10 ) )
= ( insert_nat @ A_10 @ bot_bot_nat_o ) ) ).
thf(fact_489_singleton__conv2,axiom,
! [A_10: pname] :
( ( collect_pname @ ( fequal_pname @ A_10 ) )
= ( insert_pname @ A_10 @ bot_bot_pname_o ) ) ).
thf(fact_490_singleton__conv2,axiom,
! [A_10: x_a] :
( ( collect_a @ ( fequal_a @ A_10 ) )
= ( insert_a @ A_10 @ bot_bot_a_o ) ) ).
thf(fact_491_singleton__conv2,axiom,
! [A_10: nat > $o] :
( ( collect_nat_o @ ( fequal_nat_o @ A_10 ) )
= ( insert_nat_o @ A_10 @ bot_bot_nat_o_o ) ) ).
thf(fact_492_singleton__conv2,axiom,
! [A_10: pname > $o] :
( ( collect_pname_o @ ( fequal_pname_o @ A_10 ) )
= ( insert_pname_o @ A_10 @ bot_bot_pname_o_o ) ) ).
thf(fact_493_singleton__conv2,axiom,
! [A_10: x_a > $o] :
( ( collect_a_o @ ( fequal_a_o @ A_10 ) )
= ( insert_a_o @ A_10 @ bot_bot_a_o_o ) ) ).
thf(fact_494_subset__singletonD,axiom,
! [A_9: nat > $o,X_12: nat] :
( ( ord_less_eq_nat_o @ A_9 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
=> ( ( A_9 = bot_bot_nat_o )
| ( A_9
= ( insert_nat @ X_12 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_495_subset__singletonD,axiom,
! [A_9: pname > $o,X_12: pname] :
( ( ord_less_eq_pname_o @ A_9 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
=> ( ( A_9 = bot_bot_pname_o )
| ( A_9
= ( insert_pname @ X_12 @ bot_bot_pname_o ) ) ) ) ).
thf(fact_496_subset__singletonD,axiom,
! [A_9: x_a > $o,X_12: x_a] :
( ( ord_less_eq_a_o @ A_9 @ ( insert_a @ X_12 @ bot_bot_a_o ) )
=> ( ( A_9 = bot_bot_a_o )
| ( A_9
= ( insert_a @ X_12 @ bot_bot_a_o ) ) ) ) ).
thf(fact_497_image__constant,axiom,
! [C_6: x_a,X_11: pname,A_8: pname > $o] :
( ( member_pname @ X_11 @ A_8 )
=> ( ( image_pname_a
@ ^ [X_1: pname] : C_6
@ A_8 )
= ( insert_a @ C_6 @ bot_bot_a_o ) ) ) ).
thf(fact_498_image__constant__conv,axiom,
! [C_5: x_a,A_7: pname > $o] :
( ( ( A_7 = bot_bot_pname_o )
=> ( ( image_pname_a
@ ^ [X_1: pname] : C_5
@ A_7 )
= bot_bot_a_o ) )
& ( ( A_7 != bot_bot_pname_o )
=> ( ( image_pname_a
@ ^ [X_1: pname] : C_5
@ A_7 )
= ( insert_a @ C_5 @ bot_bot_a_o ) ) ) ) ).
thf(fact_499_linorder__le__cases,axiom,
! [X_10: nat,Y_9: nat] :
( ~ ( ord_less_eq_nat @ X_10 @ Y_9 )
=> ( ord_less_eq_nat @ Y_9 @ X_10 ) ) ).
thf(fact_500_xt1_I6_J,axiom,
! [Z_2: $o,Y_8: $o,X_9: $o] :
( ( ord_less_eq_o @ Y_8 @ X_9 )
=> ( ( ord_less_eq_o @ Z_2 @ Y_8 )
=> ( ord_less_eq_o @ Z_2 @ X_9 ) ) ) ).
thf(fact_501_xt1_I6_J,axiom,
! [Z_2: nat > $o,Y_8: nat > $o,X_9: nat > $o] :
( ( ord_less_eq_nat_o @ Y_8 @ X_9 )
=> ( ( ord_less_eq_nat_o @ Z_2 @ Y_8 )
=> ( ord_less_eq_nat_o @ Z_2 @ X_9 ) ) ) ).
thf(fact_502_xt1_I6_J,axiom,
! [Z_2: pname > $o,Y_8: pname > $o,X_9: pname > $o] :
( ( ord_less_eq_pname_o @ Y_8 @ X_9 )
=> ( ( ord_less_eq_pname_o @ Z_2 @ Y_8 )
=> ( ord_less_eq_pname_o @ Z_2 @ X_9 ) ) ) ).
thf(fact_503_xt1_I6_J,axiom,
! [Z_2: nat,Y_8: nat,X_9: nat] :
( ( ord_less_eq_nat @ Y_8 @ X_9 )
=> ( ( ord_less_eq_nat @ Z_2 @ Y_8 )
=> ( ord_less_eq_nat @ Z_2 @ X_9 ) ) ) ).
thf(fact_504_xt1_I6_J,axiom,
! [Z_2: x_a > $o,Y_8: x_a > $o,X_9: x_a > $o] :
( ( ord_less_eq_a_o @ Y_8 @ X_9 )
=> ( ( ord_less_eq_a_o @ Z_2 @ Y_8 )
=> ( ord_less_eq_a_o @ Z_2 @ X_9 ) ) ) ).
thf(fact_505_xt1_I5_J,axiom,
! [Y_7: $o,X_8: $o] :
( ( ord_less_eq_o @ Y_7 @ X_8 )
=> ( ( ord_less_eq_o @ X_8 @ Y_7 )
=> ( X_8
<=> Y_7 ) ) ) ).
thf(fact_506_xt1_I5_J,axiom,
! [Y_7: nat > $o,X_8: nat > $o] :
( ( ord_less_eq_nat_o @ Y_7 @ X_8 )
=> ( ( ord_less_eq_nat_o @ X_8 @ Y_7 )
=> ( X_8 = Y_7 ) ) ) ).
thf(fact_507_xt1_I5_J,axiom,
! [Y_7: pname > $o,X_8: pname > $o] :
( ( ord_less_eq_pname_o @ Y_7 @ X_8 )
=> ( ( ord_less_eq_pname_o @ X_8 @ Y_7 )
=> ( X_8 = Y_7 ) ) ) ).
thf(fact_508_xt1_I5_J,axiom,
! [Y_7: nat,X_8: nat] :
( ( ord_less_eq_nat @ Y_7 @ X_8 )
=> ( ( ord_less_eq_nat @ X_8 @ Y_7 )
=> ( X_8 = Y_7 ) ) ) ).
thf(fact_509_xt1_I5_J,axiom,
! [Y_7: x_a > $o,X_8: x_a > $o] :
( ( ord_less_eq_a_o @ Y_7 @ X_8 )
=> ( ( ord_less_eq_a_o @ X_8 @ Y_7 )
=> ( X_8 = Y_7 ) ) ) ).
thf(fact_510_order__trans,axiom,
! [Z_1: $o,X_7: $o,Y_6: $o] :
( ( ord_less_eq_o @ X_7 @ Y_6 )
=> ( ( ord_less_eq_o @ Y_6 @ Z_1 )
=> ( ord_less_eq_o @ X_7 @ Z_1 ) ) ) ).
thf(fact_511_order__trans,axiom,
! [Z_1: nat > $o,X_7: nat > $o,Y_6: nat > $o] :
( ( ord_less_eq_nat_o @ X_7 @ Y_6 )
=> ( ( ord_less_eq_nat_o @ Y_6 @ Z_1 )
=> ( ord_less_eq_nat_o @ X_7 @ Z_1 ) ) ) ).
thf(fact_512_order__trans,axiom,
! [Z_1: pname > $o,X_7: pname > $o,Y_6: pname > $o] :
( ( ord_less_eq_pname_o @ X_7 @ Y_6 )
=> ( ( ord_less_eq_pname_o @ Y_6 @ Z_1 )
=> ( ord_less_eq_pname_o @ X_7 @ Z_1 ) ) ) ).
thf(fact_513_order__trans,axiom,
! [Z_1: nat,X_7: nat,Y_6: nat] :
( ( ord_less_eq_nat @ X_7 @ Y_6 )
=> ( ( ord_less_eq_nat @ Y_6 @ Z_1 )
=> ( ord_less_eq_nat @ X_7 @ Z_1 ) ) ) ).
thf(fact_514_order__trans,axiom,
! [Z_1: x_a > $o,X_7: x_a > $o,Y_6: x_a > $o] :
( ( ord_less_eq_a_o @ X_7 @ Y_6 )
=> ( ( ord_less_eq_a_o @ Y_6 @ Z_1 )
=> ( ord_less_eq_a_o @ X_7 @ Z_1 ) ) ) ).
thf(fact_515_order__antisym,axiom,
! [X_6: $o,Y_5: $o] :
( ( ord_less_eq_o @ X_6 @ Y_5 )
=> ( ( ord_less_eq_o @ Y_5 @ X_6 )
=> ( X_6
<=> Y_5 ) ) ) ).
thf(fact_516_order__antisym,axiom,
! [X_6: nat > $o,Y_5: nat > $o] :
( ( ord_less_eq_nat_o @ X_6 @ Y_5 )
=> ( ( ord_less_eq_nat_o @ Y_5 @ X_6 )
=> ( X_6 = Y_5 ) ) ) ).
thf(fact_517_order__antisym,axiom,
! [X_6: pname > $o,Y_5: pname > $o] :
( ( ord_less_eq_pname_o @ X_6 @ Y_5 )
=> ( ( ord_less_eq_pname_o @ Y_5 @ X_6 )
=> ( X_6 = Y_5 ) ) ) ).
thf(fact_518_order__antisym,axiom,
! [X_6: nat,Y_5: nat] :
( ( ord_less_eq_nat @ X_6 @ Y_5 )
=> ( ( ord_less_eq_nat @ Y_5 @ X_6 )
=> ( X_6 = Y_5 ) ) ) ).
thf(fact_519_order__antisym,axiom,
! [X_6: x_a > $o,Y_5: x_a > $o] :
( ( ord_less_eq_a_o @ X_6 @ Y_5 )
=> ( ( ord_less_eq_a_o @ Y_5 @ X_6 )
=> ( X_6 = Y_5 ) ) ) ).
thf(fact_520_xt1_I4_J,axiom,
! [C_4: $o,B_4: $o,A_6: $o] :
( ( ord_less_eq_o @ B_4 @ A_6 )
=> ( ( B_4
<=> C_4 )
=> ( ord_less_eq_o @ C_4 @ A_6 ) ) ) ).
thf(fact_521_xt1_I4_J,axiom,
! [C_4: nat > $o,B_4: nat > $o,A_6: nat > $o] :
( ( ord_less_eq_nat_o @ B_4 @ A_6 )
=> ( ( B_4 = C_4 )
=> ( ord_less_eq_nat_o @ C_4 @ A_6 ) ) ) ).
thf(fact_522_xt1_I4_J,axiom,
! [C_4: pname > $o,B_4: pname > $o,A_6: pname > $o] :
( ( ord_less_eq_pname_o @ B_4 @ A_6 )
=> ( ( B_4 = C_4 )
=> ( ord_less_eq_pname_o @ C_4 @ A_6 ) ) ) ).
thf(fact_523_xt1_I4_J,axiom,
! [C_4: nat,B_4: nat,A_6: nat] :
( ( ord_less_eq_nat @ B_4 @ A_6 )
=> ( ( B_4 = C_4 )
=> ( ord_less_eq_nat @ C_4 @ A_6 ) ) ) ).
thf(fact_524_xt1_I4_J,axiom,
! [C_4: x_a > $o,B_4: x_a > $o,A_6: x_a > $o] :
( ( ord_less_eq_a_o @ B_4 @ A_6 )
=> ( ( B_4 = C_4 )
=> ( ord_less_eq_a_o @ C_4 @ A_6 ) ) ) ).
thf(fact_525_ord__le__eq__trans,axiom,
! [C_3: $o,A_5: $o,B_3: $o] :
( ( ord_less_eq_o @ A_5 @ B_3 )
=> ( ( B_3
<=> C_3 )
=> ( ord_less_eq_o @ A_5 @ C_3 ) ) ) ).
thf(fact_526_ord__le__eq__trans,axiom,
! [C_3: nat > $o,A_5: nat > $o,B_3: nat > $o] :
( ( ord_less_eq_nat_o @ A_5 @ B_3 )
=> ( ( B_3 = C_3 )
=> ( ord_less_eq_nat_o @ A_5 @ C_3 ) ) ) ).
thf(fact_527_ord__le__eq__trans,axiom,
! [C_3: pname > $o,A_5: pname > $o,B_3: pname > $o] :
( ( ord_less_eq_pname_o @ A_5 @ B_3 )
=> ( ( B_3 = C_3 )
=> ( ord_less_eq_pname_o @ A_5 @ C_3 ) ) ) ).
thf(fact_528_ord__le__eq__trans,axiom,
! [C_3: nat,A_5: nat,B_3: nat] :
( ( ord_less_eq_nat @ A_5 @ B_3 )
=> ( ( B_3 = C_3 )
=> ( ord_less_eq_nat @ A_5 @ C_3 ) ) ) ).
thf(fact_529_ord__le__eq__trans,axiom,
! [C_3: x_a > $o,A_5: x_a > $o,B_3: x_a > $o] :
( ( ord_less_eq_a_o @ A_5 @ B_3 )
=> ( ( B_3 = C_3 )
=> ( ord_less_eq_a_o @ A_5 @ C_3 ) ) ) ).
thf(fact_530_xt1_I3_J,axiom,
! [C_2: $o,B_2: $o,A_4: $o] :
( ( A_4
<=> B_2 )
=> ( ( ord_less_eq_o @ C_2 @ B_2 )
=> ( ord_less_eq_o @ C_2 @ A_4 ) ) ) ).
thf(fact_531_xt1_I3_J,axiom,
! [C_2: nat > $o,A_4: nat > $o,B_2: nat > $o] :
( ( A_4 = B_2 )
=> ( ( ord_less_eq_nat_o @ C_2 @ B_2 )
=> ( ord_less_eq_nat_o @ C_2 @ A_4 ) ) ) ).
thf(fact_532_xt1_I3_J,axiom,
! [C_2: pname > $o,A_4: pname > $o,B_2: pname > $o] :
( ( A_4 = B_2 )
=> ( ( ord_less_eq_pname_o @ C_2 @ B_2 )
=> ( ord_less_eq_pname_o @ C_2 @ A_4 ) ) ) ).
thf(fact_533_xt1_I3_J,axiom,
! [C_2: nat,A_4: nat,B_2: nat] :
( ( A_4 = B_2 )
=> ( ( ord_less_eq_nat @ C_2 @ B_2 )
=> ( ord_less_eq_nat @ C_2 @ A_4 ) ) ) ).
thf(fact_534_xt1_I3_J,axiom,
! [C_2: x_a > $o,A_4: x_a > $o,B_2: x_a > $o] :
( ( A_4 = B_2 )
=> ( ( ord_less_eq_a_o @ C_2 @ B_2 )
=> ( ord_less_eq_a_o @ C_2 @ A_4 ) ) ) ).
thf(fact_535_ord__eq__le__trans,axiom,
! [C_1: $o,B_1: $o,A_3: $o] :
( ( A_3
<=> B_1 )
=> ( ( ord_less_eq_o @ B_1 @ C_1 )
=> ( ord_less_eq_o @ A_3 @ C_1 ) ) ) ).
thf(fact_536_ord__eq__le__trans,axiom,
! [C_1: nat > $o,A_3: nat > $o,B_1: nat > $o] :
( ( A_3 = B_1 )
=> ( ( ord_less_eq_nat_o @ B_1 @ C_1 )
=> ( ord_less_eq_nat_o @ A_3 @ C_1 ) ) ) ).
thf(fact_537_ord__eq__le__trans,axiom,
! [C_1: pname > $o,A_3: pname > $o,B_1: pname > $o] :
( ( A_3 = B_1 )
=> ( ( ord_less_eq_pname_o @ B_1 @ C_1 )
=> ( ord_less_eq_pname_o @ A_3 @ C_1 ) ) ) ).
thf(fact_538_ord__eq__le__trans,axiom,
! [C_1: nat,A_3: nat,B_1: nat] :
( ( A_3 = B_1 )
=> ( ( ord_less_eq_nat @ B_1 @ C_1 )
=> ( ord_less_eq_nat @ A_3 @ C_1 ) ) ) ).
thf(fact_539_ord__eq__le__trans,axiom,
! [C_1: x_a > $o,A_3: x_a > $o,B_1: x_a > $o] :
( ( A_3 = B_1 )
=> ( ( ord_less_eq_a_o @ B_1 @ C_1 )
=> ( ord_less_eq_a_o @ A_3 @ C_1 ) ) ) ).
thf(fact_540_order__antisym__conv,axiom,
! [Y_4: $o,X_5: $o] :
( ( ord_less_eq_o @ Y_4 @ X_5 )
=> ( ( ord_less_eq_o @ X_5 @ Y_4 )
<=> ( X_5
<=> Y_4 ) ) ) ).
thf(fact_541_order__antisym__conv,axiom,
! [Y_4: nat > $o,X_5: nat > $o] :
( ( ord_less_eq_nat_o @ Y_4 @ X_5 )
=> ( ( ord_less_eq_nat_o @ X_5 @ Y_4 )
<=> ( X_5 = Y_4 ) ) ) ).
thf(fact_542_order__antisym__conv,axiom,
! [Y_4: pname > $o,X_5: pname > $o] :
( ( ord_less_eq_pname_o @ Y_4 @ X_5 )
=> ( ( ord_less_eq_pname_o @ X_5 @ Y_4 )
<=> ( X_5 = Y_4 ) ) ) ).
thf(fact_543_order__antisym__conv,axiom,
! [Y_4: nat,X_5: nat] :
( ( ord_less_eq_nat @ Y_4 @ X_5 )
=> ( ( ord_less_eq_nat @ X_5 @ Y_4 )
<=> ( X_5 = Y_4 ) ) ) ).
thf(fact_544_order__antisym__conv,axiom,
! [Y_4: x_a > $o,X_5: x_a > $o] :
( ( ord_less_eq_a_o @ Y_4 @ X_5 )
=> ( ( ord_less_eq_a_o @ X_5 @ Y_4 )
<=> ( X_5 = Y_4 ) ) ) ).
thf(fact_545_order__eq__refl,axiom,
! [Y_3: $o,X_4: $o] :
( ( X_4
<=> Y_3 )
=> ( ord_less_eq_o @ X_4 @ Y_3 ) ) ).
thf(fact_546_order__eq__refl,axiom,
! [X_4: nat > $o,Y_3: nat > $o] :
( ( X_4 = Y_3 )
=> ( ord_less_eq_nat_o @ X_4 @ Y_3 ) ) ).
thf(fact_547_order__eq__refl,axiom,
! [X_4: pname > $o,Y_3: pname > $o] :
( ( X_4 = Y_3 )
=> ( ord_less_eq_pname_o @ X_4 @ Y_3 ) ) ).
thf(fact_548_order__eq__refl,axiom,
! [X_4: nat,Y_3: nat] :
( ( X_4 = Y_3 )
=> ( ord_less_eq_nat @ X_4 @ Y_3 ) ) ).
thf(fact_549_order__eq__refl,axiom,
! [X_4: x_a > $o,Y_3: x_a > $o] :
( ( X_4 = Y_3 )
=> ( ord_less_eq_a_o @ X_4 @ Y_3 ) ) ).
thf(fact_550_order__eq__iff,axiom,
! [Y_2: $o,X_3: $o] :
( ( X_3
<=> Y_2 )
<=> ( ( ord_less_eq_o @ X_3 @ Y_2 )
& ( ord_less_eq_o @ Y_2 @ X_3 ) ) ) ).
thf(fact_551_order__eq__iff,axiom,
! [X_3: nat > $o,Y_2: nat > $o] :
( ( X_3 = Y_2 )
<=> ( ( ord_less_eq_nat_o @ X_3 @ Y_2 )
& ( ord_less_eq_nat_o @ Y_2 @ X_3 ) ) ) ).
thf(fact_552_order__eq__iff,axiom,
! [X_3: pname > $o,Y_2: pname > $o] :
( ( X_3 = Y_2 )
<=> ( ( ord_less_eq_pname_o @ X_3 @ Y_2 )
& ( ord_less_eq_pname_o @ Y_2 @ X_3 ) ) ) ).
thf(fact_553_order__eq__iff,axiom,
! [X_3: nat,Y_2: nat] :
( ( X_3 = Y_2 )
<=> ( ( ord_less_eq_nat @ X_3 @ Y_2 )
& ( ord_less_eq_nat @ Y_2 @ X_3 ) ) ) ).
thf(fact_554_order__eq__iff,axiom,
! [X_3: x_a > $o,Y_2: x_a > $o] :
( ( X_3 = Y_2 )
<=> ( ( ord_less_eq_a_o @ X_3 @ Y_2 )
& ( ord_less_eq_a_o @ Y_2 @ X_3 ) ) ) ).
thf(fact_555_linorder__linear,axiom,
! [X_2: nat,Y_1: nat] :
( ( ord_less_eq_nat @ X_2 @ Y_1 )
| ( ord_less_eq_nat @ Y_1 @ X_2 ) ) ).
thf(fact_556_finite__subset__induct,axiom,
! [P_2: ( nat > $o ) > $o,A_1: nat > $o,F_3: nat > $o] :
( ( finite_finite_nat @ F_3 )
=> ( ( ord_less_eq_nat_o @ F_3 @ A_1 )
=> ( ( P_2 @ bot_bot_nat_o )
=> ( ! [A_2: nat,F_2: nat > $o] :
( ( finite_finite_nat @ F_2 )
=> ( ( member_nat @ A_2 @ A_1 )
=> ( ~ ( member_nat @ A_2 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_nat @ A_2 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_3 ) ) ) ) ) ).
thf(fact_557_finite__subset__induct,axiom,
! [P_2: ( pname > $o ) > $o,A_1: pname > $o,F_3: pname > $o] :
( ( finite_finite_pname @ F_3 )
=> ( ( ord_less_eq_pname_o @ F_3 @ A_1 )
=> ( ( P_2 @ bot_bot_pname_o )
=> ( ! [A_2: pname,F_2: pname > $o] :
( ( finite_finite_pname @ F_2 )
=> ( ( member_pname @ A_2 @ A_1 )
=> ( ~ ( member_pname @ A_2 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_pname @ A_2 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_3 ) ) ) ) ) ).
thf(fact_558_finite__subset__induct,axiom,
! [P_2: ( x_a > $o ) > $o,A_1: x_a > $o,F_3: x_a > $o] :
( ( finite_finite_a @ F_3 )
=> ( ( ord_less_eq_a_o @ F_3 @ A_1 )
=> ( ( P_2 @ bot_bot_a_o )
=> ( ! [A_2: x_a,F_2: x_a > $o] :
( ( finite_finite_a @ F_2 )
=> ( ( member_a @ A_2 @ A_1 )
=> ( ~ ( member_a @ A_2 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_a @ A_2 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_3 ) ) ) ) ) ).
thf(fact_559_finite__subset__induct,axiom,
! [P_2: ( ( nat > $o ) > $o ) > $o,A_1: ( nat > $o ) > $o,F_3: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_3 )
=> ( ( ord_less_eq_nat_o_o @ F_3 @ A_1 )
=> ( ( P_2 @ bot_bot_nat_o_o )
=> ( ! [A_2: nat > $o,F_2: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_2 )
=> ( ( member_nat_o @ A_2 @ A_1 )
=> ( ~ ( member_nat_o @ A_2 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_nat_o @ A_2 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_3 ) ) ) ) ) ).
thf(fact_560_finite__subset__induct,axiom,
! [P_2: ( ( pname > $o ) > $o ) > $o,A_1: ( pname > $o ) > $o,F_3: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_3 )
=> ( ( ord_le1205211808me_o_o @ F_3 @ A_1 )
=> ( ( P_2 @ bot_bot_pname_o_o )
=> ( ! [A_2: pname > $o,F_2: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_2 )
=> ( ( member_pname_o @ A_2 @ A_1 )
=> ( ~ ( member_pname_o @ A_2 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_pname_o @ A_2 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_3 ) ) ) ) ) ).
thf(fact_561_finite__subset__induct,axiom,
! [P_2: ( ( x_a > $o ) > $o ) > $o,A_1: ( x_a > $o ) > $o,F_3: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_3 )
=> ( ( ord_less_eq_a_o_o @ F_3 @ A_1 )
=> ( ( P_2 @ bot_bot_a_o_o )
=> ( ! [A_2: x_a > $o,F_2: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_2 )
=> ( ( member_a_o @ A_2 @ A_1 )
=> ( ~ ( member_a_o @ A_2 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_a_o @ A_2 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_3 ) ) ) ) ) ).
thf(fact_562_assms_I2_J,axiom,
! [Pn: pname,G: x_a > $o] :
( ( p @ ( insert_a @ ( mgt_call @ Pn ) @ G ) @ ( insert_a @ ( mgt @ ( the_com @ ( body @ Pn ) ) ) @ bot_bot_a_o ) )
=> ( p @ G @ ( insert_a @ ( mgt_call @ Pn ) @ bot_bot_a_o ) ) ) ).
thf(fact_563_finite__induct,axiom,
! [P_1: ( ( nat > $o ) > $o ) > $o,F_1: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_1 )
=> ( ( P_1 @ bot_bot_nat_o_o )
=> ( ! [X_1: nat > $o,F_2: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_2 )
=> ( ~ ( member_nat_o @ X_1 @ F_2 )
=> ( ( P_1 @ F_2 )
=> ( P_1 @ ( insert_nat_o @ X_1 @ F_2 ) ) ) ) )
=> ( P_1 @ F_1 ) ) ) ) ).
thf(fact_564_finite__induct,axiom,
! [P_1: ( ( pname > $o ) > $o ) > $o,F_1: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_1 )
=> ( ( P_1 @ bot_bot_pname_o_o )
=> ( ! [X_1: pname > $o,F_2: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_2 )
=> ( ~ ( member_pname_o @ X_1 @ F_2 )
=> ( ( P_1 @ F_2 )
=> ( P_1 @ ( insert_pname_o @ X_1 @ F_2 ) ) ) ) )
=> ( P_1 @ F_1 ) ) ) ) ).
thf(fact_565_finite__induct,axiom,
! [P_1: ( ( x_a > $o ) > $o ) > $o,F_1: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_1 )
=> ( ( P_1 @ bot_bot_a_o_o )
=> ( ! [X_1: x_a > $o,F_2: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_2 )
=> ( ~ ( member_a_o @ X_1 @ F_2 )
=> ( ( P_1 @ F_2 )
=> ( P_1 @ ( insert_a_o @ X_1 @ F_2 ) ) ) ) )
=> ( P_1 @ F_1 ) ) ) ) ).
thf(fact_566_finite__less__ub,axiom,
! [U: nat,F: nat > nat] :
( ! [N_2: nat] : ( ord_less_eq_nat @ N_2 @ ( F @ N_2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N_2: nat] : ( ord_less_eq_nat @ ( F @ N_2 ) @ U ) ) ) ) ).
thf(fact_567_assms_I4_J,axiom,
! [Pn: pname] :
( ( member_pname @ Pn @ u )
=> ( wt @ ( the_com @ ( body @ Pn ) ) ) ) ).
thf(fact_568_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
thf(fact_569_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
thf(fact_570_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( nat_case_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
thf(fact_571_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_572_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_573_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_574_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus_nat @ M @ K )
= ( plus_plus_nat @ N @ K ) )
<=> ( M = N ) ) ).
thf(fact_575_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus_nat @ K @ M )
= ( plus_plus_nat @ K @ N ) )
<=> ( M = N ) ) ).
thf(fact_576_nat__add__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K ) ) ) ).
thf(fact_577_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z ) ) ) ).
thf(fact_578_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_579_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_580_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_581_diff__diff__left,axiom,
! [I: nat,J_1: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J_1 ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J_1 @ K ) ) ) ).
thf(fact_582_diff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_583_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_584_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_585_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_586_le__iff__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ? [K_1: nat] :
( N
= ( plus_plus_nat @ M @ K_1 ) ) ) ).
thf(fact_587_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_588_trans__le__add1,axiom,
! [M: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J_1 @ M ) ) ) ).
thf(fact_589_trans__le__add2,axiom,
! [M: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J_1 ) ) ) ).
thf(fact_590_add__le__mono1,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J_1 @ K ) ) ) ).
thf(fact_591_add__le__mono,axiom,
! [K: nat,L: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J_1 @ L ) ) ) ) ).
thf(fact_592_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
thf(fact_593_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_594_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
thf(fact_595_diff__add__assoc2,axiom,
! [I: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J_1 @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J_1 @ K ) @ I ) ) ) ).
thf(fact_596_add__diff__assoc2,axiom,
! [I: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J_1 @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J_1 @ I ) @ K ) ) ) ).
thf(fact_597_diff__add__assoc,axiom,
! [I: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J_1 ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J_1 @ K ) ) ) ) ).
thf(fact_598_le__imp__diff__is__add,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ( ( minus_minus_nat @ J_1 @ I )
= K )
<=> ( J_1
= ( plus_plus_nat @ K @ I ) ) ) ) ).
thf(fact_599_le__add__diff__inverse2,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N ) @ N )
= M ) ) ).
thf(fact_600_le__diff__conv2,axiom,
! [I: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J_1 @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J_1 ) ) ) ).
thf(fact_601_add__diff__assoc,axiom,
! [I: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J_1 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J_1 ) @ K ) ) ) ).
thf(fact_602_le__add__diff__inverse,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_603_le__add__diff,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) ) ) ).
thf(fact_604_le__diff__conv,axiom,
! [J_1: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J_1 @ K ) @ I )
<=> ( ord_less_eq_nat @ J_1 @ ( plus_plus_nat @ I @ K ) ) ) ).
thf(fact_605_diff__diff__right,axiom,
! [I: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J_1 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J_1 ) ) ) ).
thf(fact_606_Suc__eq__plus1,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ N @ one_one_nat ) ) ).
thf(fact_607_Suc__eq__plus1__left,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ one_one_nat @ N ) ) ).
thf(fact_608_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J_1 @ K ) ) @ M )
= ( minus_minus_nat @ ( suc @ J_1 ) @ ( plus_plus_nat @ K @ M ) ) ) ) ).
thf(fact_609_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J_1: nat] :
( ( ord_less_eq_nat @ K @ J_1 )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J_1 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( suc @ J_1 ) ) ) ) ).
thf(fact_610_termination__basic__simps_I3_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_611_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_612_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
thf(fact_613_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
thf(fact_614_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_2: nat] : ( ord_less_nat @ N_2 @ K ) ) ) ).
thf(fact_615_termination__basic__simps_I1_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_616_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_nat @ X @ Z )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
thf(fact_617_add__lessD1,axiom,
! [I: nat,J_1: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J_1 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
thf(fact_618_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_619_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J_1 @ L ) ) ) ) ).
thf(fact_620_add__less__mono1,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J_1 @ K ) ) ) ).
thf(fact_621_trans__less__add2,axiom,
! [M: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J_1 ) ) ) ).
thf(fact_622_trans__less__add1,axiom,
! [M: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J_1 @ M ) ) ) ).
thf(fact_623_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_624_not__add__less2,axiom,
! [J_1: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J_1 @ I ) @ I ) ).
thf(fact_625_not__add__less1,axiom,
! [I: nat,J_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J_1 ) @ I ) ).
thf(fact_626_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_627_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_628_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
thf(fact_629_less__trans__Suc,axiom,
! [K: nat,I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ( ord_less_nat @ J_1 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
thf(fact_630_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
thf(fact_631_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_632_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
thf(fact_633_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
<=> ( N = M ) ) ) ).
thf(fact_634_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_635_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_636_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
<=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
thf(fact_637_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_638_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_639_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_640_le__eq__less__or__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_641_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_642_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_643_less__imp__diff__less,axiom,
! [N: nat,J_1: nat,K: nat] :
( ( ord_less_nat @ J_1 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J_1 @ N ) @ K ) ) ).
thf(fact_644_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_645_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_646_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_647_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_648_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_649_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_650_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_651_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( P @ N @ M ) )
=> ( ( ( M = N )
=> ( P @ N @ M ) )
=> ( ( ( ord_less_nat @ N @ M )
=> ( P @ N @ M ) )
=> ( P @ N @ M ) ) ) ) ).
thf(fact_652_finite__nat__set__iff__bounded,axiom,
! [N_1: nat > $o] :
( ( finite_finite_nat @ N_1 )
<=> ? [M_1: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_1 )
=> ( ord_less_nat @ X_1 @ M_1 ) ) ) ).
thf(fact_653_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_nat @ I_1 @ N ) ) )
= N ) ).
thf(fact_654_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( P @ K_1 ) @ ( ord_less_nat @ K_1 @ I ) ) ) ) ).
thf(fact_655_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
thf(fact_656_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
thf(fact_657_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ? [K_1: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_658_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_659_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_660_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_661_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_662_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_663_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
<=> ( N = M ) ) ) ).
thf(fact_664_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_665_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
thf(fact_666_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_667_less__diff__conv,axiom,
! [I: nat,J_1: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J_1 @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J_1 ) ) ).
thf(fact_668_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
thf(fact_669_less__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_nat @ M @ N ) ) ) ) ).
thf(fact_670_less__eq__Suc__le__raw,axiom,
! [X_1: nat] :
( ( ord_less_nat @ X_1 )
= ( ord_less_eq_nat @ ( suc @ X_1 ) ) ) ).
thf(fact_671_mono__nat__linear__lb,axiom,
! [M: nat,K: nat,F: nat > nat] :
( ! [M_1: nat,N_2: nat] :
( ( ord_less_nat @ M_1 @ N_2 )
=> ( ord_less_nat @ ( F @ M_1 ) @ ( F @ N_2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_672_inc__induct,axiom,
! [P: nat > $o,I: nat,J_1: nat] :
( ( ord_less_eq_nat @ I @ J_1 )
=> ( ( P @ J_1 )
=> ( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ( P @ ( suc @ I_1 ) )
=> ( P @ I_1 ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_673_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K_1: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_674_bounded__nat__set__is__finite,axiom,
! [N: nat,N_1: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ N_1 )
=> ( ord_less_nat @ X_1 @ N ) )
=> ( finite_finite_nat @ N_1 ) ) ).
thf(fact_675_less__mono__imp__le__mono,axiom,
! [I: nat,J_1: nat,F: nat > nat] :
( ! [I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ ( F @ I_1 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J_1 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J_1 ) ) ) ) ).
thf(fact_676_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
thf(fact_677_lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
thf(fact_678_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_679_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_680_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
thf(fact_681_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_682_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_683_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_684_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_685_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_686_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_687_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
thf(fact_688_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_689_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_690_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_691_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_692_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_693_nat_Osimps_I3_J,axiom,
! [Nat_1: nat] :
( ( suc @ Nat_1 )
!= zero_zero_nat ) ).
thf(fact_694_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_695_nat_Osimps_I2_J,axiom,
! [Nat: nat] :
( zero_zero_nat
!= ( suc @ Nat ) ) ).
thf(fact_696_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_697_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
thf(fact_698_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_699_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
%----Helper facts (12)
thf(help_fequal_1_1_fequal_000t__a_T,axiom,
! [X: x_a,Y: x_a] :
( ~ ( fequal_a @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000t__a_T,axiom,
! [X: x_a,Y: x_a] :
( ( X != Y )
| ( fequal_a @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ ( fequal_nat @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| ( fequal_nat @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
! [X: pname,Y: pname] :
( ~ ( fequal_pname @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
! [X: pname,Y: pname] :
( ( X != Y )
| ( fequal_pname @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000_062_It__a_M_Eo_J_T,axiom,
! [X: x_a > $o,Y: x_a > $o] :
( ~ ( fequal_a_o @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000_062_It__a_M_Eo_J_T,axiom,
! [X: x_a > $o,Y: x_a > $o] :
( ( X != Y )
| ( fequal_a_o @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T,axiom,
! [X: nat > $o,Y: nat > $o] :
( ~ ( fequal_nat_o @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( X != Y )
| ( fequal_nat_o @ X @ Y ) ) ).
thf(help_fequal_1_1_fequal_000_062_Itc__Com__Opname_M_Eo_J_T,axiom,
! [X: pname > $o,Y: pname > $o] :
( ~ ( fequal_pname_o @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000_062_Itc__Com__Opname_M_Eo_J_T,axiom,
! [X: pname > $o,Y: pname > $o] :
( ( X != Y )
| ( fequal_pname_o @ X @ Y ) ) ).
%----Conjectures (7)
thf(conj_0,hypothesis,
finite_finite_pname @ u ).
thf(conj_1,hypothesis,
ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ) ).
thf(conj_2,hypothesis,
ord_less_eq_nat @ ( suc @ na ) @ ( finite_card_a @ ( image_pname_a @ mgt_call @ u ) ) ).
thf(conj_3,hypothesis,
( ( finite_card_a @ g )
= ( minus_minus_nat @ ( finite_card_a @ ( image_pname_a @ mgt_call @ u ) ) @ ( suc @ na ) ) ) ).
thf(conj_4,hypothesis,
member_pname @ pn @ u ).
thf(conj_5,hypothesis,
~ ( member_a @ ( mgt_call @ pn ) @ g ) ).
thf(conj_6,conjecture,
ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ) ).
%------------------------------------------------------------------------------