TPTP Problem File: SWW473^3.p
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%------------------------------------------------------------------------------
% File : SWW473^3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 383, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_1000_thf_l383 [Bla11]
% Status : ContradictoryAxioms
% Rating : 0.50 v8.2.0, 0.46 v8.1.0, 0.45 v7.5.0, 0.43 v7.4.0, 0.33 v7.3.0, 0.56 v7.2.0, 0.38 v7.1.0, 0.62 v7.0.0, 0.71 v6.4.0, 0.83 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 : 1390 ( 234 unt; 175 typ; 0 def)
% Number of atoms : 4436 ( 862 equ; 70 cnn)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 10367 ( 301 ~; 68 |; 270 &;8078 @)
% ( 253 <=>;1397 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 2454 (2454 >; 0 *; 0 +; 0 <<)
% Number of symbols : 176 ( 169 usr; 15 con; 0-3 aty)
% Number of variables : 2984 ( 93 ^;2786 !; 105 ?;2984 :)
% SPC : TH0_CAX_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:59:45
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
thf(ty_ty_t__a,type,
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thf(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,
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%----Explicit typings (169)
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thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Int__Oint_M_Eo_J,type,
image_pname_int_o: ( pname > int > $o ) > ( pname > $o ) > ( int > $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__Int__Oint,type,
image_pname_int: ( pname > int ) > ( pname > $o ) > int > $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__Int__Oint_000_062_It__a_M_Eo_J,type,
image_int_a_o: ( int > x_a > $o ) > ( int > $o ) > ( x_a > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_Itc__Com__Opname_M_Eo_J,type,
image_int_pname_o: ( int > pname > $o ) > ( int > $o ) > ( pname > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_Itc__Int__Oint_M_Eo_J,type,
image_int_int_o: ( int > int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_Itc__Nat__Onat_M_Eo_J,type,
image_int_nat_o: ( int > nat > $o ) > ( int > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000t__a,type,
image_int_a: ( int > x_a ) > ( int > $o ) > x_a > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Com__Opname,type,
image_int_pname: ( int > pname ) > ( int > $o ) > pname > $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__Int__Oint_M_Eo_J,type,
image_nat_int_o: ( nat > int > $o ) > ( nat > $o ) > ( int > $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__Int__Oint,type,
image_nat_int: ( nat > int ) > ( nat > $o ) > int > $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__Int__Oint_M_Eo_J,type,
insert_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > ( int > $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__Int__Oint,type,
insert_int: int > ( int > $o ) > int > $o ).
thf(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
insert_nat: nat > ( nat > $o ) > nat > $o ).
thf(sy_c_Set_Othe__elem_000t__a,type,
the_elem_a: ( x_a > $o ) > x_a ).
thf(sy_c_Set_Othe__elem_000tc__Int__Oint,type,
the_elem_int: ( int > $o ) > int ).
thf(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
the_elem_nat: ( nat > $o ) > nat ).
thf(sy_c_fequal_000t__a,type,
fequal_a: x_a > x_a > $o ).
thf(sy_c_fequal_000tc__Int__Oint,type,
fequal_int: int > int > $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__Int__Oint_M_Eo_J,type,
member_int_o: ( int > $o ) > ( ( int > $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__Int__Oint,type,
member_int: int > ( int > $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 (1199)
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_161: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_161 )
=> ( finite229719499nt_o_o
@ ( collect_int_o_o
@ ^ [B_26: ( int > $o ) > $o] : ( ord_less_eq_int_o_o @ B_26 @ A_161 ) ) ) ) ).
thf(fact_2_finite__Collect__subsets,axiom,
! [A_161: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_161 )
=> ( finite1676163439at_o_o
@ ( collect_nat_o_o
@ ^ [B_26: ( nat > $o ) > $o] : ( ord_less_eq_nat_o_o @ B_26 @ A_161 ) ) ) ) ).
thf(fact_3_finite__Collect__subsets,axiom,
! [A_161: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_161 )
=> ( finite1066544169me_o_o
@ ( collect_pname_o_o
@ ^ [B_26: ( pname > $o ) > $o] : ( ord_le1205211808me_o_o @ B_26 @ A_161 ) ) ) ) ).
thf(fact_4_finite__Collect__subsets,axiom,
! [A_161: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_161 )
=> ( finite_finite_a_o_o
@ ( collect_a_o_o
@ ^ [B_26: ( x_a > $o ) > $o] : ( ord_less_eq_a_o_o @ B_26 @ A_161 ) ) ) ) ).
thf(fact_5_finite__Collect__subsets,axiom,
! [A_161: x_a > $o] :
( ( finite_finite_a @ A_161 )
=> ( finite_finite_a_o
@ ( collect_a_o
@ ^ [B_26: x_a > $o] : ( ord_less_eq_a_o @ B_26 @ A_161 ) ) ) ) ).
thf(fact_6_finite__Collect__subsets,axiom,
! [A_161: pname > $o] :
( ( finite_finite_pname @ A_161 )
=> ( finite297249702name_o
@ ( collect_pname_o
@ ^ [B_26: pname > $o] : ( ord_less_eq_pname_o @ B_26 @ A_161 ) ) ) ) ).
thf(fact_7_finite__Collect__subsets,axiom,
! [A_161: nat > $o] :
( ( finite_finite_nat @ A_161 )
=> ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [B_26: nat > $o] : ( ord_less_eq_nat_o @ B_26 @ A_161 ) ) ) ) ).
thf(fact_8_finite__Collect__subsets,axiom,
! [A_161: int > $o] :
( ( finite_finite_int @ A_161 )
=> ( finite_finite_int_o
@ ( collect_int_o
@ ^ [B_26: int > $o] : ( ord_less_eq_int_o @ B_26 @ A_161 ) ) ) ) ).
thf(fact_9_finite__imageI,axiom,
! [H: pname > int > $o,F_42: pname > $o] :
( ( finite_finite_pname @ F_42 )
=> ( finite_finite_int_o @ ( image_pname_int_o @ H @ F_42 ) ) ) ).
thf(fact_10_finite__imageI,axiom,
! [H: pname > nat > $o,F_42: pname > $o] :
( ( finite_finite_pname @ F_42 )
=> ( finite_finite_nat_o @ ( image_pname_nat_o @ H @ F_42 ) ) ) ).
thf(fact_11_finite__imageI,axiom,
! [H: pname > pname > $o,F_42: pname > $o] :
( ( finite_finite_pname @ F_42 )
=> ( finite297249702name_o @ ( image_pname_pname_o @ H @ F_42 ) ) ) ).
thf(fact_12_finite__imageI,axiom,
! [H: pname > x_a > $o,F_42: pname > $o] :
( ( finite_finite_pname @ F_42 )
=> ( finite_finite_a_o @ ( image_pname_a_o @ H @ F_42 ) ) ) ).
thf(fact_13_finite__imageI,axiom,
! [H: nat > x_a,F_42: nat > $o] :
( ( finite_finite_nat @ F_42 )
=> ( finite_finite_a @ ( image_nat_a @ H @ F_42 ) ) ) ).
thf(fact_14_finite__imageI,axiom,
! [H: nat > int > $o,F_42: nat > $o] :
( ( finite_finite_nat @ F_42 )
=> ( finite_finite_int_o @ ( image_nat_int_o @ H @ F_42 ) ) ) ).
thf(fact_15_finite__imageI,axiom,
! [H: nat > nat > $o,F_42: nat > $o] :
( ( finite_finite_nat @ F_42 )
=> ( finite_finite_nat_o @ ( image_nat_nat_o @ H @ F_42 ) ) ) ).
thf(fact_16_finite__imageI,axiom,
! [H: nat > pname > $o,F_42: nat > $o] :
( ( finite_finite_nat @ F_42 )
=> ( finite297249702name_o @ ( image_nat_pname_o @ H @ F_42 ) ) ) ).
thf(fact_17_finite__imageI,axiom,
! [H: nat > x_a > $o,F_42: nat > $o] :
( ( finite_finite_nat @ F_42 )
=> ( finite_finite_a_o @ ( image_nat_a_o @ H @ F_42 ) ) ) ).
thf(fact_18_finite__imageI,axiom,
! [H: int > x_a,F_42: int > $o] :
( ( finite_finite_int @ F_42 )
=> ( finite_finite_a @ ( image_int_a @ H @ F_42 ) ) ) ).
thf(fact_19_finite__imageI,axiom,
! [H: int > int > $o,F_42: int > $o] :
( ( finite_finite_int @ F_42 )
=> ( finite_finite_int_o @ ( image_int_int_o @ H @ F_42 ) ) ) ).
thf(fact_20_finite__imageI,axiom,
! [H: int > nat > $o,F_42: int > $o] :
( ( finite_finite_int @ F_42 )
=> ( finite_finite_nat_o @ ( image_int_nat_o @ H @ F_42 ) ) ) ).
thf(fact_21_finite__imageI,axiom,
! [H: int > pname > $o,F_42: int > $o] :
( ( finite_finite_int @ F_42 )
=> ( finite297249702name_o @ ( image_int_pname_o @ H @ F_42 ) ) ) ).
thf(fact_22_finite__imageI,axiom,
! [H: int > x_a > $o,F_42: int > $o] :
( ( finite_finite_int @ F_42 )
=> ( finite_finite_a_o @ ( image_int_a_o @ H @ F_42 ) ) ) ).
thf(fact_23_finite__imageI,axiom,
! [H: x_a > pname,F_42: x_a > $o] :
( ( finite_finite_a @ F_42 )
=> ( finite_finite_pname @ ( image_a_pname @ H @ F_42 ) ) ) ).
thf(fact_24_finite__imageI,axiom,
! [H: ( int > $o ) > pname,F_42: ( int > $o ) > $o] :
( ( finite_finite_int_o @ F_42 )
=> ( finite_finite_pname @ ( image_int_o_pname @ H @ F_42 ) ) ) ).
thf(fact_25_finite__imageI,axiom,
! [H: ( nat > $o ) > pname,F_42: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_42 )
=> ( finite_finite_pname @ ( image_nat_o_pname @ H @ F_42 ) ) ) ).
thf(fact_26_finite__imageI,axiom,
! [H: ( pname > $o ) > pname,F_42: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_42 )
=> ( finite_finite_pname @ ( image_pname_o_pname @ H @ F_42 ) ) ) ).
thf(fact_27_finite__imageI,axiom,
! [H: ( x_a > $o ) > pname,F_42: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_42 )
=> ( finite_finite_pname @ ( image_a_o_pname @ H @ F_42 ) ) ) ).
thf(fact_28_finite__imageI,axiom,
! [H: x_a > nat,F_42: x_a > $o] :
( ( finite_finite_a @ F_42 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F_42 ) ) ) ).
thf(fact_29_finite__imageI,axiom,
! [H: ( int > $o ) > nat,F_42: ( int > $o ) > $o] :
( ( finite_finite_int_o @ F_42 )
=> ( finite_finite_nat @ ( image_int_o_nat @ H @ F_42 ) ) ) ).
thf(fact_30_finite__imageI,axiom,
! [H: ( nat > $o ) > nat,F_42: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_42 )
=> ( finite_finite_nat @ ( image_nat_o_nat @ H @ F_42 ) ) ) ).
thf(fact_31_finite__imageI,axiom,
! [H: ( pname > $o ) > nat,F_42: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_42 )
=> ( finite_finite_nat @ ( image_pname_o_nat @ H @ F_42 ) ) ) ).
thf(fact_32_finite__imageI,axiom,
! [H: ( x_a > $o ) > nat,F_42: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_42 )
=> ( finite_finite_nat @ ( image_a_o_nat @ H @ F_42 ) ) ) ).
thf(fact_33_finite__imageI,axiom,
! [H: x_a > int,F_42: x_a > $o] :
( ( finite_finite_a @ F_42 )
=> ( finite_finite_int @ ( image_a_int @ H @ F_42 ) ) ) ).
thf(fact_34_finite__imageI,axiom,
! [H: ( int > $o ) > int,F_42: ( int > $o ) > $o] :
( ( finite_finite_int_o @ F_42 )
=> ( finite_finite_int @ ( image_int_o_int @ H @ F_42 ) ) ) ).
thf(fact_35_finite__imageI,axiom,
! [H: ( nat > $o ) > int,F_42: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ F_42 )
=> ( finite_finite_int @ ( image_nat_o_int @ H @ F_42 ) ) ) ).
thf(fact_36_finite__imageI,axiom,
! [H: ( pname > $o ) > int,F_42: ( pname > $o ) > $o] :
( ( finite297249702name_o @ F_42 )
=> ( finite_finite_int @ ( image_pname_o_int @ H @ F_42 ) ) ) ).
thf(fact_37_finite__imageI,axiom,
! [H: ( x_a > $o ) > int,F_42: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ F_42 )
=> ( finite_finite_int @ ( image_a_o_int @ H @ F_42 ) ) ) ).
thf(fact_38_finite__imageI,axiom,
! [H: pname > x_a,F_42: pname > $o] :
( ( finite_finite_pname @ F_42 )
=> ( finite_finite_a @ ( image_pname_a @ H @ F_42 ) ) ) ).
thf(fact_39_finite__imageI,axiom,
! [H: nat > int,F_42: nat > $o] :
( ( finite_finite_nat @ F_42 )
=> ( finite_finite_int @ ( image_nat_int @ H @ F_42 ) ) ) ).
thf(fact_40_finite_OinsertI,axiom,
! [A_160: int > $o,A_159: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_159 )
=> ( finite_finite_int_o @ ( insert_int_o @ A_160 @ A_159 ) ) ) ).
thf(fact_41_finite_OinsertI,axiom,
! [A_160: nat > $o,A_159: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_159 )
=> ( finite_finite_nat_o @ ( insert_nat_o @ A_160 @ A_159 ) ) ) ).
thf(fact_42_finite_OinsertI,axiom,
! [A_160: pname > $o,A_159: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_159 )
=> ( finite297249702name_o @ ( insert_pname_o @ A_160 @ A_159 ) ) ) ).
thf(fact_43_finite_OinsertI,axiom,
! [A_160: x_a > $o,A_159: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_159 )
=> ( finite_finite_a_o @ ( insert_a_o @ A_160 @ A_159 ) ) ) ).
thf(fact_44_finite_OinsertI,axiom,
! [A_160: pname,A_159: pname > $o] :
( ( finite_finite_pname @ A_159 )
=> ( finite_finite_pname @ ( insert_pname @ A_160 @ A_159 ) ) ) ).
thf(fact_45_finite_OinsertI,axiom,
! [A_160: nat,A_159: nat > $o] :
( ( finite_finite_nat @ A_159 )
=> ( finite_finite_nat @ ( insert_nat @ A_160 @ A_159 ) ) ) ).
thf(fact_46_finite_OinsertI,axiom,
! [A_160: int,A_159: int > $o] :
( ( finite_finite_int @ A_159 )
=> ( finite_finite_int @ ( insert_int @ A_160 @ A_159 ) ) ) ).
thf(fact_47_finite_OinsertI,axiom,
! [A_160: x_a,A_159: x_a > $o] :
( ( finite_finite_a @ A_159 )
=> ( finite_finite_a @ ( insert_a @ A_160 @ A_159 ) ) ) ).
thf(fact_48_card__image__le,axiom,
! [F_41: pname > pname,A_158: pname > $o] :
( ( finite_finite_pname @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ ( image_pname_pname @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
thf(fact_49_card__image__le,axiom,
! [F_41: x_a > x_a,A_158: x_a > $o] :
( ( finite_finite_a @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
thf(fact_50_card__image__le,axiom,
! [F_41: ( int > $o ) > x_a,A_158: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_int_o_a @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
thf(fact_51_card__image__le,axiom,
! [F_41: ( nat > $o ) > x_a,A_158: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_o_a @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
thf(fact_52_card__image__le,axiom,
! [F_41: ( pname > $o ) > x_a,A_158: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_pname_o_a @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
thf(fact_53_card__image__le,axiom,
! [F_41: ( x_a > $o ) > x_a,A_158: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_o_a @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
thf(fact_54_card__image__le,axiom,
! [F_41: pname > nat,A_158: pname > $o] :
( ( finite_finite_pname @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_pname_nat @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
thf(fact_55_card__image__le,axiom,
! [F_41: x_a > nat,A_158: x_a > $o] :
( ( finite_finite_a @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_nat @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
thf(fact_56_card__image__le,axiom,
! [F_41: ( int > $o ) > nat,A_158: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_int_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
thf(fact_57_card__image__le,axiom,
! [F_41: ( nat > $o ) > nat,A_158: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
thf(fact_58_card__image__le,axiom,
! [F_41: ( pname > $o ) > nat,A_158: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_pname_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
thf(fact_59_card__image__le,axiom,
! [F_41: ( x_a > $o ) > nat,A_158: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
thf(fact_60_card__image__le,axiom,
! [F_41: pname > int,A_158: pname > $o] :
( ( finite_finite_pname @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_pname_int @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
thf(fact_61_card__image__le,axiom,
! [F_41: x_a > int,A_158: x_a > $o] :
( ( finite_finite_a @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_a_int @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
thf(fact_62_card__image__le,axiom,
! [F_41: ( int > $o ) > int,A_158: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_int_o_int @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
thf(fact_63_card__image__le,axiom,
! [F_41: ( nat > $o ) > int,A_158: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_o_int @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
thf(fact_64_card__image__le,axiom,
! [F_41: ( pname > $o ) > int,A_158: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_pname_o_int @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
thf(fact_65_card__image__le,axiom,
! [F_41: ( x_a > $o ) > int,A_158: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_a_o_int @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
thf(fact_66_card__image__le,axiom,
! [F_41: x_a > pname,A_158: x_a > $o] :
( ( finite_finite_a @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ ( image_a_pname @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
thf(fact_67_card__image__le,axiom,
! [F_41: nat > pname,A_158: nat > $o] :
( ( finite_finite_nat @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ ( image_nat_pname @ F_41 @ A_158 ) ) @ ( finite_card_nat @ A_158 ) ) ) ).
thf(fact_68_card__image__le,axiom,
! [F_41: int > pname,A_158: int > $o] :
( ( finite_finite_int @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ ( image_int_pname @ F_41 @ A_158 ) ) @ ( finite_card_int @ A_158 ) ) ) ).
thf(fact_69_card__image__le,axiom,
! [F_41: pname > x_a,A_158: pname > $o] :
( ( finite_finite_pname @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_pname_a @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
thf(fact_70_card__image__le,axiom,
! [F_41: nat > int,A_158: nat > $o] :
( ( finite_finite_nat @ A_158 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_int @ F_41 @ A_158 ) ) @ ( finite_card_nat @ A_158 ) ) ) ).
thf(fact_71_card__mono,axiom,
! [A_157: ( int > $o ) > $o,B_89: ( int > $o ) > $o] :
( ( finite_finite_int_o @ B_89 )
=> ( ( ord_less_eq_int_o_o @ A_157 @ B_89 )
=> ( ord_less_eq_nat @ ( finite_card_int_o @ A_157 ) @ ( finite_card_int_o @ B_89 ) ) ) ) ).
thf(fact_72_card__mono,axiom,
! [A_157: ( nat > $o ) > $o,B_89: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_89 )
=> ( ( ord_less_eq_nat_o_o @ A_157 @ B_89 )
=> ( ord_less_eq_nat @ ( finite_card_nat_o @ A_157 ) @ ( finite_card_nat_o @ B_89 ) ) ) ) ).
thf(fact_73_card__mono,axiom,
! [A_157: ( pname > $o ) > $o,B_89: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_89 )
=> ( ( ord_le1205211808me_o_o @ A_157 @ B_89 )
=> ( ord_less_eq_nat @ ( finite_card_pname_o @ A_157 ) @ ( finite_card_pname_o @ B_89 ) ) ) ) ).
thf(fact_74_card__mono,axiom,
! [A_157: ( x_a > $o ) > $o,B_89: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_89 )
=> ( ( ord_less_eq_a_o_o @ A_157 @ B_89 )
=> ( ord_less_eq_nat @ ( finite_card_a_o @ A_157 ) @ ( finite_card_a_o @ B_89 ) ) ) ) ).
thf(fact_75_card__mono,axiom,
! [A_157: pname > $o,B_89: pname > $o] :
( ( finite_finite_pname @ B_89 )
=> ( ( ord_less_eq_pname_o @ A_157 @ B_89 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ A_157 ) @ ( finite_card_pname @ B_89 ) ) ) ) ).
thf(fact_76_card__mono,axiom,
! [A_157: x_a > $o,B_89: x_a > $o] :
( ( finite_finite_a @ B_89 )
=> ( ( ord_less_eq_a_o @ A_157 @ B_89 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A_157 ) @ ( finite_card_a @ B_89 ) ) ) ) ).
thf(fact_77_card__mono,axiom,
! [A_157: nat > $o,B_89: nat > $o] :
( ( finite_finite_nat @ B_89 )
=> ( ( ord_less_eq_nat_o @ A_157 @ B_89 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A_157 ) @ ( finite_card_nat @ B_89 ) ) ) ) ).
thf(fact_78_card__mono,axiom,
! [A_157: int > $o,B_89: int > $o] :
( ( finite_finite_int @ B_89 )
=> ( ( ord_less_eq_int_o @ A_157 @ B_89 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A_157 ) @ ( finite_card_int @ B_89 ) ) ) ) ).
thf(fact_79_card__seteq,axiom,
! [A_156: ( int > $o ) > $o,B_88: ( int > $o ) > $o] :
( ( finite_finite_int_o @ B_88 )
=> ( ( ord_less_eq_int_o_o @ A_156 @ B_88 )
=> ( ( ord_less_eq_nat @ ( finite_card_int_o @ B_88 ) @ ( finite_card_int_o @ A_156 ) )
=> ( A_156 = B_88 ) ) ) ) ).
thf(fact_80_card__seteq,axiom,
! [A_156: ( nat > $o ) > $o,B_88: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_88 )
=> ( ( ord_less_eq_nat_o_o @ A_156 @ B_88 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_o @ B_88 ) @ ( finite_card_nat_o @ A_156 ) )
=> ( A_156 = B_88 ) ) ) ) ).
thf(fact_81_card__seteq,axiom,
! [A_156: ( pname > $o ) > $o,B_88: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_88 )
=> ( ( ord_le1205211808me_o_o @ A_156 @ B_88 )
=> ( ( ord_less_eq_nat @ ( finite_card_pname_o @ B_88 ) @ ( finite_card_pname_o @ A_156 ) )
=> ( A_156 = B_88 ) ) ) ) ).
thf(fact_82_card__seteq,axiom,
! [A_156: ( x_a > $o ) > $o,B_88: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_88 )
=> ( ( ord_less_eq_a_o_o @ A_156 @ B_88 )
=> ( ( ord_less_eq_nat @ ( finite_card_a_o @ B_88 ) @ ( finite_card_a_o @ A_156 ) )
=> ( A_156 = B_88 ) ) ) ) ).
thf(fact_83_card__seteq,axiom,
! [A_156: pname > $o,B_88: pname > $o] :
( ( finite_finite_pname @ B_88 )
=> ( ( ord_less_eq_pname_o @ A_156 @ B_88 )
=> ( ( ord_less_eq_nat @ ( finite_card_pname @ B_88 ) @ ( finite_card_pname @ A_156 ) )
=> ( A_156 = B_88 ) ) ) ) ).
thf(fact_84_card__seteq,axiom,
! [A_156: x_a > $o,B_88: x_a > $o] :
( ( finite_finite_a @ B_88 )
=> ( ( ord_less_eq_a_o @ A_156 @ B_88 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B_88 ) @ ( finite_card_a @ A_156 ) )
=> ( A_156 = B_88 ) ) ) ) ).
thf(fact_85_card__seteq,axiom,
! [A_156: nat > $o,B_88: nat > $o] :
( ( finite_finite_nat @ B_88 )
=> ( ( ord_less_eq_nat_o @ A_156 @ B_88 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B_88 ) @ ( finite_card_nat @ A_156 ) )
=> ( A_156 = B_88 ) ) ) ) ).
thf(fact_86_card__seteq,axiom,
! [A_156: int > $o,B_88: int > $o] :
( ( finite_finite_int @ B_88 )
=> ( ( ord_less_eq_int_o @ A_156 @ B_88 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B_88 ) @ ( finite_card_int @ A_156 ) )
=> ( A_156 = B_88 ) ) ) ) ).
thf(fact_87_card__insert__le,axiom,
! [X_53: int > $o,A_155: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_155 )
=> ( ord_less_eq_nat @ ( finite_card_int_o @ A_155 ) @ ( finite_card_int_o @ ( insert_int_o @ X_53 @ A_155 ) ) ) ) ).
thf(fact_88_card__insert__le,axiom,
! [X_53: nat > $o,A_155: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_155 )
=> ( ord_less_eq_nat @ ( finite_card_nat_o @ A_155 ) @ ( finite_card_nat_o @ ( insert_nat_o @ X_53 @ A_155 ) ) ) ) ).
thf(fact_89_card__insert__le,axiom,
! [X_53: pname > $o,A_155: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_155 )
=> ( ord_less_eq_nat @ ( finite_card_pname_o @ A_155 ) @ ( finite_card_pname_o @ ( insert_pname_o @ X_53 @ A_155 ) ) ) ) ).
thf(fact_90_card__insert__le,axiom,
! [X_53: x_a > $o,A_155: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_155 )
=> ( ord_less_eq_nat @ ( finite_card_a_o @ A_155 ) @ ( finite_card_a_o @ ( insert_a_o @ X_53 @ A_155 ) ) ) ) ).
thf(fact_91_card__insert__le,axiom,
! [X_53: pname,A_155: pname > $o] :
( ( finite_finite_pname @ A_155 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ A_155 ) @ ( finite_card_pname @ ( insert_pname @ X_53 @ A_155 ) ) ) ) ).
thf(fact_92_card__insert__le,axiom,
! [X_53: nat,A_155: nat > $o] :
( ( finite_finite_nat @ A_155 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A_155 ) @ ( finite_card_nat @ ( insert_nat @ X_53 @ A_155 ) ) ) ) ).
thf(fact_93_card__insert__le,axiom,
! [X_53: int,A_155: int > $o] :
( ( finite_finite_int @ A_155 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A_155 ) @ ( finite_card_int @ ( insert_int @ X_53 @ A_155 ) ) ) ) ).
thf(fact_94_card__insert__le,axiom,
! [X_53: x_a,A_155: x_a > $o] :
( ( finite_finite_a @ A_155 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A_155 ) @ ( finite_card_a @ ( insert_a @ X_53 @ A_155 ) ) ) ) ).
thf(fact_95_card__insert__if,axiom,
! [X_52: int > $o,A_154: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_154 )
=> ( ( ( member_int_o @ X_52 @ A_154 )
=> ( ( finite_card_int_o @ ( insert_int_o @ X_52 @ A_154 ) )
= ( finite_card_int_o @ A_154 ) ) )
& ( ~ ( member_int_o @ X_52 @ A_154 )
=> ( ( finite_card_int_o @ ( insert_int_o @ X_52 @ A_154 ) )
= ( suc @ ( finite_card_int_o @ A_154 ) ) ) ) ) ) ).
thf(fact_96_card__insert__if,axiom,
! [X_52: nat > $o,A_154: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_154 )
=> ( ( ( member_nat_o @ X_52 @ A_154 )
=> ( ( finite_card_nat_o @ ( insert_nat_o @ X_52 @ A_154 ) )
= ( finite_card_nat_o @ A_154 ) ) )
& ( ~ ( member_nat_o @ X_52 @ A_154 )
=> ( ( finite_card_nat_o @ ( insert_nat_o @ X_52 @ A_154 ) )
= ( suc @ ( finite_card_nat_o @ A_154 ) ) ) ) ) ) ).
thf(fact_97_card__insert__if,axiom,
! [X_52: pname > $o,A_154: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_154 )
=> ( ( ( member_pname_o @ X_52 @ A_154 )
=> ( ( finite_card_pname_o @ ( insert_pname_o @ X_52 @ A_154 ) )
= ( finite_card_pname_o @ A_154 ) ) )
& ( ~ ( member_pname_o @ X_52 @ A_154 )
=> ( ( finite_card_pname_o @ ( insert_pname_o @ X_52 @ A_154 ) )
= ( suc @ ( finite_card_pname_o @ A_154 ) ) ) ) ) ) ).
thf(fact_98_card__insert__if,axiom,
! [X_52: x_a > $o,A_154: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_154 )
=> ( ( ( member_a_o @ X_52 @ A_154 )
=> ( ( finite_card_a_o @ ( insert_a_o @ X_52 @ A_154 ) )
= ( finite_card_a_o @ A_154 ) ) )
& ( ~ ( member_a_o @ X_52 @ A_154 )
=> ( ( finite_card_a_o @ ( insert_a_o @ X_52 @ A_154 ) )
= ( suc @ ( finite_card_a_o @ A_154 ) ) ) ) ) ) ).
thf(fact_99_card__insert__if,axiom,
! [X_52: pname,A_154: pname > $o] :
( ( finite_finite_pname @ A_154 )
=> ( ( ( member_pname @ X_52 @ A_154 )
=> ( ( finite_card_pname @ ( insert_pname @ X_52 @ A_154 ) )
= ( finite_card_pname @ A_154 ) ) )
& ( ~ ( member_pname @ X_52 @ A_154 )
=> ( ( finite_card_pname @ ( insert_pname @ X_52 @ A_154 ) )
= ( suc @ ( finite_card_pname @ A_154 ) ) ) ) ) ) ).
thf(fact_100_card__insert__if,axiom,
! [X_52: nat,A_154: nat > $o] :
( ( finite_finite_nat @ A_154 )
=> ( ( ( member_nat @ X_52 @ A_154 )
=> ( ( finite_card_nat @ ( insert_nat @ X_52 @ A_154 ) )
= ( finite_card_nat @ A_154 ) ) )
& ( ~ ( member_nat @ X_52 @ A_154 )
=> ( ( finite_card_nat @ ( insert_nat @ X_52 @ A_154 ) )
= ( suc @ ( finite_card_nat @ A_154 ) ) ) ) ) ) ).
thf(fact_101_card__insert__if,axiom,
! [X_52: int,A_154: int > $o] :
( ( finite_finite_int @ A_154 )
=> ( ( ( member_int @ X_52 @ A_154 )
=> ( ( finite_card_int @ ( insert_int @ X_52 @ A_154 ) )
= ( finite_card_int @ A_154 ) ) )
& ( ~ ( member_int @ X_52 @ A_154 )
=> ( ( finite_card_int @ ( insert_int @ X_52 @ A_154 ) )
= ( suc @ ( finite_card_int @ A_154 ) ) ) ) ) ) ).
thf(fact_102_card__insert__if,axiom,
! [X_52: x_a,A_154: x_a > $o] :
( ( finite_finite_a @ A_154 )
=> ( ( ( member_a @ X_52 @ A_154 )
=> ( ( finite_card_a @ ( insert_a @ X_52 @ A_154 ) )
= ( finite_card_a @ A_154 ) ) )
& ( ~ ( member_a @ X_52 @ A_154 )
=> ( ( finite_card_a @ ( insert_a @ X_52 @ A_154 ) )
= ( suc @ ( finite_card_a @ A_154 ) ) ) ) ) ) ).
thf(fact_103_card__insert__disjoint,axiom,
! [X_51: int > $o,A_153: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_153 )
=> ( ~ ( member_int_o @ X_51 @ A_153 )
=> ( ( finite_card_int_o @ ( insert_int_o @ X_51 @ A_153 ) )
= ( suc @ ( finite_card_int_o @ A_153 ) ) ) ) ) ).
thf(fact_104_card__insert__disjoint,axiom,
! [X_51: nat > $o,A_153: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_153 )
=> ( ~ ( member_nat_o @ X_51 @ A_153 )
=> ( ( finite_card_nat_o @ ( insert_nat_o @ X_51 @ A_153 ) )
= ( suc @ ( finite_card_nat_o @ A_153 ) ) ) ) ) ).
thf(fact_105_card__insert__disjoint,axiom,
! [X_51: pname > $o,A_153: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_153 )
=> ( ~ ( member_pname_o @ X_51 @ A_153 )
=> ( ( finite_card_pname_o @ ( insert_pname_o @ X_51 @ A_153 ) )
= ( suc @ ( finite_card_pname_o @ A_153 ) ) ) ) ) ).
thf(fact_106_card__insert__disjoint,axiom,
! [X_51: x_a > $o,A_153: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_153 )
=> ( ~ ( member_a_o @ X_51 @ A_153 )
=> ( ( finite_card_a_o @ ( insert_a_o @ X_51 @ A_153 ) )
= ( suc @ ( finite_card_a_o @ A_153 ) ) ) ) ) ).
thf(fact_107_card__insert__disjoint,axiom,
! [X_51: pname,A_153: pname > $o] :
( ( finite_finite_pname @ A_153 )
=> ( ~ ( member_pname @ X_51 @ A_153 )
=> ( ( finite_card_pname @ ( insert_pname @ X_51 @ A_153 ) )
= ( suc @ ( finite_card_pname @ A_153 ) ) ) ) ) ).
thf(fact_108_card__insert__disjoint,axiom,
! [X_51: nat,A_153: nat > $o] :
( ( finite_finite_nat @ A_153 )
=> ( ~ ( member_nat @ X_51 @ A_153 )
=> ( ( finite_card_nat @ ( insert_nat @ X_51 @ A_153 ) )
= ( suc @ ( finite_card_nat @ A_153 ) ) ) ) ) ).
thf(fact_109_card__insert__disjoint,axiom,
! [X_51: int,A_153: int > $o] :
( ( finite_finite_int @ A_153 )
=> ( ~ ( member_int @ X_51 @ A_153 )
=> ( ( finite_card_int @ ( insert_int @ X_51 @ A_153 ) )
= ( suc @ ( finite_card_int @ A_153 ) ) ) ) ) ).
thf(fact_110_card__insert__disjoint,axiom,
! [X_51: x_a,A_153: x_a > $o] :
( ( finite_finite_a @ A_153 )
=> ( ~ ( member_a @ X_51 @ A_153 )
=> ( ( finite_card_a @ ( insert_a @ X_51 @ A_153 ) )
= ( suc @ ( finite_card_a @ A_153 ) ) ) ) ) ).
thf(fact_111_finite__Collect__conjI,axiom,
! [Q_3: x_a > $o,P_13: x_a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P_13 ) )
| ( finite_finite_a @ ( collect_a @ Q_3 ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
thf(fact_112_finite__Collect__conjI,axiom,
! [Q_3: ( int > $o ) > $o,P_13: ( int > $o ) > $o] :
( ( ( finite_finite_int_o @ ( collect_int_o @ P_13 ) )
| ( finite_finite_int_o @ ( collect_int_o @ Q_3 ) ) )
=> ( finite_finite_int_o
@ ( collect_int_o
@ ^ [X_1: int > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
thf(fact_113_finite__Collect__conjI,axiom,
! [Q_3: ( nat > $o ) > $o,P_13: ( nat > $o ) > $o] :
( ( ( finite_finite_nat_o @ ( collect_nat_o @ P_13 ) )
| ( finite_finite_nat_o @ ( collect_nat_o @ Q_3 ) ) )
=> ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
thf(fact_114_finite__Collect__conjI,axiom,
! [Q_3: ( pname > $o ) > $o,P_13: ( pname > $o ) > $o] :
( ( ( finite297249702name_o @ ( collect_pname_o @ P_13 ) )
| ( finite297249702name_o @ ( collect_pname_o @ Q_3 ) ) )
=> ( finite297249702name_o
@ ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
thf(fact_115_finite__Collect__conjI,axiom,
! [Q_3: ( x_a > $o ) > $o,P_13: ( x_a > $o ) > $o] :
( ( ( finite_finite_a_o @ ( collect_a_o @ P_13 ) )
| ( finite_finite_a_o @ ( collect_a_o @ Q_3 ) ) )
=> ( finite_finite_a_o
@ ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
thf(fact_116_finite__Collect__conjI,axiom,
! [Q_3: pname > $o,P_13: pname > $o] :
( ( ( finite_finite_pname @ ( collect_pname @ P_13 ) )
| ( finite_finite_pname @ ( collect_pname @ Q_3 ) ) )
=> ( finite_finite_pname
@ ( collect_pname
@ ^ [X_1: pname] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
thf(fact_117_finite__Collect__conjI,axiom,
! [Q_3: nat > $o,P_13: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P_13 ) )
| ( finite_finite_nat @ ( collect_nat @ Q_3 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
thf(fact_118_finite__Collect__conjI,axiom,
! [Q_3: int > $o,P_13: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P_13 ) )
| ( finite_finite_int @ ( collect_int @ Q_3 ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
thf(fact_119_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_120_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
thf(fact_121_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
thf(fact_122_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_123_nat_Oinject,axiom,
! [Nat_4: nat,Nat_1: nat] :
( ( ( suc @ Nat_4 )
= ( suc @ Nat_1 ) )
<=> ( Nat_4 = Nat_1 ) ) ).
thf(fact_124_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
thf(fact_125_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
thf(fact_126_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_127_le__trans,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I_1 @ K ) ) ) ).
thf(fact_128_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_129_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_130_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_131_diff__commute,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ K ) @ J ) ) ).
thf(fact_132_finite__Collect__disjI,axiom,
! [P_12: x_a > $o,Q_2: x_a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X_1: x_a] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
<=> ( ( finite_finite_a @ ( collect_a @ P_12 ) )
& ( finite_finite_a @ ( collect_a @ Q_2 ) ) ) ) ).
thf(fact_133_finite__Collect__disjI,axiom,
! [P_12: ( int > $o ) > $o,Q_2: ( int > $o ) > $o] :
( ( finite_finite_int_o
@ ( collect_int_o
@ ^ [X_1: int > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
<=> ( ( finite_finite_int_o @ ( collect_int_o @ P_12 ) )
& ( finite_finite_int_o @ ( collect_int_o @ Q_2 ) ) ) ) ).
thf(fact_134_finite__Collect__disjI,axiom,
! [P_12: ( nat > $o ) > $o,Q_2: ( nat > $o ) > $o] :
( ( finite_finite_nat_o
@ ( collect_nat_o
@ ^ [X_1: nat > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
<=> ( ( finite_finite_nat_o @ ( collect_nat_o @ P_12 ) )
& ( finite_finite_nat_o @ ( collect_nat_o @ Q_2 ) ) ) ) ).
thf(fact_135_finite__Collect__disjI,axiom,
! [P_12: ( pname > $o ) > $o,Q_2: ( pname > $o ) > $o] :
( ( finite297249702name_o
@ ( collect_pname_o
@ ^ [X_1: pname > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
<=> ( ( finite297249702name_o @ ( collect_pname_o @ P_12 ) )
& ( finite297249702name_o @ ( collect_pname_o @ Q_2 ) ) ) ) ).
thf(fact_136_finite__Collect__disjI,axiom,
! [P_12: ( x_a > $o ) > $o,Q_2: ( x_a > $o ) > $o] :
( ( finite_finite_a_o
@ ( collect_a_o
@ ^ [X_1: x_a > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
<=> ( ( finite_finite_a_o @ ( collect_a_o @ P_12 ) )
& ( finite_finite_a_o @ ( collect_a_o @ Q_2 ) ) ) ) ).
thf(fact_137_finite__Collect__disjI,axiom,
! [P_12: pname > $o,Q_2: pname > $o] :
( ( finite_finite_pname
@ ( collect_pname
@ ^ [X_1: pname] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
<=> ( ( finite_finite_pname @ ( collect_pname @ P_12 ) )
& ( finite_finite_pname @ ( collect_pname @ Q_2 ) ) ) ) ).
thf(fact_138_finite__Collect__disjI,axiom,
! [P_12: nat > $o,Q_2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X_1: nat] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
<=> ( ( finite_finite_nat @ ( collect_nat @ P_12 ) )
& ( finite_finite_nat @ ( collect_nat @ Q_2 ) ) ) ) ).
thf(fact_139_finite__Collect__disjI,axiom,
! [P_12: int > $o,Q_2: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X_1: int] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
<=> ( ( finite_finite_int @ ( collect_int @ P_12 ) )
& ( finite_finite_int @ ( collect_int @ Q_2 ) ) ) ) ).
thf(fact_140_finite__insert,axiom,
! [A_152: int > $o,A_151: ( int > $o ) > $o] :
( ( finite_finite_int_o @ ( insert_int_o @ A_152 @ A_151 ) )
<=> ( finite_finite_int_o @ A_151 ) ) ).
thf(fact_141_finite__insert,axiom,
! [A_152: nat > $o,A_151: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ ( insert_nat_o @ A_152 @ A_151 ) )
<=> ( finite_finite_nat_o @ A_151 ) ) ).
thf(fact_142_finite__insert,axiom,
! [A_152: pname > $o,A_151: ( pname > $o ) > $o] :
( ( finite297249702name_o @ ( insert_pname_o @ A_152 @ A_151 ) )
<=> ( finite297249702name_o @ A_151 ) ) ).
thf(fact_143_finite__insert,axiom,
! [A_152: x_a > $o,A_151: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ ( insert_a_o @ A_152 @ A_151 ) )
<=> ( finite_finite_a_o @ A_151 ) ) ).
thf(fact_144_finite__insert,axiom,
! [A_152: pname,A_151: pname > $o] :
( ( finite_finite_pname @ ( insert_pname @ A_152 @ A_151 ) )
<=> ( finite_finite_pname @ A_151 ) ) ).
thf(fact_145_finite__insert,axiom,
! [A_152: nat,A_151: nat > $o] :
( ( finite_finite_nat @ ( insert_nat @ A_152 @ A_151 ) )
<=> ( finite_finite_nat @ A_151 ) ) ).
thf(fact_146_finite__insert,axiom,
! [A_152: int,A_151: int > $o] :
( ( finite_finite_int @ ( insert_int @ A_152 @ A_151 ) )
<=> ( finite_finite_int @ A_151 ) ) ).
thf(fact_147_finite__insert,axiom,
! [A_152: x_a,A_151: x_a > $o] :
( ( finite_finite_a @ ( insert_a @ A_152 @ A_151 ) )
<=> ( finite_finite_a @ A_151 ) ) ).
thf(fact_148_finite__subset,axiom,
! [A_150: ( int > $o ) > $o,B_87: ( int > $o ) > $o] :
( ( ord_less_eq_int_o_o @ A_150 @ B_87 )
=> ( ( finite_finite_int_o @ B_87 )
=> ( finite_finite_int_o @ A_150 ) ) ) ).
thf(fact_149_finite__subset,axiom,
! [A_150: ( nat > $o ) > $o,B_87: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ A_150 @ B_87 )
=> ( ( finite_finite_nat_o @ B_87 )
=> ( finite_finite_nat_o @ A_150 ) ) ) ).
thf(fact_150_finite__subset,axiom,
! [A_150: ( pname > $o ) > $o,B_87: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ A_150 @ B_87 )
=> ( ( finite297249702name_o @ B_87 )
=> ( finite297249702name_o @ A_150 ) ) ) ).
thf(fact_151_finite__subset,axiom,
! [A_150: ( x_a > $o ) > $o,B_87: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ A_150 @ B_87 )
=> ( ( finite_finite_a_o @ B_87 )
=> ( finite_finite_a_o @ A_150 ) ) ) ).
thf(fact_152_finite__subset,axiom,
! [A_150: x_a > $o,B_87: x_a > $o] :
( ( ord_less_eq_a_o @ A_150 @ B_87 )
=> ( ( finite_finite_a @ B_87 )
=> ( finite_finite_a @ A_150 ) ) ) ).
thf(fact_153_finite__subset,axiom,
! [A_150: pname > $o,B_87: pname > $o] :
( ( ord_less_eq_pname_o @ A_150 @ B_87 )
=> ( ( finite_finite_pname @ B_87 )
=> ( finite_finite_pname @ A_150 ) ) ) ).
thf(fact_154_finite__subset,axiom,
! [A_150: nat > $o,B_87: nat > $o] :
( ( ord_less_eq_nat_o @ A_150 @ B_87 )
=> ( ( finite_finite_nat @ B_87 )
=> ( finite_finite_nat @ A_150 ) ) ) ).
thf(fact_155_finite__subset,axiom,
! [A_150: int > $o,B_87: int > $o] :
( ( ord_less_eq_int_o @ A_150 @ B_87 )
=> ( ( finite_finite_int @ B_87 )
=> ( finite_finite_int @ A_150 ) ) ) ).
thf(fact_156_rev__finite__subset,axiom,
! [A_149: ( int > $o ) > $o,B_86: ( int > $o ) > $o] :
( ( finite_finite_int_o @ B_86 )
=> ( ( ord_less_eq_int_o_o @ A_149 @ B_86 )
=> ( finite_finite_int_o @ A_149 ) ) ) ).
thf(fact_157_rev__finite__subset,axiom,
! [A_149: ( nat > $o ) > $o,B_86: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_86 )
=> ( ( ord_less_eq_nat_o_o @ A_149 @ B_86 )
=> ( finite_finite_nat_o @ A_149 ) ) ) ).
thf(fact_158_rev__finite__subset,axiom,
! [A_149: ( pname > $o ) > $o,B_86: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_86 )
=> ( ( ord_le1205211808me_o_o @ A_149 @ B_86 )
=> ( finite297249702name_o @ A_149 ) ) ) ).
thf(fact_159_rev__finite__subset,axiom,
! [A_149: ( x_a > $o ) > $o,B_86: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_86 )
=> ( ( ord_less_eq_a_o_o @ A_149 @ B_86 )
=> ( finite_finite_a_o @ A_149 ) ) ) ).
thf(fact_160_rev__finite__subset,axiom,
! [A_149: x_a > $o,B_86: x_a > $o] :
( ( finite_finite_a @ B_86 )
=> ( ( ord_less_eq_a_o @ A_149 @ B_86 )
=> ( finite_finite_a @ A_149 ) ) ) ).
thf(fact_161_rev__finite__subset,axiom,
! [A_149: pname > $o,B_86: pname > $o] :
( ( finite_finite_pname @ B_86 )
=> ( ( ord_less_eq_pname_o @ A_149 @ B_86 )
=> ( finite_finite_pname @ A_149 ) ) ) ).
thf(fact_162_rev__finite__subset,axiom,
! [A_149: nat > $o,B_86: nat > $o] :
( ( finite_finite_nat @ B_86 )
=> ( ( ord_less_eq_nat_o @ A_149 @ B_86 )
=> ( finite_finite_nat @ A_149 ) ) ) ).
thf(fact_163_rev__finite__subset,axiom,
! [A_149: int > $o,B_86: int > $o] :
( ( finite_finite_int @ B_86 )
=> ( ( ord_less_eq_int_o @ A_149 @ B_86 )
=> ( finite_finite_int @ A_149 ) ) ) ).
thf(fact_164_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_165_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_166_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_167_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_168_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_169_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_eq_nat @ M @ N )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_170_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
thf(fact_171_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_172_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_173_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_174_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_175_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_176_diff__diff__cancel,axiom,
! [I_1: nat,N: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I_1 ) )
= I_1 ) ) ).
thf(fact_177_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_178_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_179_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_180_finite__surj,axiom,
! [B_85: x_a > $o,F_40: x_a > x_a,A_148: x_a > $o] :
( ( finite_finite_a @ A_148 )
=> ( ( ord_less_eq_a_o @ B_85 @ ( image_a_a @ F_40 @ A_148 ) )
=> ( finite_finite_a @ B_85 ) ) ) ).
thf(fact_181_finite__surj,axiom,
! [B_85: x_a > $o,F_40: ( int > $o ) > x_a,A_148: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_148 )
=> ( ( ord_less_eq_a_o @ B_85 @ ( image_int_o_a @ F_40 @ A_148 ) )
=> ( finite_finite_a @ B_85 ) ) ) ).
thf(fact_182_finite__surj,axiom,
! [B_85: x_a > $o,F_40: ( nat > $o ) > x_a,A_148: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_148 )
=> ( ( ord_less_eq_a_o @ B_85 @ ( image_nat_o_a @ F_40 @ A_148 ) )
=> ( finite_finite_a @ B_85 ) ) ) ).
thf(fact_183_finite__surj,axiom,
! [B_85: x_a > $o,F_40: ( pname > $o ) > x_a,A_148: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_148 )
=> ( ( ord_less_eq_a_o @ B_85 @ ( image_pname_o_a @ F_40 @ A_148 ) )
=> ( finite_finite_a @ B_85 ) ) ) ).
thf(fact_184_finite__surj,axiom,
! [B_85: x_a > $o,F_40: ( x_a > $o ) > x_a,A_148: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_148 )
=> ( ( ord_less_eq_a_o @ B_85 @ ( image_a_o_a @ F_40 @ A_148 ) )
=> ( finite_finite_a @ B_85 ) ) ) ).
thf(fact_185_finite__surj,axiom,
! [B_85: ( int > $o ) > $o,F_40: pname > int > $o,A_148: pname > $o] :
( ( finite_finite_pname @ A_148 )
=> ( ( ord_less_eq_int_o_o @ B_85 @ ( image_pname_int_o @ F_40 @ A_148 ) )
=> ( finite_finite_int_o @ B_85 ) ) ) ).
thf(fact_186_finite__surj,axiom,
! [B_85: ( nat > $o ) > $o,F_40: pname > nat > $o,A_148: pname > $o] :
( ( finite_finite_pname @ A_148 )
=> ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_pname_nat_o @ F_40 @ A_148 ) )
=> ( finite_finite_nat_o @ B_85 ) ) ) ).
thf(fact_187_finite__surj,axiom,
! [B_85: ( pname > $o ) > $o,F_40: pname > pname > $o,A_148: pname > $o] :
( ( finite_finite_pname @ A_148 )
=> ( ( ord_le1205211808me_o_o @ B_85 @ ( image_pname_pname_o @ F_40 @ A_148 ) )
=> ( finite297249702name_o @ B_85 ) ) ) ).
thf(fact_188_finite__surj,axiom,
! [B_85: ( x_a > $o ) > $o,F_40: pname > x_a > $o,A_148: pname > $o] :
( ( finite_finite_pname @ A_148 )
=> ( ( ord_less_eq_a_o_o @ B_85 @ ( image_pname_a_o @ F_40 @ A_148 ) )
=> ( finite_finite_a_o @ B_85 ) ) ) ).
thf(fact_189_finite__surj,axiom,
! [B_85: pname > $o,F_40: pname > pname,A_148: pname > $o] :
( ( finite_finite_pname @ A_148 )
=> ( ( ord_less_eq_pname_o @ B_85 @ ( image_pname_pname @ F_40 @ A_148 ) )
=> ( finite_finite_pname @ B_85 ) ) ) ).
thf(fact_190_finite__surj,axiom,
! [B_85: x_a > $o,F_40: nat > x_a,A_148: nat > $o] :
( ( finite_finite_nat @ A_148 )
=> ( ( ord_less_eq_a_o @ B_85 @ ( image_nat_a @ F_40 @ A_148 ) )
=> ( finite_finite_a @ B_85 ) ) ) ).
thf(fact_191_finite__surj,axiom,
! [B_85: ( int > $o ) > $o,F_40: nat > int > $o,A_148: nat > $o] :
( ( finite_finite_nat @ A_148 )
=> ( ( ord_less_eq_int_o_o @ B_85 @ ( image_nat_int_o @ F_40 @ A_148 ) )
=> ( finite_finite_int_o @ B_85 ) ) ) ).
thf(fact_192_finite__surj,axiom,
! [B_85: ( nat > $o ) > $o,F_40: nat > nat > $o,A_148: nat > $o] :
( ( finite_finite_nat @ A_148 )
=> ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_nat_nat_o @ F_40 @ A_148 ) )
=> ( finite_finite_nat_o @ B_85 ) ) ) ).
thf(fact_193_finite__surj,axiom,
! [B_85: ( pname > $o ) > $o,F_40: nat > pname > $o,A_148: nat > $o] :
( ( finite_finite_nat @ A_148 )
=> ( ( ord_le1205211808me_o_o @ B_85 @ ( image_nat_pname_o @ F_40 @ A_148 ) )
=> ( finite297249702name_o @ B_85 ) ) ) ).
thf(fact_194_finite__surj,axiom,
! [B_85: ( x_a > $o ) > $o,F_40: nat > x_a > $o,A_148: nat > $o] :
( ( finite_finite_nat @ A_148 )
=> ( ( ord_less_eq_a_o_o @ B_85 @ ( image_nat_a_o @ F_40 @ A_148 ) )
=> ( finite_finite_a_o @ B_85 ) ) ) ).
thf(fact_195_finite__surj,axiom,
! [B_85: pname > $o,F_40: nat > pname,A_148: nat > $o] :
( ( finite_finite_nat @ A_148 )
=> ( ( ord_less_eq_pname_o @ B_85 @ ( image_nat_pname @ F_40 @ A_148 ) )
=> ( finite_finite_pname @ B_85 ) ) ) ).
thf(fact_196_finite__surj,axiom,
! [B_85: x_a > $o,F_40: int > x_a,A_148: int > $o] :
( ( finite_finite_int @ A_148 )
=> ( ( ord_less_eq_a_o @ B_85 @ ( image_int_a @ F_40 @ A_148 ) )
=> ( finite_finite_a @ B_85 ) ) ) ).
thf(fact_197_finite__surj,axiom,
! [B_85: ( int > $o ) > $o,F_40: int > int > $o,A_148: int > $o] :
( ( finite_finite_int @ A_148 )
=> ( ( ord_less_eq_int_o_o @ B_85 @ ( image_int_int_o @ F_40 @ A_148 ) )
=> ( finite_finite_int_o @ B_85 ) ) ) ).
thf(fact_198_finite__surj,axiom,
! [B_85: ( nat > $o ) > $o,F_40: int > nat > $o,A_148: int > $o] :
( ( finite_finite_int @ A_148 )
=> ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_int_nat_o @ F_40 @ A_148 ) )
=> ( finite_finite_nat_o @ B_85 ) ) ) ).
thf(fact_199_finite__surj,axiom,
! [B_85: ( pname > $o ) > $o,F_40: int > pname > $o,A_148: int > $o] :
( ( finite_finite_int @ A_148 )
=> ( ( ord_le1205211808me_o_o @ B_85 @ ( image_int_pname_o @ F_40 @ A_148 ) )
=> ( finite297249702name_o @ B_85 ) ) ) ).
thf(fact_200_finite__surj,axiom,
! [B_85: ( x_a > $o ) > $o,F_40: int > x_a > $o,A_148: int > $o] :
( ( finite_finite_int @ A_148 )
=> ( ( ord_less_eq_a_o_o @ B_85 @ ( image_int_a_o @ F_40 @ A_148 ) )
=> ( finite_finite_a_o @ B_85 ) ) ) ).
thf(fact_201_finite__surj,axiom,
! [B_85: pname > $o,F_40: int > pname,A_148: int > $o] :
( ( finite_finite_int @ A_148 )
=> ( ( ord_less_eq_pname_o @ B_85 @ ( image_int_pname @ F_40 @ A_148 ) )
=> ( finite_finite_pname @ B_85 ) ) ) ).
thf(fact_202_finite__surj,axiom,
! [B_85: pname > $o,F_40: x_a > pname,A_148: x_a > $o] :
( ( finite_finite_a @ A_148 )
=> ( ( ord_less_eq_pname_o @ B_85 @ ( image_a_pname @ F_40 @ A_148 ) )
=> ( finite_finite_pname @ B_85 ) ) ) ).
thf(fact_203_finite__surj,axiom,
! [B_85: pname > $o,F_40: ( int > $o ) > pname,A_148: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_148 )
=> ( ( ord_less_eq_pname_o @ B_85 @ ( image_int_o_pname @ F_40 @ A_148 ) )
=> ( finite_finite_pname @ B_85 ) ) ) ).
thf(fact_204_finite__surj,axiom,
! [B_85: pname > $o,F_40: ( nat > $o ) > pname,A_148: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_148 )
=> ( ( ord_less_eq_pname_o @ B_85 @ ( image_nat_o_pname @ F_40 @ A_148 ) )
=> ( finite_finite_pname @ B_85 ) ) ) ).
thf(fact_205_finite__surj,axiom,
! [B_85: pname > $o,F_40: ( pname > $o ) > pname,A_148: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_148 )
=> ( ( ord_less_eq_pname_o @ B_85 @ ( image_pname_o_pname @ F_40 @ A_148 ) )
=> ( finite_finite_pname @ B_85 ) ) ) ).
thf(fact_206_finite__surj,axiom,
! [B_85: pname > $o,F_40: ( x_a > $o ) > pname,A_148: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_148 )
=> ( ( ord_less_eq_pname_o @ B_85 @ ( image_a_o_pname @ F_40 @ A_148 ) )
=> ( finite_finite_pname @ B_85 ) ) ) ).
thf(fact_207_finite__surj,axiom,
! [B_85: nat > $o,F_40: x_a > nat,A_148: x_a > $o] :
( ( finite_finite_a @ A_148 )
=> ( ( ord_less_eq_nat_o @ B_85 @ ( image_a_nat @ F_40 @ A_148 ) )
=> ( finite_finite_nat @ B_85 ) ) ) ).
thf(fact_208_finite__surj,axiom,
! [B_85: nat > $o,F_40: ( int > $o ) > nat,A_148: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_148 )
=> ( ( ord_less_eq_nat_o @ B_85 @ ( image_int_o_nat @ F_40 @ A_148 ) )
=> ( finite_finite_nat @ B_85 ) ) ) ).
thf(fact_209_finite__surj,axiom,
! [B_85: nat > $o,F_40: ( nat > $o ) > nat,A_148: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_148 )
=> ( ( ord_less_eq_nat_o @ B_85 @ ( image_nat_o_nat @ F_40 @ A_148 ) )
=> ( finite_finite_nat @ B_85 ) ) ) ).
thf(fact_210_finite__surj,axiom,
! [B_85: nat > $o,F_40: ( pname > $o ) > nat,A_148: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_148 )
=> ( ( ord_less_eq_nat_o @ B_85 @ ( image_pname_o_nat @ F_40 @ A_148 ) )
=> ( finite_finite_nat @ B_85 ) ) ) ).
thf(fact_211_finite__surj,axiom,
! [B_85: nat > $o,F_40: ( x_a > $o ) > nat,A_148: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_148 )
=> ( ( ord_less_eq_nat_o @ B_85 @ ( image_a_o_nat @ F_40 @ A_148 ) )
=> ( finite_finite_nat @ B_85 ) ) ) ).
thf(fact_212_finite__surj,axiom,
! [B_85: int > $o,F_40: x_a > int,A_148: x_a > $o] :
( ( finite_finite_a @ A_148 )
=> ( ( ord_less_eq_int_o @ B_85 @ ( image_a_int @ F_40 @ A_148 ) )
=> ( finite_finite_int @ B_85 ) ) ) ).
thf(fact_213_finite__surj,axiom,
! [B_85: int > $o,F_40: ( int > $o ) > int,A_148: ( int > $o ) > $o] :
( ( finite_finite_int_o @ A_148 )
=> ( ( ord_less_eq_int_o @ B_85 @ ( image_int_o_int @ F_40 @ A_148 ) )
=> ( finite_finite_int @ B_85 ) ) ) ).
thf(fact_214_finite__surj,axiom,
! [B_85: int > $o,F_40: ( nat > $o ) > int,A_148: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ A_148 )
=> ( ( ord_less_eq_int_o @ B_85 @ ( image_nat_o_int @ F_40 @ A_148 ) )
=> ( finite_finite_int @ B_85 ) ) ) ).
thf(fact_215_finite__surj,axiom,
! [B_85: int > $o,F_40: ( pname > $o ) > int,A_148: ( pname > $o ) > $o] :
( ( finite297249702name_o @ A_148 )
=> ( ( ord_less_eq_int_o @ B_85 @ ( image_pname_o_int @ F_40 @ A_148 ) )
=> ( finite_finite_int @ B_85 ) ) ) ).
thf(fact_216_finite__surj,axiom,
! [B_85: int > $o,F_40: ( x_a > $o ) > int,A_148: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ A_148 )
=> ( ( ord_less_eq_int_o @ B_85 @ ( image_a_o_int @ F_40 @ A_148 ) )
=> ( finite_finite_int @ B_85 ) ) ) ).
thf(fact_217_finite__surj,axiom,
! [B_85: x_a > $o,F_40: pname > x_a,A_148: pname > $o] :
( ( finite_finite_pname @ A_148 )
=> ( ( ord_less_eq_a_o @ B_85 @ ( image_pname_a @ F_40 @ A_148 ) )
=> ( finite_finite_a @ B_85 ) ) ) ).
thf(fact_218_finite__surj,axiom,
! [B_85: int > $o,F_40: nat > int,A_148: nat > $o] :
( ( finite_finite_nat @ A_148 )
=> ( ( ord_less_eq_int_o @ B_85 @ ( image_nat_int @ F_40 @ A_148 ) )
=> ( finite_finite_int @ B_85 ) ) ) ).
thf(fact_219_finite__subset__image,axiom,
! [F_39: ( int > $o ) > x_a,A_147: ( int > $o ) > $o,B_84: x_a > $o] :
( ( finite_finite_a @ B_84 )
=> ( ( ord_less_eq_a_o @ B_84 @ ( image_int_o_a @ F_39 @ A_147 ) )
=> ? [C_34: ( int > $o ) > $o] :
( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
& ( finite_finite_int_o @ C_34 )
& ( B_84
= ( image_int_o_a @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_220_finite__subset__image,axiom,
! [F_39: ( nat > $o ) > x_a,A_147: ( nat > $o ) > $o,B_84: x_a > $o] :
( ( finite_finite_a @ B_84 )
=> ( ( ord_less_eq_a_o @ B_84 @ ( image_nat_o_a @ F_39 @ A_147 ) )
=> ? [C_34: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
& ( finite_finite_nat_o @ C_34 )
& ( B_84
= ( image_nat_o_a @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_221_finite__subset__image,axiom,
! [F_39: ( pname > $o ) > x_a,A_147: ( pname > $o ) > $o,B_84: x_a > $o] :
( ( finite_finite_a @ B_84 )
=> ( ( ord_less_eq_a_o @ B_84 @ ( image_pname_o_a @ F_39 @ A_147 ) )
=> ? [C_34: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
& ( finite297249702name_o @ C_34 )
& ( B_84
= ( image_pname_o_a @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_222_finite__subset__image,axiom,
! [F_39: ( x_a > $o ) > x_a,A_147: ( x_a > $o ) > $o,B_84: x_a > $o] :
( ( finite_finite_a @ B_84 )
=> ( ( ord_less_eq_a_o @ B_84 @ ( image_a_o_a @ F_39 @ A_147 ) )
=> ? [C_34: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
& ( finite_finite_a_o @ C_34 )
& ( B_84
= ( image_a_o_a @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_223_finite__subset__image,axiom,
! [F_39: x_a > x_a,A_147: x_a > $o,B_84: x_a > $o] :
( ( finite_finite_a @ B_84 )
=> ( ( ord_less_eq_a_o @ B_84 @ ( image_a_a @ F_39 @ A_147 ) )
=> ? [C_34: x_a > $o] :
( ( ord_less_eq_a_o @ C_34 @ A_147 )
& ( finite_finite_a @ C_34 )
& ( B_84
= ( image_a_a @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_224_finite__subset__image,axiom,
! [F_39: x_a > int > $o,A_147: x_a > $o,B_84: ( int > $o ) > $o] :
( ( finite_finite_int_o @ B_84 )
=> ( ( ord_less_eq_int_o_o @ B_84 @ ( image_a_int_o @ F_39 @ A_147 ) )
=> ? [C_34: x_a > $o] :
( ( ord_less_eq_a_o @ C_34 @ A_147 )
& ( finite_finite_a @ C_34 )
& ( B_84
= ( image_a_int_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_225_finite__subset__image,axiom,
! [F_39: x_a > nat > $o,A_147: x_a > $o,B_84: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_84 )
=> ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_a_nat_o @ F_39 @ A_147 ) )
=> ? [C_34: x_a > $o] :
( ( ord_less_eq_a_o @ C_34 @ A_147 )
& ( finite_finite_a @ C_34 )
& ( B_84
= ( image_a_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_226_finite__subset__image,axiom,
! [F_39: x_a > pname > $o,A_147: x_a > $o,B_84: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_84 )
=> ( ( ord_le1205211808me_o_o @ B_84 @ ( image_a_pname_o @ F_39 @ A_147 ) )
=> ? [C_34: x_a > $o] :
( ( ord_less_eq_a_o @ C_34 @ A_147 )
& ( finite_finite_a @ C_34 )
& ( B_84
= ( image_a_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_227_finite__subset__image,axiom,
! [F_39: x_a > x_a > $o,A_147: x_a > $o,B_84: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_84 )
=> ( ( ord_less_eq_a_o_o @ B_84 @ ( image_a_a_o @ F_39 @ A_147 ) )
=> ? [C_34: x_a > $o] :
( ( ord_less_eq_a_o @ C_34 @ A_147 )
& ( finite_finite_a @ C_34 )
& ( B_84
= ( image_a_a_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_228_finite__subset__image,axiom,
! [F_39: x_a > pname,A_147: x_a > $o,B_84: pname > $o] :
( ( finite_finite_pname @ B_84 )
=> ( ( ord_less_eq_pname_o @ B_84 @ ( image_a_pname @ F_39 @ A_147 ) )
=> ? [C_34: x_a > $o] :
( ( ord_less_eq_a_o @ C_34 @ A_147 )
& ( finite_finite_a @ C_34 )
& ( B_84
= ( image_a_pname @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_229_finite__subset__image,axiom,
! [F_39: ( int > $o ) > pname,A_147: ( int > $o ) > $o,B_84: pname > $o] :
( ( finite_finite_pname @ B_84 )
=> ( ( ord_less_eq_pname_o @ B_84 @ ( image_int_o_pname @ F_39 @ A_147 ) )
=> ? [C_34: ( int > $o ) > $o] :
( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
& ( finite_finite_int_o @ C_34 )
& ( B_84
= ( image_int_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_230_finite__subset__image,axiom,
! [F_39: ( nat > $o ) > pname,A_147: ( nat > $o ) > $o,B_84: pname > $o] :
( ( finite_finite_pname @ B_84 )
=> ( ( ord_less_eq_pname_o @ B_84 @ ( image_nat_o_pname @ F_39 @ A_147 ) )
=> ? [C_34: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
& ( finite_finite_nat_o @ C_34 )
& ( B_84
= ( image_nat_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_231_finite__subset__image,axiom,
! [F_39: ( pname > $o ) > pname,A_147: ( pname > $o ) > $o,B_84: pname > $o] :
( ( finite_finite_pname @ B_84 )
=> ( ( ord_less_eq_pname_o @ B_84 @ ( image_pname_o_pname @ F_39 @ A_147 ) )
=> ? [C_34: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
& ( finite297249702name_o @ C_34 )
& ( B_84
= ( image_pname_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_232_finite__subset__image,axiom,
! [F_39: ( x_a > $o ) > pname,A_147: ( x_a > $o ) > $o,B_84: pname > $o] :
( ( finite_finite_pname @ B_84 )
=> ( ( ord_less_eq_pname_o @ B_84 @ ( image_a_o_pname @ F_39 @ A_147 ) )
=> ? [C_34: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
& ( finite_finite_a_o @ C_34 )
& ( B_84
= ( image_a_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_233_finite__subset__image,axiom,
! [F_39: x_a > nat,A_147: x_a > $o,B_84: nat > $o] :
( ( finite_finite_nat @ B_84 )
=> ( ( ord_less_eq_nat_o @ B_84 @ ( image_a_nat @ F_39 @ A_147 ) )
=> ? [C_34: x_a > $o] :
( ( ord_less_eq_a_o @ C_34 @ A_147 )
& ( finite_finite_a @ C_34 )
& ( B_84
= ( image_a_nat @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_234_finite__subset__image,axiom,
! [F_39: ( int > $o ) > nat,A_147: ( int > $o ) > $o,B_84: nat > $o] :
( ( finite_finite_nat @ B_84 )
=> ( ( ord_less_eq_nat_o @ B_84 @ ( image_int_o_nat @ F_39 @ A_147 ) )
=> ? [C_34: ( int > $o ) > $o] :
( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
& ( finite_finite_int_o @ C_34 )
& ( B_84
= ( image_int_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_235_finite__subset__image,axiom,
! [F_39: ( nat > $o ) > nat,A_147: ( nat > $o ) > $o,B_84: nat > $o] :
( ( finite_finite_nat @ B_84 )
=> ( ( ord_less_eq_nat_o @ B_84 @ ( image_nat_o_nat @ F_39 @ A_147 ) )
=> ? [C_34: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
& ( finite_finite_nat_o @ C_34 )
& ( B_84
= ( image_nat_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_236_finite__subset__image,axiom,
! [F_39: ( pname > $o ) > nat,A_147: ( pname > $o ) > $o,B_84: nat > $o] :
( ( finite_finite_nat @ B_84 )
=> ( ( ord_less_eq_nat_o @ B_84 @ ( image_pname_o_nat @ F_39 @ A_147 ) )
=> ? [C_34: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
& ( finite297249702name_o @ C_34 )
& ( B_84
= ( image_pname_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_237_finite__subset__image,axiom,
! [F_39: ( x_a > $o ) > nat,A_147: ( x_a > $o ) > $o,B_84: nat > $o] :
( ( finite_finite_nat @ B_84 )
=> ( ( ord_less_eq_nat_o @ B_84 @ ( image_a_o_nat @ F_39 @ A_147 ) )
=> ? [C_34: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
& ( finite_finite_a_o @ C_34 )
& ( B_84
= ( image_a_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_238_finite__subset__image,axiom,
! [F_39: pname > nat,A_147: pname > $o,B_84: nat > $o] :
( ( finite_finite_nat @ B_84 )
=> ( ( ord_less_eq_nat_o @ B_84 @ ( image_pname_nat @ F_39 @ A_147 ) )
=> ? [C_34: pname > $o] :
( ( ord_less_eq_pname_o @ C_34 @ A_147 )
& ( finite_finite_pname @ C_34 )
& ( B_84
= ( image_pname_nat @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_239_finite__subset__image,axiom,
! [F_39: x_a > int,A_147: x_a > $o,B_84: int > $o] :
( ( finite_finite_int @ B_84 )
=> ( ( ord_less_eq_int_o @ B_84 @ ( image_a_int @ F_39 @ A_147 ) )
=> ? [C_34: x_a > $o] :
( ( ord_less_eq_a_o @ C_34 @ A_147 )
& ( finite_finite_a @ C_34 )
& ( B_84
= ( image_a_int @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_240_finite__subset__image,axiom,
! [F_39: ( int > $o ) > int,A_147: ( int > $o ) > $o,B_84: int > $o] :
( ( finite_finite_int @ B_84 )
=> ( ( ord_less_eq_int_o @ B_84 @ ( image_int_o_int @ F_39 @ A_147 ) )
=> ? [C_34: ( int > $o ) > $o] :
( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
& ( finite_finite_int_o @ C_34 )
& ( B_84
= ( image_int_o_int @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_241_finite__subset__image,axiom,
! [F_39: ( nat > $o ) > int,A_147: ( nat > $o ) > $o,B_84: int > $o] :
( ( finite_finite_int @ B_84 )
=> ( ( ord_less_eq_int_o @ B_84 @ ( image_nat_o_int @ F_39 @ A_147 ) )
=> ? [C_34: ( nat > $o ) > $o] :
( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
& ( finite_finite_nat_o @ C_34 )
& ( B_84
= ( image_nat_o_int @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_242_finite__subset__image,axiom,
! [F_39: ( pname > $o ) > int,A_147: ( pname > $o ) > $o,B_84: int > $o] :
( ( finite_finite_int @ B_84 )
=> ( ( ord_less_eq_int_o @ B_84 @ ( image_pname_o_int @ F_39 @ A_147 ) )
=> ? [C_34: ( pname > $o ) > $o] :
( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
& ( finite297249702name_o @ C_34 )
& ( B_84
= ( image_pname_o_int @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_243_finite__subset__image,axiom,
! [F_39: ( x_a > $o ) > int,A_147: ( x_a > $o ) > $o,B_84: int > $o] :
( ( finite_finite_int @ B_84 )
=> ( ( ord_less_eq_int_o @ B_84 @ ( image_a_o_int @ F_39 @ A_147 ) )
=> ? [C_34: ( x_a > $o ) > $o] :
( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
& ( finite_finite_a_o @ C_34 )
& ( B_84
= ( image_a_o_int @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_244_finite__subset__image,axiom,
! [F_39: pname > int,A_147: pname > $o,B_84: int > $o] :
( ( finite_finite_int @ B_84 )
=> ( ( ord_less_eq_int_o @ B_84 @ ( image_pname_int @ F_39 @ A_147 ) )
=> ? [C_34: pname > $o] :
( ( ord_less_eq_pname_o @ C_34 @ A_147 )
& ( finite_finite_pname @ C_34 )
& ( B_84
= ( image_pname_int @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_245_finite__subset__image,axiom,
! [F_39: pname > int > $o,A_147: pname > $o,B_84: ( int > $o ) > $o] :
( ( finite_finite_int_o @ B_84 )
=> ( ( ord_less_eq_int_o_o @ B_84 @ ( image_pname_int_o @ F_39 @ A_147 ) )
=> ? [C_34: pname > $o] :
( ( ord_less_eq_pname_o @ C_34 @ A_147 )
& ( finite_finite_pname @ C_34 )
& ( B_84
= ( image_pname_int_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_246_finite__subset__image,axiom,
! [F_39: pname > nat > $o,A_147: pname > $o,B_84: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_84 )
=> ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_pname_nat_o @ F_39 @ A_147 ) )
=> ? [C_34: pname > $o] :
( ( ord_less_eq_pname_o @ C_34 @ A_147 )
& ( finite_finite_pname @ C_34 )
& ( B_84
= ( image_pname_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_247_finite__subset__image,axiom,
! [F_39: pname > pname > $o,A_147: pname > $o,B_84: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_84 )
=> ( ( ord_le1205211808me_o_o @ B_84 @ ( image_pname_pname_o @ F_39 @ A_147 ) )
=> ? [C_34: pname > $o] :
( ( ord_less_eq_pname_o @ C_34 @ A_147 )
& ( finite_finite_pname @ C_34 )
& ( B_84
= ( image_pname_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_248_finite__subset__image,axiom,
! [F_39: pname > x_a > $o,A_147: pname > $o,B_84: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_84 )
=> ( ( ord_less_eq_a_o_o @ B_84 @ ( image_pname_a_o @ F_39 @ A_147 ) )
=> ? [C_34: pname > $o] :
( ( ord_less_eq_pname_o @ C_34 @ A_147 )
& ( finite_finite_pname @ C_34 )
& ( B_84
= ( image_pname_a_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_249_finite__subset__image,axiom,
! [F_39: pname > pname,A_147: pname > $o,B_84: pname > $o] :
( ( finite_finite_pname @ B_84 )
=> ( ( ord_less_eq_pname_o @ B_84 @ ( image_pname_pname @ F_39 @ A_147 ) )
=> ? [C_34: pname > $o] :
( ( ord_less_eq_pname_o @ C_34 @ A_147 )
& ( finite_finite_pname @ C_34 )
& ( B_84
= ( image_pname_pname @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_250_finite__subset__image,axiom,
! [F_39: nat > x_a,A_147: nat > $o,B_84: x_a > $o] :
( ( finite_finite_a @ B_84 )
=> ( ( ord_less_eq_a_o @ B_84 @ ( image_nat_a @ F_39 @ A_147 ) )
=> ? [C_34: nat > $o] :
( ( ord_less_eq_nat_o @ C_34 @ A_147 )
& ( finite_finite_nat @ C_34 )
& ( B_84
= ( image_nat_a @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_251_finite__subset__image,axiom,
! [F_39: nat > int > $o,A_147: nat > $o,B_84: ( int > $o ) > $o] :
( ( finite_finite_int_o @ B_84 )
=> ( ( ord_less_eq_int_o_o @ B_84 @ ( image_nat_int_o @ F_39 @ A_147 ) )
=> ? [C_34: nat > $o] :
( ( ord_less_eq_nat_o @ C_34 @ A_147 )
& ( finite_finite_nat @ C_34 )
& ( B_84
= ( image_nat_int_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_252_finite__subset__image,axiom,
! [F_39: nat > nat > $o,A_147: nat > $o,B_84: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_84 )
=> ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_nat_nat_o @ F_39 @ A_147 ) )
=> ? [C_34: nat > $o] :
( ( ord_less_eq_nat_o @ C_34 @ A_147 )
& ( finite_finite_nat @ C_34 )
& ( B_84
= ( image_nat_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_253_finite__subset__image,axiom,
! [F_39: nat > pname > $o,A_147: nat > $o,B_84: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_84 )
=> ( ( ord_le1205211808me_o_o @ B_84 @ ( image_nat_pname_o @ F_39 @ A_147 ) )
=> ? [C_34: nat > $o] :
( ( ord_less_eq_nat_o @ C_34 @ A_147 )
& ( finite_finite_nat @ C_34 )
& ( B_84
= ( image_nat_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_254_finite__subset__image,axiom,
! [F_39: nat > x_a > $o,A_147: nat > $o,B_84: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_84 )
=> ( ( ord_less_eq_a_o_o @ B_84 @ ( image_nat_a_o @ F_39 @ A_147 ) )
=> ? [C_34: nat > $o] :
( ( ord_less_eq_nat_o @ C_34 @ A_147 )
& ( finite_finite_nat @ C_34 )
& ( B_84
= ( image_nat_a_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_255_finite__subset__image,axiom,
! [F_39: nat > pname,A_147: nat > $o,B_84: pname > $o] :
( ( finite_finite_pname @ B_84 )
=> ( ( ord_less_eq_pname_o @ B_84 @ ( image_nat_pname @ F_39 @ A_147 ) )
=> ? [C_34: nat > $o] :
( ( ord_less_eq_nat_o @ C_34 @ A_147 )
& ( finite_finite_nat @ C_34 )
& ( B_84
= ( image_nat_pname @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_256_finite__subset__image,axiom,
! [F_39: int > x_a,A_147: int > $o,B_84: x_a > $o] :
( ( finite_finite_a @ B_84 )
=> ( ( ord_less_eq_a_o @ B_84 @ ( image_int_a @ F_39 @ A_147 ) )
=> ? [C_34: int > $o] :
( ( ord_less_eq_int_o @ C_34 @ A_147 )
& ( finite_finite_int @ C_34 )
& ( B_84
= ( image_int_a @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_257_finite__subset__image,axiom,
! [F_39: int > int > $o,A_147: int > $o,B_84: ( int > $o ) > $o] :
( ( finite_finite_int_o @ B_84 )
=> ( ( ord_less_eq_int_o_o @ B_84 @ ( image_int_int_o @ F_39 @ A_147 ) )
=> ? [C_34: int > $o] :
( ( ord_less_eq_int_o @ C_34 @ A_147 )
& ( finite_finite_int @ C_34 )
& ( B_84
= ( image_int_int_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_258_finite__subset__image,axiom,
! [F_39: int > nat > $o,A_147: int > $o,B_84: ( nat > $o ) > $o] :
( ( finite_finite_nat_o @ B_84 )
=> ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_int_nat_o @ F_39 @ A_147 ) )
=> ? [C_34: int > $o] :
( ( ord_less_eq_int_o @ C_34 @ A_147 )
& ( finite_finite_int @ C_34 )
& ( B_84
= ( image_int_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_259_finite__subset__image,axiom,
! [F_39: int > pname > $o,A_147: int > $o,B_84: ( pname > $o ) > $o] :
( ( finite297249702name_o @ B_84 )
=> ( ( ord_le1205211808me_o_o @ B_84 @ ( image_int_pname_o @ F_39 @ A_147 ) )
=> ? [C_34: int > $o] :
( ( ord_less_eq_int_o @ C_34 @ A_147 )
& ( finite_finite_int @ C_34 )
& ( B_84
= ( image_int_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_260_finite__subset__image,axiom,
! [F_39: int > x_a > $o,A_147: int > $o,B_84: ( x_a > $o ) > $o] :
( ( finite_finite_a_o @ B_84 )
=> ( ( ord_less_eq_a_o_o @ B_84 @ ( image_int_a_o @ F_39 @ A_147 ) )
=> ? [C_34: int > $o] :
( ( ord_less_eq_int_o @ C_34 @ A_147 )
& ( finite_finite_int @ C_34 )
& ( B_84
= ( image_int_a_o @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_261_finite__subset__image,axiom,
! [F_39: int > pname,A_147: int > $o,B_84: pname > $o] :
( ( finite_finite_pname @ B_84 )
=> ( ( ord_less_eq_pname_o @ B_84 @ ( image_int_pname @ F_39 @ A_147 ) )
=> ? [C_34: int > $o] :
( ( ord_less_eq_int_o @ C_34 @ A_147 )
& ( finite_finite_int @ C_34 )
& ( B_84
= ( image_int_pname @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_262_finite__subset__image,axiom,
! [F_39: pname > x_a,A_147: pname > $o,B_84: x_a > $o] :
( ( finite_finite_a @ B_84 )
=> ( ( ord_less_eq_a_o @ B_84 @ ( image_pname_a @ F_39 @ A_147 ) )
=> ? [C_34: pname > $o] :
( ( ord_less_eq_pname_o @ C_34 @ A_147 )
& ( finite_finite_pname @ C_34 )
& ( B_84
= ( image_pname_a @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_263_finite__subset__image,axiom,
! [F_39: nat > int,A_147: nat > $o,B_84: int > $o] :
( ( finite_finite_int @ B_84 )
=> ( ( ord_less_eq_int_o @ B_84 @ ( image_nat_int @ F_39 @ A_147 ) )
=> ? [C_34: nat > $o] :
( ( ord_less_eq_nat_o @ C_34 @ A_147 )
& ( finite_finite_nat @ C_34 )
& ( B_84
= ( image_nat_int @ F_39 @ C_34 ) ) ) ) ) ).
thf(fact_264_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_38: nat > pname > $o] :
( ! [N_1: nat] : ( ord_less_eq_pname_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_pname_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
thf(fact_265_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_38: nat > $o] :
( ! [N_1: nat] : ( ord_less_eq_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
thf(fact_266_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_38: nat > x_a > $o] :
( ! [N_1: nat] : ( ord_less_eq_a_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_a_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
thf(fact_267_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_38: nat > nat] :
( ! [N_1: nat] : ( ord_less_eq_nat @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_nat @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
thf(fact_268_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_38: nat > int] :
( ! [N_1: nat] : ( ord_less_eq_int @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_int @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
thf(fact_269_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_38: nat > nat > $o] :
( ! [N_1: nat] : ( ord_less_eq_nat_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_nat_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
thf(fact_270_lift__Suc__mono__le,axiom,
! [N_4: nat,N_3: nat,F_38: nat > int > $o] :
( ! [N_1: nat] : ( ord_less_eq_int_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N_4 @ N_3 )
=> ( ord_less_eq_int_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
thf(fact_271_pigeonhole__infinite,axiom,
! [F_37: nat > int > $o,A_146: nat > $o] :
( ~ ( finite_finite_nat @ A_146 )
=> ( ( finite_finite_int_o @ ( image_nat_int_o @ F_37 @ A_146 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_146 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_37: nat] :
( (&) @ ( member_nat @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_272_pigeonhole__infinite,axiom,
! [F_37: nat > nat > $o,A_146: nat > $o] :
( ~ ( finite_finite_nat @ A_146 )
=> ( ( finite_finite_nat_o @ ( image_nat_nat_o @ F_37 @ A_146 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_146 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_37: nat] :
( (&) @ ( member_nat @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_273_pigeonhole__infinite,axiom,
! [F_37: nat > pname > $o,A_146: nat > $o] :
( ~ ( finite_finite_nat @ A_146 )
=> ( ( finite297249702name_o @ ( image_nat_pname_o @ F_37 @ A_146 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_146 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_37: nat] :
( (&) @ ( member_nat @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_274_pigeonhole__infinite,axiom,
! [F_37: nat > x_a > $o,A_146: nat > $o] :
( ~ ( finite_finite_nat @ A_146 )
=> ( ( finite_finite_a_o @ ( image_nat_a_o @ F_37 @ A_146 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_146 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_37: nat] :
( (&) @ ( member_nat @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_275_pigeonhole__infinite,axiom,
! [F_37: int > x_a,A_146: int > $o] :
( ~ ( finite_finite_int @ A_146 )
=> ( ( finite_finite_a @ ( image_int_a @ F_37 @ A_146 ) )
=> ? [X_1: int] :
( ( member_int @ X_1 @ A_146 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A_37: int] :
( (&) @ ( member_int @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_276_pigeonhole__infinite,axiom,
! [F_37: int > int > $o,A_146: int > $o] :
( ~ ( finite_finite_int @ A_146 )
=> ( ( finite_finite_int_o @ ( image_int_int_o @ F_37 @ A_146 ) )
=> ? [X_1: int] :
( ( member_int @ X_1 @ A_146 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A_37: int] :
( (&) @ ( member_int @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_277_pigeonhole__infinite,axiom,
! [F_37: int > nat > $o,A_146: int > $o] :
( ~ ( finite_finite_int @ A_146 )
=> ( ( finite_finite_nat_o @ ( image_int_nat_o @ F_37 @ A_146 ) )
=> ? [X_1: int] :
( ( member_int @ X_1 @ A_146 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A_37: int] :
( (&) @ ( member_int @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_278_pigeonhole__infinite,axiom,
! [F_37: int > pname > $o,A_146: int > $o] :
( ~ ( finite_finite_int @ A_146 )
=> ( ( finite297249702name_o @ ( image_int_pname_o @ F_37 @ A_146 ) )
=> ? [X_1: int] :
( ( member_int @ X_1 @ A_146 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A_37: int] :
( (&) @ ( member_int @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_279_pigeonhole__infinite,axiom,
! [F_37: int > x_a > $o,A_146: int > $o] :
( ~ ( finite_finite_int @ A_146 )
=> ( ( finite_finite_a_o @ ( image_int_a_o @ F_37 @ A_146 ) )
=> ? [X_1: int] :
( ( member_int @ X_1 @ A_146 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A_37: int] :
( (&) @ ( member_int @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_280_pigeonhole__infinite,axiom,
! [F_37: pname > x_a,A_146: pname > $o] :
( ~ ( finite_finite_pname @ A_146 )
=> ( ( finite_finite_a @ ( image_pname_a @ F_37 @ A_146 ) )
=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_146 )
& ~ ( finite_finite_pname
@ ( collect_pname
@ ^ [A_37: pname] :
( (&) @ ( member_pname @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_281_pigeonhole__infinite,axiom,
! [F_37: nat > int,A_146: nat > $o] :
( ~ ( finite_finite_nat @ A_146 )
=> ( ( finite_finite_int @ ( image_nat_int @ F_37 @ A_146 ) )
=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_146 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A_37: nat] :
( (&) @ ( member_nat @ A_37 @ A_146 )
@ ( ( F_37 @ A_37 )
= ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_282_image__eqI,axiom,
! [A_145: nat > $o,B_83: int,F_36: nat > int,X_50: nat] :
( ( B_83
= ( F_36 @ X_50 ) )
=> ( ( member_nat @ X_50 @ A_145 )
=> ( member_int @ B_83 @ ( image_nat_int @ F_36 @ A_145 ) ) ) ) ).
thf(fact_283_image__eqI,axiom,
! [A_145: pname > $o,B_83: x_a,F_36: pname > x_a,X_50: pname] :
( ( B_83
= ( F_36 @ X_50 ) )
=> ( ( member_pname @ X_50 @ A_145 )
=> ( member_a @ B_83 @ ( image_pname_a @ F_36 @ A_145 ) ) ) ) ).
thf(fact_284_equalityI,axiom,
! [A_144: int > $o,B_82: int > $o] :
( ( ord_less_eq_int_o @ A_144 @ B_82 )
=> ( ( ord_less_eq_int_o @ B_82 @ A_144 )
=> ( A_144 = B_82 ) ) ) ).
thf(fact_285_equalityI,axiom,
! [A_144: nat > $o,B_82: nat > $o] :
( ( ord_less_eq_nat_o @ A_144 @ B_82 )
=> ( ( ord_less_eq_nat_o @ B_82 @ A_144 )
=> ( A_144 = B_82 ) ) ) ).
thf(fact_286_equalityI,axiom,
! [A_144: x_a > $o,B_82: x_a > $o] :
( ( ord_less_eq_a_o @ A_144 @ B_82 )
=> ( ( ord_less_eq_a_o @ B_82 @ A_144 )
=> ( A_144 = B_82 ) ) ) ).
thf(fact_287_subsetD,axiom,
! [C_33: int,A_143: int > $o,B_81: int > $o] :
( ( ord_less_eq_int_o @ A_143 @ B_81 )
=> ( ( member_int @ C_33 @ A_143 )
=> ( member_int @ C_33 @ B_81 ) ) ) ).
thf(fact_288_subsetD,axiom,
! [C_33: nat,A_143: nat > $o,B_81: nat > $o] :
( ( ord_less_eq_nat_o @ A_143 @ B_81 )
=> ( ( member_nat @ C_33 @ A_143 )
=> ( member_nat @ C_33 @ B_81 ) ) ) ).
thf(fact_289_subsetD,axiom,
! [C_33: x_a,A_143: x_a > $o,B_81: x_a > $o] :
( ( ord_less_eq_a_o @ A_143 @ B_81 )
=> ( ( member_a @ C_33 @ A_143 )
=> ( member_a @ C_33 @ B_81 ) ) ) ).
thf(fact_290_subsetD,axiom,
! [C_33: pname,A_143: pname > $o,B_81: pname > $o] :
( ( ord_less_eq_pname_o @ A_143 @ B_81 )
=> ( ( member_pname @ C_33 @ A_143 )
=> ( member_pname @ C_33 @ B_81 ) ) ) ).
thf(fact_291_insertCI,axiom,
! [B_80: x_a,A_142: x_a,B_79: x_a > $o] :
( ( ~ ( member_a @ A_142 @ B_79 )
=> ( A_142 = B_80 ) )
=> ( member_a @ A_142 @ ( insert_a @ B_80 @ B_79 ) ) ) ).
thf(fact_292_insertCI,axiom,
! [B_80: int,A_142: int,B_79: int > $o] :
( ( ~ ( member_int @ A_142 @ B_79 )
=> ( A_142 = B_80 ) )
=> ( member_int @ A_142 @ ( insert_int @ B_80 @ B_79 ) ) ) ).
thf(fact_293_insertCI,axiom,
! [B_80: nat,A_142: nat,B_79: nat > $o] :
( ( ~ ( member_nat @ A_142 @ B_79 )
=> ( A_142 = B_80 ) )
=> ( member_nat @ A_142 @ ( insert_nat @ B_80 @ B_79 ) ) ) ).
thf(fact_294_insertCI,axiom,
! [B_80: pname,A_142: pname,B_79: pname > $o] :
( ( ~ ( member_pname @ A_142 @ B_79 )
=> ( A_142 = B_80 ) )
=> ( member_pname @ A_142 @ ( insert_pname @ B_80 @ B_79 ) ) ) ).
thf(fact_295_insertE,axiom,
! [A_141: x_a,B_78: x_a,A_140: x_a > $o] :
( ( member_a @ A_141 @ ( insert_a @ B_78 @ A_140 ) )
=> ( ( A_141 != B_78 )
=> ( member_a @ A_141 @ A_140 ) ) ) ).
thf(fact_296_insertE,axiom,
! [A_141: int,B_78: int,A_140: int > $o] :
( ( member_int @ A_141 @ ( insert_int @ B_78 @ A_140 ) )
=> ( ( A_141 != B_78 )
=> ( member_int @ A_141 @ A_140 ) ) ) ).
thf(fact_297_insertE,axiom,
! [A_141: nat,B_78: nat,A_140: nat > $o] :
( ( member_nat @ A_141 @ ( insert_nat @ B_78 @ A_140 ) )
=> ( ( A_141 != B_78 )
=> ( member_nat @ A_141 @ A_140 ) ) ) ).
thf(fact_298_insertE,axiom,
! [A_141: pname,B_78: pname,A_140: pname > $o] :
( ( member_pname @ A_141 @ ( insert_pname @ B_78 @ A_140 ) )
=> ( ( A_141 != B_78 )
=> ( member_pname @ A_141 @ A_140 ) ) ) ).
thf(fact_299_zero__induct__lemma,axiom,
! [I_1: nat,P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ ( minus_minus_nat @ K @ I_1 ) ) ) ) ).
thf(fact_300_Suc__le__D,axiom,
! [N: nat,M_3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M_3 )
=> ? [M_1: nat] :
( M_3
= ( suc @ M_1 ) ) ) ).
thf(fact_301_order__refl,axiom,
! [X_49: int > $o] : ( ord_less_eq_int_o @ X_49 @ X_49 ) ).
thf(fact_302_order__refl,axiom,
! [X_49: nat > $o] : ( ord_less_eq_nat_o @ X_49 @ X_49 ) ).
thf(fact_303_order__refl,axiom,
! [X_49: int] : ( ord_less_eq_int @ X_49 @ X_49 ) ).
thf(fact_304_order__refl,axiom,
! [X_49: nat] : ( ord_less_eq_nat @ X_49 @ X_49 ) ).
thf(fact_305_order__refl,axiom,
! [X_49: x_a > $o] : ( ord_less_eq_a_o @ X_49 @ X_49 ) ).
thf(fact_306_linorder__linear,axiom,
! [X_48: int,Y_12: int] :
( ( ord_less_eq_int @ X_48 @ Y_12 )
| ( ord_less_eq_int @ Y_12 @ X_48 ) ) ).
thf(fact_307_linorder__linear,axiom,
! [X_48: nat,Y_12: nat] :
( ( ord_less_eq_nat @ X_48 @ Y_12 )
| ( ord_less_eq_nat @ Y_12 @ X_48 ) ) ).
thf(fact_308_order__eq__iff,axiom,
! [X_47: int > $o,Y_11: int > $o] :
( ( X_47 = Y_11 )
<=> ( ( ord_less_eq_int_o @ X_47 @ Y_11 )
& ( ord_less_eq_int_o @ Y_11 @ X_47 ) ) ) ).
thf(fact_309_order__eq__iff,axiom,
! [X_47: nat > $o,Y_11: nat > $o] :
( ( X_47 = Y_11 )
<=> ( ( ord_less_eq_nat_o @ X_47 @ Y_11 )
& ( ord_less_eq_nat_o @ Y_11 @ X_47 ) ) ) ).
thf(fact_310_order__eq__iff,axiom,
! [X_47: int,Y_11: int] :
( ( X_47 = Y_11 )
<=> ( ( ord_less_eq_int @ X_47 @ Y_11 )
& ( ord_less_eq_int @ Y_11 @ X_47 ) ) ) ).
thf(fact_311_order__eq__iff,axiom,
! [X_47: nat,Y_11: nat] :
( ( X_47 = Y_11 )
<=> ( ( ord_less_eq_nat @ X_47 @ Y_11 )
& ( ord_less_eq_nat @ Y_11 @ X_47 ) ) ) ).
thf(fact_312_order__eq__iff,axiom,
! [X_47: x_a > $o,Y_11: x_a > $o] :
( ( X_47 = Y_11 )
<=> ( ( ord_less_eq_a_o @ X_47 @ Y_11 )
& ( ord_less_eq_a_o @ Y_11 @ X_47 ) ) ) ).
thf(fact_313_order__eq__refl,axiom,
! [X_46: int > $o,Y_10: int > $o] :
( ( X_46 = Y_10 )
=> ( ord_less_eq_int_o @ X_46 @ Y_10 ) ) ).
thf(fact_314_order__eq__refl,axiom,
! [X_46: nat > $o,Y_10: nat > $o] :
( ( X_46 = Y_10 )
=> ( ord_less_eq_nat_o @ X_46 @ Y_10 ) ) ).
thf(fact_315_order__eq__refl,axiom,
! [X_46: int,Y_10: int] :
( ( X_46 = Y_10 )
=> ( ord_less_eq_int @ X_46 @ Y_10 ) ) ).
thf(fact_316_order__eq__refl,axiom,
! [X_46: nat,Y_10: nat] :
( ( X_46 = Y_10 )
=> ( ord_less_eq_nat @ X_46 @ Y_10 ) ) ).
thf(fact_317_order__eq__refl,axiom,
! [X_46: x_a > $o,Y_10: x_a > $o] :
( ( X_46 = Y_10 )
=> ( ord_less_eq_a_o @ X_46 @ Y_10 ) ) ).
thf(fact_318_order__antisym__conv,axiom,
! [Y_9: int > $o,X_45: int > $o] :
( ( ord_less_eq_int_o @ Y_9 @ X_45 )
=> ( ( ord_less_eq_int_o @ X_45 @ Y_9 )
<=> ( X_45 = Y_9 ) ) ) ).
thf(fact_319_order__antisym__conv,axiom,
! [Y_9: nat > $o,X_45: nat > $o] :
( ( ord_less_eq_nat_o @ Y_9 @ X_45 )
=> ( ( ord_less_eq_nat_o @ X_45 @ Y_9 )
<=> ( X_45 = Y_9 ) ) ) ).
thf(fact_320_order__antisym__conv,axiom,
! [Y_9: int,X_45: int] :
( ( ord_less_eq_int @ Y_9 @ X_45 )
=> ( ( ord_less_eq_int @ X_45 @ Y_9 )
<=> ( X_45 = Y_9 ) ) ) ).
thf(fact_321_order__antisym__conv,axiom,
! [Y_9: nat,X_45: nat] :
( ( ord_less_eq_nat @ Y_9 @ X_45 )
=> ( ( ord_less_eq_nat @ X_45 @ Y_9 )
<=> ( X_45 = Y_9 ) ) ) ).
thf(fact_322_order__antisym__conv,axiom,
! [Y_9: x_a > $o,X_45: x_a > $o] :
( ( ord_less_eq_a_o @ Y_9 @ X_45 )
=> ( ( ord_less_eq_a_o @ X_45 @ Y_9 )
<=> ( X_45 = Y_9 ) ) ) ).
thf(fact_323_ord__eq__le__trans,axiom,
! [C_32: int > $o,A_139: int > $o,B_77: int > $o] :
( ( A_139 = B_77 )
=> ( ( ord_less_eq_int_o @ B_77 @ C_32 )
=> ( ord_less_eq_int_o @ A_139 @ C_32 ) ) ) ).
thf(fact_324_ord__eq__le__trans,axiom,
! [C_32: nat > $o,A_139: nat > $o,B_77: nat > $o] :
( ( A_139 = B_77 )
=> ( ( ord_less_eq_nat_o @ B_77 @ C_32 )
=> ( ord_less_eq_nat_o @ A_139 @ C_32 ) ) ) ).
thf(fact_325_ord__eq__le__trans,axiom,
! [C_32: int,A_139: int,B_77: int] :
( ( A_139 = B_77 )
=> ( ( ord_less_eq_int @ B_77 @ C_32 )
=> ( ord_less_eq_int @ A_139 @ C_32 ) ) ) ).
thf(fact_326_ord__eq__le__trans,axiom,
! [C_32: nat,A_139: nat,B_77: nat] :
( ( A_139 = B_77 )
=> ( ( ord_less_eq_nat @ B_77 @ C_32 )
=> ( ord_less_eq_nat @ A_139 @ C_32 ) ) ) ).
thf(fact_327_ord__eq__le__trans,axiom,
! [C_32: x_a > $o,A_139: x_a > $o,B_77: x_a > $o] :
( ( A_139 = B_77 )
=> ( ( ord_less_eq_a_o @ B_77 @ C_32 )
=> ( ord_less_eq_a_o @ A_139 @ C_32 ) ) ) ).
thf(fact_328_xt1_I3_J,axiom,
! [C_31: int > $o,A_138: int > $o,B_76: int > $o] :
( ( A_138 = B_76 )
=> ( ( ord_less_eq_int_o @ C_31 @ B_76 )
=> ( ord_less_eq_int_o @ C_31 @ A_138 ) ) ) ).
thf(fact_329_xt1_I3_J,axiom,
! [C_31: nat > $o,A_138: nat > $o,B_76: nat > $o] :
( ( A_138 = B_76 )
=> ( ( ord_less_eq_nat_o @ C_31 @ B_76 )
=> ( ord_less_eq_nat_o @ C_31 @ A_138 ) ) ) ).
thf(fact_330_xt1_I3_J,axiom,
! [C_31: int,A_138: int,B_76: int] :
( ( A_138 = B_76 )
=> ( ( ord_less_eq_int @ C_31 @ B_76 )
=> ( ord_less_eq_int @ C_31 @ A_138 ) ) ) ).
thf(fact_331_xt1_I3_J,axiom,
! [C_31: nat,A_138: nat,B_76: nat] :
( ( A_138 = B_76 )
=> ( ( ord_less_eq_nat @ C_31 @ B_76 )
=> ( ord_less_eq_nat @ C_31 @ A_138 ) ) ) ).
thf(fact_332_xt1_I3_J,axiom,
! [C_31: x_a > $o,A_138: x_a > $o,B_76: x_a > $o] :
( ( A_138 = B_76 )
=> ( ( ord_less_eq_a_o @ C_31 @ B_76 )
=> ( ord_less_eq_a_o @ C_31 @ A_138 ) ) ) ).
thf(fact_333_ord__le__eq__trans,axiom,
! [C_30: int > $o,A_137: int > $o,B_75: int > $o] :
( ( ord_less_eq_int_o @ A_137 @ B_75 )
=> ( ( B_75 = C_30 )
=> ( ord_less_eq_int_o @ A_137 @ C_30 ) ) ) ).
thf(fact_334_ord__le__eq__trans,axiom,
! [C_30: nat > $o,A_137: nat > $o,B_75: nat > $o] :
( ( ord_less_eq_nat_o @ A_137 @ B_75 )
=> ( ( B_75 = C_30 )
=> ( ord_less_eq_nat_o @ A_137 @ C_30 ) ) ) ).
thf(fact_335_ord__le__eq__trans,axiom,
! [C_30: int,A_137: int,B_75: int] :
( ( ord_less_eq_int @ A_137 @ B_75 )
=> ( ( B_75 = C_30 )
=> ( ord_less_eq_int @ A_137 @ C_30 ) ) ) ).
thf(fact_336_ord__le__eq__trans,axiom,
! [C_30: nat,A_137: nat,B_75: nat] :
( ( ord_less_eq_nat @ A_137 @ B_75 )
=> ( ( B_75 = C_30 )
=> ( ord_less_eq_nat @ A_137 @ C_30 ) ) ) ).
thf(fact_337_ord__le__eq__trans,axiom,
! [C_30: x_a > $o,A_137: x_a > $o,B_75: x_a > $o] :
( ( ord_less_eq_a_o @ A_137 @ B_75 )
=> ( ( B_75 = C_30 )
=> ( ord_less_eq_a_o @ A_137 @ C_30 ) ) ) ).
thf(fact_338_xt1_I4_J,axiom,
! [C_29: int > $o,B_74: int > $o,A_136: int > $o] :
( ( ord_less_eq_int_o @ B_74 @ A_136 )
=> ( ( B_74 = C_29 )
=> ( ord_less_eq_int_o @ C_29 @ A_136 ) ) ) ).
thf(fact_339_xt1_I4_J,axiom,
! [C_29: nat > $o,B_74: nat > $o,A_136: nat > $o] :
( ( ord_less_eq_nat_o @ B_74 @ A_136 )
=> ( ( B_74 = C_29 )
=> ( ord_less_eq_nat_o @ C_29 @ A_136 ) ) ) ).
thf(fact_340_xt1_I4_J,axiom,
! [C_29: int,B_74: int,A_136: int] :
( ( ord_less_eq_int @ B_74 @ A_136 )
=> ( ( B_74 = C_29 )
=> ( ord_less_eq_int @ C_29 @ A_136 ) ) ) ).
thf(fact_341_xt1_I4_J,axiom,
! [C_29: nat,B_74: nat,A_136: nat] :
( ( ord_less_eq_nat @ B_74 @ A_136 )
=> ( ( B_74 = C_29 )
=> ( ord_less_eq_nat @ C_29 @ A_136 ) ) ) ).
thf(fact_342_xt1_I4_J,axiom,
! [C_29: x_a > $o,B_74: x_a > $o,A_136: x_a > $o] :
( ( ord_less_eq_a_o @ B_74 @ A_136 )
=> ( ( B_74 = C_29 )
=> ( ord_less_eq_a_o @ C_29 @ A_136 ) ) ) ).
thf(fact_343_order__antisym,axiom,
! [X_44: int > $o,Y_8: int > $o] :
( ( ord_less_eq_int_o @ X_44 @ Y_8 )
=> ( ( ord_less_eq_int_o @ Y_8 @ X_44 )
=> ( X_44 = Y_8 ) ) ) ).
thf(fact_344_order__antisym,axiom,
! [X_44: nat > $o,Y_8: nat > $o] :
( ( ord_less_eq_nat_o @ X_44 @ Y_8 )
=> ( ( ord_less_eq_nat_o @ Y_8 @ X_44 )
=> ( X_44 = Y_8 ) ) ) ).
thf(fact_345_order__antisym,axiom,
! [X_44: int,Y_8: int] :
( ( ord_less_eq_int @ X_44 @ Y_8 )
=> ( ( ord_less_eq_int @ Y_8 @ X_44 )
=> ( X_44 = Y_8 ) ) ) ).
thf(fact_346_order__antisym,axiom,
! [X_44: nat,Y_8: nat] :
( ( ord_less_eq_nat @ X_44 @ Y_8 )
=> ( ( ord_less_eq_nat @ Y_8 @ X_44 )
=> ( X_44 = Y_8 ) ) ) ).
thf(fact_347_order__antisym,axiom,
! [X_44: x_a > $o,Y_8: x_a > $o] :
( ( ord_less_eq_a_o @ X_44 @ Y_8 )
=> ( ( ord_less_eq_a_o @ Y_8 @ X_44 )
=> ( X_44 = Y_8 ) ) ) ).
thf(fact_348_order__trans,axiom,
! [Z_6: int > $o,X_43: int > $o,Y_7: int > $o] :
( ( ord_less_eq_int_o @ X_43 @ Y_7 )
=> ( ( ord_less_eq_int_o @ Y_7 @ Z_6 )
=> ( ord_less_eq_int_o @ X_43 @ Z_6 ) ) ) ).
thf(fact_349_order__trans,axiom,
! [Z_6: nat > $o,X_43: nat > $o,Y_7: nat > $o] :
( ( ord_less_eq_nat_o @ X_43 @ Y_7 )
=> ( ( ord_less_eq_nat_o @ Y_7 @ Z_6 )
=> ( ord_less_eq_nat_o @ X_43 @ Z_6 ) ) ) ).
thf(fact_350_order__trans,axiom,
! [Z_6: int,X_43: int,Y_7: int] :
( ( ord_less_eq_int @ X_43 @ Y_7 )
=> ( ( ord_less_eq_int @ Y_7 @ Z_6 )
=> ( ord_less_eq_int @ X_43 @ Z_6 ) ) ) ).
thf(fact_351_order__trans,axiom,
! [Z_6: nat,X_43: nat,Y_7: nat] :
( ( ord_less_eq_nat @ X_43 @ Y_7 )
=> ( ( ord_less_eq_nat @ Y_7 @ Z_6 )
=> ( ord_less_eq_nat @ X_43 @ Z_6 ) ) ) ).
thf(fact_352_order__trans,axiom,
! [Z_6: x_a > $o,X_43: x_a > $o,Y_7: x_a > $o] :
( ( ord_less_eq_a_o @ X_43 @ Y_7 )
=> ( ( ord_less_eq_a_o @ Y_7 @ Z_6 )
=> ( ord_less_eq_a_o @ X_43 @ Z_6 ) ) ) ).
thf(fact_353_xt1_I5_J,axiom,
! [Y_6: int > $o,X_42: int > $o] :
( ( ord_less_eq_int_o @ Y_6 @ X_42 )
=> ( ( ord_less_eq_int_o @ X_42 @ Y_6 )
=> ( X_42 = Y_6 ) ) ) ).
thf(fact_354_xt1_I5_J,axiom,
! [Y_6: nat > $o,X_42: nat > $o] :
( ( ord_less_eq_nat_o @ Y_6 @ X_42 )
=> ( ( ord_less_eq_nat_o @ X_42 @ Y_6 )
=> ( X_42 = Y_6 ) ) ) ).
thf(fact_355_xt1_I5_J,axiom,
! [Y_6: int,X_42: int] :
( ( ord_less_eq_int @ Y_6 @ X_42 )
=> ( ( ord_less_eq_int @ X_42 @ Y_6 )
=> ( X_42 = Y_6 ) ) ) ).
thf(fact_356_xt1_I5_J,axiom,
! [Y_6: nat,X_42: nat] :
( ( ord_less_eq_nat @ Y_6 @ X_42 )
=> ( ( ord_less_eq_nat @ X_42 @ Y_6 )
=> ( X_42 = Y_6 ) ) ) ).
thf(fact_357_xt1_I5_J,axiom,
! [Y_6: x_a > $o,X_42: x_a > $o] :
( ( ord_less_eq_a_o @ Y_6 @ X_42 )
=> ( ( ord_less_eq_a_o @ X_42 @ Y_6 )
=> ( X_42 = Y_6 ) ) ) ).
thf(fact_358_xt1_I6_J,axiom,
! [Z_5: int > $o,Y_5: int > $o,X_41: int > $o] :
( ( ord_less_eq_int_o @ Y_5 @ X_41 )
=> ( ( ord_less_eq_int_o @ Z_5 @ Y_5 )
=> ( ord_less_eq_int_o @ Z_5 @ X_41 ) ) ) ).
thf(fact_359_xt1_I6_J,axiom,
! [Z_5: nat > $o,Y_5: nat > $o,X_41: nat > $o] :
( ( ord_less_eq_nat_o @ Y_5 @ X_41 )
=> ( ( ord_less_eq_nat_o @ Z_5 @ Y_5 )
=> ( ord_less_eq_nat_o @ Z_5 @ X_41 ) ) ) ).
thf(fact_360_xt1_I6_J,axiom,
! [Z_5: int,Y_5: int,X_41: int] :
( ( ord_less_eq_int @ Y_5 @ X_41 )
=> ( ( ord_less_eq_int @ Z_5 @ Y_5 )
=> ( ord_less_eq_int @ Z_5 @ X_41 ) ) ) ).
thf(fact_361_xt1_I6_J,axiom,
! [Z_5: nat,Y_5: nat,X_41: nat] :
( ( ord_less_eq_nat @ Y_5 @ X_41 )
=> ( ( ord_less_eq_nat @ Z_5 @ Y_5 )
=> ( ord_less_eq_nat @ Z_5 @ X_41 ) ) ) ).
thf(fact_362_xt1_I6_J,axiom,
! [Z_5: x_a > $o,Y_5: x_a > $o,X_41: x_a > $o] :
( ( ord_less_eq_a_o @ Y_5 @ X_41 )
=> ( ( ord_less_eq_a_o @ Z_5 @ Y_5 )
=> ( ord_less_eq_a_o @ Z_5 @ X_41 ) ) ) ).
thf(fact_363_linorder__le__cases,axiom,
! [X_40: int,Y_4: int] :
( ~ ( ord_less_eq_int @ X_40 @ Y_4 )
=> ( ord_less_eq_int @ Y_4 @ X_40 ) ) ).
thf(fact_364_linorder__le__cases,axiom,
! [X_40: nat,Y_4: nat] :
( ~ ( ord_less_eq_nat @ X_40 @ Y_4 )
=> ( ord_less_eq_nat @ Y_4 @ X_40 ) ) ).
thf(fact_365_insertI1,axiom,
! [A_135: x_a,B_73: x_a > $o] : ( member_a @ A_135 @ ( insert_a @ A_135 @ B_73 ) ) ).
thf(fact_366_insertI1,axiom,
! [A_135: int,B_73: int > $o] : ( member_int @ A_135 @ ( insert_int @ A_135 @ B_73 ) ) ).
thf(fact_367_insertI1,axiom,
! [A_135: nat,B_73: nat > $o] : ( member_nat @ A_135 @ ( insert_nat @ A_135 @ B_73 ) ) ).
thf(fact_368_insertI1,axiom,
! [A_135: pname,B_73: pname > $o] : ( member_pname @ A_135 @ ( insert_pname @ A_135 @ B_73 ) ) ).
thf(fact_369_insert__compr,axiom,
! [A_134: x_a,B_72: x_a > $o] :
( ( insert_a @ A_134 @ B_72 )
= ( collect_a
@ ^ [X_1: x_a] : ( (|) @ ( X_1 = A_134 ) @ ( member_a @ X_1 @ B_72 ) ) ) ) ).
thf(fact_370_insert__compr,axiom,
! [A_134: int,B_72: int > $o] :
( ( insert_int @ A_134 @ B_72 )
= ( collect_int
@ ^ [X_1: int] : ( (|) @ ( X_1 = A_134 ) @ ( member_int @ X_1 @ B_72 ) ) ) ) ).
thf(fact_371_insert__compr,axiom,
! [A_134: nat,B_72: nat > $o] :
( ( insert_nat @ A_134 @ B_72 )
= ( collect_nat
@ ^ [X_1: nat] : ( (|) @ ( X_1 = A_134 ) @ ( member_nat @ X_1 @ B_72 ) ) ) ) ).
thf(fact_372_insert__compr,axiom,
! [A_134: pname,B_72: pname > $o] :
( ( insert_pname @ A_134 @ B_72 )
= ( collect_pname
@ ^ [X_1: pname] : ( (|) @ ( X_1 = A_134 ) @ ( member_pname @ X_1 @ B_72 ) ) ) ) ).
thf(fact_373_insert__Collect,axiom,
! [A_133: x_a,P_11: x_a > $o] :
( ( insert_a @ A_133 @ ( collect_a @ P_11 ) )
= ( collect_a
@ ^ [U_1: x_a] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
thf(fact_374_insert__Collect,axiom,
! [A_133: int,P_11: int > $o] :
( ( insert_int @ A_133 @ ( collect_int @ P_11 ) )
= ( collect_int
@ ^ [U_1: int] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
thf(fact_375_insert__Collect,axiom,
! [A_133: nat,P_11: nat > $o] :
( ( insert_nat @ A_133 @ ( collect_nat @ P_11 ) )
= ( collect_nat
@ ^ [U_1: nat] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
thf(fact_376_mem__def,axiom,
! [X_39: int,A_132: int > $o] :
( ( member_int @ X_39 @ A_132 )
<=> ( A_132 @ X_39 ) ) ).
thf(fact_377_mem__def,axiom,
! [X_39: nat,A_132: nat > $o] :
( ( member_nat @ X_39 @ A_132 )
<=> ( A_132 @ X_39 ) ) ).
thf(fact_378_mem__def,axiom,
! [X_39: x_a,A_132: x_a > $o] :
( ( member_a @ X_39 @ A_132 )
<=> ( A_132 @ X_39 ) ) ).
thf(fact_379_mem__def,axiom,
! [X_39: pname,A_132: pname > $o] :
( ( member_pname @ X_39 @ A_132 )
<=> ( A_132 @ X_39 ) ) ).
thf(fact_380_Collect__def,axiom,
! [P_10: int > $o] :
( ( collect_int @ P_10 )
= P_10 ) ).
thf(fact_381_Collect__def,axiom,
! [P_10: nat > $o] :
( ( collect_nat @ P_10 )
= P_10 ) ).
thf(fact_382_insert__absorb2,axiom,
! [X_38: x_a,A_131: x_a > $o] :
( ( insert_a @ X_38 @ ( insert_a @ X_38 @ A_131 ) )
= ( insert_a @ X_38 @ A_131 ) ) ).
thf(fact_383_insert__commute,axiom,
! [X_37: x_a,Y_3: x_a,A_130: x_a > $o] :
( ( insert_a @ X_37 @ ( insert_a @ Y_3 @ A_130 ) )
= ( insert_a @ Y_3 @ ( insert_a @ X_37 @ A_130 ) ) ) ).
thf(fact_384_insert__iff,axiom,
! [A_129: x_a,B_71: x_a,A_128: x_a > $o] :
( ( member_a @ A_129 @ ( insert_a @ B_71 @ A_128 ) )
<=> ( ( A_129 = B_71 )
| ( member_a @ A_129 @ A_128 ) ) ) ).
thf(fact_385_insert__iff,axiom,
! [A_129: int,B_71: int,A_128: int > $o] :
( ( member_int @ A_129 @ ( insert_int @ B_71 @ A_128 ) )
<=> ( ( A_129 = B_71 )
| ( member_int @ A_129 @ A_128 ) ) ) ).
thf(fact_386_insert__iff,axiom,
! [A_129: nat,B_71: nat,A_128: nat > $o] :
( ( member_nat @ A_129 @ ( insert_nat @ B_71 @ A_128 ) )
<=> ( ( A_129 = B_71 )
| ( member_nat @ A_129 @ A_128 ) ) ) ).
thf(fact_387_insert__iff,axiom,
! [A_129: pname,B_71: pname,A_128: pname > $o] :
( ( member_pname @ A_129 @ ( insert_pname @ B_71 @ A_128 ) )
<=> ( ( A_129 = B_71 )
| ( member_pname @ A_129 @ A_128 ) ) ) ).
thf(fact_388_insert__code,axiom,
! [Y_2: x_a,A_127: x_a > $o,X_36: x_a] :
( ( insert_a @ Y_2 @ A_127 @ X_36 )
<=> ( ( Y_2 = X_36 )
| ( A_127 @ X_36 ) ) ) ).
thf(fact_389_insert__ident,axiom,
! [B_70: x_a > $o,X_35: x_a,A_126: x_a > $o] :
( ~ ( member_a @ X_35 @ A_126 )
=> ( ~ ( member_a @ X_35 @ B_70 )
=> ( ( ( insert_a @ X_35 @ A_126 )
= ( insert_a @ X_35 @ B_70 ) )
<=> ( A_126 = B_70 ) ) ) ) ).
thf(fact_390_insert__ident,axiom,
! [B_70: int > $o,X_35: int,A_126: int > $o] :
( ~ ( member_int @ X_35 @ A_126 )
=> ( ~ ( member_int @ X_35 @ B_70 )
=> ( ( ( insert_int @ X_35 @ A_126 )
= ( insert_int @ X_35 @ B_70 ) )
<=> ( A_126 = B_70 ) ) ) ) ).
thf(fact_391_insert__ident,axiom,
! [B_70: nat > $o,X_35: nat,A_126: nat > $o] :
( ~ ( member_nat @ X_35 @ A_126 )
=> ( ~ ( member_nat @ X_35 @ B_70 )
=> ( ( ( insert_nat @ X_35 @ A_126 )
= ( insert_nat @ X_35 @ B_70 ) )
<=> ( A_126 = B_70 ) ) ) ) ).
thf(fact_392_insert__ident,axiom,
! [B_70: pname > $o,X_35: pname,A_126: pname > $o] :
( ~ ( member_pname @ X_35 @ A_126 )
=> ( ~ ( member_pname @ X_35 @ B_70 )
=> ( ( ( insert_pname @ X_35 @ A_126 )
= ( insert_pname @ X_35 @ B_70 ) )
<=> ( A_126 = B_70 ) ) ) ) ).
thf(fact_393_insertI2,axiom,
! [B_69: x_a,A_125: x_a,B_68: x_a > $o] :
( ( member_a @ A_125 @ B_68 )
=> ( member_a @ A_125 @ ( insert_a @ B_69 @ B_68 ) ) ) ).
thf(fact_394_insertI2,axiom,
! [B_69: int,A_125: int,B_68: int > $o] :
( ( member_int @ A_125 @ B_68 )
=> ( member_int @ A_125 @ ( insert_int @ B_69 @ B_68 ) ) ) ).
thf(fact_395_insertI2,axiom,
! [B_69: nat,A_125: nat,B_68: nat > $o] :
( ( member_nat @ A_125 @ B_68 )
=> ( member_nat @ A_125 @ ( insert_nat @ B_69 @ B_68 ) ) ) ).
thf(fact_396_insertI2,axiom,
! [B_69: pname,A_125: pname,B_68: pname > $o] :
( ( member_pname @ A_125 @ B_68 )
=> ( member_pname @ A_125 @ ( insert_pname @ B_69 @ B_68 ) ) ) ).
thf(fact_397_insert__absorb,axiom,
! [A_124: x_a,A_123: x_a > $o] :
( ( member_a @ A_124 @ A_123 )
=> ( ( insert_a @ A_124 @ A_123 )
= A_123 ) ) ).
thf(fact_398_insert__absorb,axiom,
! [A_124: int,A_123: int > $o] :
( ( member_int @ A_124 @ A_123 )
=> ( ( insert_int @ A_124 @ A_123 )
= A_123 ) ) ).
thf(fact_399_insert__absorb,axiom,
! [A_124: nat,A_123: nat > $o] :
( ( member_nat @ A_124 @ A_123 )
=> ( ( insert_nat @ A_124 @ A_123 )
= A_123 ) ) ).
thf(fact_400_insert__absorb,axiom,
! [A_124: pname,A_123: pname > $o] :
( ( member_pname @ A_124 @ A_123 )
=> ( ( insert_pname @ A_124 @ A_123 )
= A_123 ) ) ).
thf(fact_401_subset__refl,axiom,
! [A_122: int > $o] : ( ord_less_eq_int_o @ A_122 @ A_122 ) ).
thf(fact_402_subset__refl,axiom,
! [A_122: nat > $o] : ( ord_less_eq_nat_o @ A_122 @ A_122 ) ).
thf(fact_403_subset__refl,axiom,
! [A_122: x_a > $o] : ( ord_less_eq_a_o @ A_122 @ A_122 ) ).
thf(fact_404_set__eq__subset,axiom,
! [A_121: int > $o,B_67: int > $o] :
( ( A_121 = B_67 )
<=> ( ( ord_less_eq_int_o @ A_121 @ B_67 )
& ( ord_less_eq_int_o @ B_67 @ A_121 ) ) ) ).
thf(fact_405_set__eq__subset,axiom,
! [A_121: nat > $o,B_67: nat > $o] :
( ( A_121 = B_67 )
<=> ( ( ord_less_eq_nat_o @ A_121 @ B_67 )
& ( ord_less_eq_nat_o @ B_67 @ A_121 ) ) ) ).
thf(fact_406_set__eq__subset,axiom,
! [A_121: x_a > $o,B_67: x_a > $o] :
( ( A_121 = B_67 )
<=> ( ( ord_less_eq_a_o @ A_121 @ B_67 )
& ( ord_less_eq_a_o @ B_67 @ A_121 ) ) ) ).
thf(fact_407_equalityD1,axiom,
! [A_120: int > $o,B_66: int > $o] :
( ( A_120 = B_66 )
=> ( ord_less_eq_int_o @ A_120 @ B_66 ) ) ).
thf(fact_408_equalityD1,axiom,
! [A_120: nat > $o,B_66: nat > $o] :
( ( A_120 = B_66 )
=> ( ord_less_eq_nat_o @ A_120 @ B_66 ) ) ).
thf(fact_409_equalityD1,axiom,
! [A_120: x_a > $o,B_66: x_a > $o] :
( ( A_120 = B_66 )
=> ( ord_less_eq_a_o @ A_120 @ B_66 ) ) ).
thf(fact_410_equalityD2,axiom,
! [A_119: int > $o,B_65: int > $o] :
( ( A_119 = B_65 )
=> ( ord_less_eq_int_o @ B_65 @ A_119 ) ) ).
thf(fact_411_equalityD2,axiom,
! [A_119: nat > $o,B_65: nat > $o] :
( ( A_119 = B_65 )
=> ( ord_less_eq_nat_o @ B_65 @ A_119 ) ) ).
thf(fact_412_equalityD2,axiom,
! [A_119: x_a > $o,B_65: x_a > $o] :
( ( A_119 = B_65 )
=> ( ord_less_eq_a_o @ B_65 @ A_119 ) ) ).
thf(fact_413_in__mono,axiom,
! [X_34: int,A_118: int > $o,B_64: int > $o] :
( ( ord_less_eq_int_o @ A_118 @ B_64 )
=> ( ( member_int @ X_34 @ A_118 )
=> ( member_int @ X_34 @ B_64 ) ) ) ).
thf(fact_414_in__mono,axiom,
! [X_34: nat,A_118: nat > $o,B_64: nat > $o] :
( ( ord_less_eq_nat_o @ A_118 @ B_64 )
=> ( ( member_nat @ X_34 @ A_118 )
=> ( member_nat @ X_34 @ B_64 ) ) ) ).
thf(fact_415_in__mono,axiom,
! [X_34: x_a,A_118: x_a > $o,B_64: x_a > $o] :
( ( ord_less_eq_a_o @ A_118 @ B_64 )
=> ( ( member_a @ X_34 @ A_118 )
=> ( member_a @ X_34 @ B_64 ) ) ) ).
thf(fact_416_in__mono,axiom,
! [X_34: pname,A_118: pname > $o,B_64: pname > $o] :
( ( ord_less_eq_pname_o @ A_118 @ B_64 )
=> ( ( member_pname @ X_34 @ A_118 )
=> ( member_pname @ X_34 @ B_64 ) ) ) ).
thf(fact_417_set__rev__mp,axiom,
! [B_63: int > $o,X_33: int,A_117: int > $o] :
( ( member_int @ X_33 @ A_117 )
=> ( ( ord_less_eq_int_o @ A_117 @ B_63 )
=> ( member_int @ X_33 @ B_63 ) ) ) ).
thf(fact_418_set__rev__mp,axiom,
! [B_63: nat > $o,X_33: nat,A_117: nat > $o] :
( ( member_nat @ X_33 @ A_117 )
=> ( ( ord_less_eq_nat_o @ A_117 @ B_63 )
=> ( member_nat @ X_33 @ B_63 ) ) ) ).
thf(fact_419_set__rev__mp,axiom,
! [B_63: x_a > $o,X_33: x_a,A_117: x_a > $o] :
( ( member_a @ X_33 @ A_117 )
=> ( ( ord_less_eq_a_o @ A_117 @ B_63 )
=> ( member_a @ X_33 @ B_63 ) ) ) ).
thf(fact_420_set__rev__mp,axiom,
! [B_63: pname > $o,X_33: pname,A_117: pname > $o] :
( ( member_pname @ X_33 @ A_117 )
=> ( ( ord_less_eq_pname_o @ A_117 @ B_63 )
=> ( member_pname @ X_33 @ B_63 ) ) ) ).
thf(fact_421_set__mp,axiom,
! [X_32: int,A_116: int > $o,B_62: int > $o] :
( ( ord_less_eq_int_o @ A_116 @ B_62 )
=> ( ( member_int @ X_32 @ A_116 )
=> ( member_int @ X_32 @ B_62 ) ) ) ).
thf(fact_422_set__mp,axiom,
! [X_32: nat,A_116: nat > $o,B_62: nat > $o] :
( ( ord_less_eq_nat_o @ A_116 @ B_62 )
=> ( ( member_nat @ X_32 @ A_116 )
=> ( member_nat @ X_32 @ B_62 ) ) ) ).
thf(fact_423_set__mp,axiom,
! [X_32: x_a,A_116: x_a > $o,B_62: x_a > $o] :
( ( ord_less_eq_a_o @ A_116 @ B_62 )
=> ( ( member_a @ X_32 @ A_116 )
=> ( member_a @ X_32 @ B_62 ) ) ) ).
thf(fact_424_set__mp,axiom,
! [X_32: pname,A_116: pname > $o,B_62: pname > $o] :
( ( ord_less_eq_pname_o @ A_116 @ B_62 )
=> ( ( member_pname @ X_32 @ A_116 )
=> ( member_pname @ X_32 @ B_62 ) ) ) ).
thf(fact_425_subset__trans,axiom,
! [C_28: int > $o,A_115: int > $o,B_61: int > $o] :
( ( ord_less_eq_int_o @ A_115 @ B_61 )
=> ( ( ord_less_eq_int_o @ B_61 @ C_28 )
=> ( ord_less_eq_int_o @ A_115 @ C_28 ) ) ) ).
thf(fact_426_subset__trans,axiom,
! [C_28: nat > $o,A_115: nat > $o,B_61: nat > $o] :
( ( ord_less_eq_nat_o @ A_115 @ B_61 )
=> ( ( ord_less_eq_nat_o @ B_61 @ C_28 )
=> ( ord_less_eq_nat_o @ A_115 @ C_28 ) ) ) ).
thf(fact_427_subset__trans,axiom,
! [C_28: x_a > $o,A_115: x_a > $o,B_61: x_a > $o] :
( ( ord_less_eq_a_o @ A_115 @ B_61 )
=> ( ( ord_less_eq_a_o @ B_61 @ C_28 )
=> ( ord_less_eq_a_o @ A_115 @ C_28 ) ) ) ).
thf(fact_428_equalityE,axiom,
! [A_114: int > $o,B_60: int > $o] :
( ( A_114 = B_60 )
=> ~ ( ( ord_less_eq_int_o @ A_114 @ B_60 )
=> ~ ( ord_less_eq_int_o @ B_60 @ A_114 ) ) ) ).
thf(fact_429_equalityE,axiom,
! [A_114: nat > $o,B_60: nat > $o] :
( ( A_114 = B_60 )
=> ~ ( ( ord_less_eq_nat_o @ A_114 @ B_60 )
=> ~ ( ord_less_eq_nat_o @ B_60 @ A_114 ) ) ) ).
thf(fact_430_equalityE,axiom,
! [A_114: x_a > $o,B_60: x_a > $o] :
( ( A_114 = B_60 )
=> ~ ( ( ord_less_eq_a_o @ A_114 @ B_60 )
=> ~ ( ord_less_eq_a_o @ B_60 @ A_114 ) ) ) ).
thf(fact_431_image__iff,axiom,
! [Z_4: int,F_35: nat > int,A_113: nat > $o] :
( ( member_int @ Z_4 @ ( image_nat_int @ F_35 @ A_113 ) )
<=> ? [X_1: nat] :
( ( member_nat @ X_1 @ A_113 )
& ( Z_4
= ( F_35 @ X_1 ) ) ) ) ).
thf(fact_432_image__iff,axiom,
! [Z_4: x_a,F_35: pname > x_a,A_113: pname > $o] :
( ( member_a @ Z_4 @ ( image_pname_a @ F_35 @ A_113 ) )
<=> ? [X_1: pname] :
( ( member_pname @ X_1 @ A_113 )
& ( Z_4
= ( F_35 @ X_1 ) ) ) ) ).
thf(fact_433_imageI,axiom,
! [F_34: nat > int,X_31: nat,A_112: nat > $o] :
( ( member_nat @ X_31 @ A_112 )
=> ( member_int @ ( F_34 @ X_31 ) @ ( image_nat_int @ F_34 @ A_112 ) ) ) ).
thf(fact_434_imageI,axiom,
! [F_34: pname > x_a,X_31: pname,A_112: pname > $o] :
( ( member_pname @ X_31 @ A_112 )
=> ( member_a @ ( F_34 @ X_31 ) @ ( image_pname_a @ F_34 @ A_112 ) ) ) ).
thf(fact_435_rev__image__eqI,axiom,
! [B_59: int,F_33: nat > int,X_30: nat,A_111: nat > $o] :
( ( member_nat @ X_30 @ A_111 )
=> ( ( B_59
= ( F_33 @ X_30 ) )
=> ( member_int @ B_59 @ ( image_nat_int @ F_33 @ A_111 ) ) ) ) ).
thf(fact_436_rev__image__eqI,axiom,
! [B_59: x_a,F_33: pname > x_a,X_30: pname,A_111: pname > $o] :
( ( member_pname @ X_30 @ A_111 )
=> ( ( B_59
= ( F_33 @ X_30 ) )
=> ( member_a @ B_59 @ ( image_pname_a @ F_33 @ A_111 ) ) ) ) ).
thf(fact_437_insert__compr__raw,axiom,
! [X_1: x_a,Xa: x_a > $o] :
( ( insert_a @ X_1 @ Xa )
= ( collect_a
@ ^ [Y_1: x_a] : ( (|) @ ( Y_1 = X_1 ) @ ( member_a @ Y_1 @ Xa ) ) ) ) ).
thf(fact_438_insert__compr__raw,axiom,
! [X_1: int,Xa: int > $o] :
( ( insert_int @ X_1 @ Xa )
= ( collect_int
@ ^ [Y_1: int] : ( (|) @ ( Y_1 = X_1 ) @ ( member_int @ Y_1 @ Xa ) ) ) ) ).
thf(fact_439_insert__compr__raw,axiom,
! [X_1: nat,Xa: nat > $o] :
( ( insert_nat @ X_1 @ Xa )
= ( collect_nat
@ ^ [Y_1: nat] : ( (|) @ ( Y_1 = X_1 ) @ ( member_nat @ Y_1 @ Xa ) ) ) ) ).
thf(fact_440_insert__compr__raw,axiom,
! [X_1: pname,Xa: pname > $o] :
( ( insert_pname @ X_1 @ Xa )
= ( collect_pname
@ ^ [Y_1: pname] : ( (|) @ ( Y_1 = X_1 ) @ ( member_pname @ Y_1 @ Xa ) ) ) ) ).
thf(fact_441_le__fun__def,axiom,
! [F_32: int > $o,G_4: int > $o] :
( ( ord_less_eq_int_o @ F_32 @ G_4 )
<=> ! [X_1: int] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
thf(fact_442_le__fun__def,axiom,
! [F_32: nat > $o,G_4: nat > $o] :
( ( ord_less_eq_nat_o @ F_32 @ G_4 )
<=> ! [X_1: nat] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
thf(fact_443_le__fun__def,axiom,
! [F_32: x_a > $o,G_4: x_a > $o] :
( ( ord_less_eq_a_o @ F_32 @ G_4 )
<=> ! [X_1: x_a] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
thf(fact_444_le__funD,axiom,
! [X_29: int,F_31: int > $o,G_3: int > $o] :
( ( ord_less_eq_int_o @ F_31 @ G_3 )
=> ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
thf(fact_445_le__funD,axiom,
! [X_29: nat,F_31: nat > $o,G_3: nat > $o] :
( ( ord_less_eq_nat_o @ F_31 @ G_3 )
=> ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
thf(fact_446_le__funD,axiom,
! [X_29: x_a,F_31: x_a > $o,G_3: x_a > $o] :
( ( ord_less_eq_a_o @ F_31 @ G_3 )
=> ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
thf(fact_447_le__funE,axiom,
! [X_28: int,F_30: int > $o,G_2: int > $o] :
( ( ord_less_eq_int_o @ F_30 @ G_2 )
=> ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
thf(fact_448_le__funE,axiom,
! [X_28: nat,F_30: nat > $o,G_2: nat > $o] :
( ( ord_less_eq_nat_o @ F_30 @ G_2 )
=> ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
thf(fact_449_le__funE,axiom,
! [X_28: x_a,F_30: x_a > $o,G_2: x_a > $o] :
( ( ord_less_eq_a_o @ F_30 @ G_2 )
=> ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
thf(fact_450_subset__insertI,axiom,
! [B_58: x_a > $o,A_110: x_a] : ( ord_less_eq_a_o @ B_58 @ ( insert_a @ A_110 @ B_58 ) ) ).
thf(fact_451_subset__insertI,axiom,
! [B_58: int > $o,A_110: int] : ( ord_less_eq_int_o @ B_58 @ ( insert_int @ A_110 @ B_58 ) ) ).
thf(fact_452_subset__insertI,axiom,
! [B_58: nat > $o,A_110: nat] : ( ord_less_eq_nat_o @ B_58 @ ( insert_nat @ A_110 @ B_58 ) ) ).
thf(fact_453_insert__subset,axiom,
! [X_27: x_a,A_109: x_a > $o,B_57: x_a > $o] :
( ( ord_less_eq_a_o @ ( insert_a @ X_27 @ A_109 ) @ B_57 )
<=> ( ( member_a @ X_27 @ B_57 )
& ( ord_less_eq_a_o @ A_109 @ B_57 ) ) ) ).
thf(fact_454_insert__subset,axiom,
! [X_27: int,A_109: int > $o,B_57: int > $o] :
( ( ord_less_eq_int_o @ ( insert_int @ X_27 @ A_109 ) @ B_57 )
<=> ( ( member_int @ X_27 @ B_57 )
& ( ord_less_eq_int_o @ A_109 @ B_57 ) ) ) ).
thf(fact_455_insert__subset,axiom,
! [X_27: nat,A_109: nat > $o,B_57: nat > $o] :
( ( ord_less_eq_nat_o @ ( insert_nat @ X_27 @ A_109 ) @ B_57 )
<=> ( ( member_nat @ X_27 @ B_57 )
& ( ord_less_eq_nat_o @ A_109 @ B_57 ) ) ) ).
thf(fact_456_insert__subset,axiom,
! [X_27: pname,A_109: pname > $o,B_57: pname > $o] :
( ( ord_less_eq_pname_o @ ( insert_pname @ X_27 @ A_109 ) @ B_57 )
<=> ( ( member_pname @ X_27 @ B_57 )
& ( ord_less_eq_pname_o @ A_109 @ B_57 ) ) ) ).
thf(fact_457_subset__insert,axiom,
! [B_56: x_a > $o,X_26: x_a,A_108: x_a > $o] :
( ~ ( member_a @ X_26 @ A_108 )
=> ( ( ord_less_eq_a_o @ A_108 @ ( insert_a @ X_26 @ B_56 ) )
<=> ( ord_less_eq_a_o @ A_108 @ B_56 ) ) ) ).
thf(fact_458_subset__insert,axiom,
! [B_56: int > $o,X_26: int,A_108: int > $o] :
( ~ ( member_int @ X_26 @ A_108 )
=> ( ( ord_less_eq_int_o @ A_108 @ ( insert_int @ X_26 @ B_56 ) )
<=> ( ord_less_eq_int_o @ A_108 @ B_56 ) ) ) ).
thf(fact_459_subset__insert,axiom,
! [B_56: nat > $o,X_26: nat,A_108: nat > $o] :
( ~ ( member_nat @ X_26 @ A_108 )
=> ( ( ord_less_eq_nat_o @ A_108 @ ( insert_nat @ X_26 @ B_56 ) )
<=> ( ord_less_eq_nat_o @ A_108 @ B_56 ) ) ) ).
thf(fact_460_subset__insert,axiom,
! [B_56: pname > $o,X_26: pname,A_108: pname > $o] :
( ~ ( member_pname @ X_26 @ A_108 )
=> ( ( ord_less_eq_pname_o @ A_108 @ ( insert_pname @ X_26 @ B_56 ) )
<=> ( ord_less_eq_pname_o @ A_108 @ B_56 ) ) ) ).
thf(fact_461_subset__insertI2,axiom,
! [B_55: x_a,A_107: x_a > $o,B_54: x_a > $o] :
( ( ord_less_eq_a_o @ A_107 @ B_54 )
=> ( ord_less_eq_a_o @ A_107 @ ( insert_a @ B_55 @ B_54 ) ) ) ).
thf(fact_462_subset__insertI2,axiom,
! [B_55: int,A_107: int > $o,B_54: int > $o] :
( ( ord_less_eq_int_o @ A_107 @ B_54 )
=> ( ord_less_eq_int_o @ A_107 @ ( insert_int @ B_55 @ B_54 ) ) ) ).
thf(fact_463_subset__insertI2,axiom,
! [B_55: nat,A_107: nat > $o,B_54: nat > $o] :
( ( ord_less_eq_nat_o @ A_107 @ B_54 )
=> ( ord_less_eq_nat_o @ A_107 @ ( insert_nat @ B_55 @ B_54 ) ) ) ).
thf(fact_464_insert__mono,axiom,
! [A_106: x_a,C_27: x_a > $o,D_7: x_a > $o] :
( ( ord_less_eq_a_o @ C_27 @ D_7 )
=> ( ord_less_eq_a_o @ ( insert_a @ A_106 @ C_27 ) @ ( insert_a @ A_106 @ D_7 ) ) ) ).
thf(fact_465_insert__mono,axiom,
! [A_106: int,C_27: int > $o,D_7: int > $o] :
( ( ord_less_eq_int_o @ C_27 @ D_7 )
=> ( ord_less_eq_int_o @ ( insert_int @ A_106 @ C_27 ) @ ( insert_int @ A_106 @ D_7 ) ) ) ).
thf(fact_466_insert__mono,axiom,
! [A_106: nat,C_27: nat > $o,D_7: nat > $o] :
( ( ord_less_eq_nat_o @ C_27 @ D_7 )
=> ( ord_less_eq_nat_o @ ( insert_nat @ A_106 @ C_27 ) @ ( insert_nat @ A_106 @ D_7 ) ) ) ).
thf(fact_467_image__insert,axiom,
! [F_29: nat > int,A_105: nat,B_53: nat > $o] :
( ( image_nat_int @ F_29 @ ( insert_nat @ A_105 @ B_53 ) )
= ( insert_int @ ( F_29 @ A_105 ) @ ( image_nat_int @ F_29 @ B_53 ) ) ) ).
thf(fact_468_image__insert,axiom,
! [F_29: pname > x_a,A_105: pname,B_53: pname > $o] :
( ( image_pname_a @ F_29 @ ( insert_pname @ A_105 @ B_53 ) )
= ( insert_a @ ( F_29 @ A_105 ) @ ( image_pname_a @ F_29 @ B_53 ) ) ) ).
thf(fact_469_insert__image,axiom,
! [F_28: nat > int,X_25: nat,A_104: nat > $o] :
( ( member_nat @ X_25 @ A_104 )
=> ( ( insert_int @ ( F_28 @ X_25 ) @ ( image_nat_int @ F_28 @ A_104 ) )
= ( image_nat_int @ F_28 @ A_104 ) ) ) ).
thf(fact_470_insert__image,axiom,
! [F_28: pname > x_a,X_25: pname,A_104: pname > $o] :
( ( member_pname @ X_25 @ A_104 )
=> ( ( insert_a @ ( F_28 @ X_25 ) @ ( image_pname_a @ F_28 @ A_104 ) )
= ( image_pname_a @ F_28 @ A_104 ) ) ) ).
thf(fact_471_subset__image__iff,axiom,
! [B_52: int > $o,F_27: nat > int,A_103: nat > $o] :
( ( ord_less_eq_int_o @ B_52 @ ( image_nat_int @ F_27 @ A_103 ) )
<=> ? [AA: nat > $o] :
( ( ord_less_eq_nat_o @ AA @ A_103 )
& ( B_52
= ( image_nat_int @ F_27 @ AA ) ) ) ) ).
thf(fact_472_subset__image__iff,axiom,
! [B_52: x_a > $o,F_27: pname > x_a,A_103: pname > $o] :
( ( ord_less_eq_a_o @ B_52 @ ( image_pname_a @ F_27 @ A_103 ) )
<=> ? [AA: pname > $o] :
( ( ord_less_eq_pname_o @ AA @ A_103 )
& ( B_52
= ( image_pname_a @ F_27 @ AA ) ) ) ) ).
thf(fact_473_image__mono,axiom,
! [F_26: nat > int,A_102: nat > $o,B_51: nat > $o] :
( ( ord_less_eq_nat_o @ A_102 @ B_51 )
=> ( ord_less_eq_int_o @ ( image_nat_int @ F_26 @ A_102 ) @ ( image_nat_int @ F_26 @ B_51 ) ) ) ).
thf(fact_474_image__mono,axiom,
! [F_26: pname > x_a,A_102: pname > $o,B_51: pname > $o] :
( ( ord_less_eq_pname_o @ A_102 @ B_51 )
=> ( ord_less_eq_a_o @ ( image_pname_a @ F_26 @ A_102 ) @ ( image_pname_a @ F_26 @ B_51 ) ) ) ).
thf(fact_475_imageE,axiom,
! [B_50: int,F_25: nat > int,A_101: nat > $o] :
( ( member_int @ B_50 @ ( image_nat_int @ F_25 @ A_101 ) )
=> ~ ! [X_1: nat] :
( ( B_50
= ( F_25 @ X_1 ) )
=> ~ ( member_nat @ X_1 @ A_101 ) ) ) ).
thf(fact_476_imageE,axiom,
! [B_50: x_a,F_25: pname > x_a,A_101: pname > $o] :
( ( member_a @ B_50 @ ( image_pname_a @ F_25 @ A_101 ) )
=> ~ ! [X_1: pname] :
( ( B_50
= ( F_25 @ X_1 ) )
=> ~ ( member_pname @ X_1 @ A_101 ) ) ) ).
thf(fact_477_subsetI,axiom,
! [B_49: int > $o,A_100: int > $o] :
( ! [X_1: int] :
( ( member_int @ X_1 @ A_100 )
=> ( member_int @ X_1 @ B_49 ) )
=> ( ord_less_eq_int_o @ A_100 @ B_49 ) ) ).
thf(fact_478_subsetI,axiom,
! [B_49: nat > $o,A_100: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_100 )
=> ( member_nat @ X_1 @ B_49 ) )
=> ( ord_less_eq_nat_o @ A_100 @ B_49 ) ) ).
thf(fact_479_subsetI,axiom,
! [B_49: x_a > $o,A_100: x_a > $o] :
( ! [X_1: x_a] :
( ( member_a @ X_1 @ A_100 )
=> ( member_a @ X_1 @ B_49 ) )
=> ( ord_less_eq_a_o @ A_100 @ B_49 ) ) ).
thf(fact_480_subsetI,axiom,
! [B_49: pname > $o,A_100: pname > $o] :
( ! [X_1: pname] :
( ( member_pname @ X_1 @ A_100 )
=> ( member_pname @ X_1 @ B_49 ) )
=> ( ord_less_eq_pname_o @ A_100 @ B_49 ) ) ).
thf(fact_481_image__subsetI,axiom,
! [F_24: nat > int,B_48: int > $o,A_99: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ A_99 )
=> ( member_int @ ( F_24 @ X_1 ) @ B_48 ) )
=> ( ord_less_eq_int_o @ ( image_nat_int @ F_24 @ A_99 ) @ B_48 ) ) ).
thf(fact_482_image__subsetI,axiom,
! [F_24: pname > x_a,B_48: x_a > $o,A_99: pname > $o] :
( ! [X_1: pname] :
( ( member_pname @ X_1 @ A_99 )
=> ( member_a @ ( F_24 @ X_1 ) @ B_48 ) )
=> ( ord_less_eq_a_o @ ( image_pname_a @ F_24 @ A_99 ) @ B_48 ) ) ).
thf(fact_483_le__funI,axiom,
! [F_23: int > $o,G_1: int > $o] :
( ! [X_1: int] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
=> ( ord_less_eq_int_o @ F_23 @ G_1 ) ) ).
thf(fact_484_le__funI,axiom,
! [F_23: nat > $o,G_1: nat > $o] :
( ! [X_1: nat] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
=> ( ord_less_eq_nat_o @ F_23 @ G_1 ) ) ).
thf(fact_485_le__funI,axiom,
! [F_23: x_a > $o,G_1: x_a > $o] :
( ! [X_1: x_a] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
=> ( ord_less_eq_a_o @ F_23 @ G_1 ) ) ).
thf(fact_486_finite__nat__set__iff__bounded__le,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M_1: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_eq_nat @ X_1 @ M_1 ) ) ) ).
thf(fact_487_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_488_diff__eq__diff__less__eq,axiom,
! [A_98: int,B_47: int,C_26: int,D_6: int] :
( ( ( minus_minus_int @ A_98 @ B_47 )
= ( minus_minus_int @ C_26 @ D_6 ) )
=> ( ( ord_less_eq_int @ A_98 @ B_47 )
<=> ( ord_less_eq_int @ C_26 @ D_6 ) ) ) ).
thf(fact_489_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_490_emptyE,axiom,
! [A_97: int] :
~ ( member_int @ A_97 @ bot_bot_int_o ) ).
thf(fact_491_emptyE,axiom,
! [A_97: nat] :
~ ( member_nat @ A_97 @ bot_bot_nat_o ) ).
thf(fact_492_emptyE,axiom,
! [A_97: x_a] :
~ ( member_a @ A_97 @ bot_bot_a_o ) ).
thf(fact_493_emptyE,axiom,
! [A_97: pname] :
~ ( member_pname @ A_97 @ bot_bot_pname_o ) ).
thf(fact_494_finite__Diff,axiom,
! [B_46: int > $o,A_96: int > $o] :
( ( finite_finite_int @ A_96 )
=> ( finite_finite_int @ ( minus_minus_int_o @ A_96 @ B_46 ) ) ) ).
thf(fact_495_finite__Diff,axiom,
! [B_46: nat > $o,A_96: nat > $o] :
( ( finite_finite_nat @ A_96 )
=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_96 @ B_46 ) ) ) ).
thf(fact_496_finite__Diff,axiom,
! [B_46: pname > $o,A_96: pname > $o] :
( ( finite_finite_pname @ A_96 )
=> ( finite_finite_pname @ ( minus_minus_pname_o @ A_96 @ B_46 ) ) ) ).
thf(fact_497_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_int_o ).
thf(fact_498_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_nat_o ).
thf(fact_499_finite_OemptyI,axiom,
finite_finite_pname @ bot_bot_pname_o ).
thf(fact_500_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_a_o ).
thf(fact_501_empty__subsetI,axiom,
! [A_95: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_95 ) ).
thf(fact_502_empty__subsetI,axiom,
! [A_95: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_95 ) ).
thf(fact_503_empty__subsetI,axiom,
! [A_95: x_a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A_95 ) ).
thf(fact_504_equals0D,axiom,
! [A_94: int,A_93: int > $o] :
( ( A_93 = bot_bot_int_o )
=> ~ ( member_int @ A_94 @ A_93 ) ) ).
thf(fact_505_equals0D,axiom,
! [A_94: nat,A_93: nat > $o] :
( ( A_93 = bot_bot_nat_o )
=> ~ ( member_nat @ A_94 @ A_93 ) ) ).
thf(fact_506_equals0D,axiom,
! [A_94: x_a,A_93: x_a > $o] :
( ( A_93 = bot_bot_a_o )
=> ~ ( member_a @ A_94 @ A_93 ) ) ).
thf(fact_507_equals0D,axiom,
! [A_94: pname,A_93: pname > $o] :
( ( A_93 = bot_bot_pname_o )
=> ~ ( member_pname @ A_94 @ A_93 ) ) ).
thf(fact_508_Collect__empty__eq,axiom,
! [P_9: int > $o] :
( ( ( collect_int @ P_9 )
= bot_bot_int_o )
<=> ! [X_1: int] :
~ ( P_9 @ X_1 ) ) ).
thf(fact_509_Collect__empty__eq,axiom,
! [P_9: nat > $o] :
( ( ( collect_nat @ P_9 )
= bot_bot_nat_o )
<=> ! [X_1: nat] :
~ ( P_9 @ X_1 ) ) ).
thf(fact_510_Collect__empty__eq,axiom,
! [P_9: x_a > $o] :
( ( ( collect_a @ P_9 )
= bot_bot_a_o )
<=> ! [X_1: x_a] :
~ ( P_9 @ X_1 ) ) ).
thf(fact_511_Diff__cancel,axiom,
! [A_92: nat > $o] :
( ( minus_minus_nat_o @ A_92 @ A_92 )
= bot_bot_nat_o ) ).
thf(fact_512_Diff__cancel,axiom,
! [A_92: int > $o] :
( ( minus_minus_int_o @ A_92 @ A_92 )
= bot_bot_int_o ) ).
thf(fact_513_Diff__cancel,axiom,
! [A_92: x_a > $o] :
( ( minus_minus_a_o @ A_92 @ A_92 )
= bot_bot_a_o ) ).
thf(fact_514_Diff__empty,axiom,
! [A_91: nat > $o] :
( ( minus_minus_nat_o @ A_91 @ bot_bot_nat_o )
= A_91 ) ).
thf(fact_515_Diff__empty,axiom,
! [A_91: int > $o] :
( ( minus_minus_int_o @ A_91 @ bot_bot_int_o )
= A_91 ) ).
thf(fact_516_Diff__empty,axiom,
! [A_91: x_a > $o] :
( ( minus_minus_a_o @ A_91 @ bot_bot_a_o )
= A_91 ) ).
thf(fact_517_empty__iff,axiom,
! [C_25: int] :
~ ( member_int @ C_25 @ bot_bot_int_o ) ).
thf(fact_518_empty__iff,axiom,
! [C_25: nat] :
~ ( member_nat @ C_25 @ bot_bot_nat_o ) ).
thf(fact_519_empty__iff,axiom,
! [C_25: x_a] :
~ ( member_a @ C_25 @ bot_bot_a_o ) ).
thf(fact_520_empty__iff,axiom,
! [C_25: pname] :
~ ( member_pname @ C_25 @ bot_bot_pname_o ) ).
thf(fact_521_empty__Collect__eq,axiom,
! [P_8: int > $o] :
( ( bot_bot_int_o
= ( collect_int @ P_8 ) )
<=> ! [X_1: int] :
~ ( P_8 @ X_1 ) ) ).
thf(fact_522_empty__Collect__eq,axiom,
! [P_8: nat > $o] :
( ( bot_bot_nat_o
= ( collect_nat @ P_8 ) )
<=> ! [X_1: nat] :
~ ( P_8 @ X_1 ) ) ).
thf(fact_523_empty__Collect__eq,axiom,
! [P_8: x_a > $o] :
( ( bot_bot_a_o
= ( collect_a @ P_8 ) )
<=> ! [X_1: x_a] :
~ ( P_8 @ X_1 ) ) ).
thf(fact_524_empty__Diff,axiom,
! [A_90: nat > $o] :
( ( minus_minus_nat_o @ bot_bot_nat_o @ A_90 )
= bot_bot_nat_o ) ).
thf(fact_525_empty__Diff,axiom,
! [A_90: int > $o] :
( ( minus_minus_int_o @ bot_bot_int_o @ A_90 )
= bot_bot_int_o ) ).
thf(fact_526_empty__Diff,axiom,
! [A_90: x_a > $o] :
( ( minus_minus_a_o @ bot_bot_a_o @ A_90 )
= bot_bot_a_o ) ).
thf(fact_527_ex__in__conv,axiom,
! [A_89: int > $o] :
( ? [X_1: int] : ( member_int @ X_1 @ A_89 )
<=> ( A_89 != bot_bot_int_o ) ) ).
thf(fact_528_ex__in__conv,axiom,
! [A_89: nat > $o] :
( ? [X_1: nat] : ( member_nat @ X_1 @ A_89 )
<=> ( A_89 != bot_bot_nat_o ) ) ).
thf(fact_529_ex__in__conv,axiom,
! [A_89: x_a > $o] :
( ? [X_1: x_a] : ( member_a @ X_1 @ A_89 )
<=> ( A_89 != bot_bot_a_o ) ) ).
thf(fact_530_ex__in__conv,axiom,
! [A_89: pname > $o] :
( ? [X_1: pname] : ( member_pname @ X_1 @ A_89 )
<=> ( A_89 != bot_bot_pname_o ) ) ).
thf(fact_531_all__not__in__conv,axiom,
! [A_88: int > $o] :
( ! [X_1: int] :
~ ( member_int @ X_1 @ A_88 )
<=> ( A_88 = bot_bot_int_o ) ) ).
thf(fact_532_all__not__in__conv,axiom,
! [A_88: nat > $o] :
( ! [X_1: nat] :
~ ( member_nat @ X_1 @ A_88 )
<=> ( A_88 = bot_bot_nat_o ) ) ).
thf(fact_533_all__not__in__conv,axiom,
! [A_88: x_a > $o] :
( ! [X_1: x_a] :
~ ( member_a @ X_1 @ A_88 )
<=> ( A_88 = bot_bot_a_o ) ) ).
thf(fact_534_all__not__in__conv,axiom,
! [A_88: pname > $o] :
( ! [X_1: pname] :
~ ( member_pname @ X_1 @ A_88 )
<=> ( A_88 = bot_bot_pname_o ) ) ).
thf(fact_535_bot__apply,axiom,
! [X_24: nat] :
( ( bot_bot_nat_o @ X_24 )
<=> bot_bot_o ) ).
thf(fact_536_bot__apply,axiom,
! [X_24: int] :
( ( bot_bot_int_o @ X_24 )
<=> bot_bot_o ) ).
thf(fact_537_bot__apply,axiom,
! [X_24: x_a] :
( ( bot_bot_a_o @ X_24 )
<=> bot_bot_o ) ).
thf(fact_538_empty__def,axiom,
( bot_bot_int_o
= ( collect_int
@ ^ [X_1: int] : $false ) ) ).
thf(fact_539_empty__def,axiom,
( bot_bot_nat_o
= ( collect_nat
@ ^ [X_1: nat] : $false ) ) ).
thf(fact_540_empty__def,axiom,
( bot_bot_a_o
= ( collect_a
@ ^ [X_1: x_a] : $false ) ) ).
thf(fact_541_bot__fun__def,axiom,
! [X_1: nat] :
( ( bot_bot_nat_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_542_bot__fun__def,axiom,
! [X_1: int] :
( ( bot_bot_int_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_543_bot__fun__def,axiom,
! [X_1: x_a] :
( ( bot_bot_a_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_544_insert__Diff,axiom,
! [A_87: x_a,A_86: x_a > $o] :
( ( member_a @ A_87 @ A_86 )
=> ( ( insert_a @ A_87 @ ( minus_minus_a_o @ A_86 @ ( insert_a @ A_87 @ bot_bot_a_o ) ) )
= A_86 ) ) ).
thf(fact_545_insert__Diff,axiom,
! [A_87: int,A_86: int > $o] :
( ( member_int @ A_87 @ A_86 )
=> ( ( insert_int @ A_87 @ ( minus_minus_int_o @ A_86 @ ( insert_int @ A_87 @ bot_bot_int_o ) ) )
= A_86 ) ) ).
thf(fact_546_insert__Diff,axiom,
! [A_87: nat,A_86: nat > $o] :
( ( member_nat @ A_87 @ A_86 )
=> ( ( insert_nat @ A_87 @ ( minus_minus_nat_o @ A_86 @ ( insert_nat @ A_87 @ bot_bot_nat_o ) ) )
= A_86 ) ) ).
thf(fact_547_insert__Diff,axiom,
! [A_87: pname,A_86: pname > $o] :
( ( member_pname @ A_87 @ A_86 )
=> ( ( insert_pname @ A_87 @ ( minus_minus_pname_o @ A_86 @ ( insert_pname @ A_87 @ bot_bot_pname_o ) ) )
= A_86 ) ) ).
thf(fact_548_Diff__insert__absorb,axiom,
! [X_23: x_a,A_85: x_a > $o] :
( ~ ( member_a @ X_23 @ A_85 )
=> ( ( minus_minus_a_o @ ( insert_a @ X_23 @ A_85 ) @ ( insert_a @ X_23 @ bot_bot_a_o ) )
= A_85 ) ) ).
thf(fact_549_Diff__insert__absorb,axiom,
! [X_23: int,A_85: int > $o] :
( ~ ( member_int @ X_23 @ A_85 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_23 @ A_85 ) @ ( insert_int @ X_23 @ bot_bot_int_o ) )
= A_85 ) ) ).
thf(fact_550_Diff__insert__absorb,axiom,
! [X_23: nat,A_85: nat > $o] :
( ~ ( member_nat @ X_23 @ A_85 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_23 @ A_85 ) @ ( insert_nat @ X_23 @ bot_bot_nat_o ) )
= A_85 ) ) ).
thf(fact_551_Diff__insert__absorb,axiom,
! [X_23: pname,A_85: pname > $o] :
( ~ ( member_pname @ X_23 @ A_85 )
=> ( ( minus_minus_pname_o @ ( insert_pname @ X_23 @ A_85 ) @ ( insert_pname @ X_23 @ bot_bot_pname_o ) )
= A_85 ) ) ).
thf(fact_552_insert__Diff__single,axiom,
! [A_84: x_a,A_83: x_a > $o] :
( ( insert_a @ A_84 @ ( minus_minus_a_o @ A_83 @ ( insert_a @ A_84 @ bot_bot_a_o ) ) )
= ( insert_a @ A_84 @ A_83 ) ) ).
thf(fact_553_insert__Diff__single,axiom,
! [A_84: nat,A_83: nat > $o] :
( ( insert_nat @ A_84 @ ( minus_minus_nat_o @ A_83 @ ( insert_nat @ A_84 @ bot_bot_nat_o ) ) )
= ( insert_nat @ A_84 @ A_83 ) ) ).
thf(fact_554_insert__Diff__single,axiom,
! [A_84: int,A_83: int > $o] :
( ( insert_int @ A_84 @ ( minus_minus_int_o @ A_83 @ ( insert_int @ A_84 @ bot_bot_int_o ) ) )
= ( insert_int @ A_84 @ A_83 ) ) ).
thf(fact_555_Diff__insert2,axiom,
! [A_82: x_a > $o,A_81: x_a,B_45: x_a > $o] :
( ( minus_minus_a_o @ A_82 @ ( insert_a @ A_81 @ B_45 ) )
= ( minus_minus_a_o @ ( minus_minus_a_o @ A_82 @ ( insert_a @ A_81 @ bot_bot_a_o ) ) @ B_45 ) ) ).
thf(fact_556_Diff__insert2,axiom,
! [A_82: nat > $o,A_81: nat,B_45: nat > $o] :
( ( minus_minus_nat_o @ A_82 @ ( insert_nat @ A_81 @ B_45 ) )
= ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_82 @ ( insert_nat @ A_81 @ bot_bot_nat_o ) ) @ B_45 ) ) ).
thf(fact_557_Diff__insert2,axiom,
! [A_82: int > $o,A_81: int,B_45: int > $o] :
( ( minus_minus_int_o @ A_82 @ ( insert_int @ A_81 @ B_45 ) )
= ( minus_minus_int_o @ ( minus_minus_int_o @ A_82 @ ( insert_int @ A_81 @ bot_bot_int_o ) ) @ B_45 ) ) ).
thf(fact_558_Diff__insert,axiom,
! [A_80: x_a > $o,A_79: x_a,B_44: x_a > $o] :
( ( minus_minus_a_o @ A_80 @ ( insert_a @ A_79 @ B_44 ) )
= ( minus_minus_a_o @ ( minus_minus_a_o @ A_80 @ B_44 ) @ ( insert_a @ A_79 @ bot_bot_a_o ) ) ) ).
thf(fact_559_Diff__insert,axiom,
! [A_80: nat > $o,A_79: nat,B_44: nat > $o] :
( ( minus_minus_nat_o @ A_80 @ ( insert_nat @ A_79 @ B_44 ) )
= ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_80 @ B_44 ) @ ( insert_nat @ A_79 @ bot_bot_nat_o ) ) ) ).
thf(fact_560_Diff__insert,axiom,
! [A_80: int > $o,A_79: int,B_44: int > $o] :
( ( minus_minus_int_o @ A_80 @ ( insert_int @ A_79 @ B_44 ) )
= ( minus_minus_int_o @ ( minus_minus_int_o @ A_80 @ B_44 ) @ ( insert_int @ A_79 @ bot_bot_int_o ) ) ) ).
thf(fact_561_diff__single__insert,axiom,
! [A_78: x_a > $o,X_22: x_a,B_43: x_a > $o] :
( ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_78 @ ( insert_a @ X_22 @ bot_bot_a_o ) ) @ B_43 )
=> ( ( member_a @ X_22 @ A_78 )
=> ( ord_less_eq_a_o @ A_78 @ ( insert_a @ X_22 @ B_43 ) ) ) ) ).
thf(fact_562_diff__single__insert,axiom,
! [A_78: int > $o,X_22: int,B_43: int > $o] :
( ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_78 @ ( insert_int @ X_22 @ bot_bot_int_o ) ) @ B_43 )
=> ( ( member_int @ X_22 @ A_78 )
=> ( ord_less_eq_int_o @ A_78 @ ( insert_int @ X_22 @ B_43 ) ) ) ) ).
thf(fact_563_diff__single__insert,axiom,
! [A_78: nat > $o,X_22: nat,B_43: nat > $o] :
( ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_78 @ ( insert_nat @ X_22 @ bot_bot_nat_o ) ) @ B_43 )
=> ( ( member_nat @ X_22 @ A_78 )
=> ( ord_less_eq_nat_o @ A_78 @ ( insert_nat @ X_22 @ B_43 ) ) ) ) ).
thf(fact_564_diff__single__insert,axiom,
! [A_78: pname > $o,X_22: pname,B_43: pname > $o] :
( ( ord_less_eq_pname_o @ ( minus_minus_pname_o @ A_78 @ ( insert_pname @ X_22 @ bot_bot_pname_o ) ) @ B_43 )
=> ( ( member_pname @ X_22 @ A_78 )
=> ( ord_less_eq_pname_o @ A_78 @ ( insert_pname @ X_22 @ B_43 ) ) ) ) ).
thf(fact_565_subset__insert__iff,axiom,
! [A_77: x_a > $o,X_21: x_a,B_42: x_a > $o] :
( ( ord_less_eq_a_o @ A_77 @ ( insert_a @ X_21 @ B_42 ) )
<=> ( ( ( member_a @ X_21 @ A_77 )
=> ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_77 @ ( insert_a @ X_21 @ bot_bot_a_o ) ) @ B_42 ) )
& ( ~ ( member_a @ X_21 @ A_77 )
=> ( ord_less_eq_a_o @ A_77 @ B_42 ) ) ) ) ).
thf(fact_566_subset__insert__iff,axiom,
! [A_77: int > $o,X_21: int,B_42: int > $o] :
( ( ord_less_eq_int_o @ A_77 @ ( insert_int @ X_21 @ B_42 ) )
<=> ( ( ( member_int @ X_21 @ A_77 )
=> ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_77 @ ( insert_int @ X_21 @ bot_bot_int_o ) ) @ B_42 ) )
& ( ~ ( member_int @ X_21 @ A_77 )
=> ( ord_less_eq_int_o @ A_77 @ B_42 ) ) ) ) ).
thf(fact_567_subset__insert__iff,axiom,
! [A_77: nat > $o,X_21: nat,B_42: nat > $o] :
( ( ord_less_eq_nat_o @ A_77 @ ( insert_nat @ X_21 @ B_42 ) )
<=> ( ( ( member_nat @ X_21 @ A_77 )
=> ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_77 @ ( insert_nat @ X_21 @ bot_bot_nat_o ) ) @ B_42 ) )
& ( ~ ( member_nat @ X_21 @ A_77 )
=> ( ord_less_eq_nat_o @ A_77 @ B_42 ) ) ) ) ).
thf(fact_568_subset__insert__iff,axiom,
! [A_77: pname > $o,X_21: pname,B_42: pname > $o] :
( ( ord_less_eq_pname_o @ A_77 @ ( insert_pname @ X_21 @ B_42 ) )
<=> ( ( ( member_pname @ X_21 @ A_77 )
=> ( ord_less_eq_pname_o @ ( minus_minus_pname_o @ A_77 @ ( insert_pname @ X_21 @ bot_bot_pname_o ) ) @ B_42 ) )
& ( ~ ( member_pname @ X_21 @ A_77 )
=> ( ord_less_eq_pname_o @ A_77 @ B_42 ) ) ) ) ).
thf(fact_569_finite__Diff2,axiom,
! [A_76: int > $o,B_41: int > $o] :
( ( finite_finite_int @ B_41 )
=> ( ( finite_finite_int @ ( minus_minus_int_o @ A_76 @ B_41 ) )
<=> ( finite_finite_int @ A_76 ) ) ) ).
thf(fact_570_finite__Diff2,axiom,
! [A_76: nat > $o,B_41: nat > $o] :
( ( finite_finite_nat @ B_41 )
=> ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_76 @ B_41 ) )
<=> ( finite_finite_nat @ A_76 ) ) ) ).
thf(fact_571_finite__Diff2,axiom,
! [A_76: pname > $o,B_41: pname > $o] :
( ( finite_finite_pname @ B_41 )
=> ( ( finite_finite_pname @ ( minus_minus_pname_o @ A_76 @ B_41 ) )
<=> ( finite_finite_pname @ A_76 ) ) ) ).
thf(fact_572_insert__Diff1,axiom,
! [A_75: x_a > $o,X_20: x_a,B_40: x_a > $o] :
( ( member_a @ X_20 @ B_40 )
=> ( ( minus_minus_a_o @ ( insert_a @ X_20 @ A_75 ) @ B_40 )
= ( minus_minus_a_o @ A_75 @ B_40 ) ) ) ).
thf(fact_573_insert__Diff1,axiom,
! [A_75: int > $o,X_20: int,B_40: int > $o] :
( ( member_int @ X_20 @ B_40 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_20 @ A_75 ) @ B_40 )
= ( minus_minus_int_o @ A_75 @ B_40 ) ) ) ).
thf(fact_574_insert__Diff1,axiom,
! [A_75: nat > $o,X_20: nat,B_40: nat > $o] :
( ( member_nat @ X_20 @ B_40 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_20 @ A_75 ) @ B_40 )
= ( minus_minus_nat_o @ A_75 @ B_40 ) ) ) ).
thf(fact_575_insert__Diff1,axiom,
! [A_75: pname > $o,X_20: pname,B_40: pname > $o] :
( ( member_pname @ X_20 @ B_40 )
=> ( ( minus_minus_pname_o @ ( insert_pname @ X_20 @ A_75 ) @ B_40 )
= ( minus_minus_pname_o @ A_75 @ B_40 ) ) ) ).
thf(fact_576_insert__Diff__if,axiom,
! [A_74: x_a > $o,X_19: x_a,B_39: x_a > $o] :
( ( ( member_a @ X_19 @ B_39 )
=> ( ( minus_minus_a_o @ ( insert_a @ X_19 @ A_74 ) @ B_39 )
= ( minus_minus_a_o @ A_74 @ B_39 ) ) )
& ( ~ ( member_a @ X_19 @ B_39 )
=> ( ( minus_minus_a_o @ ( insert_a @ X_19 @ A_74 ) @ B_39 )
= ( insert_a @ X_19 @ ( minus_minus_a_o @ A_74 @ B_39 ) ) ) ) ) ).
thf(fact_577_insert__Diff__if,axiom,
! [A_74: int > $o,X_19: int,B_39: int > $o] :
( ( ( member_int @ X_19 @ B_39 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_19 @ A_74 ) @ B_39 )
= ( minus_minus_int_o @ A_74 @ B_39 ) ) )
& ( ~ ( member_int @ X_19 @ B_39 )
=> ( ( minus_minus_int_o @ ( insert_int @ X_19 @ A_74 ) @ B_39 )
= ( insert_int @ X_19 @ ( minus_minus_int_o @ A_74 @ B_39 ) ) ) ) ) ).
thf(fact_578_insert__Diff__if,axiom,
! [A_74: nat > $o,X_19: nat,B_39: nat > $o] :
( ( ( member_nat @ X_19 @ B_39 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_19 @ A_74 ) @ B_39 )
= ( minus_minus_nat_o @ A_74 @ B_39 ) ) )
& ( ~ ( member_nat @ X_19 @ B_39 )
=> ( ( minus_minus_nat_o @ ( insert_nat @ X_19 @ A_74 ) @ B_39 )
= ( insert_nat @ X_19 @ ( minus_minus_nat_o @ A_74 @ B_39 ) ) ) ) ) ).
thf(fact_579_insert__Diff__if,axiom,
! [A_74: pname > $o,X_19: pname,B_39: pname > $o] :
( ( ( member_pname @ X_19 @ B_39 )
=> ( ( minus_minus_pname_o @ ( insert_pname @ X_19 @ A_74 ) @ B_39 )
= ( minus_minus_pname_o @ A_74 @ B_39 ) ) )
& ( ~ ( member_pname @ X_19 @ B_39 )
=> ( ( minus_minus_pname_o @ ( insert_pname @ X_19 @ A_74 ) @ B_39 )
= ( insert_pname @ X_19 @ ( minus_minus_pname_o @ A_74 @ B_39 ) ) ) ) ) ).
thf(fact_580_double__diff,axiom,
! [C_24: int > $o,A_73: int > $o,B_38: int > $o] :
( ( ord_less_eq_int_o @ A_73 @ B_38 )
=> ( ( ord_less_eq_int_o @ B_38 @ C_24 )
=> ( ( minus_minus_int_o @ B_38 @ ( minus_minus_int_o @ C_24 @ A_73 ) )
= A_73 ) ) ) ).
thf(fact_581_double__diff,axiom,
! [C_24: nat > $o,A_73: nat > $o,B_38: nat > $o] :
( ( ord_less_eq_nat_o @ A_73 @ B_38 )
=> ( ( ord_less_eq_nat_o @ B_38 @ C_24 )
=> ( ( minus_minus_nat_o @ B_38 @ ( minus_minus_nat_o @ C_24 @ A_73 ) )
= A_73 ) ) ) ).
thf(fact_582_double__diff,axiom,
! [C_24: x_a > $o,A_73: x_a > $o,B_38: x_a > $o] :
( ( ord_less_eq_a_o @ A_73 @ B_38 )
=> ( ( ord_less_eq_a_o @ B_38 @ C_24 )
=> ( ( minus_minus_a_o @ B_38 @ ( minus_minus_a_o @ C_24 @ A_73 ) )
= A_73 ) ) ) ).
thf(fact_583_Diff__mono,axiom,
! [D_5: int > $o,B_37: int > $o,A_72: int > $o,C_23: int > $o] :
( ( ord_less_eq_int_o @ A_72 @ C_23 )
=> ( ( ord_less_eq_int_o @ D_5 @ B_37 )
=> ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_72 @ B_37 ) @ ( minus_minus_int_o @ C_23 @ D_5 ) ) ) ) ).
thf(fact_584_Diff__mono,axiom,
! [D_5: nat > $o,B_37: nat > $o,A_72: nat > $o,C_23: nat > $o] :
( ( ord_less_eq_nat_o @ A_72 @ C_23 )
=> ( ( ord_less_eq_nat_o @ D_5 @ B_37 )
=> ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_72 @ B_37 ) @ ( minus_minus_nat_o @ C_23 @ D_5 ) ) ) ) ).
thf(fact_585_Diff__mono,axiom,
! [D_5: x_a > $o,B_37: x_a > $o,A_72: x_a > $o,C_23: x_a > $o] :
( ( ord_less_eq_a_o @ A_72 @ C_23 )
=> ( ( ord_less_eq_a_o @ D_5 @ B_37 )
=> ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_72 @ B_37 ) @ ( minus_minus_a_o @ C_23 @ D_5 ) ) ) ) ).
thf(fact_586_Diff__subset,axiom,
! [A_71: int > $o,B_36: int > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_71 @ B_36 ) @ A_71 ) ).
thf(fact_587_Diff__subset,axiom,
! [A_71: nat > $o,B_36: nat > $o] : ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_71 @ B_36 ) @ A_71 ) ).
thf(fact_588_Diff__subset,axiom,
! [A_71: x_a > $o,B_36: x_a > $o] : ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_71 @ B_36 ) @ A_71 ) ).
thf(fact_589_singleton__inject,axiom,
! [A_70: x_a,B_35: x_a] :
( ( ( insert_a @ A_70 @ bot_bot_a_o )
= ( insert_a @ B_35 @ bot_bot_a_o ) )
=> ( A_70 = B_35 ) ) ).
thf(fact_590_singleton__inject,axiom,
! [A_70: nat,B_35: nat] :
( ( ( insert_nat @ A_70 @ bot_bot_nat_o )
= ( insert_nat @ B_35 @ bot_bot_nat_o ) )
=> ( A_70 = B_35 ) ) ).
thf(fact_591_singleton__inject,axiom,
! [A_70: int,B_35: int] :
( ( ( insert_int @ A_70 @ bot_bot_int_o )
= ( insert_int @ B_35 @ bot_bot_int_o ) )
=> ( A_70 = B_35 ) ) ).
thf(fact_592_singletonE,axiom,
! [B_34: x_a,A_69: x_a] :
( ( member_a @ B_34 @ ( insert_a @ A_69 @ bot_bot_a_o ) )
=> ( B_34 = A_69 ) ) ).
thf(fact_593_singletonE,axiom,
! [B_34: int,A_69: int] :
( ( member_int @ B_34 @ ( insert_int @ A_69 @ bot_bot_int_o ) )
=> ( B_34 = A_69 ) ) ).
thf(fact_594_singletonE,axiom,
! [B_34: nat,A_69: nat] :
( ( member_nat @ B_34 @ ( insert_nat @ A_69 @ bot_bot_nat_o ) )
=> ( B_34 = A_69 ) ) ).
thf(fact_595_singletonE,axiom,
! [B_34: pname,A_69: pname] :
( ( member_pname @ B_34 @ ( insert_pname @ A_69 @ bot_bot_pname_o ) )
=> ( B_34 = A_69 ) ) ).
thf(fact_596_doubleton__eq__iff,axiom,
! [A_68: x_a,B_33: x_a,C_22: x_a,D_4: x_a] :
( ( ( insert_a @ A_68 @ ( insert_a @ B_33 @ bot_bot_a_o ) )
= ( insert_a @ C_22 @ ( insert_a @ D_4 @ bot_bot_a_o ) ) )
<=> ( ( ( A_68 = C_22 )
& ( B_33 = D_4 ) )
| ( ( A_68 = D_4 )
& ( B_33 = C_22 ) ) ) ) ).
thf(fact_597_doubleton__eq__iff,axiom,
! [A_68: nat,B_33: nat,C_22: nat,D_4: nat] :
( ( ( insert_nat @ A_68 @ ( insert_nat @ B_33 @ bot_bot_nat_o ) )
= ( insert_nat @ C_22 @ ( insert_nat @ D_4 @ bot_bot_nat_o ) ) )
<=> ( ( ( A_68 = C_22 )
& ( B_33 = D_4 ) )
| ( ( A_68 = D_4 )
& ( B_33 = C_22 ) ) ) ) ).
thf(fact_598_doubleton__eq__iff,axiom,
! [A_68: int,B_33: int,C_22: int,D_4: int] :
( ( ( insert_int @ A_68 @ ( insert_int @ B_33 @ bot_bot_int_o ) )
= ( insert_int @ C_22 @ ( insert_int @ D_4 @ bot_bot_int_o ) ) )
<=> ( ( ( A_68 = C_22 )
& ( B_33 = D_4 ) )
| ( ( A_68 = D_4 )
& ( B_33 = C_22 ) ) ) ) ).
thf(fact_599_singleton__iff,axiom,
! [B_32: x_a,A_67: x_a] :
( ( member_a @ B_32 @ ( insert_a @ A_67 @ bot_bot_a_o ) )
<=> ( B_32 = A_67 ) ) ).
thf(fact_600_singleton__iff,axiom,
! [B_32: int,A_67: int] :
( ( member_int @ B_32 @ ( insert_int @ A_67 @ bot_bot_int_o ) )
<=> ( B_32 = A_67 ) ) ).
thf(fact_601_singleton__iff,axiom,
! [B_32: nat,A_67: nat] :
( ( member_nat @ B_32 @ ( insert_nat @ A_67 @ bot_bot_nat_o ) )
<=> ( B_32 = A_67 ) ) ).
thf(fact_602_singleton__iff,axiom,
! [B_32: pname,A_67: pname] :
( ( member_pname @ B_32 @ ( insert_pname @ A_67 @ bot_bot_pname_o ) )
<=> ( B_32 = A_67 ) ) ).
thf(fact_603_insert__not__empty,axiom,
! [A_66: x_a,A_65: x_a > $o] :
( ( insert_a @ A_66 @ A_65 )
!= bot_bot_a_o ) ).
thf(fact_604_insert__not__empty,axiom,
! [A_66: nat,A_65: nat > $o] :
( ( insert_nat @ A_66 @ A_65 )
!= bot_bot_nat_o ) ).
thf(fact_605_insert__not__empty,axiom,
! [A_66: int,A_65: int > $o] :
( ( insert_int @ A_66 @ A_65 )
!= bot_bot_int_o ) ).
thf(fact_606_empty__not__insert,axiom,
! [A_64: x_a,A_63: x_a > $o] :
( bot_bot_a_o
!= ( insert_a @ A_64 @ A_63 ) ) ).
thf(fact_607_empty__not__insert,axiom,
! [A_64: nat,A_63: nat > $o] :
( bot_bot_nat_o
!= ( insert_nat @ A_64 @ A_63 ) ) ).
thf(fact_608_empty__not__insert,axiom,
! [A_64: int,A_63: int > $o] :
( bot_bot_int_o
!= ( insert_int @ A_64 @ A_63 ) ) ).
thf(fact_609_subset__empty,axiom,
! [A_62: nat > $o] :
( ( ord_less_eq_nat_o @ A_62 @ bot_bot_nat_o )
<=> ( A_62 = bot_bot_nat_o ) ) ).
thf(fact_610_subset__empty,axiom,
! [A_62: int > $o] :
( ( ord_less_eq_int_o @ A_62 @ bot_bot_int_o )
<=> ( A_62 = bot_bot_int_o ) ) ).
thf(fact_611_subset__empty,axiom,
! [A_62: x_a > $o] :
( ( ord_less_eq_a_o @ A_62 @ bot_bot_a_o )
<=> ( A_62 = bot_bot_a_o ) ) ).
thf(fact_612_image__is__empty,axiom,
! [F_22: nat > int,A_61: nat > $o] :
( ( ( image_nat_int @ F_22 @ A_61 )
= bot_bot_int_o )
<=> ( A_61 = bot_bot_nat_o ) ) ).
thf(fact_613_image__is__empty,axiom,
! [F_22: pname > x_a,A_61: pname > $o] :
( ( ( image_pname_a @ F_22 @ A_61 )
= bot_bot_a_o )
<=> ( A_61 = bot_bot_pname_o ) ) ).
thf(fact_614_image__empty,axiom,
! [F_21: nat > int] :
( ( image_nat_int @ F_21 @ bot_bot_nat_o )
= bot_bot_int_o ) ).
thf(fact_615_image__empty,axiom,
! [F_21: pname > x_a] :
( ( image_pname_a @ F_21 @ bot_bot_pname_o )
= bot_bot_a_o ) ).
thf(fact_616_empty__is__image,axiom,
! [F_20: nat > int,A_60: nat > $o] :
( ( bot_bot_int_o
= ( image_nat_int @ F_20 @ A_60 ) )
<=> ( A_60 = bot_bot_nat_o ) ) ).
thf(fact_617_empty__is__image,axiom,
! [F_20: pname > x_a,A_60: pname > $o] :
( ( bot_bot_a_o
= ( image_pname_a @ F_20 @ A_60 ) )
<=> ( A_60 = bot_bot_pname_o ) ) ).
thf(fact_618_le__bot,axiom,
! [A_59: nat > $o] :
( ( ord_less_eq_nat_o @ A_59 @ bot_bot_nat_o )
=> ( A_59 = bot_bot_nat_o ) ) ).
thf(fact_619_le__bot,axiom,
! [A_59: int > $o] :
( ( ord_less_eq_int_o @ A_59 @ bot_bot_int_o )
=> ( A_59 = bot_bot_int_o ) ) ).
thf(fact_620_le__bot,axiom,
! [A_59: nat] :
( ( ord_less_eq_nat @ A_59 @ bot_bot_nat )
=> ( A_59 = bot_bot_nat ) ) ).
thf(fact_621_le__bot,axiom,
! [A_59: x_a > $o] :
( ( ord_less_eq_a_o @ A_59 @ bot_bot_a_o )
=> ( A_59 = bot_bot_a_o ) ) ).
thf(fact_622_bot__unique,axiom,
! [A_58: nat > $o] :
( ( ord_less_eq_nat_o @ A_58 @ bot_bot_nat_o )
<=> ( A_58 = bot_bot_nat_o ) ) ).
thf(fact_623_bot__unique,axiom,
! [A_58: int > $o] :
( ( ord_less_eq_int_o @ A_58 @ bot_bot_int_o )
<=> ( A_58 = bot_bot_int_o ) ) ).
thf(fact_624_bot__unique,axiom,
! [A_58: nat] :
( ( ord_less_eq_nat @ A_58 @ bot_bot_nat )
<=> ( A_58 = bot_bot_nat ) ) ).
thf(fact_625_bot__unique,axiom,
! [A_58: x_a > $o] :
( ( ord_less_eq_a_o @ A_58 @ bot_bot_a_o )
<=> ( A_58 = bot_bot_a_o ) ) ).
thf(fact_626_bot__least,axiom,
! [A_57: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_57 ) ).
thf(fact_627_bot__least,axiom,
! [A_57: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_57 ) ).
thf(fact_628_bot__least,axiom,
! [A_57: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_57 ) ).
thf(fact_629_bot__least,axiom,
! [A_57: x_a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A_57 ) ).
thf(fact_630_Collect__conv__if,axiom,
! [P_7: x_a > $o,A_56: x_a] :
( ( ( P_7 @ A_56 )
=> ( ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
= ( insert_a @ A_56 @ bot_bot_a_o ) ) )
& ( ~ ( P_7 @ A_56 )
=> ( ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
= bot_bot_a_o ) ) ) ).
thf(fact_631_Collect__conv__if,axiom,
! [P_7: int > $o,A_56: int] :
( ( ( P_7 @ A_56 )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
= ( insert_int @ A_56 @ bot_bot_int_o ) ) )
& ( ~ ( P_7 @ A_56 )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
= bot_bot_int_o ) ) ) ).
thf(fact_632_Collect__conv__if,axiom,
! [P_7: nat > $o,A_56: nat] :
( ( ( P_7 @ A_56 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
= ( insert_nat @ A_56 @ bot_bot_nat_o ) ) )
& ( ~ ( P_7 @ A_56 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_633_Collect__conv__if2,axiom,
! [P_6: x_a > $o,A_55: x_a] :
( ( ( P_6 @ A_55 )
=> ( ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
= ( insert_a @ A_55 @ bot_bot_a_o ) ) )
& ( ~ ( P_6 @ A_55 )
=> ( ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
= bot_bot_a_o ) ) ) ).
thf(fact_634_Collect__conv__if2,axiom,
! [P_6: int > $o,A_55: int] :
( ( ( P_6 @ A_55 )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
= ( insert_int @ A_55 @ bot_bot_int_o ) ) )
& ( ~ ( P_6 @ A_55 )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
= bot_bot_int_o ) ) ) ).
thf(fact_635_Collect__conv__if2,axiom,
! [P_6: nat > $o,A_55: nat] :
( ( ( P_6 @ A_55 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
= ( insert_nat @ A_55 @ bot_bot_nat_o ) ) )
& ( ~ ( P_6 @ A_55 )
=> ( ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
= bot_bot_nat_o ) ) ) ).
thf(fact_636_singleton__conv,axiom,
! [A_54: x_a] :
( ( collect_a
@ ^ [X_1: x_a] : X_1 = A_54 )
= ( insert_a @ A_54 @ bot_bot_a_o ) ) ).
thf(fact_637_singleton__conv,axiom,
! [A_54: int] :
( ( collect_int
@ ^ [X_1: int] : X_1 = A_54 )
= ( insert_int @ A_54 @ bot_bot_int_o ) ) ).
thf(fact_638_singleton__conv,axiom,
! [A_54: nat] :
( ( collect_nat
@ ^ [X_1: nat] : X_1 = A_54 )
= ( insert_nat @ A_54 @ bot_bot_nat_o ) ) ).
thf(fact_639_singleton__conv2,axiom,
! [A_53: x_a] :
( ( collect_a @ ( fequal_a @ A_53 ) )
= ( insert_a @ A_53 @ bot_bot_a_o ) ) ).
thf(fact_640_singleton__conv2,axiom,
! [A_53: int] :
( ( collect_int @ ( fequal_int @ A_53 ) )
= ( insert_int @ A_53 @ bot_bot_int_o ) ) ).
thf(fact_641_singleton__conv2,axiom,
! [A_53: nat] :
( ( collect_nat @ ( fequal_nat @ A_53 ) )
= ( insert_nat @ A_53 @ bot_bot_nat_o ) ) ).
thf(fact_642_card__Suc__Diff1,axiom,
! [X_18: x_a,A_52: x_a > $o] :
( ( finite_finite_a @ A_52 )
=> ( ( member_a @ X_18 @ A_52 )
=> ( ( suc @ ( finite_card_a @ ( minus_minus_a_o @ A_52 @ ( insert_a @ X_18 @ bot_bot_a_o ) ) ) )
= ( finite_card_a @ A_52 ) ) ) ) ).
thf(fact_643_card__Suc__Diff1,axiom,
! [X_18: int,A_52: int > $o] :
( ( finite_finite_int @ A_52 )
=> ( ( member_int @ X_18 @ A_52 )
=> ( ( suc @ ( finite_card_int @ ( minus_minus_int_o @ A_52 @ ( insert_int @ X_18 @ bot_bot_int_o ) ) ) )
= ( finite_card_int @ A_52 ) ) ) ) ).
thf(fact_644_card__Suc__Diff1,axiom,
! [X_18: nat,A_52: nat > $o] :
( ( finite_finite_nat @ A_52 )
=> ( ( member_nat @ X_18 @ A_52 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_nat_o @ A_52 @ ( insert_nat @ X_18 @ bot_bot_nat_o ) ) ) )
= ( finite_card_nat @ A_52 ) ) ) ) ).
thf(fact_645_card__Suc__Diff1,axiom,
! [X_18: pname,A_52: pname > $o] :
( ( finite_finite_pname @ A_52 )
=> ( ( member_pname @ X_18 @ A_52 )
=> ( ( suc @ ( finite_card_pname @ ( minus_minus_pname_o @ A_52 @ ( insert_pname @ X_18 @ bot_bot_pname_o ) ) ) )
= ( finite_card_pname @ A_52 ) ) ) ) ).
thf(fact_646_card__insert,axiom,
! [X_17: x_a,A_51: x_a > $o] :
( ( finite_finite_a @ A_51 )
=> ( ( finite_card_a @ ( insert_a @ X_17 @ A_51 ) )
= ( suc @ ( finite_card_a @ ( minus_minus_a_o @ A_51 @ ( insert_a @ X_17 @ bot_bot_a_o ) ) ) ) ) ) ).
thf(fact_647_card__insert,axiom,
! [X_17: int,A_51: int > $o] :
( ( finite_finite_int @ A_51 )
=> ( ( finite_card_int @ ( insert_int @ X_17 @ A_51 ) )
= ( suc @ ( finite_card_int @ ( minus_minus_int_o @ A_51 @ ( insert_int @ X_17 @ bot_bot_int_o ) ) ) ) ) ) ).
thf(fact_648_card__insert,axiom,
! [X_17: nat,A_51: nat > $o] :
( ( finite_finite_nat @ A_51 )
=> ( ( finite_card_nat @ ( insert_nat @ X_17 @ A_51 ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_nat_o @ A_51 @ ( insert_nat @ X_17 @ bot_bot_nat_o ) ) ) ) ) ) ).
thf(fact_649_card__insert,axiom,
! [X_17: pname,A_51: pname > $o] :
( ( finite_finite_pname @ A_51 )
=> ( ( finite_card_pname @ ( insert_pname @ X_17 @ A_51 ) )
= ( suc @ ( finite_card_pname @ ( minus_minus_pname_o @ A_51 @ ( insert_pname @ X_17 @ bot_bot_pname_o ) ) ) ) ) ) ).
thf(fact_650_card__Diff1__le,axiom,
! [X_16: x_a,A_50: x_a > $o] :
( ( finite_finite_a @ A_50 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_50 @ ( insert_a @ X_16 @ bot_bot_a_o ) ) ) @ ( finite_card_a @ A_50 ) ) ) ).
thf(fact_651_card__Diff1__le,axiom,
! [X_16: int,A_50: int > $o] :
( ( finite_finite_int @ A_50 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_50 @ ( insert_int @ X_16 @ bot_bot_int_o ) ) ) @ ( finite_card_int @ A_50 ) ) ) ).
thf(fact_652_card__Diff1__le,axiom,
! [X_16: nat,A_50: nat > $o] :
( ( finite_finite_nat @ A_50 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_50 @ ( insert_nat @ X_16 @ bot_bot_nat_o ) ) ) @ ( finite_card_nat @ A_50 ) ) ) ).
thf(fact_653_card__Diff1__le,axiom,
! [X_16: pname,A_50: pname > $o] :
( ( finite_finite_pname @ A_50 )
=> ( ord_less_eq_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_50 @ ( insert_pname @ X_16 @ bot_bot_pname_o ) ) ) @ ( finite_card_pname @ A_50 ) ) ) ).
thf(fact_654_finite__Diff__insert,axiom,
! [A_49: x_a > $o,A_48: x_a,B_31: x_a > $o] :
( ( finite_finite_a @ ( minus_minus_a_o @ A_49 @ ( insert_a @ A_48 @ B_31 ) ) )
<=> ( finite_finite_a @ ( minus_minus_a_o @ A_49 @ B_31 ) ) ) ).
thf(fact_655_finite__Diff__insert,axiom,
! [A_49: int > $o,A_48: int,B_31: int > $o] :
( ( finite_finite_int @ ( minus_minus_int_o @ A_49 @ ( insert_int @ A_48 @ B_31 ) ) )
<=> ( finite_finite_int @ ( minus_minus_int_o @ A_49 @ B_31 ) ) ) ).
thf(fact_656_finite__Diff__insert,axiom,
! [A_49: nat > $o,A_48: nat,B_31: nat > $o] :
( ( finite_finite_nat @ ( minus_minus_nat_o @ A_49 @ ( insert_nat @ A_48 @ B_31 ) ) )
<=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_49 @ B_31 ) ) ) ).
thf(fact_657_finite__Diff__insert,axiom,
! [A_49: pname > $o,A_48: pname,B_31: pname > $o] :
( ( finite_finite_pname @ ( minus_minus_pname_o @ A_49 @ ( insert_pname @ A_48 @ B_31 ) ) )
<=> ( finite_finite_pname @ ( minus_minus_pname_o @ A_49 @ B_31 ) ) ) ).
thf(fact_658_image__diff__subset,axiom,
! [F_19: nat > int,A_47: nat > $o,B_30: nat > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ ( image_nat_int @ F_19 @ A_47 ) @ ( image_nat_int @ F_19 @ B_30 ) ) @ ( image_nat_int @ F_19 @ ( minus_minus_nat_o @ A_47 @ B_30 ) ) ) ).
thf(fact_659_image__diff__subset,axiom,
! [F_19: pname > x_a,A_47: pname > $o,B_30: pname > $o] : ( ord_less_eq_a_o @ ( minus_minus_a_o @ ( image_pname_a @ F_19 @ A_47 ) @ ( image_pname_a @ F_19 @ B_30 ) ) @ ( image_pname_a @ F_19 @ ( minus_minus_pname_o @ A_47 @ B_30 ) ) ) ).
thf(fact_660_subset__singletonD,axiom,
! [A_46: x_a > $o,X_15: x_a] :
( ( ord_less_eq_a_o @ A_46 @ ( insert_a @ X_15 @ bot_bot_a_o ) )
=> ( ( A_46 = bot_bot_a_o )
| ( A_46
= ( insert_a @ X_15 @ bot_bot_a_o ) ) ) ) ).
thf(fact_661_subset__singletonD,axiom,
! [A_46: nat > $o,X_15: nat] :
( ( ord_less_eq_nat_o @ A_46 @ ( insert_nat @ X_15 @ bot_bot_nat_o ) )
=> ( ( A_46 = bot_bot_nat_o )
| ( A_46
= ( insert_nat @ X_15 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_662_subset__singletonD,axiom,
! [A_46: int > $o,X_15: int] :
( ( ord_less_eq_int_o @ A_46 @ ( insert_int @ X_15 @ bot_bot_int_o ) )
=> ( ( A_46 = bot_bot_int_o )
| ( A_46
= ( insert_int @ X_15 @ bot_bot_int_o ) ) ) ) ).
thf(fact_663_nat__case__Suc,axiom,
! [F1: nat,F2: nat > nat,Nat_3: nat] :
( ( nat_case_nat @ F1 @ F2 @ ( suc @ Nat_3 ) )
= ( F2 @ Nat_3 ) ) ).
thf(fact_664_nat__case__Suc,axiom,
! [F1: $o,F2: nat > $o,Nat_3: nat] :
( ( nat_case_o @ F1 @ F2 @ ( suc @ Nat_3 ) )
<=> ( F2 @ Nat_3 ) ) ).
thf(fact_665_image__constant,axiom,
! [C_21: int,X_14: nat,A_45: nat > $o] :
( ( member_nat @ X_14 @ A_45 )
=> ( ( image_nat_int
@ ^ [X_1: nat] : C_21
@ A_45 )
= ( insert_int @ C_21 @ bot_bot_int_o ) ) ) ).
thf(fact_666_image__constant,axiom,
! [C_21: x_a,X_14: pname,A_45: pname > $o] :
( ( member_pname @ X_14 @ A_45 )
=> ( ( image_pname_a
@ ^ [X_1: pname] : C_21
@ A_45 )
= ( insert_a @ C_21 @ bot_bot_a_o ) ) ) ).
thf(fact_667_image__constant__conv,axiom,
! [C_20: int,A_44: nat > $o] :
( ( ( A_44 = bot_bot_nat_o )
=> ( ( image_nat_int
@ ^ [X_1: nat] : C_20
@ A_44 )
= bot_bot_int_o ) )
& ( ( A_44 != bot_bot_nat_o )
=> ( ( image_nat_int
@ ^ [X_1: nat] : C_20
@ A_44 )
= ( insert_int @ C_20 @ bot_bot_int_o ) ) ) ) ).
thf(fact_668_image__constant__conv,axiom,
! [C_20: x_a,A_44: pname > $o] :
( ( ( A_44 = bot_bot_pname_o )
=> ( ( image_pname_a
@ ^ [X_1: pname] : C_20
@ A_44 )
= bot_bot_a_o ) )
& ( ( A_44 != bot_bot_pname_o )
=> ( ( image_pname_a
@ ^ [X_1: pname] : C_20
@ A_44 )
= ( insert_a @ C_20 @ bot_bot_a_o ) ) ) ) ).
thf(fact_669_diff__eq__diff__eq,axiom,
! [A_43: int,B_29: int,C_19: int,D_3: int] :
( ( ( minus_minus_int @ A_43 @ B_29 )
= ( minus_minus_int @ C_19 @ D_3 ) )
=> ( ( A_43 = B_29 )
<=> ( C_19 = D_3 ) ) ) ).
thf(fact_670_card__Diff__subset,axiom,
! [A_42: int > $o,B_28: int > $o] :
( ( finite_finite_int @ B_28 )
=> ( ( ord_less_eq_int_o @ B_28 @ A_42 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_42 @ B_28 ) )
= ( minus_minus_nat @ ( finite_card_int @ A_42 ) @ ( finite_card_int @ B_28 ) ) ) ) ) ).
thf(fact_671_card__Diff__subset,axiom,
! [A_42: nat > $o,B_28: nat > $o] :
( ( finite_finite_nat @ B_28 )
=> ( ( ord_less_eq_nat_o @ B_28 @ A_42 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_42 @ B_28 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A_42 ) @ ( finite_card_nat @ B_28 ) ) ) ) ) ).
thf(fact_672_card__Diff__subset,axiom,
! [A_42: x_a > $o,B_28: x_a > $o] :
( ( finite_finite_a @ B_28 )
=> ( ( ord_less_eq_a_o @ B_28 @ A_42 )
=> ( ( finite_card_a @ ( minus_minus_a_o @ A_42 @ B_28 ) )
= ( minus_minus_nat @ ( finite_card_a @ A_42 ) @ ( finite_card_a @ B_28 ) ) ) ) ) ).
thf(fact_673_card__Diff__subset,axiom,
! [A_42: pname > $o,B_28: pname > $o] :
( ( finite_finite_pname @ B_28 )
=> ( ( ord_less_eq_pname_o @ B_28 @ A_42 )
=> ( ( finite_card_pname @ ( minus_minus_pname_o @ A_42 @ B_28 ) )
= ( minus_minus_nat @ ( finite_card_pname @ A_42 ) @ ( finite_card_pname @ B_28 ) ) ) ) ) ).
thf(fact_674_diff__card__le__card__Diff,axiom,
! [A_41: int > $o,B_27: int > $o] :
( ( finite_finite_int @ B_27 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A_41 ) @ ( finite_card_int @ B_27 ) ) @ ( finite_card_int @ ( minus_minus_int_o @ A_41 @ B_27 ) ) ) ) ).
thf(fact_675_diff__card__le__card__Diff,axiom,
! [A_41: nat > $o,B_27: nat > $o] :
( ( finite_finite_nat @ B_27 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A_41 ) @ ( finite_card_nat @ B_27 ) ) @ ( finite_card_nat @ ( minus_minus_nat_o @ A_41 @ B_27 ) ) ) ) ).
thf(fact_676_diff__card__le__card__Diff,axiom,
! [A_41: x_a > $o,B_27: x_a > $o] :
( ( finite_finite_a @ B_27 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A_41 ) @ ( finite_card_a @ B_27 ) ) @ ( finite_card_a @ ( minus_minus_a_o @ A_41 @ B_27 ) ) ) ) ).
thf(fact_677_diff__card__le__card__Diff,axiom,
! [A_41: pname > $o,B_27: pname > $o] :
( ( finite_finite_pname @ B_27 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_pname @ A_41 ) @ ( finite_card_pname @ B_27 ) ) @ ( finite_card_pname @ ( minus_minus_pname_o @ A_41 @ B_27 ) ) ) ) ).
thf(fact_678_finite__subset__induct,axiom,
! [P_5: ( x_a > $o ) > $o,A_40: x_a > $o,F_18: x_a > $o] :
( ( finite_finite_a @ F_18 )
=> ( ( ord_less_eq_a_o @ F_18 @ A_40 )
=> ( ( P_5 @ bot_bot_a_o )
=> ( ! [A_37: x_a,F_2: x_a > $o] :
( ( finite_finite_a @ F_2 )
=> ( ( member_a @ A_37 @ A_40 )
=> ( ~ ( member_a @ A_37 @ F_2 )
=> ( ( P_5 @ F_2 )
=> ( P_5 @ ( insert_a @ A_37 @ F_2 ) ) ) ) ) )
=> ( P_5 @ F_18 ) ) ) ) ) ).
thf(fact_679_finite__subset__induct,axiom,
! [P_5: ( int > $o ) > $o,A_40: int > $o,F_18: int > $o] :
( ( finite_finite_int @ F_18 )
=> ( ( ord_less_eq_int_o @ F_18 @ A_40 )
=> ( ( P_5 @ bot_bot_int_o )
=> ( ! [A_37: int,F_2: int > $o] :
( ( finite_finite_int @ F_2 )
=> ( ( member_int @ A_37 @ A_40 )
=> ( ~ ( member_int @ A_37 @ F_2 )
=> ( ( P_5 @ F_2 )
=> ( P_5 @ ( insert_int @ A_37 @ F_2 ) ) ) ) ) )
=> ( P_5 @ F_18 ) ) ) ) ) ).
thf(fact_680_finite__subset__induct,axiom,
! [P_5: ( nat > $o ) > $o,A_40: nat > $o,F_18: nat > $o] :
( ( finite_finite_nat @ F_18 )
=> ( ( ord_less_eq_nat_o @ F_18 @ A_40 )
=> ( ( P_5 @ bot_bot_nat_o )
=> ( ! [A_37: nat,F_2: nat > $o] :
( ( finite_finite_nat @ F_2 )
=> ( ( member_nat @ A_37 @ A_40 )
=> ( ~ ( member_nat @ A_37 @ F_2 )
=> ( ( P_5 @ F_2 )
=> ( P_5 @ ( insert_nat @ A_37 @ F_2 ) ) ) ) ) )
=> ( P_5 @ F_18 ) ) ) ) ) ).
thf(fact_681_finite__subset__induct,axiom,
! [P_5: ( pname > $o ) > $o,A_40: pname > $o,F_18: pname > $o] :
( ( finite_finite_pname @ F_18 )
=> ( ( ord_less_eq_pname_o @ F_18 @ A_40 )
=> ( ( P_5 @ bot_bot_pname_o )
=> ( ! [A_37: pname,F_2: pname > $o] :
( ( finite_finite_pname @ F_2 )
=> ( ( member_pname @ A_37 @ A_40 )
=> ( ~ ( member_pname @ A_37 @ F_2 )
=> ( ( P_5 @ F_2 )
=> ( P_5 @ ( insert_pname @ A_37 @ F_2 ) ) ) ) ) )
=> ( P_5 @ F_18 ) ) ) ) ) ).
thf(fact_682_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_683_finite__empty__induct,axiom,
! [P_4: ( x_a > $o ) > $o,A_39: x_a > $o] :
( ( finite_finite_a @ A_39 )
=> ( ( P_4 @ A_39 )
=> ( ! [A_37: x_a,A_38: x_a > $o] :
( ( finite_finite_a @ A_38 )
=> ( ( member_a @ A_37 @ A_38 )
=> ( ( P_4 @ A_38 )
=> ( P_4 @ ( minus_minus_a_o @ A_38 @ ( insert_a @ A_37 @ bot_bot_a_o ) ) ) ) ) )
=> ( P_4 @ bot_bot_a_o ) ) ) ) ).
thf(fact_684_finite__empty__induct,axiom,
! [P_4: ( int > $o ) > $o,A_39: int > $o] :
( ( finite_finite_int @ A_39 )
=> ( ( P_4 @ A_39 )
=> ( ! [A_37: int,A_38: int > $o] :
( ( finite_finite_int @ A_38 )
=> ( ( member_int @ A_37 @ A_38 )
=> ( ( P_4 @ A_38 )
=> ( P_4 @ ( minus_minus_int_o @ A_38 @ ( insert_int @ A_37 @ bot_bot_int_o ) ) ) ) ) )
=> ( P_4 @ bot_bot_int_o ) ) ) ) ).
thf(fact_685_finite__empty__induct,axiom,
! [P_4: ( nat > $o ) > $o,A_39: nat > $o] :
( ( finite_finite_nat @ A_39 )
=> ( ( P_4 @ A_39 )
=> ( ! [A_37: nat,A_38: nat > $o] :
( ( finite_finite_nat @ A_38 )
=> ( ( member_nat @ A_37 @ A_38 )
=> ( ( P_4 @ A_38 )
=> ( P_4 @ ( minus_minus_nat_o @ A_38 @ ( insert_nat @ A_37 @ bot_bot_nat_o ) ) ) ) ) )
=> ( P_4 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_686_finite__empty__induct,axiom,
! [P_4: ( pname > $o ) > $o,A_39: pname > $o] :
( ( finite_finite_pname @ A_39 )
=> ( ( P_4 @ A_39 )
=> ( ! [A_37: pname,A_38: pname > $o] :
( ( finite_finite_pname @ A_38 )
=> ( ( member_pname @ A_37 @ A_38 )
=> ( ( P_4 @ A_38 )
=> ( P_4 @ ( minus_minus_pname_o @ A_38 @ ( insert_pname @ A_37 @ bot_bot_pname_o ) ) ) ) ) )
=> ( P_4 @ bot_bot_pname_o ) ) ) ) ).
thf(fact_687_finite__induct,axiom,
! [P_3: ( x_a > $o ) > $o,F_17: x_a > $o] :
( ( finite_finite_a @ F_17 )
=> ( ( P_3 @ bot_bot_a_o )
=> ( ! [X_1: x_a,F_2: x_a > $o] :
( ( finite_finite_a @ F_2 )
=> ( ~ ( member_a @ X_1 @ F_2 )
=> ( ( P_3 @ F_2 )
=> ( P_3 @ ( insert_a @ X_1 @ F_2 ) ) ) ) )
=> ( P_3 @ F_17 ) ) ) ) ).
thf(fact_688_finite__induct,axiom,
! [P_3: ( int > $o ) > $o,F_17: int > $o] :
( ( finite_finite_int @ F_17 )
=> ( ( P_3 @ bot_bot_int_o )
=> ( ! [X_1: int,F_2: int > $o] :
( ( finite_finite_int @ F_2 )
=> ( ~ ( member_int @ X_1 @ F_2 )
=> ( ( P_3 @ F_2 )
=> ( P_3 @ ( insert_int @ X_1 @ F_2 ) ) ) ) )
=> ( P_3 @ F_17 ) ) ) ) ).
thf(fact_689_finite__induct,axiom,
! [P_3: ( nat > $o ) > $o,F_17: nat > $o] :
( ( finite_finite_nat @ F_17 )
=> ( ( P_3 @ bot_bot_nat_o )
=> ( ! [X_1: nat,F_2: nat > $o] :
( ( finite_finite_nat @ F_2 )
=> ( ~ ( member_nat @ X_1 @ F_2 )
=> ( ( P_3 @ F_2 )
=> ( P_3 @ ( insert_nat @ X_1 @ F_2 ) ) ) ) )
=> ( P_3 @ F_17 ) ) ) ) ).
thf(fact_690_finite__induct,axiom,
! [P_3: ( pname > $o ) > $o,F_17: pname > $o] :
( ( finite_finite_pname @ F_17 )
=> ( ( P_3 @ bot_bot_pname_o )
=> ( ! [X_1: pname,F_2: pname > $o] :
( ( finite_finite_pname @ F_2 )
=> ( ~ ( member_pname @ X_1 @ F_2 )
=> ( ( P_3 @ F_2 )
=> ( P_3 @ ( insert_pname @ X_1 @ F_2 ) ) ) ) )
=> ( P_3 @ F_17 ) ) ) ) ).
thf(fact_691_finite_Osimps,axiom,
! [A_36: x_a > $o] :
( ( finite_finite_a @ A_36 )
<=> ( ( A_36 = bot_bot_a_o )
| ? [A_38: x_a > $o,A_37: x_a] :
( ( A_36
= ( insert_a @ A_37 @ A_38 ) )
& ( finite_finite_a @ A_38 ) ) ) ) ).
thf(fact_692_finite_Osimps,axiom,
! [A_36: int > $o] :
( ( finite_finite_int @ A_36 )
<=> ( ( A_36 = bot_bot_int_o )
| ? [A_38: int > $o,A_37: int] :
( ( A_36
= ( insert_int @ A_37 @ A_38 ) )
& ( finite_finite_int @ A_38 ) ) ) ) ).
thf(fact_693_finite_Osimps,axiom,
! [A_36: nat > $o] :
( ( finite_finite_nat @ A_36 )
<=> ( ( A_36 = bot_bot_nat_o )
| ? [A_38: nat > $o,A_37: nat] :
( ( A_36
= ( insert_nat @ A_37 @ A_38 ) )
& ( finite_finite_nat @ A_38 ) ) ) ) ).
thf(fact_694_finite_Osimps,axiom,
! [A_36: pname > $o] :
( ( finite_finite_pname @ A_36 )
<=> ( ( A_36 = bot_bot_pname_o )
| ? [A_38: pname > $o,A_37: pname] :
( ( A_36
= ( insert_pname @ A_37 @ A_38 ) )
& ( finite_finite_pname @ A_38 ) ) ) ) ).
thf(fact_695_finite__less__ub,axiom,
! [U: nat,F: nat > nat] :
( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F @ N_1 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F @ N_1 ) @ U ) ) ) ) ).
thf(fact_696_the__elem__eq,axiom,
! [X_13: x_a] :
( ( the_elem_a @ ( insert_a @ X_13 @ bot_bot_a_o ) )
= X_13 ) ).
thf(fact_697_the__elem__eq,axiom,
! [X_13: nat] :
( ( the_elem_nat @ ( insert_nat @ X_13 @ bot_bot_nat_o ) )
= X_13 ) ).
thf(fact_698_the__elem__eq,axiom,
! [X_13: int] :
( ( the_elem_int @ ( insert_int @ X_13 @ bot_bot_int_o ) )
= X_13 ) ).
thf(fact_699_nonempty__iff,axiom,
! [A_35: x_a > $o] :
( ( A_35 != bot_bot_a_o )
<=> ? [X_1: x_a,B_26: x_a > $o] :
( ( A_35
= ( insert_a @ X_1 @ B_26 ) )
& ~ ( member_a @ X_1 @ B_26 ) ) ) ).
thf(fact_700_nonempty__iff,axiom,
! [A_35: int > $o] :
( ( A_35 != bot_bot_int_o )
<=> ? [X_1: int,B_26: int > $o] :
( ( A_35
= ( insert_int @ X_1 @ B_26 ) )
& ~ ( member_int @ X_1 @ B_26 ) ) ) ).
thf(fact_701_nonempty__iff,axiom,
! [A_35: nat > $o] :
( ( A_35 != bot_bot_nat_o )
<=> ? [X_1: nat,B_26: nat > $o] :
( ( A_35
= ( insert_nat @ X_1 @ B_26 ) )
& ~ ( member_nat @ X_1 @ B_26 ) ) ) ).
thf(fact_702_nonempty__iff,axiom,
! [A_35: pname > $o] :
( ( A_35 != bot_bot_pname_o )
<=> ? [X_1: pname,B_26: pname > $o] :
( ( A_35
= ( insert_pname @ X_1 @ B_26 ) )
& ~ ( member_pname @ X_1 @ B_26 ) ) ) ).
thf(fact_703_assms_I4_J,axiom,
! [Pn: pname] :
( ( member_pname @ Pn @ u )
=> ( wt @ ( the_com @ ( body @ Pn ) ) ) ) ).
thf(fact_704_DiffE,axiom,
! [C_18: int,A_34: int > $o,B_25: int > $o] :
( ( member_int @ C_18 @ ( minus_minus_int_o @ A_34 @ B_25 ) )
=> ~ ( ( member_int @ C_18 @ A_34 )
=> ( member_int @ C_18 @ B_25 ) ) ) ).
thf(fact_705_DiffE,axiom,
! [C_18: nat,A_34: nat > $o,B_25: nat > $o] :
( ( member_nat @ C_18 @ ( minus_minus_nat_o @ A_34 @ B_25 ) )
=> ~ ( ( member_nat @ C_18 @ A_34 )
=> ( member_nat @ C_18 @ B_25 ) ) ) ).
thf(fact_706_DiffE,axiom,
! [C_18: x_a,A_34: x_a > $o,B_25: x_a > $o] :
( ( member_a @ C_18 @ ( minus_minus_a_o @ A_34 @ B_25 ) )
=> ~ ( ( member_a @ C_18 @ A_34 )
=> ( member_a @ C_18 @ B_25 ) ) ) ).
thf(fact_707_DiffE,axiom,
! [C_18: pname,A_34: pname > $o,B_25: pname > $o] :
( ( member_pname @ C_18 @ ( minus_minus_pname_o @ A_34 @ B_25 ) )
=> ~ ( ( member_pname @ C_18 @ A_34 )
=> ( member_pname @ C_18 @ B_25 ) ) ) ).
thf(fact_708_DiffI,axiom,
! [B_24: int > $o,C_17: int,A_33: int > $o] :
( ( member_int @ C_17 @ A_33 )
=> ( ~ ( member_int @ C_17 @ B_24 )
=> ( member_int @ C_17 @ ( minus_minus_int_o @ A_33 @ B_24 ) ) ) ) ).
thf(fact_709_DiffI,axiom,
! [B_24: nat > $o,C_17: nat,A_33: nat > $o] :
( ( member_nat @ C_17 @ A_33 )
=> ( ~ ( member_nat @ C_17 @ B_24 )
=> ( member_nat @ C_17 @ ( minus_minus_nat_o @ A_33 @ B_24 ) ) ) ) ).
thf(fact_710_DiffI,axiom,
! [B_24: x_a > $o,C_17: x_a,A_33: x_a > $o] :
( ( member_a @ C_17 @ A_33 )
=> ( ~ ( member_a @ C_17 @ B_24 )
=> ( member_a @ C_17 @ ( minus_minus_a_o @ A_33 @ B_24 ) ) ) ) ).
thf(fact_711_DiffI,axiom,
! [B_24: pname > $o,C_17: pname,A_33: pname > $o] :
( ( member_pname @ C_17 @ A_33 )
=> ( ~ ( member_pname @ C_17 @ B_24 )
=> ( member_pname @ C_17 @ ( minus_minus_pname_o @ A_33 @ B_24 ) ) ) ) ).
thf(fact_712_DiffD2,axiom,
! [C_16: int,A_32: int > $o,B_23: int > $o] :
( ( member_int @ C_16 @ ( minus_minus_int_o @ A_32 @ B_23 ) )
=> ~ ( member_int @ C_16 @ B_23 ) ) ).
thf(fact_713_DiffD2,axiom,
! [C_16: nat,A_32: nat > $o,B_23: nat > $o] :
( ( member_nat @ C_16 @ ( minus_minus_nat_o @ A_32 @ B_23 ) )
=> ~ ( member_nat @ C_16 @ B_23 ) ) ).
thf(fact_714_DiffD2,axiom,
! [C_16: x_a,A_32: x_a > $o,B_23: x_a > $o] :
( ( member_a @ C_16 @ ( minus_minus_a_o @ A_32 @ B_23 ) )
=> ~ ( member_a @ C_16 @ B_23 ) ) ).
thf(fact_715_DiffD2,axiom,
! [C_16: pname,A_32: pname > $o,B_23: pname > $o] :
( ( member_pname @ C_16 @ ( minus_minus_pname_o @ A_32 @ B_23 ) )
=> ~ ( member_pname @ C_16 @ B_23 ) ) ).
thf(fact_716_DiffD1,axiom,
! [C_15: int,A_31: int > $o,B_22: int > $o] :
( ( member_int @ C_15 @ ( minus_minus_int_o @ A_31 @ B_22 ) )
=> ( member_int @ C_15 @ A_31 ) ) ).
thf(fact_717_DiffD1,axiom,
! [C_15: nat,A_31: nat > $o,B_22: nat > $o] :
( ( member_nat @ C_15 @ ( minus_minus_nat_o @ A_31 @ B_22 ) )
=> ( member_nat @ C_15 @ A_31 ) ) ).
thf(fact_718_DiffD1,axiom,
! [C_15: x_a,A_31: x_a > $o,B_22: x_a > $o] :
( ( member_a @ C_15 @ ( minus_minus_a_o @ A_31 @ B_22 ) )
=> ( member_a @ C_15 @ A_31 ) ) ).
thf(fact_719_DiffD1,axiom,
! [C_15: pname,A_31: pname > $o,B_22: pname > $o] :
( ( member_pname @ C_15 @ ( minus_minus_pname_o @ A_31 @ B_22 ) )
=> ( member_pname @ C_15 @ A_31 ) ) ).
thf(fact_720_Diff__iff,axiom,
! [C_14: int,A_30: int > $o,B_21: int > $o] :
( ( member_int @ C_14 @ ( minus_minus_int_o @ A_30 @ B_21 ) )
<=> ( ( member_int @ C_14 @ A_30 )
& ~ ( member_int @ C_14 @ B_21 ) ) ) ).
thf(fact_721_Diff__iff,axiom,
! [C_14: nat,A_30: nat > $o,B_21: nat > $o] :
( ( member_nat @ C_14 @ ( minus_minus_nat_o @ A_30 @ B_21 ) )
<=> ( ( member_nat @ C_14 @ A_30 )
& ~ ( member_nat @ C_14 @ B_21 ) ) ) ).
thf(fact_722_Diff__iff,axiom,
! [C_14: x_a,A_30: x_a > $o,B_21: x_a > $o] :
( ( member_a @ C_14 @ ( minus_minus_a_o @ A_30 @ B_21 ) )
<=> ( ( member_a @ C_14 @ A_30 )
& ~ ( member_a @ C_14 @ B_21 ) ) ) ).
thf(fact_723_Diff__iff,axiom,
! [C_14: pname,A_30: pname > $o,B_21: pname > $o] :
( ( member_pname @ C_14 @ ( minus_minus_pname_o @ A_30 @ B_21 ) )
<=> ( ( member_pname @ C_14 @ A_30 )
& ~ ( member_pname @ C_14 @ B_21 ) ) ) ).
thf(fact_724_set__diff__eq,axiom,
! [A_29: int > $o,B_20: int > $o] :
( ( minus_minus_int_o @ A_29 @ B_20 )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( member_int @ X_1 @ A_29 ) @ ( (~) @ ( member_int @ X_1 @ B_20 ) ) ) ) ) ).
thf(fact_725_set__diff__eq,axiom,
! [A_29: nat > $o,B_20: nat > $o] :
( ( minus_minus_nat_o @ A_29 @ B_20 )
= ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( member_nat @ X_1 @ A_29 ) @ ( (~) @ ( member_nat @ X_1 @ B_20 ) ) ) ) ) ).
thf(fact_726_set__diff__eq,axiom,
! [A_29: x_a > $o,B_20: x_a > $o] :
( ( minus_minus_a_o @ A_29 @ B_20 )
= ( collect_a
@ ^ [X_1: x_a] : ( (&) @ ( member_a @ X_1 @ A_29 ) @ ( (~) @ ( member_a @ X_1 @ B_20 ) ) ) ) ) ).
thf(fact_727_set__diff__eq,axiom,
! [A_29: pname > $o,B_20: pname > $o] :
( ( minus_minus_pname_o @ A_29 @ B_20 )
= ( collect_pname
@ ^ [X_1: pname] : ( (&) @ ( member_pname @ X_1 @ A_29 ) @ ( (~) @ ( member_pname @ X_1 @ B_20 ) ) ) ) ) ).
thf(fact_728_folding__one_Oinsert__remove,axiom,
! [X_12: x_a,A_28: x_a > $o,F_16: x_a > x_a > x_a,F_15: ( x_a > $o ) > x_a] :
( ( finite_folding_one_a @ F_16 @ F_15 )
=> ( ( finite_finite_a @ A_28 )
=> ( ( ( ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) )
= bot_bot_a_o )
=> ( ( F_15 @ ( insert_a @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) )
!= bot_bot_a_o )
=> ( ( F_15 @ ( insert_a @ X_12 @ A_28 ) )
= ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) ) ) ) ) ) ) ) ) ).
thf(fact_729_folding__one_Oinsert__remove,axiom,
! [X_12: int,A_28: int > $o,F_16: int > int > int,F_15: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_16 @ F_15 )
=> ( ( finite_finite_int @ A_28 )
=> ( ( ( ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) )
= bot_bot_int_o )
=> ( ( F_15 @ ( insert_int @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) )
!= bot_bot_int_o )
=> ( ( F_15 @ ( insert_int @ X_12 @ A_28 ) )
= ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ).
thf(fact_730_folding__one_Oinsert__remove,axiom,
! [X_12: nat,A_28: nat > $o,F_16: nat > nat > nat,F_15: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_16 @ F_15 )
=> ( ( finite_finite_nat @ A_28 )
=> ( ( ( ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_15 @ ( insert_nat @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_15 @ ( insert_nat @ X_12 @ A_28 ) )
= ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
thf(fact_731_folding__one_Oinsert__remove,axiom,
! [X_12: pname,A_28: pname > $o,F_16: pname > pname > pname,F_15: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_16 @ F_15 )
=> ( ( finite_finite_pname @ A_28 )
=> ( ( ( ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
= bot_bot_pname_o )
=> ( ( F_15 @ ( insert_pname @ X_12 @ A_28 ) )
= X_12 ) )
& ( ( ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
!= bot_bot_pname_o )
=> ( ( F_15 @ ( insert_pname @ X_12 @ A_28 ) )
= ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) ).
thf(fact_732_folding__one_Oremove,axiom,
! [X_11: x_a,A_27: x_a > $o,F_14: x_a > x_a > x_a,F_13: ( x_a > $o ) > x_a] :
( ( finite_folding_one_a @ F_14 @ F_13 )
=> ( ( finite_finite_a @ A_27 )
=> ( ( member_a @ X_11 @ A_27 )
=> ( ( ( ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) )
= bot_bot_a_o )
=> ( ( F_13 @ A_27 )
= X_11 ) )
& ( ( ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) )
!= bot_bot_a_o )
=> ( ( F_13 @ A_27 )
= ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_733_folding__one_Oremove,axiom,
! [X_11: int,A_27: int > $o,F_14: int > int > int,F_13: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_14 @ F_13 )
=> ( ( finite_finite_int @ A_27 )
=> ( ( member_int @ X_11 @ A_27 )
=> ( ( ( ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) )
= bot_bot_int_o )
=> ( ( F_13 @ A_27 )
= X_11 ) )
& ( ( ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) )
!= bot_bot_int_o )
=> ( ( F_13 @ A_27 )
= ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_734_folding__one_Oremove,axiom,
! [X_11: nat,A_27: nat > $o,F_14: nat > nat > nat,F_13: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_14 @ F_13 )
=> ( ( finite_finite_nat @ A_27 )
=> ( ( member_nat @ X_11 @ A_27 )
=> ( ( ( ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) )
= bot_bot_nat_o )
=> ( ( F_13 @ A_27 )
= X_11 ) )
& ( ( ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) )
!= bot_bot_nat_o )
=> ( ( F_13 @ A_27 )
= ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_735_folding__one_Oremove,axiom,
! [X_11: pname,A_27: pname > $o,F_14: pname > pname > pname,F_13: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_14 @ F_13 )
=> ( ( finite_finite_pname @ A_27 )
=> ( ( member_pname @ X_11 @ A_27 )
=> ( ( ( ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) )
= bot_bot_pname_o )
=> ( ( F_13 @ A_27 )
= X_11 ) )
& ( ( ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) )
!= bot_bot_pname_o )
=> ( ( F_13 @ A_27 )
= ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_736_card__Diff__singleton__if,axiom,
! [X_10: x_a,A_26: x_a > $o] :
( ( finite_finite_a @ A_26 )
=> ( ( ( member_a @ X_10 @ A_26 )
=> ( ( finite_card_a @ ( minus_minus_a_o @ A_26 @ ( insert_a @ X_10 @ bot_bot_a_o ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A_26 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X_10 @ A_26 )
=> ( ( finite_card_a @ ( minus_minus_a_o @ A_26 @ ( insert_a @ X_10 @ bot_bot_a_o ) ) )
= ( finite_card_a @ A_26 ) ) ) ) ) ).
thf(fact_737_card__Diff__singleton__if,axiom,
! [X_10: int,A_26: int > $o] :
( ( finite_finite_int @ A_26 )
=> ( ( ( member_int @ X_10 @ A_26 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_26 @ ( insert_int @ X_10 @ bot_bot_int_o ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A_26 ) @ one_one_nat ) ) )
& ( ~ ( member_int @ X_10 @ A_26 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_26 @ ( insert_int @ X_10 @ bot_bot_int_o ) ) )
= ( finite_card_int @ A_26 ) ) ) ) ) ).
thf(fact_738_card__Diff__singleton__if,axiom,
! [X_10: nat,A_26: nat > $o] :
( ( finite_finite_nat @ A_26 )
=> ( ( ( member_nat @ X_10 @ A_26 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_26 @ ( insert_nat @ X_10 @ bot_bot_nat_o ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A_26 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X_10 @ A_26 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_26 @ ( insert_nat @ X_10 @ bot_bot_nat_o ) ) )
= ( finite_card_nat @ A_26 ) ) ) ) ) ).
thf(fact_739_card__Diff__singleton__if,axiom,
! [X_10: pname,A_26: pname > $o] :
( ( finite_finite_pname @ A_26 )
=> ( ( ( member_pname @ X_10 @ A_26 )
=> ( ( finite_card_pname @ ( minus_minus_pname_o @ A_26 @ ( insert_pname @ X_10 @ bot_bot_pname_o ) ) )
= ( minus_minus_nat @ ( finite_card_pname @ A_26 ) @ one_one_nat ) ) )
& ( ~ ( member_pname @ X_10 @ A_26 )
=> ( ( finite_card_pname @ ( minus_minus_pname_o @ A_26 @ ( insert_pname @ X_10 @ bot_bot_pname_o ) ) )
= ( finite_card_pname @ A_26 ) ) ) ) ) ).
thf(fact_740_card__Diff__singleton,axiom,
! [X_9: x_a,A_25: x_a > $o] :
( ( finite_finite_a @ A_25 )
=> ( ( member_a @ X_9 @ A_25 )
=> ( ( finite_card_a @ ( minus_minus_a_o @ A_25 @ ( insert_a @ X_9 @ bot_bot_a_o ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A_25 ) @ one_one_nat ) ) ) ) ).
thf(fact_741_card__Diff__singleton,axiom,
! [X_9: int,A_25: int > $o] :
( ( finite_finite_int @ A_25 )
=> ( ( member_int @ X_9 @ A_25 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_25 @ ( insert_int @ X_9 @ bot_bot_int_o ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A_25 ) @ one_one_nat ) ) ) ) ).
thf(fact_742_card__Diff__singleton,axiom,
! [X_9: nat,A_25: nat > $o] :
( ( finite_finite_nat @ A_25 )
=> ( ( member_nat @ X_9 @ A_25 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_25 @ ( insert_nat @ X_9 @ bot_bot_nat_o ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A_25 ) @ one_one_nat ) ) ) ) ).
thf(fact_743_card__Diff__singleton,axiom,
! [X_9: pname,A_25: pname > $o] :
( ( finite_finite_pname @ A_25 )
=> ( ( member_pname @ X_9 @ A_25 )
=> ( ( finite_card_pname @ ( minus_minus_pname_o @ A_25 @ ( insert_pname @ X_9 @ bot_bot_pname_o ) ) )
= ( minus_minus_nat @ ( finite_card_pname @ A_25 ) @ one_one_nat ) ) ) ) ).
thf(fact_744_one__reorient,axiom,
! [X_8: int] :
( ( one_one_int = X_8 )
<=> ( X_8 = one_one_int ) ) ).
thf(fact_745_one__reorient,axiom,
! [X_8: nat] :
( ( one_one_nat = X_8 )
<=> ( X_8 = one_one_nat ) ) ).
thf(fact_746_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
thf(fact_747_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_748_folding__one_Osingleton,axiom,
! [X_7: x_a,F_12: x_a > x_a > x_a,F_11: ( x_a > $o ) > x_a] :
( ( finite_folding_one_a @ F_12 @ F_11 )
=> ( ( F_11 @ ( insert_a @ X_7 @ bot_bot_a_o ) )
= X_7 ) ) ).
thf(fact_749_folding__one_Osingleton,axiom,
! [X_7: nat,F_12: nat > nat > nat,F_11: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_12 @ F_11 )
=> ( ( F_11 @ ( insert_nat @ X_7 @ bot_bot_nat_o ) )
= X_7 ) ) ).
thf(fact_750_folding__one_Osingleton,axiom,
! [X_7: int,F_12: int > int > int,F_11: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_12 @ F_11 )
=> ( ( F_11 @ ( insert_int @ X_7 @ bot_bot_int_o ) )
= X_7 ) ) ).
thf(fact_751_card__Diff__insert,axiom,
! [B_19: x_a > $o,A_24: x_a,A_23: x_a > $o] :
( ( finite_finite_a @ A_23 )
=> ( ( member_a @ A_24 @ A_23 )
=> ( ~ ( member_a @ A_24 @ B_19 )
=> ( ( finite_card_a @ ( minus_minus_a_o @ A_23 @ ( insert_a @ A_24 @ B_19 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_752_card__Diff__insert,axiom,
! [B_19: int > $o,A_24: int,A_23: int > $o] :
( ( finite_finite_int @ A_23 )
=> ( ( member_int @ A_24 @ A_23 )
=> ( ~ ( member_int @ A_24 @ B_19 )
=> ( ( finite_card_int @ ( minus_minus_int_o @ A_23 @ ( insert_int @ A_24 @ B_19 ) ) )
= ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_753_card__Diff__insert,axiom,
! [B_19: nat > $o,A_24: nat,A_23: nat > $o] :
( ( finite_finite_nat @ A_23 )
=> ( ( member_nat @ A_24 @ A_23 )
=> ( ~ ( member_nat @ A_24 @ B_19 )
=> ( ( finite_card_nat @ ( minus_minus_nat_o @ A_23 @ ( insert_nat @ A_24 @ B_19 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_754_card__Diff__insert,axiom,
! [B_19: pname > $o,A_24: pname,A_23: pname > $o] :
( ( finite_finite_pname @ A_23 )
=> ( ( member_pname @ A_24 @ A_23 )
=> ( ~ ( member_pname @ A_24 @ B_19 )
=> ( ( finite_card_pname @ ( minus_minus_pname_o @ A_23 @ ( insert_pname @ A_24 @ B_19 ) ) )
= ( minus_minus_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_755_folding__one_Oinsert,axiom,
! [X_6: x_a,A_22: x_a > $o,F_10: x_a > x_a > x_a,F_9: ( x_a > $o ) > x_a] :
( ( finite_folding_one_a @ F_10 @ F_9 )
=> ( ( finite_finite_a @ A_22 )
=> ( ~ ( member_a @ X_6 @ A_22 )
=> ( ( A_22 != bot_bot_a_o )
=> ( ( F_9 @ ( insert_a @ X_6 @ A_22 ) )
= ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
thf(fact_756_folding__one_Oinsert,axiom,
! [X_6: int,A_22: int > $o,F_10: int > int > int,F_9: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_10 @ F_9 )
=> ( ( finite_finite_int @ A_22 )
=> ( ~ ( member_int @ X_6 @ A_22 )
=> ( ( A_22 != bot_bot_int_o )
=> ( ( F_9 @ ( insert_int @ X_6 @ A_22 ) )
= ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
thf(fact_757_folding__one_Oinsert,axiom,
! [X_6: nat,A_22: nat > $o,F_10: nat > nat > nat,F_9: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_10 @ F_9 )
=> ( ( finite_finite_nat @ A_22 )
=> ( ~ ( member_nat @ X_6 @ A_22 )
=> ( ( A_22 != bot_bot_nat_o )
=> ( ( F_9 @ ( insert_nat @ X_6 @ A_22 ) )
= ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
thf(fact_758_folding__one_Oinsert,axiom,
! [X_6: pname,A_22: pname > $o,F_10: pname > pname > pname,F_9: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_10 @ F_9 )
=> ( ( finite_finite_pname @ A_22 )
=> ( ~ ( member_pname @ X_6 @ A_22 )
=> ( ( A_22 != bot_bot_pname_o )
=> ( ( F_9 @ ( insert_pname @ X_6 @ A_22 ) )
= ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
thf(fact_759_folding__one_Oclosed,axiom,
! [A_21: x_a > $o,F_8: x_a > x_a > x_a,F_7: ( x_a > $o ) > x_a] :
( ( finite_folding_one_a @ F_8 @ F_7 )
=> ( ( finite_finite_a @ A_21 )
=> ( ( A_21 != bot_bot_a_o )
=> ( ! [X_1: x_a,Y_1: x_a] : ( member_a @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_a @ X_1 @ ( insert_a @ Y_1 @ bot_bot_a_o ) ) )
=> ( member_a @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
thf(fact_760_folding__one_Oclosed,axiom,
! [A_21: int > $o,F_8: int > int > int,F_7: ( int > $o ) > int] :
( ( finite1626084323ne_int @ F_8 @ F_7 )
=> ( ( finite_finite_int @ A_21 )
=> ( ( A_21 != bot_bot_int_o )
=> ( ! [X_1: int,Y_1: int] : ( member_int @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_int @ X_1 @ ( insert_int @ Y_1 @ bot_bot_int_o ) ) )
=> ( member_int @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
thf(fact_761_folding__one_Oclosed,axiom,
! [A_21: nat > $o,F_8: nat > nat > nat,F_7: ( nat > $o ) > nat] :
( ( finite988810631ne_nat @ F_8 @ F_7 )
=> ( ( finite_finite_nat @ A_21 )
=> ( ( A_21 != bot_bot_nat_o )
=> ( ! [X_1: nat,Y_1: nat] : ( member_nat @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_nat @ X_1 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
=> ( member_nat @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
thf(fact_762_folding__one_Oclosed,axiom,
! [A_21: pname > $o,F_8: pname > pname > pname,F_7: ( pname > $o ) > pname] :
( ( finite1282449217_pname @ F_8 @ F_7 )
=> ( ( finite_finite_pname @ A_21 )
=> ( ( A_21 != bot_bot_pname_o )
=> ( ! [X_1: pname,Y_1: pname] : ( member_pname @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_pname @ X_1 @ ( insert_pname @ Y_1 @ bot_bot_pname_o ) ) )
=> ( member_pname @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
thf(fact_763_card_Oremove,axiom,
! [X_5: x_a,A_20: x_a > $o] :
( ( finite_finite_a @ A_20 )
=> ( ( member_a @ X_5 @ A_20 )
=> ( ( finite_card_a @ A_20 )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_20 @ ( insert_a @ X_5 @ bot_bot_a_o ) ) ) ) ) ) ) ).
thf(fact_764_card_Oremove,axiom,
! [X_5: int,A_20: int > $o] :
( ( finite_finite_int @ A_20 )
=> ( ( member_int @ X_5 @ A_20 )
=> ( ( finite_card_int @ A_20 )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_20 @ ( insert_int @ X_5 @ bot_bot_int_o ) ) ) ) ) ) ) ).
thf(fact_765_card_Oremove,axiom,
! [X_5: nat,A_20: nat > $o] :
( ( finite_finite_nat @ A_20 )
=> ( ( member_nat @ X_5 @ A_20 )
=> ( ( finite_card_nat @ A_20 )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_20 @ ( insert_nat @ X_5 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
thf(fact_766_card_Oremove,axiom,
! [X_5: pname,A_20: pname > $o] :
( ( finite_finite_pname @ A_20 )
=> ( ( member_pname @ X_5 @ A_20 )
=> ( ( finite_card_pname @ A_20 )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_20 @ ( insert_pname @ X_5 @ bot_bot_pname_o ) ) ) ) ) ) ) ).
thf(fact_767_card_Oinsert__remove,axiom,
! [X_4: x_a,A_19: x_a > $o] :
( ( finite_finite_a @ A_19 )
=> ( ( finite_card_a @ ( insert_a @ X_4 @ A_19 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_19 @ ( insert_a @ X_4 @ bot_bot_a_o ) ) ) ) ) ) ).
thf(fact_768_card_Oinsert__remove,axiom,
! [X_4: int,A_19: int > $o] :
( ( finite_finite_int @ A_19 )
=> ( ( finite_card_int @ ( insert_int @ X_4 @ A_19 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_19 @ ( insert_int @ X_4 @ bot_bot_int_o ) ) ) ) ) ) ).
thf(fact_769_card_Oinsert__remove,axiom,
! [X_4: nat,A_19: nat > $o] :
( ( finite_finite_nat @ A_19 )
=> ( ( finite_card_nat @ ( insert_nat @ X_4 @ A_19 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_19 @ ( insert_nat @ X_4 @ bot_bot_nat_o ) ) ) ) ) ) ).
thf(fact_770_card_Oinsert__remove,axiom,
! [X_4: pname,A_19: pname > $o] :
( ( finite_finite_pname @ A_19 )
=> ( ( finite_card_pname @ ( insert_pname @ X_4 @ A_19 ) )
= ( plus_plus_nat @ one_one_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_19 @ ( insert_pname @ X_4 @ bot_bot_pname_o ) ) ) ) ) ) ).
thf(fact_771_folding__one__idem_Osubset__idem,axiom,
! [B_18: int > $o,A_18: int > $o,F_6: int > int > int,F_5: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_6 @ F_5 )
=> ( ( finite_finite_int @ A_18 )
=> ( ( B_18 != bot_bot_int_o )
=> ( ( ord_less_eq_int_o @ B_18 @ A_18 )
=> ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
= ( F_5 @ A_18 ) ) ) ) ) ) ).
thf(fact_772_folding__one__idem_Osubset__idem,axiom,
! [B_18: nat > $o,A_18: nat > $o,F_6: nat > nat > nat,F_5: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_6 @ F_5 )
=> ( ( finite_finite_nat @ A_18 )
=> ( ( B_18 != bot_bot_nat_o )
=> ( ( ord_less_eq_nat_o @ B_18 @ A_18 )
=> ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
= ( F_5 @ A_18 ) ) ) ) ) ) ).
thf(fact_773_folding__one__idem_Osubset__idem,axiom,
! [B_18: pname > $o,A_18: pname > $o,F_6: pname > pname > pname,F_5: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_6 @ F_5 )
=> ( ( finite_finite_pname @ A_18 )
=> ( ( B_18 != bot_bot_pname_o )
=> ( ( ord_less_eq_pname_o @ B_18 @ A_18 )
=> ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
= ( F_5 @ A_18 ) ) ) ) ) ) ).
thf(fact_774_folding__one__idem_Osubset__idem,axiom,
! [B_18: x_a > $o,A_18: x_a > $o,F_6: x_a > x_a > x_a,F_5: ( x_a > $o ) > x_a] :
( ( finite1819937229idem_a @ F_6 @ F_5 )
=> ( ( finite_finite_a @ A_18 )
=> ( ( B_18 != bot_bot_a_o )
=> ( ( ord_less_eq_a_o @ B_18 @ A_18 )
=> ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
= ( F_5 @ A_18 ) ) ) ) ) ) ).
thf(fact_775_folding__one__idem_Oinsert__idem,axiom,
! [X_3: x_a,A_17: x_a > $o,F_4: x_a > x_a > x_a,F_3: ( x_a > $o ) > x_a] :
( ( finite1819937229idem_a @ F_4 @ F_3 )
=> ( ( finite_finite_a @ A_17 )
=> ( ( A_17 != bot_bot_a_o )
=> ( ( F_3 @ ( insert_a @ X_3 @ A_17 ) )
= ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
thf(fact_776_folding__one__idem_Oinsert__idem,axiom,
! [X_3: int,A_17: int > $o,F_4: int > int > int,F_3: ( int > $o ) > int] :
( ( finite1432773856em_int @ F_4 @ F_3 )
=> ( ( finite_finite_int @ A_17 )
=> ( ( A_17 != bot_bot_int_o )
=> ( ( F_3 @ ( insert_int @ X_3 @ A_17 ) )
= ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
thf(fact_777_folding__one__idem_Oinsert__idem,axiom,
! [X_3: nat,A_17: nat > $o,F_4: nat > nat > nat,F_3: ( nat > $o ) > nat] :
( ( finite795500164em_nat @ F_4 @ F_3 )
=> ( ( finite_finite_nat @ A_17 )
=> ( ( A_17 != bot_bot_nat_o )
=> ( ( F_3 @ ( insert_nat @ X_3 @ A_17 ) )
= ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
thf(fact_778_folding__one__idem_Oinsert__idem,axiom,
! [X_3: pname,A_17: pname > $o,F_4: pname > pname > pname,F_3: ( pname > $o ) > pname] :
( ( finite89670078_pname @ F_4 @ F_3 )
=> ( ( finite_finite_pname @ A_17 )
=> ( ( A_17 != bot_bot_pname_o )
=> ( ( F_3 @ ( insert_pname @ X_3 @ A_17 ) )
= ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
thf(fact_779_finite__ne__induct,axiom,
! [P_2: ( x_a > $o ) > $o,F_1: x_a > $o] :
( ( finite_finite_a @ F_1 )
=> ( ( F_1 != bot_bot_a_o )
=> ( ! [X_1: x_a] : ( P_2 @ ( insert_a @ X_1 @ bot_bot_a_o ) )
=> ( ! [X_1: x_a,F_2: x_a > $o] :
( ( finite_finite_a @ F_2 )
=> ( ( F_2 != bot_bot_a_o )
=> ( ~ ( member_a @ X_1 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_a @ X_1 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_1 ) ) ) ) ) ).
thf(fact_780_finite__ne__induct,axiom,
! [P_2: ( int > $o ) > $o,F_1: int > $o] :
( ( finite_finite_int @ F_1 )
=> ( ( F_1 != bot_bot_int_o )
=> ( ! [X_1: int] : ( P_2 @ ( insert_int @ X_1 @ bot_bot_int_o ) )
=> ( ! [X_1: int,F_2: int > $o] :
( ( finite_finite_int @ F_2 )
=> ( ( F_2 != bot_bot_int_o )
=> ( ~ ( member_int @ X_1 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_int @ X_1 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_1 ) ) ) ) ) ).
thf(fact_781_finite__ne__induct,axiom,
! [P_2: ( nat > $o ) > $o,F_1: nat > $o] :
( ( finite_finite_nat @ F_1 )
=> ( ( F_1 != bot_bot_nat_o )
=> ( ! [X_1: nat] : ( P_2 @ ( insert_nat @ X_1 @ bot_bot_nat_o ) )
=> ( ! [X_1: nat,F_2: nat > $o] :
( ( finite_finite_nat @ F_2 )
=> ( ( F_2 != bot_bot_nat_o )
=> ( ~ ( member_nat @ X_1 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_nat @ X_1 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_1 ) ) ) ) ) ).
thf(fact_782_finite__ne__induct,axiom,
! [P_2: ( pname > $o ) > $o,F_1: pname > $o] :
( ( finite_finite_pname @ F_1 )
=> ( ( F_1 != bot_bot_pname_o )
=> ( ! [X_1: pname] : ( P_2 @ ( insert_pname @ X_1 @ bot_bot_pname_o ) )
=> ( ! [X_1: pname,F_2: pname > $o] :
( ( finite_finite_pname @ F_2 )
=> ( ( F_2 != bot_bot_pname_o )
=> ( ~ ( member_pname @ X_1 @ F_2 )
=> ( ( P_2 @ F_2 )
=> ( P_2 @ ( insert_pname @ X_1 @ F_2 ) ) ) ) ) )
=> ( P_2 @ F_1 ) ) ) ) ) ).
thf(fact_783_the__elem__def,axiom,
! [X_2: x_a > $o] :
( ( the_elem_a @ X_2 )
= ( the_a
@ ^ [X_1: x_a] :
( X_2
= ( insert_a @ X_1 @ bot_bot_a_o ) ) ) ) ).
thf(fact_784_the__elem__def,axiom,
! [X_2: nat > $o] :
( ( the_elem_nat @ X_2 )
= ( the_nat
@ ^ [X_1: nat] :
( X_2
= ( insert_nat @ X_1 @ bot_bot_nat_o ) ) ) ) ).
thf(fact_785_the__elem__def,axiom,
! [X_2: int > $o] :
( ( the_elem_int @ X_2 )
= ( the_int
@ ^ [X_1: int] :
( X_2
= ( insert_int @ X_1 @ bot_bot_int_o ) ) ) ) ).
thf(fact_786_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_16: int,B_17: int,C_13: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_16 @ B_17 ) @ C_13 )
= ( plus_plus_int @ A_16 @ ( plus_plus_int @ B_17 @ C_13 ) ) ) ).
thf(fact_787_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_16: nat,B_17: nat,C_13: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_16 @ B_17 ) @ C_13 )
= ( plus_plus_nat @ A_16 @ ( plus_plus_nat @ B_17 @ C_13 ) ) ) ).
thf(fact_788_add__left__cancel,axiom,
! [A_15: int,B_16: int,C_12: int] :
( ( ( plus_plus_int @ A_15 @ B_16 )
= ( plus_plus_int @ A_15 @ C_12 ) )
<=> ( B_16 = C_12 ) ) ).
thf(fact_789_add__left__cancel,axiom,
! [A_15: nat,B_16: nat,C_12: nat] :
( ( ( plus_plus_nat @ A_15 @ B_16 )
= ( plus_plus_nat @ A_15 @ C_12 ) )
<=> ( B_16 = C_12 ) ) ).
thf(fact_790_add__right__cancel,axiom,
! [B_15: int,A_14: int,C_11: int] :
( ( ( plus_plus_int @ B_15 @ A_14 )
= ( plus_plus_int @ C_11 @ A_14 ) )
<=> ( B_15 = C_11 ) ) ).
thf(fact_791_add__right__cancel,axiom,
! [B_15: nat,A_14: nat,C_11: nat] :
( ( ( plus_plus_nat @ B_15 @ A_14 )
= ( plus_plus_nat @ C_11 @ A_14 ) )
<=> ( B_15 = C_11 ) ) ).
thf(fact_792_add__left__imp__eq,axiom,
! [A_13: int,B_14: int,C_10: int] :
( ( ( plus_plus_int @ A_13 @ B_14 )
= ( plus_plus_int @ A_13 @ C_10 ) )
=> ( B_14 = C_10 ) ) ).
thf(fact_793_add__left__imp__eq,axiom,
! [A_13: nat,B_14: nat,C_10: nat] :
( ( ( plus_plus_nat @ A_13 @ B_14 )
= ( plus_plus_nat @ A_13 @ C_10 ) )
=> ( B_14 = C_10 ) ) ).
thf(fact_794_add__imp__eq,axiom,
! [A_12: int,B_13: int,C_9: int] :
( ( ( plus_plus_int @ A_12 @ B_13 )
= ( plus_plus_int @ A_12 @ C_9 ) )
=> ( B_13 = C_9 ) ) ).
thf(fact_795_add__imp__eq,axiom,
! [A_12: nat,B_13: nat,C_9: nat] :
( ( ( plus_plus_nat @ A_12 @ B_13 )
= ( plus_plus_nat @ A_12 @ C_9 ) )
=> ( B_13 = C_9 ) ) ).
thf(fact_796_add__right__imp__eq,axiom,
! [B_12: int,A_11: int,C_8: int] :
( ( ( plus_plus_int @ B_12 @ A_11 )
= ( plus_plus_int @ C_8 @ A_11 ) )
=> ( B_12 = C_8 ) ) ).
thf(fact_797_add__right__imp__eq,axiom,
! [B_12: nat,A_11: nat,C_8: nat] :
( ( ( plus_plus_nat @ B_12 @ A_11 )
= ( plus_plus_nat @ C_8 @ A_11 ) )
=> ( B_12 = C_8 ) ) ).
thf(fact_798_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_799_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_800_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_801_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_802_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_803_add__le__cancel__right,axiom,
! [A_10: int,C_7: int,B_11: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_10 @ C_7 ) @ ( plus_plus_int @ B_11 @ C_7 ) )
<=> ( ord_less_eq_int @ A_10 @ B_11 ) ) ).
thf(fact_804_add__le__cancel__right,axiom,
! [A_10: nat,C_7: nat,B_11: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_10 @ C_7 ) @ ( plus_plus_nat @ B_11 @ C_7 ) )
<=> ( ord_less_eq_nat @ A_10 @ B_11 ) ) ).
thf(fact_805_add__le__cancel__left,axiom,
! [C_6: int,A_9: int,B_10: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_6 @ A_9 ) @ ( plus_plus_int @ C_6 @ B_10 ) )
<=> ( ord_less_eq_int @ A_9 @ B_10 ) ) ).
thf(fact_806_add__le__cancel__left,axiom,
! [C_6: nat,A_9: nat,B_10: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_6 @ A_9 ) @ ( plus_plus_nat @ C_6 @ B_10 ) )
<=> ( ord_less_eq_nat @ A_9 @ B_10 ) ) ).
thf(fact_807_add__right__mono,axiom,
! [C_5: int,A_8: int,B_9: int] :
( ( ord_less_eq_int @ A_8 @ B_9 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_8 @ C_5 ) @ ( plus_plus_int @ B_9 @ C_5 ) ) ) ).
thf(fact_808_add__right__mono,axiom,
! [C_5: nat,A_8: nat,B_9: nat] :
( ( ord_less_eq_nat @ A_8 @ B_9 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_8 @ C_5 ) @ ( plus_plus_nat @ B_9 @ C_5 ) ) ) ).
thf(fact_809_add__left__mono,axiom,
! [C_4: int,A_7: int,B_8: int] :
( ( ord_less_eq_int @ A_7 @ B_8 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_4 @ A_7 ) @ ( plus_plus_int @ C_4 @ B_8 ) ) ) ).
thf(fact_810_add__left__mono,axiom,
! [C_4: nat,A_7: nat,B_8: nat] :
( ( ord_less_eq_nat @ A_7 @ B_8 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_4 @ A_7 ) @ ( plus_plus_nat @ C_4 @ B_8 ) ) ) ).
thf(fact_811_add__mono,axiom,
! [C_3: int,D_2: int,A_6: int,B_7: int] :
( ( ord_less_eq_int @ A_6 @ B_7 )
=> ( ( ord_less_eq_int @ C_3 @ D_2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_6 @ C_3 ) @ ( plus_plus_int @ B_7 @ D_2 ) ) ) ) ).
thf(fact_812_add__mono,axiom,
! [C_3: nat,D_2: nat,A_6: nat,B_7: nat] :
( ( ord_less_eq_nat @ A_6 @ B_7 )
=> ( ( ord_less_eq_nat @ C_3 @ D_2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_6 @ C_3 ) @ ( plus_plus_nat @ B_7 @ D_2 ) ) ) ) ).
thf(fact_813_add__le__imp__le__right,axiom,
! [A_5: int,C_2: int,B_6: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_5 @ C_2 ) @ ( plus_plus_int @ B_6 @ C_2 ) )
=> ( ord_less_eq_int @ A_5 @ B_6 ) ) ).
thf(fact_814_add__le__imp__le__right,axiom,
! [A_5: nat,C_2: nat,B_6: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_5 @ C_2 ) @ ( plus_plus_nat @ B_6 @ C_2 ) )
=> ( ord_less_eq_nat @ A_5 @ B_6 ) ) ).
thf(fact_815_add__le__imp__le__left,axiom,
! [C_1: int,A_4: int,B_5: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_1 @ A_4 ) @ ( plus_plus_int @ C_1 @ B_5 ) )
=> ( ord_less_eq_int @ A_4 @ B_5 ) ) ).
thf(fact_816_add__le__imp__le__left,axiom,
! [C_1: nat,A_4: nat,B_5: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_1 @ A_4 ) @ ( plus_plus_nat @ C_1 @ B_5 ) )
=> ( ord_less_eq_nat @ A_4 @ B_5 ) ) ).
thf(fact_817_diff__add__cancel,axiom,
! [A_3: int,B_4: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_3 @ B_4 ) @ B_4 )
= A_3 ) ).
thf(fact_818_add__diff__cancel,axiom,
! [A_2: int,B_3: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_2 @ B_3 ) @ B_3 )
= A_2 ) ).
thf(fact_819_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_820_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_821_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_822_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_823_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_824_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_825_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_826_trans__le__add1,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_827_trans__le__add2,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_828_add__le__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_829_add__le__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_830_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_831_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_832_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_833_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_834_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_835_diff__diff__left,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
= ( minus_minus_nat @ I_1 @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_836_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_837_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_838_diff__diff__right,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
thf(fact_839_le__diff__conv,axiom,
! [J: nat,K: nat,I_1: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I_1 @ K ) ) ) ).
thf(fact_840_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_841_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_842_add__diff__assoc,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K ) ) ) ).
thf(fact_843_le__diff__conv2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
thf(fact_844_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_845_le__imp__diff__is__add,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ( minus_minus_nat @ J @ I_1 )
= K )
<=> ( J
= ( plus_plus_nat @ K @ I_1 ) ) ) ) ).
thf(fact_846_diff__add__assoc,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
= ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
thf(fact_847_add__diff__assoc2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K ) ) ) ).
thf(fact_848_diff__add__assoc2,axiom,
! [I_1: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 ) ) ) ).
thf(fact_849_Suc__eq__plus1__left,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ one_one_nat @ N ) ) ).
thf(fact_850_Suc__eq__plus1,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ N @ one_one_nat ) ) ).
thf(fact_851_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( suc @ J ) ) ) ) ).
thf(fact_852_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ M )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ M ) ) ) ) ).
thf(fact_853_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_854_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_855_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
thf(fact_856_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
thf(fact_857_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_nat @ N_1 @ K ) ) ) ).
thf(fact_858_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_859_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_860_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_861_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_862_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_863_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_864_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_865_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
<=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
thf(fact_866_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_867_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_868_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_869_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
thf(fact_870_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_871_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
thf(fact_872_less__trans__Suc,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I_1 ) @ K ) ) ) ).
thf(fact_873_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
thf(fact_874_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_875_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_876_not__add__less1,axiom,
! [I_1: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ I_1 ) ).
thf(fact_877_not__add__less2,axiom,
! [J: nat,I_1: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I_1 ) @ I_1 ) ).
thf(fact_878_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_879_trans__less__add1,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_880_trans__less__add2,axiom,
! [M: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_881_add__less__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_882_add__less__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_883_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_884_add__lessD1,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
=> ( ord_less_nat @ I_1 @ K ) ) ).
thf(fact_885_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_886_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_887_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_888_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_889_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_890_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_891_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_892_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_893_less__imp__diff__less,axiom,
! [N: nat,J: nat,K: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
thf(fact_894_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_895_finite__nat__set__iff__bounded,axiom,
! [N_2: nat > $o] :
( ( finite_finite_nat @ N_2 )
<=> ? [M_1: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_nat @ X_1 @ M_1 ) ) ) ).
thf(fact_896_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
thf(fact_897_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I_1: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( P @ K_1 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) ) ).
thf(fact_898_less__add__Suc1,axiom,
! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ I_1 @ M ) ) ) ).
thf(fact_899_less__add__Suc2,axiom,
! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ M @ I_1 ) ) ) ).
thf(fact_900_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_901_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_902_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_903_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_904_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_905_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_906_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_907_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_908_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
thf(fact_909_less__diff__conv,axiom,
! [I_1: nat,J: nat,K: nat] :
( ( ord_less_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ).
thf(fact_910_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_911_diff__less__mono,axiom,
! [C: nat,A_1: nat,B_2: nat] :
( ( ord_less_nat @ A_1 @ B_2 )
=> ( ( ord_less_eq_nat @ C @ A_1 )
=> ( ord_less_nat @ ( minus_minus_nat @ A_1 @ C ) @ ( minus_minus_nat @ B_2 @ C ) ) ) ) ).
thf(fact_912_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_913_less__eq__Suc__le__raw,axiom,
! [X_1: nat] :
( ( ord_less_nat @ X_1 )
= ( ord_less_eq_nat @ ( suc @ X_1 ) ) ) ).
thf(fact_914_mono__nat__linear__lb,axiom,
! [M: nat,K: nat,F: nat > nat] :
( ! [M_1: nat,N_1: nat] :
( ( ord_less_nat @ M_1 @ N_1 )
=> ( ord_less_nat @ ( F @ M_1 ) @ ( F @ N_1 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_915_inc__induct,axiom,
! [P: nat > $o,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( P @ J )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_916_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_917_bounded__nat__set__is__finite,axiom,
! [N: nat,N_2: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ N_2 )
=> ( ord_less_nat @ X_1 @ N ) )
=> ( finite_finite_nat @ N_2 ) ) ).
thf(fact_918_less__mono__imp__le__mono,axiom,
! [I_1: nat,J: nat,F: nat > nat] :
( ! [I: nat,J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J_1 ) ) )
=> ( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( F @ I_1 ) @ ( F @ J ) ) ) ) ).
thf(fact_919_lessE,axiom,
! [I_1: nat,K: nat] :
( ( ord_less_nat @ I_1 @ K )
=> ( ( K
!= ( suc @ I_1 ) )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( K
!= ( suc @ J_1 ) ) ) ) ) ).
thf(fact_920_Suc__lessE,axiom,
! [I_1: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I_1 ) @ K )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( K
!= ( suc @ J_1 ) ) ) ) ).
thf(fact_921_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_922_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_923_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
thf(fact_924_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_925_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_926_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_927_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_928_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_929_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_930_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
thf(fact_931_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_932_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
thf(fact_933_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_934_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_935_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_936_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_937_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_938_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_939_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_940_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_941_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_942_nat_Osimps_I3_J,axiom,
! [Nat_2: nat] :
( ( suc @ Nat_2 )
!= zero_zero_nat ) ).
thf(fact_943_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_944_nat_Osimps_I2_J,axiom,
! [Nat_1: nat] :
( zero_zero_nat
!= ( suc @ Nat_1 ) ) ).
thf(fact_945_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_946_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
<=> ? [M_1: nat] :
( N
= ( suc @ M_1 ) ) ) ).
thf(fact_947_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_948_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( M = zero_zero_nat )
| ? [J_1: nat] :
( ( M
= ( suc @ J_1 ) )
& ( ord_less_nat @ J_1 @ N ) ) ) ) ).
thf(fact_949_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_950_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_951_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_952_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_953_diff__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
thf(fact_954_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_955_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_956_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_957_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_958_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [K_1: nat] : K_1
@ ( minus_minus_nat @ M @ N ) ) ) ).
thf(fact_959_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
thf(fact_960_diff__Suc__less,axiom,
! [I_1: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I_1 ) ) @ N ) ) ).
thf(fact_961_nat__diff__split,axiom,
! [P: nat > $o,A_1: nat,B_2: nat] :
( ( P @ ( minus_minus_nat @ A_1 @ B_2 ) )
<=> ( ( ( ord_less_nat @ A_1 @ B_2 )
=> ( P @ zero_zero_nat ) )
& ! [D_1: nat] :
( ( A_1
= ( plus_plus_nat @ B_2 @ D_1 ) )
=> ( P @ D_1 ) ) ) ) ).
thf(fact_962_nat__diff__split__asm,axiom,
! [P: nat > $o,A_1: nat,B_2: nat] :
( ( P @ ( minus_minus_nat @ A_1 @ B_2 ) )
<=> ~ ( ( ( ord_less_nat @ A_1 @ B_2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D_1: nat] :
( ( A_1
= ( plus_plus_nat @ B_2 @ D_1 ) )
& ~ ( P @ D_1 ) ) ) ) ).
thf(fact_963_card__less__Suc,axiom,
! [I_1: nat,M_2: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_2 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_2 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) ) ) ) ).
thf(fact_964_card__less,axiom,
! [I_1: nat,M_2: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) )
!= zero_zero_nat ) ) ).
thf(fact_965_card__less__Suc2,axiom,
! [I_1: nat,M_2: nat > $o] :
( ~ ( member_nat @ zero_zero_nat @ M_2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_2 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) ) ) ) ).
thf(fact_966_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
thf(fact_967_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
thf(fact_968_add__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N )
= N ) )
& ( ( M != zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N )
= ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
thf(fact_969_ex__least__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_nat @ K_1 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K_1 )
=> ~ ( P @ I ) )
& ( P @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
thf(fact_970_ex__least__nat__le,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K_1 )
=> ~ ( P @ I ) )
& ( P @ K_1 ) ) ) ) ).
thf(fact_971_less__imp__add__positive,axiom,
! [I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ? [K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ( plus_plus_nat @ I_1 @ K_1 )
= J ) ) ) ).
thf(fact_972_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M_1: nat] :
( N
= ( suc @ M_1 ) ) ) ).
thf(fact_973_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
thf(fact_974_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_975_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M_1: nat] :
( N
= ( suc @ M_1 ) ) ) ).
thf(fact_976_nat__induct,axiom,
! [N: nat,P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [N_1: nat] :
( ( P @ N_1 )
=> ( P @ ( suc @ N_1 ) ) )
=> ( P @ N ) ) ) ).
thf(fact_977_expand__Suc,axiom,
! [V: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
=> ( ( number_number_of_nat @ V )
= ( suc @ ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat ) ) ) ) ).
thf(fact_978_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_979_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_980_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_981_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_982_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_983_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( M = N ) ) ).
thf(fact_984_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_985_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_986_mult__le__mono,axiom,
! [K: nat,L: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
thf(fact_987_mult__le__mono2,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ).
thf(fact_988_mult__le__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
thf(fact_989_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_990_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_991_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_992_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_993_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_994_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_995_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_996_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_997_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
<=> ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_998_mult__less__mono2,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
thf(fact_999_mult__less__mono1,axiom,
! [K: nat,I_1: nat,J: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
thf(fact_1000_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1001_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1002_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_1003_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1004_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_1005_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_1006_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1007_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
thf(fact_1008_n__less__m__mult__n,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
thf(fact_1009_n__less__n__mult__m,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
thf(fact_1010_one__less__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
thf(fact_1011_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
<=> ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
thf(fact_1012_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1013_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1014_mult__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( times_times_nat @ M @ N )
= zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( times_times_nat @ M @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
thf(fact_1015_nat__less__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1016_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_1017_nat__mult__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( times_times_nat @ M @ N ) @ K )
= ( times_times_nat @ M @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_1018_less__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_1019_zmult__1,axiom,
! [Z: int] :
( ( times_times_int @ one_one_int @ Z )
= Z ) ).
thf(fact_1020_zmult__1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ one_one_int )
= Z ) ).
thf(fact_1021_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_1022_times__numeral__code_I5_J,axiom,
! [V: int,W: int] :
( ( times_times_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( times_times_int @ V @ W ) ) ) ).
thf(fact_1023_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( times_times_int @ ( times_times_int @ Z1 @ Z2 ) @ Z3 )
= ( times_times_int @ Z1 @ ( times_times_int @ Z2 @ Z3 ) ) ) ).
thf(fact_1024_less__eq__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_1025_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_1026_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_1027_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_1028_plus__numeral__code_I9_J,axiom,
! [V: int,W: int] :
( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) ) ).
thf(fact_1029_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_1030_zmult__zless__mono2,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I_1 ) @ ( times_times_int @ K @ J ) ) ) ) ).
thf(fact_1031_pos__zmult__eq__1__iff,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
<=> ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
thf(fact_1032_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
thf(fact_1033_zadd__0__right,axiom,
! [Z: int] :
( ( plus_plus_int @ Z @ zero_zero_int )
= Z ) ).
thf(fact_1034_zadd__0,axiom,
! [Z: int] :
( ( plus_plus_int @ zero_zero_int @ Z )
= Z ) ).
thf(fact_1035_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_1036_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
<=> ( ord_less_int @ zero_zero_int @ Z ) ) ).
thf(fact_1037_less__bin__lemma,axiom,
! [K: int,L: int] :
( ( ord_less_int @ K @ L )
<=> ( ord_less_int @ ( minus_minus_int @ K @ L ) @ zero_zero_int ) ) ).
thf(fact_1038_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
thf(fact_1039_odd__less__0,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
<=> ( ord_less_int @ Z @ zero_zero_int ) ) ).
thf(fact_1040_zadd__left__mono,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_eq_int @ I_1 @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I_1 ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_1041_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ Z1 @ Z2 ) @ Z3 )
= ( plus_plus_int @ Z1 @ ( plus_plus_int @ Z2 @ Z3 ) ) ) ).
thf(fact_1042_zadd__left__commute,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z ) )
= ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z ) ) ) ).
thf(fact_1043_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_1044_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_1045_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_1046_zle__trans,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_eq_int @ I_1 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I_1 @ K ) ) ) ).
thf(fact_1047_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
thf(fact_1048_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_1049_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_1050_zless__le,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ Z @ W )
<=> ( ( ord_less_eq_int @ Z @ W )
& ( Z != W ) ) ) ).
thf(fact_1051_zadd__strict__right__mono,axiom,
! [K: int,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ord_less_int @ ( plus_plus_int @ I_1 @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
thf(fact_1052_zadd__zless__mono,axiom,
! [Z_3: int,Z: int,W_1: int,W: int] :
( ( ord_less_int @ W_1 @ W )
=> ( ( ord_less_eq_int @ Z_3 @ Z )
=> ( ord_less_int @ ( plus_plus_int @ W_1 @ Z_3 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
thf(fact_1053_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_1054_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
<=> ( ord_less_int @ W @ Z ) ) ).
thf(fact_1055_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
thf(fact_1056_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
<=> ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
thf(fact_1057_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
thf(fact_1058_left__add__mult__distrib,axiom,
! [I_1: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I_1 @ J ) @ U ) @ K ) ) ).
thf(fact_1059_nat__mult__eq__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( M = N ) ) ) ).
thf(fact_1060_nat__mult__less__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1061_nat__mult__le__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1062_nat__le__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1063_nat__diff__add__eq1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1064_nat__eq__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M )
= N ) ) ) ).
thf(fact_1065_nat__le__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1066_nat__diff__add__eq2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1067_nat__eq__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
( ( ord_less_eq_nat @ I_1 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
thf(fact_1068_nat__less__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
( ( ord_less_eq_nat @ J @ I_1 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1069_zdiv__mono2__neg__lemma,axiom,
! [B_2: int,Q_1: int,R_1: int,B_1: int,Q: int,R: int] :
( ( ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) @ zero_zero_int )
=> ( ( ord_less_int @ R_1 @ B_2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ B_2 )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ) ) ).
thf(fact_1070_unique__quotient__lemma__neg,axiom,
! [B_2: int,Q: int,R: int,Q_1: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q ) @ R ) @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( ord_less_int @ B_2 @ R_1 )
=> ( ( ord_less_int @ B_2 @ R )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ).
thf(fact_1071_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_1072_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_1073_self__quotient__aux1,axiom,
! [R_1: int,Q_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( A_1
= ( plus_plus_int @ R_1 @ ( times_times_int @ A_1 @ Q_1 ) ) )
=> ( ( ord_less_int @ R_1 @ A_1 )
=> ( ord_less_eq_int @ one_one_int @ Q_1 ) ) ) ) ).
thf(fact_1074_self__quotient__aux2,axiom,
! [R_1: int,Q_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( A_1
= ( plus_plus_int @ R_1 @ ( times_times_int @ A_1 @ Q_1 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ord_less_eq_int @ Q_1 @ one_one_int ) ) ) ) ).
thf(fact_1075_q__pos__lemma,axiom,
! [B_1: int,Q: int,R: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
=> ( ( ord_less_int @ R @ B_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ zero_zero_int @ Q ) ) ) ) ).
thf(fact_1076_q__neg__lemma,axiom,
! [B_1: int,Q: int,R: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ Q @ zero_zero_int ) ) ) ) ).
thf(fact_1077_unique__quotient__lemma,axiom,
! [B_2: int,Q: int,R: int,Q_1: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q ) @ R ) @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B_2 )
=> ( ( ord_less_int @ R_1 @ B_2 )
=> ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ).
thf(fact_1078_zdiv__mono2__lemma,axiom,
! [B_2: int,Q_1: int,R_1: int,B_1: int,Q: int,R: int] :
( ( ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
=> ( ( ord_less_int @ R @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ B_2 )
=> ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ) ) ).
thf(fact_1079_int__less__induct,axiom,
! [P: int > $o,I_1: int,K: int] :
( ( ord_less_int @ I_1 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1080_int__le__induct,axiom,
! [P: int > $o,I_1: int,K: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1081_int__gr__induct,axiom,
! [P: int > $o,K: int,I_1: int] :
( ( ord_less_int @ K @ I_1 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1082_int__ge__induct,axiom,
! [P: int > $o,K: int,I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1083_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_1084_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_1085_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_1086_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_1087_transfer__nat__int__set__cong,axiom,
! [P_1: int > $o,P: int > $o] :
( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P_1 @ X_1 ) ) ) ) ) ).
thf(fact_1088_decr__mult__lemma,axiom,
! [K: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_1089_conj__le__cong,axiom,
! [P_1: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_1 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
& P_1 ) ) ) ).
thf(fact_1090_imp__le__cong,axiom,
! [P_1: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_1 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P_1 ) ) ) ).
thf(fact_1091_incr__mult__lemma,axiom,
! [K: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
thf(fact_1092_int__induct,axiom,
! [I_1: int,P: int > $o,K: int] :
( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) ) ) ).
thf(fact_1093_minusinfinity,axiom,
! [P: int > $o,P1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K_1: int] :
( ( P1 @ X_1 )
<=> ( P1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
=> ( ? [Z_2: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z_2 )
=> ( ( P @ X_1 )
<=> ( P1 @ X_1 ) ) )
=> ( ( ex @ P1 )
=> ( ex @ P ) ) ) ) ) ).
thf(fact_1094_plusinfinity,axiom,
! [P: int > $o,P_1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K_1: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
=> ( ? [Z_2: int] :
! [X_1: int] :
( ( ord_less_int @ Z_2 @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ( ex @ P_1 )
=> ( ex @ P ) ) ) ) ) ).
thf(fact_1095_tsub__def,axiom,
! [Y: int,X: int] :
( ( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) )
& ( ~ ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= zero_zero_int ) ) ) ).
thf(fact_1096_zero__zle__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_1097_int__less__0__conv,axiom,
! [K: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) ).
thf(fact_1098_int__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_1099_int__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_1100_int__eq__0__conv,axiom,
! [N: nat] :
( ( ( semiri1621563631at_int @ N )
= zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_1101_zless__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1102_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1103_zadd__int,axiom,
! [M: nat,N: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_1104_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
thf(fact_1105_zle__iff__zadd,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ Z )
<=> ? [N_1: nat] :
( Z
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_1106_zmult__int,axiom,
! [M: nat,N: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_1107_int__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_1108_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X: nat,Y: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( times_times_nat @ X @ Y ) ) ) ).
thf(fact_1109_transfer__int__nat__relations_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_nat @ X @ Y ) ) ).
thf(fact_1110_transfer__int__nat__relations_I3_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_1111_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X: nat,Y: nat] :
( ( nat_tsub @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) ) ).
thf(fact_1112_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X: nat,Y: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ X @ Y ) ) ) ).
thf(fact_1113_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A: nat > $o] :
( ( finite_card_nat @ A )
= ( finite_card_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_1114_transfer__nat__int__set__relations_I1_J,axiom,
! [A: nat > $o] :
( ( finite_finite_nat @ A )
<=> ( finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_1115_transfer__int__nat__numerals_I2_J,axiom,
( one_one_int
= ( semiri1621563631at_int @ one_one_nat ) ) ).
thf(fact_1116_transfer__int__nat__numerals_I1_J,axiom,
( zero_zero_int
= ( semiri1621563631at_int @ zero_zero_nat ) ) ).
thf(fact_1117_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
! [P: int > $o] :
( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
= ( image_nat_int @ semiri1621563631at_int
@ ( collect_nat
@ ^ [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ) ) ).
thf(fact_1118_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ Z ) ) ).
thf(fact_1119_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: int > $o] :
( ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( P @ X_1 ) )
<=> ? [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_1120_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: int > $o] :
( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P @ X_1 ) )
<=> ! [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_1121_int__le__0__conv,axiom,
! [N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_1122_int__Suc0__eq__1,axiom,
( ( semiri1621563631at_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
thf(fact_1123_zless__iff__Suc__zadd,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
<=> ? [N_1: nat] :
( Z
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_1124_int__Suc,axiom,
! [M: nat] :
( ( semiri1621563631at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_1125_zdiff__int,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_1126_zero__less__int__conv,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_1127_zmult__zless__mono2__lemma,axiom,
! [K: nat,I_1: int,J: int] :
( ( ord_less_int @ I_1 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ I_1 ) @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ J ) ) ) ) ).
thf(fact_1128_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( nat_tsub @ X @ Y ) ) ) ) ).
thf(fact_1129_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) )
<=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
thf(fact_1130_tsub__eq,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_1131_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
& ( K
= ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_1132_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1621563631at_int @ M )
= ( semiri1621563631at_int @ N ) )
<=> ( M = N ) ) ).
thf(fact_1133_transfer__nat__int__set__relations_I4_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_nat_o @ A @ B )
<=> ( ord_less_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_1134_transfer__nat__int__set__relations_I5_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
<=> ( ord_less_eq_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_1135_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
( bot_bot_int_o
= ( image_nat_int @ semiri1621563631at_int @ bot_bot_nat_o ) ) ).
thf(fact_1136_transfer__nat__int__set__relations_I3_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( A = B )
<=> ( ( image_nat_int @ semiri1621563631at_int @ A )
= ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_1137_transfer__nat__int__set__relations_I2_J,axiom,
! [X: nat,A: nat > $o] :
( ( member_nat @ X @ A )
<=> ( member_int @ ( semiri1621563631at_int @ X ) @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_1138_int__if__cong,axiom,
! [X: nat,Y: nat,P: $o] :
( ( P
=> ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) )
& ( ~ P
=> ( ( semiri1621563631at_int @ Y )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) ) ) ).
thf(fact_1139_transfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_1140_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N_1: nat] :
( K
!= ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_1141_nonneg__eq__int,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ~ ! [M_1: nat] :
( Z
!= ( semiri1621563631at_int @ M_1 ) ) ) ).
thf(fact_1142_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N_1: nat] :
( K
= ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_1143_int__diff__cases,axiom,
! [Z: int] :
~ ! [M_1: nat,N_1: nat] :
( Z
!= ( minus_minus_int @ ( semiri1621563631at_int @ M_1 ) @ ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_1144_decr__lemma,axiom,
! [X: int,Z: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
thf(fact_1145_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
<=> ( Z = zero_zero_int ) ) ).
thf(fact_1146_neg__def,axiom,
! [Z_1: int] :
( ( nat_neg @ Z_1 )
<=> ( ord_less_int @ Z_1 @ zero_zero_int ) ) ).
thf(fact_1147_not__neg__eq__ge__0,axiom,
! [X: int] :
( ~ ( nat_neg @ X )
<=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
thf(fact_1148_not__neg__1,axiom,
~ ( nat_neg @ one_one_int ) ).
thf(fact_1149_not__neg__0,axiom,
~ ( nat_neg @ zero_zero_int ) ).
thf(fact_1150_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
thf(fact_1151_not__neg__int,axiom,
! [N: nat] :
~ ( nat_neg @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_1152_abs__int__eq,axiom,
! [M: nat] :
( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
= ( semiri1621563631at_int @ M ) ) ).
thf(fact_1153_neg__imp__number__of__eq__0,axiom,
! [V: int] :
( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( number_number_of_nat @ V )
= zero_zero_nat ) ) ).
thf(fact_1154_eq__nat__number__of,axiom,
! [V: int,V_1: int] :
( ( ( number_number_of_nat @ V )
= ( number_number_of_nat @ V_1 ) )
<=> ( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ord_less_eq_int @ ( number_number_of_int @ V_1 ) @ zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( number_number_of_int @ V )
= zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( V = V_1 ) ) ) ) ) ) ).
thf(fact_1155_nat__number__of__add__left,axiom,
! [V_1: int,K: nat,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V ) @ K ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) @ K ) ) ) ) ) ) ).
thf(fact_1156_int__nat__number__of,axiom,
! [V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
= zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
= ( number_number_of_int @ V ) ) ) ) ).
thf(fact_1157_incr__lemma,axiom,
! [Z: int,X: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
thf(fact_1158_int__val__lemma,axiom,
! [K: int,F: nat > int,N: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
thf(fact_1159_nat0__intermed__int__val,axiom,
! [K: int,F: nat > int,N: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
thf(fact_1160_nat__intermed__int__val,axiom,
! [K: int,F: nat > int,N: nat,M: nat] :
( ! [I: nat] :
( ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_nat @ I @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ) ).
thf(fact_1161_Suc__nat__number__of__add,axiom,
! [N: nat,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ N ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( succ @ V ) ) @ N ) ) ) ) ).
thf(fact_1162_succ__def,axiom,
! [K: int] :
( ( succ @ K )
= ( plus_plus_int @ K @ one_one_int ) ) ).
thf(fact_1163_Suc__nat__number__of,axiom,
! [V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( number_number_of_nat @ V ) )
= one_one_nat ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( number_number_of_nat @ V ) )
= ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
thf(fact_1164_nat__number__of__Bit1,axiom,
! [W: int] :
( ( ( nat_neg @ ( number_number_of_int @ W ) )
=> ( ( number_number_of_nat @ ( bit1 @ W ) )
= zero_zero_nat ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ W ) )
=> ( ( number_number_of_nat @ ( bit1 @ W ) )
= ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ W ) @ ( number_number_of_nat @ W ) ) ) ) ) ) ).
thf(fact_1165_nat__1__add__number__of,axiom,
! [V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
= one_one_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
= ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
thf(fact_1166_succ__Pls,axiom,
( ( succ @ pls )
= ( bit1 @ pls ) ) ).
thf(fact_1167_neg__number__of__Bit1,axiom,
! [W: int] :
( ( nat_neg @ ( number_number_of_int @ ( bit1 @ W ) ) )
<=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
thf(fact_1168_not__neg__number__of__Pls,axiom,
~ ( nat_neg @ ( number_number_of_int @ pls ) ) ).
thf(fact_1169_transfer__int__nat__numerals_I4_J,axiom,
( ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) )
= ( semiri1621563631at_int @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1170_numeral__1__eq__Suc__0,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_1171_numeral__3__eq__3,axiom,
( ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_1172_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_1173_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(fact_1174_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_1175_one__is__num__one,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_1176_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_1177_Bit1__def,axiom,
! [K: int] :
( ( bit1 @ K )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) ).
thf(fact_1178_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_1179_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_1180_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_1181_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( ( bit1 @ K )
= ( bit1 @ L ) )
<=> ( K = L ) ) ).
thf(fact_1182_rel__simps_I46_J,axiom,
! [K: int] :
( ( bit1 @ K )
!= pls ) ).
thf(fact_1183_rel__simps_I39_J,axiom,
! [L: int] :
( pls
!= ( bit1 @ L ) ) ).
thf(fact_1184_diff__bin__simps_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ pls )
= K ) ).
thf(fact_1185_add__Pls,axiom,
! [K: int] :
( ( plus_plus_int @ pls @ K )
= K ) ).
thf(fact_1186_add__Pls__right,axiom,
! [K: int] :
( ( plus_plus_int @ K @ pls )
= K ) ).
thf(fact_1187_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_1188_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ).
thf(fact_1189_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_1190_rel__simps_I22_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_1191_rel__simps_I34_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_1192_less__eq__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_1193_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_1194_rel__simps_I12_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_1195_rel__simps_I17_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_1196_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_1197_rel__simps_I29_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_1198_rel__simps_I5_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
%----Helper facts (9)
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_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ~ ( fequal_int @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( X != Y )
| ( fequal_int @ 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 ) ) ).
%----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 ) ).
%------------------------------------------------------------------------------