TPTP Problem File: SLH0359^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Frequency_Moments/0087_Frequency_Moment_k/prob_00399_016029__20020116_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1462 ( 582 unt; 381 typ; 0 def)
% Number of atoms : 2691 (1164 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 6390 ( 448 ~; 50 |; 143 &;4782 @)
% ( 0 <=>; 967 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 77 ( 76 usr)
% Number of type conns : 587 ( 587 >; 0 *; 0 +; 0 <<)
% Number of symbols : 308 ( 305 usr; 85 con; 0-3 aty)
% Number of variables : 1971 ( 43 ^;1793 !; 135 ?;1971 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:19:24.791
%------------------------------------------------------------------------------
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image_875570014554754200it_nat: ( product_unit > nat ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Numeral____Type__Onum0,type,
image_6449127158079674652l_num0: ( product_unit > numeral_num0 ) > set_Product_unit > set_Numeral_num0 ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Numeral____Type__Onum1,type,
image_6449127158079674653l_num1: ( product_unit > numeral_num1 ) > set_Product_unit > set_Numeral_num1 ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__String__Oliteral,type,
image_5876984745897992460iteral: ( product_unit > literal ) > set_Product_unit > set_literal ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Nat__Onat,type,
image_literal_nat: ( literal > nat ) > set_literal > set_nat ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Numeral____Type__Onum0,type,
image_3608546570274595605l_num0: ( literal > numeral_num0 ) > set_literal > set_Numeral_num0 ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__String__Oliteral,type,
image_8195128725298311301iteral: ( literal > literal ) > set_literal > set_literal ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
is_emp1662574758705540307at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Numeral____Type__Onum0,type,
member_Numeral_num0: numeral_num0 > set_Numeral_num0 > $o ).
thf(sy_c_member_001t__Numeral____Type__Onum1,type,
member_Numeral_num1: numeral_num1 > set_Numeral_num1 > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__String__Oliteral,type,
member_literal: literal > set_literal > $o ).
thf(sy_v_M____,type,
m: nat ).
thf(sy_v_as,type,
as: list_nat ).
% Relevant facts (1071)
thf(fact_0_assms,axiom,
as != nil_nat ).
% assms
thf(fact_1_M__def,axiom,
( m
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ ( count_list_nat @ as ) @ ( set_nat2 @ as ) ) ) ) ).
% M_def
thf(fact_2_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat2 @ X @ A )
=> ( member_nat2 @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_3_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_4_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_5_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X @ A )
=> ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_6_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat2 @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_7_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_8_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_9_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_10_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat2 @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat2 @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_11_image__set__eqI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ B2 )
=> ( ( member_nat2 @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_nat_nat @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_12_insort__insert__key__triv,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord1921536354676448932at_nat @ F @ X @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_13_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_14_insort__insert__insort__key,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord1921536354676448932at_nat @ F @ X @ Xs )
= ( linord8961336180081300637at_nat @ F @ X @ Xs ) ) ) ).
% insort_insert_insort_key
thf(fact_15_insort__insert__key__def,axiom,
( linord1921536354676448932at_nat
= ( ^ [F2: nat > nat,X3: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ ( set_nat2 @ Xs2 ) ) ) @ Xs2 @ ( linord8961336180081300637at_nat @ F2 @ X3 @ Xs2 ) ) ) ) ).
% insort_insert_key_def
thf(fact_16_count__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_17_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_18_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_19_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_nat @ N2 @ nil_nat )
= N2 ) ).
% gen_length_code(1)
thf(fact_20_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_21_set__empty,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( ( set_Pr5648618587558075414at_nat @ Xs )
= bot_bo2099793752762293965at_nat )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% set_empty
thf(fact_22_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_23_set__empty2,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% set_empty2
thf(fact_24_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_25_mem__simps_I2_J,axiom,
! [C2: nat] :
~ ( member_nat2 @ C2 @ bot_bot_set_nat ) ).
% mem_simps(2)
thf(fact_26_mem__simps_I2_J,axiom,
! [C2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ).
% mem_simps(2)
thf(fact_27_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat2 @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_28_all__not__in__conv,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ! [X3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X3 @ A ) )
= ( A = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_29_Collect__empty__eq,axiom,
! [P: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P )
= bot_bo2099793752762293965at_nat )
= ( ! [X3: product_prod_nat_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_30_empty__Collect__eq,axiom,
! [P: product_prod_nat_nat > $o] :
( ( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ P ) )
= ( ! [X3: product_prod_nat_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_31_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_32_image__empty,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat] :
( ( image_5168914502847457605at_nat @ F @ bot_bo2099793752762293965at_nat )
= bot_bo2099793752762293965at_nat ) ).
% image_empty
thf(fact_33_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_34_empty__is__image,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( bot_bo2099793752762293965at_nat
= ( image_5168914502847457605at_nat @ F @ A ) )
= ( A = bot_bo2099793752762293965at_nat ) ) ).
% empty_is_image
thf(fact_35_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_36_image__is__empty,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
( ( ( image_5168914502847457605at_nat @ F @ A )
= bot_bo2099793752762293965at_nat )
= ( A = bot_bo2099793752762293965at_nat ) ) ).
% image_is_empty
thf(fact_37_emptyE,axiom,
! [A2: nat] :
~ ( member_nat2 @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_38_emptyE,axiom,
! [A2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_39_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_40_equals0D,axiom,
! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
( ( A = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_41_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_42_equals0I,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ! [Y2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y2 @ A )
=> ( A = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_43_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat2 @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_44_ex__in__conv,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A ) )
= ( A != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_45_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_46_empty__set,axiom,
( bot_bo2099793752762293965at_nat
= ( set_Pr5648618587558075414at_nat @ nil_Pr5478986624290739719at_nat ) ) ).
% empty_set
thf(fact_47_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_49_count__list_Osimps_I1_J,axiom,
! [Y: nat] :
( ( count_list_nat @ nil_nat @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_50_count__list__0__iff,axiom,
! [Xs: list_nat,X: nat] :
( ( ( count_list_nat @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_51_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_52_non__empty__space,axiom,
( ( as != nil_nat )
=> ( ( frequency_Moment_M_1 @ as )
!= bot_bo2099793752762293965at_nat ) ) ).
% non_empty_space
thf(fact_53_can__select__set__list__ex1,axiom,
! [P: nat > $o,A: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_54_Set_Ois__empty__def,axiom,
( is_emp1662574758705540307at_nat
= ( ^ [A3: set_Pr1261947904930325089at_nat] : ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).
% Set.is_empty_def
thf(fact_55_f__arg__min__list__f,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( F @ ( arg_min_list_nat_nat @ F @ Xs ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_56_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_57_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_58_bot__set__def,axiom,
( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% bot_set_def
thf(fact_59_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A3 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_60_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_61_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_62_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_63_fin__space,axiom,
( ( as != nil_nat )
=> ( finite6177210948735845034at_nat @ ( frequency_Moment_M_1 @ as ) ) ) ).
% fin_space
thf(fact_64_is__empty__set,axiom,
! [Xs: list_nat] :
( ( is_empty_nat @ ( set_nat2 @ Xs ) )
= ( null_nat @ Xs ) ) ).
% is_empty_set
thf(fact_65_min__list__Min,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( min_list_nat @ Xs )
= ( lattic8721135487736765967in_nat @ ( set_nat2 @ Xs ) ) ) ) ).
% min_list_Min
thf(fact_66_bot__list__def,axiom,
bot_bot_list_nat = nil_nat ).
% bot_list_def
thf(fact_67_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_68_bot__empty__eq,axiom,
( bot_bo482883023278783056_nat_o
= ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq
thf(fact_69_Collect__empty__eq__bot,axiom,
! [P: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P )
= bot_bo2099793752762293965at_nat )
= ( P = bot_bo482883023278783056_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_70_real__count__list__pos,axiom,
! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ as ) )
=> ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ ( count_list_nat @ as @ X ) ) ) ) ).
% real_count_list_pos
thf(fact_71_card__space,axiom,
( ( as != nil_nat )
=> ( ( finite711546835091564841at_nat @ ( frequency_Moment_M_1 @ as ) )
= ( size_size_list_nat @ as ) ) ) ).
% card_space
thf(fact_72_list_Osize_I1_J,axiom,
! [X: nat > nat] :
( ( size_list_nat @ X @ nil_nat )
= zero_zero_nat ) ).
% list.size(1)
thf(fact_73_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_74_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_75_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_76_not__gr__zero,axiom,
! [N2: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 ) )
= ( N2 = zero_z7100319975126383169nnreal ) ) ).
% not_gr_zero
thf(fact_77_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_78_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_79_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_80_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_81_List_Ofinite__set,axiom,
! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% List.finite_set
thf(fact_82_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_83_List_Ofinite__set,axiom,
! [Xs: list_literal] : ( finite5847741373460823677iteral @ ( set_literal2 @ Xs ) ) ).
% List.finite_set
thf(fact_84_List_Ofinite__set,axiom,
! [Xs: list_Numeral_num0] : ( finite1111429032697314573l_num0 @ ( set_Numeral_num02 @ Xs ) ) ).
% List.finite_set
thf(fact_85_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_86_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_87_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_88_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_89_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_90_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_91_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_92_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_93_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_94_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_95_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_96_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ ( semiri6283507881447550617nnreal @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_97_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_98_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_99_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_100_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ zero_z7100319975126383169nnreal ) ).
% of_nat_less_0_iff
thf(fact_101_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_102_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_103_order__trans__rules_I28_J,axiom,
! [A2: real,B: real,C2: real] :
( ( A2 = B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order_trans_rules(28)
thf(fact_104_order__trans__rules_I28_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order_trans_rules(28)
thf(fact_105_order__trans__rules_I28_J,axiom,
! [A2: int,B: int,C2: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order_trans_rules(28)
thf(fact_106_order__trans__rules_I28_J,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( A2 = B )
=> ( ( ord_le7381754540660121996nnreal @ B @ C2 )
=> ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% order_trans_rules(28)
thf(fact_107_order__trans__rules_I27_J,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order_trans_rules(27)
thf(fact_108_order__trans__rules_I27_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order_trans_rules(27)
thf(fact_109_order__trans__rules_I27_J,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order_trans_rules(27)
thf(fact_110_order__trans__rules_I27_J,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% order_trans_rules(27)
thf(fact_111_order__trans__rules_I20_J,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order_trans_rules(20)
thf(fact_112_order__trans__rules_I20_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_trans_rules(20)
thf(fact_113_order__trans__rules_I20_J,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_trans_rules(20)
thf(fact_114_order__trans__rules_I20_J,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B )
=> ~ ( ord_le7381754540660121996nnreal @ B @ A2 ) ) ).
% order_trans_rules(20)
thf(fact_115_order__trans__rules_I19_J,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_116_order__trans__rules_I19_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_117_order__trans__rules_I19_J,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_118_order__trans__rules_I19_J,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ( ( ord_le7381754540660121996nnreal @ Y @ Z )
=> ( ord_le7381754540660121996nnreal @ X @ Z ) ) ) ).
% order_trans_rules(19)
thf(fact_119_order__trans__rules_I12_J,axiom,
! [A2: real,F: real > real,B: real,C2: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_120_order__trans__rules_I12_J,axiom,
! [A2: nat,F: real > nat,B: real,C2: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_121_order__trans__rules_I12_J,axiom,
! [A2: int,F: real > int,B: real,C2: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_122_order__trans__rules_I12_J,axiom,
! [A2: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C2: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_123_order__trans__rules_I12_J,axiom,
! [A2: real,F: nat > real,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_124_order__trans__rules_I12_J,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_125_order__trans__rules_I12_J,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_126_order__trans__rules_I12_J,axiom,
! [A2: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_127_order__trans__rules_I12_J,axiom,
! [A2: real,F: int > real,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_128_order__trans__rules_I12_J,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(12)
thf(fact_129_order__trans__rules_I11_J,axiom,
! [A2: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_130_order__trans__rules_I11_J,axiom,
! [A2: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_131_order__trans__rules_I11_J,axiom,
! [A2: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_132_order__trans__rules_I11_J,axiom,
! [A2: real,B: real,F: real > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_133_order__trans__rules_I11_J,axiom,
! [A2: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_134_order__trans__rules_I11_J,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_135_order__trans__rules_I11_J,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_136_order__trans__rules_I11_J,axiom,
! [A2: nat,B: nat,F: nat > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_137_order__trans__rules_I11_J,axiom,
! [A2: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_138_order__trans__rules_I11_J,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(11)
thf(fact_139_order__trans__rules_I2_J,axiom,
! [A2: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_140_order__trans__rules_I2_J,axiom,
! [A2: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_141_order__trans__rules_I2_J,axiom,
! [A2: real,F: int > real,B: int,C2: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_142_order__trans__rules_I2_J,axiom,
! [A2: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C2 )
=> ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_143_order__trans__rules_I2_J,axiom,
! [A2: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_144_order__trans__rules_I2_J,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_145_order__trans__rules_I2_J,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_146_order__trans__rules_I2_J,axiom,
! [A2: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le7381754540660121996nnreal @ B @ C2 )
=> ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_147_order__trans__rules_I2_J,axiom,
! [A2: int,F: real > int,B: real,C2: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_148_order__trans__rules_I2_J,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_trans_rules(2)
thf(fact_149_order__trans__rules_I1_J,axiom,
! [A2: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_150_order__trans__rules_I1_J,axiom,
! [A2: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_151_order__trans__rules_I1_J,axiom,
! [A2: real,B: real,F: real > int,C2: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_152_order__trans__rules_I1_J,axiom,
! [A2: real,B: real,F: real > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C2 )
=> ( ! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_153_order__trans__rules_I1_J,axiom,
! [A2: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_154_order__trans__rules_I1_J,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_155_order__trans__rules_I1_J,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_156_order__trans__rules_I1_J,axiom,
! [A2: nat,B: nat,F: nat > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le7381754540660121996nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le7381754540660121996nnreal @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_157_order__trans__rules_I1_J,axiom,
! [A2: int,B: int,F: int > real,C2: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_158_order__trans__rules_I1_J,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_trans_rules(1)
thf(fact_159_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_160_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_161_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_162_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_163_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_164_neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% neqE
thf(fact_165_neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% neqE
thf(fact_166_neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% neqE
thf(fact_167_neqE,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( X != Y )
=> ( ~ ( ord_le7381754540660121996nnreal @ X @ Y )
=> ( ord_le7381754540660121996nnreal @ Y @ X ) ) ) ).
% neqE
thf(fact_168_neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% neq_iff
thf(fact_169_neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% neq_iff
thf(fact_170_neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% neq_iff
thf(fact_171_neq__iff,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( X != Y )
= ( ( ord_le7381754540660121996nnreal @ X @ Y )
| ( ord_le7381754540660121996nnreal @ Y @ X ) ) ) ).
% neq_iff
thf(fact_172_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y ) ) ) ).
% dense
thf(fact_173_dense,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ? [Z2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Z2 )
& ( ord_le7381754540660121996nnreal @ Z2 @ Y ) ) ) ).
% dense
thf(fact_174_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_175_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_176_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_177_less__imp__neq,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_178_less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% less_asym
thf(fact_179_less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_asym
thf(fact_180_less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_asym
thf(fact_181_less__asym,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ~ ( ord_le7381754540660121996nnreal @ Y @ X ) ) ).
% less_asym
thf(fact_182_order_Oasym,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order.asym
thf(fact_183_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_184_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_185_order_Oasym,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B )
=> ~ ( ord_le7381754540660121996nnreal @ B @ A2 ) ) ).
% order.asym
thf(fact_186_less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% less_linear
thf(fact_187_less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% less_linear
thf(fact_188_less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% less_linear
thf(fact_189_less__linear,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
| ( X = Y )
| ( ord_le7381754540660121996nnreal @ Y @ X ) ) ).
% less_linear
thf(fact_190_less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% less_irrefl
thf(fact_191_less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% less_irrefl
thf(fact_192_less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% less_irrefl
thf(fact_193_less__irrefl,axiom,
! [X: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ X @ X ) ).
% less_irrefl
thf(fact_194_less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_195_less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_196_less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_197_less__imp__not__eq,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_198_less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% less_not_sym
thf(fact_199_less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_not_sym
thf(fact_200_less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_not_sym
thf(fact_201_less__not__sym,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ~ ( ord_le7381754540660121996nnreal @ Y @ X ) ) ).
% less_not_sym
thf(fact_202_order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% order.irrefl
thf(fact_203_order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% order.irrefl
thf(fact_204_order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% order.irrefl
thf(fact_205_order_Oirrefl,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A2 @ A2 ) ).
% order.irrefl
thf(fact_206_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_207_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_208_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_209_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_210_antisym__conv3,axiom,
! [Y: extend8495563244428889912nnreal,X: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ Y @ X )
=> ( ( ~ ( ord_le7381754540660121996nnreal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_211_less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_212_less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_213_less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_214_less__imp__not__eq2,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_215_less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_216_less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_217_less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_218_less__imp__triv,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,P: $o] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ( ( ord_le7381754540660121996nnreal @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_219_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_220_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_221_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_222_linorder__cases,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ~ ( ord_le7381754540660121996nnreal @ X @ Y )
=> ( ( X != Y )
=> ( ord_le7381754540660121996nnreal @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_223_dual__order_Oasym,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_224_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_225_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_226_dual__order_Oasym,axiom,
! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A2 )
=> ~ ( ord_le7381754540660121996nnreal @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_227_less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_228_less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_229_less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_230_less__imp__not__less,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ X @ Y )
=> ~ ( ord_le7381754540660121996nnreal @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_231_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P2: nat > $o] :
? [N3: nat] :
( ( P2 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P2 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_232_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_233_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_234_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_235_linorder__less__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extend8495563244428889912nnreal] : ( P @ A4 @ A4 )
=> ( ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_236_order_Ostrict__trans,axiom,
! [A2: real,B: real,C2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_237_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_238_order_Ostrict__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_239_order_Ostrict__trans,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B )
=> ( ( ord_le7381754540660121996nnreal @ B @ C2 )
=> ( ord_le7381754540660121996nnreal @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_240_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_241_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_242_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_243_not__less__iff__gr__or__eq,axiom,
! [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ~ ( ord_le7381754540660121996nnreal @ X @ Y ) )
= ( ( ord_le7381754540660121996nnreal @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_244_dual__order_Ostrict__trans,axiom,
! [B: real,A2: real,C2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_245_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_246_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_247_dual__order_Ostrict__trans,axiom,
! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A2 )
=> ( ( ord_le7381754540660121996nnreal @ C2 @ B )
=> ( ord_le7381754540660121996nnreal @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_248_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_249_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_250_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_251_order_Ostrict__implies__not__eq,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_252_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_253_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_254_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_255_dual__order_Ostrict__implies__not__eq,axiom,
! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_256_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_257_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_258_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_259_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_260_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_261_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_le7381754540660121996nnreal @ ( semiri6283507881447550617nnreal @ M2 ) @ ( semiri6283507881447550617nnreal @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_262_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_263_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_264_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N2 ) ).
% Ex_list_of_length
thf(fact_265_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_266_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_267_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_268_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_269_zero__less__iff__neq__zero,axiom,
! [N2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 )
= ( N2 != zero_z7100319975126383169nnreal ) ) ).
% zero_less_iff_neq_zero
thf(fact_270_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_271_gr__implies__not__zero,axiom,
! [M2: extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal] :
( ( ord_le7381754540660121996nnreal @ M2 @ N2 )
=> ( N2 != zero_z7100319975126383169nnreal ) ) ).
% gr_implies_not_zero
thf(fact_272_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_273_not__less__zero,axiom,
! [N2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ N2 @ zero_z7100319975126383169nnreal ) ).
% not_less_zero
thf(fact_274_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_275_gr__zeroI,axiom,
! [N2: extend8495563244428889912nnreal] :
( ( N2 != zero_z7100319975126383169nnreal )
=> ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ N2 ) ) ).
% gr_zeroI
thf(fact_276_bot_Onot__eq__extremum,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( A2 != bot_bo2099793752762293965at_nat )
= ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_277_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_278_bot_Onot__eq__extremum,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( A2 != bot_bo841427958541957580nnreal )
= ( ord_le7381754540660121996nnreal @ bot_bo841427958541957580nnreal @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_279_bot_Oextremum__strict,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% bot.extremum_strict
thf(fact_280_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_281_bot_Oextremum__strict,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A2 @ bot_bo841427958541957580nnreal ) ).
% bot.extremum_strict
thf(fact_282_finite__list,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ? [Xs3: list_P6011104703257516679at_nat] :
( ( set_Pr5648618587558075414at_nat @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_283_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_284_finite__list,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ? [Xs3: list_literal] :
( ( set_literal2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_285_finite__list,axiom,
! [A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ? [Xs3: list_Numeral_num0] :
( ( set_Numeral_num02 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_286_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_287_eq__Nil__null,axiom,
! [Xs: list_nat] :
( ( Xs = nil_nat )
= ( null_nat @ Xs ) ) ).
% eq_Nil_null
thf(fact_288_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
% null_rec(2)
thf(fact_289_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_290_Min__gr__iff,axiom,
! [A: set_literal,X: literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( A != bot_bot_set_literal )
=> ( ( ord_less_literal @ X @ ( lattic7587878784716877781iteral @ A ) )
= ( ! [X3: literal] :
( ( member_literal @ X3 @ A )
=> ( ord_less_literal @ X @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_291_Min__gr__iff,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_real @ X @ ( lattic3629708407755379051n_real @ A ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_real @ X @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_292_Min__gr__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_293_Min__gr__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_int @ X @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_294_Min__gr__iff,axiom,
! [A: set_Ex3793607809372303086nnreal,X: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A )
=> ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( ord_le7381754540660121996nnreal @ X @ ( lattic8839003927053164919nnreal @ A ) )
= ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A )
=> ( ord_le7381754540660121996nnreal @ X @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_295_Max__less__iff,axiom,
! [A: set_literal,X: literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( A != bot_bot_set_literal )
=> ( ( ord_less_literal @ ( lattic5693287827546831171iteral @ A ) @ X )
= ( ! [X3: literal] :
( ( member_literal @ X3 @ A )
=> ( ord_less_literal @ X3 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_296_Max__less__iff,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_real @ ( lattic4275903605611617917x_real @ A ) @ X )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ( ord_less_real @ X3 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_297_Max__less__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_int @ X3 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_298_Max__less__iff,axiom,
! [A: set_Ex3793607809372303086nnreal,X: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A )
=> ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( ord_le7381754540660121996nnreal @ ( lattic933167949679527817nnreal @ A ) @ X )
= ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A )
=> ( ord_le7381754540660121996nnreal @ X3 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_299_Max__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_300_card__0__eq,axiom,
! [A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( ( finite6454714172617411597l_num1 @ A )
= zero_zero_nat )
= ( A = bot_bo5651135754355059505l_num1 ) ) ) ).
% card_0_eq
thf(fact_301_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_302_card__0__eq,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ( finite_card_literal @ A )
= zero_zero_nat )
= ( A = bot_bot_set_literal ) ) ) ).
% card_0_eq
thf(fact_303_card__0__eq,axiom,
! [A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( ( finite6454714172617411596l_num0 @ A )
= zero_zero_nat )
= ( A = bot_bo5651135750051830704l_num0 ) ) ) ).
% card_0_eq
thf(fact_304_card__0__eq,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( ( finite711546835091564841at_nat @ A )
= zero_zero_nat )
= ( A = bot_bo2099793752762293965at_nat ) ) ) ).
% card_0_eq
thf(fact_305_card_Oinfinite,axiom,
! [A: set_Numeral_num1] :
( ~ ( finite1111429032697314574l_num1 @ A )
=> ( ( finite6454714172617411597l_num1 @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_306_card_Oinfinite,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ A )
=> ( ( finite711546835091564841at_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_307_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_308_card_Oinfinite,axiom,
! [A: set_literal] :
( ~ ( finite5847741373460823677iteral @ A )
=> ( ( finite_card_literal @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_309_card_Oinfinite,axiom,
! [A: set_Numeral_num0] :
( ~ ( finite1111429032697314573l_num0 @ A )
=> ( ( finite6454714172617411596l_num0 @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_310_card_Oempty,axiom,
( ( finite6454714172617411597l_num1 @ bot_bo5651135754355059505l_num1 )
= zero_zero_nat ) ).
% card.empty
thf(fact_311_card_Oempty,axiom,
( ( finite6454714172617411596l_num0 @ bot_bo5651135750051830704l_num0 )
= zero_zero_nat ) ).
% card.empty
thf(fact_312_card_Oempty,axiom,
( ( finite_card_literal @ bot_bot_set_literal )
= zero_zero_nat ) ).
% card.empty
thf(fact_313_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_314_card_Oempty,axiom,
( ( finite711546835091564841at_nat @ bot_bo2099793752762293965at_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_315_finite__imageI,axiom,
! [F3: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_316_finite__imageI,axiom,
! [F3: set_nat,H: nat > literal] :
( ( finite_finite_nat @ F3 )
=> ( finite5847741373460823677iteral @ ( image_nat_literal @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_317_finite__imageI,axiom,
! [F3: set_nat,H: nat > numeral_num0] :
( ( finite_finite_nat @ F3 )
=> ( finite1111429032697314573l_num0 @ ( image_5550796612950789325l_num0 @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_318_finite__imageI,axiom,
! [F3: set_literal,H: literal > nat] :
( ( finite5847741373460823677iteral @ F3 )
=> ( finite_finite_nat @ ( image_literal_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_319_finite__imageI,axiom,
! [F3: set_literal,H: literal > literal] :
( ( finite5847741373460823677iteral @ F3 )
=> ( finite5847741373460823677iteral @ ( image_8195128725298311301iteral @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_320_finite__imageI,axiom,
! [F3: set_literal,H: literal > numeral_num0] :
( ( finite5847741373460823677iteral @ F3 )
=> ( finite1111429032697314573l_num0 @ ( image_3608546570274595605l_num0 @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_321_finite__imageI,axiom,
! [F3: set_Numeral_num0,H: numeral_num0 > nat] :
( ( finite1111429032697314573l_num0 @ F3 )
=> ( finite_finite_nat @ ( image_8797574156932312687m0_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_322_finite__imageI,axiom,
! [F3: set_Numeral_num0,H: numeral_num0 > literal] :
( ( finite1111429032697314573l_num0 @ F3 )
=> ( finite5847741373460823677iteral @ ( image_8737817577461598069iteral @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_323_finite__imageI,axiom,
! [F3: set_Numeral_num0,H: numeral_num0 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ F3 )
=> ( finite1111429032697314573l_num0 @ ( image_2832974300507296261l_num0 @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_324_finite__imageI,axiom,
! [F3: set_Pr1261947904930325089at_nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ F3 )
=> ( finite_finite_nat @ ( image_2486076414777270412at_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_325_Min__less__iff,axiom,
! [A: set_literal,X: literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( A != bot_bot_set_literal )
=> ( ( ord_less_literal @ ( lattic7587878784716877781iteral @ A ) @ X )
= ( ? [X3: literal] :
( ( member_literal @ X3 @ A )
& ( ord_less_literal @ X3 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_326_Min__less__iff,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_real @ ( lattic3629708407755379051n_real @ A ) @ X )
= ( ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ord_less_real @ X3 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_327_Min__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A ) @ X )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_328_Min__less__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_int @ ( lattic8718645017227715691in_int @ A ) @ X )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( ord_less_int @ X3 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_329_Min__less__iff,axiom,
! [A: set_Ex3793607809372303086nnreal,X: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A )
=> ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( ord_le7381754540660121996nnreal @ ( lattic8839003927053164919nnreal @ A ) @ X )
= ( ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A )
& ( ord_le7381754540660121996nnreal @ X3 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_330_Max__gr__iff,axiom,
! [A: set_literal,X: literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( A != bot_bot_set_literal )
=> ( ( ord_less_literal @ X @ ( lattic5693287827546831171iteral @ A ) )
= ( ? [X3: literal] :
( ( member_literal @ X3 @ A )
& ( ord_less_literal @ X @ X3 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_331_Max__gr__iff,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_real @ X @ ( lattic4275903605611617917x_real @ A ) )
= ( ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ord_less_real @ X @ X3 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_332_Max__gr__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_int @ X @ ( lattic8263393255366662781ax_int @ A ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( ord_less_int @ X @ X3 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_333_Max__gr__iff,axiom,
! [A: set_Ex3793607809372303086nnreal,X: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A )
=> ( ( A != bot_bo4854962954004695426nnreal )
=> ( ( ord_le7381754540660121996nnreal @ X @ ( lattic933167949679527817nnreal @ A ) )
= ( ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A )
& ( ord_le7381754540660121996nnreal @ X @ X3 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_334_Max__gr__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_335_card__eq__0__iff,axiom,
! [A: set_Numeral_num1] :
( ( ( finite6454714172617411597l_num1 @ A )
= zero_zero_nat )
= ( ( A = bot_bo5651135754355059505l_num1 )
| ~ ( finite1111429032697314574l_num1 @ A ) ) ) ).
% card_eq_0_iff
thf(fact_336_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_337_card__eq__0__iff,axiom,
! [A: set_literal] :
( ( ( finite_card_literal @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_literal )
| ~ ( finite5847741373460823677iteral @ A ) ) ) ).
% card_eq_0_iff
thf(fact_338_card__eq__0__iff,axiom,
! [A: set_Numeral_num0] :
( ( ( finite6454714172617411596l_num0 @ A )
= zero_zero_nat )
= ( ( A = bot_bo5651135750051830704l_num0 )
| ~ ( finite1111429032697314573l_num0 @ A ) ) ) ).
% card_eq_0_iff
thf(fact_339_card__eq__0__iff,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ( finite711546835091564841at_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bo2099793752762293965at_nat )
| ~ ( finite6177210948735845034at_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_340_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_341_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_342_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_343_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_344_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_345_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_346_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_347_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_348_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_349_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_350_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_351_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_352_finite__maxlen,axiom,
! [M: set_list_nat] :
( ( finite8100373058378681591st_nat @ M )
=> ? [N4: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ M )
=> ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_353_finite__psubset__induct,axiom,
! [A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( finite6177210948735845034at_nat @ A )
=> ( ! [A5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A5 )
=> ( ! [B4: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_354_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_355_finite__psubset__induct,axiom,
! [A: set_literal,P: set_literal > $o] :
( ( finite5847741373460823677iteral @ A )
=> ( ! [A5: set_literal] :
( ( finite5847741373460823677iteral @ A5 )
=> ( ! [B4: set_literal] :
( ( ord_less_set_literal @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_356_finite__psubset__induct,axiom,
! [A: set_Numeral_num0,P: set_Numeral_num0 > $o] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ! [A5: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A5 )
=> ( ! [B4: set_Numeral_num0] :
( ( ord_le526730871819019248l_num0 @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_357_psubset__card__mono,axiom,
! [B2: set_Numeral_num1,A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ B2 )
=> ( ( ord_le526730876122248049l_num1 @ A @ B2 )
=> ( ord_less_nat @ ( finite6454714172617411597l_num1 @ A ) @ ( finite6454714172617411597l_num1 @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_358_psubset__card__mono,axiom,
! [B2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B2 )
=> ( ( ord_le7866589430770878221at_nat @ A @ B2 )
=> ( ord_less_nat @ ( finite711546835091564841at_nat @ A ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_359_psubset__card__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_360_psubset__card__mono,axiom,
! [B2: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B2 )
=> ( ( ord_less_set_literal @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_361_psubset__card__mono,axiom,
! [B2: set_Numeral_num0,A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ B2 )
=> ( ( ord_le526730871819019248l_num0 @ A @ B2 )
=> ( ord_less_nat @ ( finite6454714172617411596l_num0 @ A ) @ ( finite6454714172617411596l_num0 @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_362_not__psubset__empty,axiom,
! [A: set_Pr1261947904930325089at_nat] :
~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% not_psubset_empty
thf(fact_363_less__list__code_I1_J,axiom,
! [Xs: list_nat] :
~ ( ord_less_list_nat @ Xs @ nil_nat ) ).
% less_list_code(1)
thf(fact_364_not__less__Nil,axiom,
! [X: list_nat] :
~ ( ord_less_list_nat @ X @ nil_nat ) ).
% not_less_Nil
thf(fact_365_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_366_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_367_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_368_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_369_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_370_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_371_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_372_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_373_card__ge__0__finite,axiom,
! [A: set_Numeral_num1] :
( ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ A ) )
=> ( finite1111429032697314574l_num1 @ A ) ) ).
% card_ge_0_finite
thf(fact_374_card__ge__0__finite,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A ) )
=> ( finite6177210948735845034at_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_375_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_376_card__ge__0__finite,axiom,
! [A: set_literal] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ A ) )
=> ( finite5847741373460823677iteral @ A ) ) ).
% card_ge_0_finite
thf(fact_377_card__ge__0__finite,axiom,
! [A: set_Numeral_num0] :
( ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ A ) )
=> ( finite1111429032697314573l_num0 @ A ) ) ).
% card_ge_0_finite
thf(fact_378_size__list__estimation,axiom,
! [X: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X ) )
=> ( ord_less_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_379_card__gt__0__iff,axiom,
! [A: set_Numeral_num1] :
( ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ A ) )
= ( ( A != bot_bo5651135754355059505l_num1 )
& ( finite1111429032697314574l_num1 @ A ) ) ) ).
% card_gt_0_iff
thf(fact_380_card__gt__0__iff,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
= ( ( A != bot_bot_set_nat )
& ( finite_finite_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_381_card__gt__0__iff,axiom,
! [A: set_literal] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ A ) )
= ( ( A != bot_bot_set_literal )
& ( finite5847741373460823677iteral @ A ) ) ) ).
% card_gt_0_iff
thf(fact_382_card__gt__0__iff,axiom,
! [A: set_Numeral_num0] :
( ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ A ) )
= ( ( A != bot_bo5651135750051830704l_num0 )
& ( finite1111429032697314573l_num0 @ A ) ) ) ).
% card_gt_0_iff
thf(fact_383_card__gt__0__iff,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A ) )
= ( ( A != bot_bo2099793752762293965at_nat )
& ( finite6177210948735845034at_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_384_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_385_rel__simps_I70_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% rel_simps(70)
thf(fact_386_rel__simps_I70_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% rel_simps(70)
thf(fact_387_rel__simps_I70_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% rel_simps(70)
thf(fact_388_rel__simps_I70_J,axiom,
~ ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ zero_z7100319975126383169nnreal ) ).
% rel_simps(70)
thf(fact_389_finite_Ointros_I1_J,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.intros(1)
thf(fact_390_finite_Ointros_I1_J,axiom,
finite5847741373460823677iteral @ bot_bot_set_literal ).
% finite.intros(1)
thf(fact_391_finite_Ointros_I1_J,axiom,
finite1111429032697314573l_num0 @ bot_bo5651135750051830704l_num0 ).
% finite.intros(1)
thf(fact_392_finite_Ointros_I1_J,axiom,
finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% finite.intros(1)
thf(fact_393_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_394_infinite__imp__nonempty,axiom,
! [S2: set_literal] :
( ~ ( finite5847741373460823677iteral @ S2 )
=> ( S2 != bot_bot_set_literal ) ) ).
% infinite_imp_nonempty
thf(fact_395_infinite__imp__nonempty,axiom,
! [S2: set_Numeral_num0] :
( ~ ( finite1111429032697314573l_num0 @ S2 )
=> ( S2 != bot_bo5651135750051830704l_num0 ) ) ).
% infinite_imp_nonempty
thf(fact_396_infinite__imp__nonempty,axiom,
! [S2: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ S2 )
=> ( S2 != bot_bo2099793752762293965at_nat ) ) ).
% infinite_imp_nonempty
thf(fact_397_infinite__growing,axiom,
! [X6: set_literal] :
( ( X6 != bot_bot_set_literal )
=> ( ! [X2: literal] :
( ( member_literal @ X2 @ X6 )
=> ? [Xa: literal] :
( ( member_literal @ Xa @ X6 )
& ( ord_less_literal @ X2 @ Xa ) ) )
=> ~ ( finite5847741373460823677iteral @ X6 ) ) ) ).
% infinite_growing
thf(fact_398_infinite__growing,axiom,
! [X6: set_real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X2: real] :
( ( member_real @ X2 @ X6 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X6 )
& ( ord_less_real @ X2 @ Xa ) ) )
=> ~ ( finite_finite_real @ X6 ) ) ) ).
% infinite_growing
thf(fact_399_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X6 )
=> ? [Xa: nat] :
( ( member_nat2 @ Xa @ X6 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_400_infinite__growing,axiom,
! [X6: set_int] :
( ( X6 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X6 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X6 )
& ( ord_less_int @ X2 @ Xa ) ) )
=> ~ ( finite_finite_int @ X6 ) ) ) ).
% infinite_growing
thf(fact_401_infinite__growing,axiom,
! [X6: set_Ex3793607809372303086nnreal] :
( ( X6 != bot_bo4854962954004695426nnreal )
=> ( ! [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ X6 )
=> ? [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ X6 )
& ( ord_le7381754540660121996nnreal @ X2 @ Xa ) ) )
=> ~ ( finite3782138982310603983nnreal @ X6 ) ) ) ).
% infinite_growing
thf(fact_402_ex__min__if__finite,axiom,
! [S2: set_literal] :
( ( finite5847741373460823677iteral @ S2 )
=> ( ( S2 != bot_bot_set_literal )
=> ? [X2: literal] :
( ( member_literal @ X2 @ S2 )
& ~ ? [Xa: literal] :
( ( member_literal @ Xa @ S2 )
& ( ord_less_literal @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_403_ex__min__if__finite,axiom,
! [S2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ S2 )
=> ( ( S2 != bot_bo2099793752762293965at_nat )
=> ? [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ S2 )
& ~ ? [Xa: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa @ S2 )
& ( ord_le1203424502768444845at_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_404_ex__min__if__finite,axiom,
! [S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ? [X2: real] :
( ( member_real @ X2 @ S2 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S2 )
& ( ord_less_real @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_405_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat2 @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_406_ex__min__if__finite,axiom,
! [S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ S2 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S2 )
& ( ord_less_int @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_407_ex__min__if__finite,axiom,
! [S2: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ S2 )
=> ( ( S2 != bot_bo4854962954004695426nnreal )
=> ? [X2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X2 @ S2 )
& ~ ? [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ S2 )
& ( ord_le7381754540660121996nnreal @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_408_Max__in,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( A != bot_bot_set_literal )
=> ( member_literal @ ( lattic5693287827546831171iteral @ A ) @ A ) ) ) ).
% Max_in
thf(fact_409_Max__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat2 @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ).
% Max_in
thf(fact_410_Min__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat2 @ ( lattic8721135487736765967in_nat @ A ) @ A ) ) ) ).
% Min_in
thf(fact_411_Min__in,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( A != bot_bot_set_literal )
=> ( member_literal @ ( lattic7587878784716877781iteral @ A ) @ A ) ) ) ).
% Min_in
thf(fact_412_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_413_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat2 @ X4 @ S2 )
& ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_414_arg__min__if__finite_I2_J,axiom,
! [S2: set_literal,F: literal > real] :
( ( finite5847741373460823677iteral @ S2 )
=> ( ( S2 != bot_bot_set_literal )
=> ~ ? [X4: literal] :
( ( member_literal @ X4 @ S2 )
& ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic1331604278971363751l_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_415_arg__min__if__finite_I2_J,axiom,
! [S2: set_Numeral_num0,F: numeral_num0 > real] :
( ( finite1111429032697314573l_num0 @ S2 )
=> ( ( S2 != bot_bo5651135750051830704l_num0 )
=> ~ ? [X4: numeral_num0] :
( ( member_Numeral_num0 @ X4 @ S2 )
& ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic46422294264233852710_real @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_416_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat2 @ X4 @ S2 )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_417_arg__min__if__finite_I2_J,axiom,
! [S2: set_literal,F: literal > nat] :
( ( finite5847741373460823677iteral @ S2 )
=> ( ( S2 != bot_bot_set_literal )
=> ~ ? [X4: literal] :
( ( member_literal @ X4 @ S2 )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic3379788727592557643al_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_418_arg__min__if__finite_I2_J,axiom,
! [S2: set_Numeral_num0,F: numeral_num0 > nat] :
( ( finite1111429032697314573l_num0 @ S2 )
=> ( ( S2 != bot_bo5651135750051830704l_num0 )
=> ~ ? [X4: numeral_num0] :
( ( member_Numeral_num0 @ X4 @ S2 )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic59464497659707227m0_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_419_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat2 @ X4 @ S2 )
& ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic7444442490073309207at_int @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_420_arg__min__if__finite_I2_J,axiom,
! [S2: set_literal,F: literal > int] :
( ( finite5847741373460823677iteral @ S2 )
=> ( ( S2 != bot_bot_set_literal )
=> ~ ? [X4: literal] :
( ( member_literal @ X4 @ S2 )
& ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic3377298257083507367al_int @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_421_arg__min__if__finite_I2_J,axiom,
! [S2: set_Numeral_num0,F: numeral_num0 > int] :
( ( finite1111429032697314573l_num0 @ S2 )
=> ( ( S2 != bot_bo5651135750051830704l_num0 )
=> ~ ? [X4: numeral_num0] :
( ( member_Numeral_num0 @ X4 @ S2 )
& ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic56974027150656951m0_int @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_422_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat2 @ X4 @ S2 )
& ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ ( lattic7087934650257931555nnreal @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_423_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_424_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_425_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_426_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat2 @ C2 @ A )
=> ( member_nat2 @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_427_int__int__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% int_int_eq
thf(fact_428_bgauge__existence__lemma,axiom,
! [S: set_nat,Q: real > nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ S )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ( Q @ D2 @ X3 ) ) ) )
= ( ! [X3: nat] :
? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ( ( member_nat2 @ X3 @ S )
=> ( Q @ D2 @ X3 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_429_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A6: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_430_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_431_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_432_finite__transitivity__chain,axiom,
! [A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ? [Y4: nat] :
( ( member_nat2 @ Y4 @ A )
& ( R @ X2 @ Y4 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_433_finite__transitivity__chain,axiom,
! [A: set_literal,R: literal > literal > $o] :
( ( finite5847741373460823677iteral @ A )
=> ( ! [X2: literal] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: literal,Y2: literal,Z2: literal] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [X2: literal] :
( ( member_literal @ X2 @ A )
=> ? [Y4: literal] :
( ( member_literal @ Y4 @ A )
& ( R @ X2 @ Y4 ) ) )
=> ( A = bot_bot_set_literal ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_434_finite__transitivity__chain,axiom,
! [A: set_Numeral_num0,R: numeral_num0 > numeral_num0 > $o] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ! [X2: numeral_num0] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: numeral_num0,Y2: numeral_num0,Z2: numeral_num0] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [X2: numeral_num0] :
( ( member_Numeral_num0 @ X2 @ A )
=> ? [Y4: numeral_num0] :
( ( member_Numeral_num0 @ Y4 @ A )
& ( R @ X2 @ Y4 ) ) )
=> ( A = bot_bo5651135750051830704l_num0 ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_435_finite__transitivity__chain,axiom,
! [A: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( finite6177210948735845034at_nat @ A )
=> ( ! [X2: product_prod_nat_nat] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A )
=> ? [Y4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y4 @ A )
& ( R @ X2 @ Y4 ) ) )
=> ( A = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_436_reals__Archimedean2,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% reals_Archimedean2
thf(fact_437_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_438_verit__comp__simplify_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_439_verit__comp__simplify_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_440_verit__comp__simplify_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_441_verit__comp__simplify_I1_J,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ A2 @ A2 ) ).
% verit_comp_simplify(1)
thf(fact_442_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ( member_nat2 @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_443_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ( member_nat2 @ N5 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_444_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A6: nat,B5: nat] :
( ( semiri1314217659103216013at_int @ A6 )
= ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_445_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_446_minus__Min__eq__Max,axiom,
! [S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( uminus_uminus_int @ ( lattic8718645017227715691in_int @ S2 ) )
= ( lattic8263393255366662781ax_int @ ( image_int_int @ uminus_uminus_int @ S2 ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_447_minus__Min__eq__Max,axiom,
! [S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( uminus_uminus_real @ ( lattic3629708407755379051n_real @ S2 ) )
= ( lattic4275903605611617917x_real @ ( image_real_real @ uminus_uminus_real @ S2 ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_448_minus__Max__eq__Min,axiom,
! [S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( uminus_uminus_int @ ( lattic8263393255366662781ax_int @ S2 ) )
= ( lattic8718645017227715691in_int @ ( image_int_int @ uminus_uminus_int @ S2 ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_449_minus__Max__eq__Min,axiom,
! [S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ( S2 != bot_bot_set_real )
=> ( ( uminus_uminus_real @ ( lattic4275903605611617917x_real @ S2 ) )
= ( lattic3629708407755379051n_real @ ( image_real_real @ uminus_uminus_real @ S2 ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_450_finite__enumerate__mono__iff,axiom,
! [S2: set_Numeral_num1,M2: nat,N2: nat] :
( ( finite1111429032697314574l_num1 @ S2 )
=> ( ( ord_less_nat @ M2 @ ( finite6454714172617411597l_num1 @ S2 ) )
=> ( ( ord_less_nat @ N2 @ ( finite6454714172617411597l_num1 @ S2 ) )
=> ( ( ord_le6405328735288452753l_num1 @ ( infini4751236147369608586l_num1 @ S2 @ M2 ) @ ( infini4751236147369608586l_num1 @ S2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_451_finite__enumerate__mono__iff,axiom,
! [S2: set_nat,M2: nat,N2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ M2 @ ( finite_card_nat @ S2 ) )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_452_card__range__greater__zero,axiom,
! [F: product_unit > numeral_num1] :
( ( finite1111429032697314574l_num1 @ ( image_6449127158079674653l_num1 @ F @ top_to1996260823553986621t_unit ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ ( image_6449127158079674653l_num1 @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% card_range_greater_zero
thf(fact_453_card__range__greater__zero,axiom,
! [F: product_unit > nat] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% card_range_greater_zero
thf(fact_454_card__range__greater__zero,axiom,
! [F: product_unit > literal] :
( ( finite5847741373460823677iteral @ ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% card_range_greater_zero
thf(fact_455_card__range__greater__zero,axiom,
! [F: product_unit > numeral_num0] :
( ( finite1111429032697314573l_num0 @ ( image_6449127158079674652l_num0 @ F @ top_to1996260823553986621t_unit ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ ( image_6449127158079674652l_num0 @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% card_range_greater_zero
thf(fact_456_card__range__greater__zero,axiom,
! [F: numeral_num1 > numeral_num1] :
( ( finite1111429032697314574l_num1 @ ( image_6783865936125884741l_num1 @ F @ top_to3689904429138878997l_num1 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ ( image_6783865936125884741l_num1 @ F @ top_to3689904429138878997l_num1 ) ) ) ) ).
% card_range_greater_zero
thf(fact_457_card__range__greater__zero,axiom,
! [F: numeral_num1 > nat] :
( ( finite_finite_nat @ ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 ) ) ) ) ).
% card_range_greater_zero
thf(fact_458_card__range__greater__zero,axiom,
! [F: numeral_num1 > literal] :
( ( finite5847741373460823677iteral @ ( image_5852747068178070836iteral @ F @ top_to3689904429138878997l_num1 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ ( image_5852747068178070836iteral @ F @ top_to3689904429138878997l_num1 ) ) ) ) ).
% card_range_greater_zero
thf(fact_459_card__range__greater__zero,axiom,
! [F: numeral_num1 > numeral_num0] :
( ( finite1111429032697314573l_num0 @ ( image_6783865936125884740l_num0 @ F @ top_to3689904429138878997l_num1 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ ( image_6783865936125884740l_num0 @ F @ top_to3689904429138878997l_num1 ) ) ) ) ).
% card_range_greater_zero
thf(fact_460_card__range__greater__zero,axiom,
! [F: numeral_num0 > numeral_num1] :
( ( finite1111429032697314574l_num1 @ ( image_2832974300507296262l_num1 @ F @ top_to3689904424835650196l_num0 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ ( image_2832974300507296262l_num1 @ F @ top_to3689904424835650196l_num0 ) ) ) ) ).
% card_range_greater_zero
thf(fact_461_card__range__greater__zero,axiom,
! [F: numeral_num0 > nat] :
( ( finite_finite_nat @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) ) ) ) ).
% card_range_greater_zero
thf(fact_462_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% neg_int_cases
thf(fact_463_sorted__list__of__multiset__empty,axiom,
( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
= nil_nat ) ).
% sorted_list_of_multiset_empty
thf(fact_464_ex__inverse__of__nat__less,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_465_UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_466_UNIV__I,axiom,
! [X: numeral_num1] : ( member_Numeral_num1 @ X @ top_to3689904429138878997l_num1 ) ).
% UNIV_I
thf(fact_467_UNIV__I,axiom,
! [X: numeral_num0] : ( member_Numeral_num0 @ X @ top_to3689904424835650196l_num0 ) ).
% UNIV_I
thf(fact_468_UNIV__I,axiom,
! [X: literal] : ( member_literal @ X @ top_top_set_literal ) ).
% UNIV_I
thf(fact_469_UNIV__I,axiom,
! [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_470_more__arith__simps_I10_J,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% more_arith_simps(10)
thf(fact_471_more__arith__simps_I10_J,axiom,
! [A2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
= A2 ) ).
% more_arith_simps(10)
thf(fact_472_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_473_neg__equal__iff__equal,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_474_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_475_neg__equal__zero,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= A2 )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_476_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_477_equal__neg__zero,axiom,
! [A2: real] :
( ( A2
= ( uminus_uminus_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_478_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_479_neg__equal__0__iff__equal,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_480_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_481_neg__0__equal__iff__equal,axiom,
! [A2: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A2 ) )
= ( zero_zero_real = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_482_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_483_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_484_neg__less__iff__less,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_485_neg__less__iff__less,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_486_finite__compl,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A )
=> ( ( finite6177210948735845034at_nat @ ( uminus6524753893492686040at_nat @ A ) )
= ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_compl
thf(fact_487_finite__compl,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ ( uminus5944136376168626660t_unit @ A ) )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_compl
thf(fact_488_finite__compl,axiom,
! [A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ A )
=> ( ( finite1111429032697314574l_num1 @ ( uminus7637779981753519036l_num1 @ A ) )
= ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_compl
thf(fact_489_finite__compl,axiom,
! [A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ A )
=> ( ( finite1111429032697314573l_num0 @ ( uminus7637779977450290235l_num0 @ A ) )
= ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_compl
thf(fact_490_finite__compl,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite5847741373460823677iteral @ ( uminus3698395583336102443iteral @ A ) )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_compl
thf(fact_491_finite__compl,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( uminus5710092332889474511et_nat @ A ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_compl
thf(fact_492_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_493_finite__Plus__UNIV__iff,axiom,
( ( finite4840195107178753492l_num1 @ top_to4509454260816870435l_num1 )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_494_finite__Plus__UNIV__iff,axiom,
( ( finite4840195102875524691l_num0 @ top_to4438420220770524450l_num0 )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_495_finite__Plus__UNIV__iff,axiom,
( ( finite2327689956804853827iteral @ top_to8707194323715505426iteral )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_496_finite__Plus__UNIV__iff,axiom,
( ( finite4401952911629260215it_nat @ top_to2894617605782473790it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_497_finite__Plus__UNIV__iff,axiom,
( ( finite3481290279640071204t_unit @ top_to7960292400817545843t_unit )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_498_finite__Plus__UNIV__iff,axiom,
( ( finite5174933885224963580l_num1 @ top_to474455691063265355l_num1 )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_499_finite__Plus__UNIV__iff,axiom,
( ( finite5174933880921734779l_num0 @ top_to403421651016919370l_num0 )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_500_finite__Plus__UNIV__iff,axiom,
( ( finite2303452279084932203iteral @ top_to6496210893225035066iteral )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_501_finite__Plus__UNIV__iff,axiom,
( ( finite4336029346108437391m1_nat @ top_to2378920326186532374m1_nat )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_502_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_503_less__neg__neg,axiom,
! [A2: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_504_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_505_neg__less__pos,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_pos
thf(fact_506_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_507_neg__0__less__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_508_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_509_neg__less__0__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_510_negative__eq__positive,axiom,
! [N2: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N2 = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_511_enumerate__mono__iff,axiom,
! [S2: set_nat,M2: nat,N2: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% enumerate_mono_iff
thf(fact_512_range__enumerate,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S2 ) @ top_top_set_nat )
= S2 ) ) ).
% range_enumerate
thf(fact_513_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_514_enumerate__Ex,axiom,
! [S2: set_nat,S: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( member_nat2 @ S @ S2 )
=> ? [N4: nat] :
( ( infini8530281810654367211te_nat @ S2 @ N4 )
= S ) ) ) ).
% enumerate_Ex
thf(fact_515_Compl__empty__eq,axiom,
( ( uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat )
= top_to4669805908274784177at_nat ) ).
% Compl_empty_eq
thf(fact_516_Compl__empty__eq,axiom,
( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
= top_to1996260823553986621t_unit ) ).
% Compl_empty_eq
thf(fact_517_Compl__empty__eq,axiom,
( ( uminus7637779981753519036l_num1 @ bot_bo5651135754355059505l_num1 )
= top_to3689904429138878997l_num1 ) ).
% Compl_empty_eq
thf(fact_518_Compl__empty__eq,axiom,
( ( uminus7637779977450290235l_num0 @ bot_bo5651135750051830704l_num0 )
= top_to3689904424835650196l_num0 ) ).
% Compl_empty_eq
thf(fact_519_Compl__empty__eq,axiom,
( ( uminus3698395583336102443iteral @ bot_bot_set_literal )
= top_top_set_literal ) ).
% Compl_empty_eq
thf(fact_520_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_521_Compl__UNIV__eq,axiom,
( ( uminus6524753893492686040at_nat @ top_to4669805908274784177at_nat )
= bot_bo2099793752762293965at_nat ) ).
% Compl_UNIV_eq
thf(fact_522_Compl__UNIV__eq,axiom,
( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Compl_UNIV_eq
thf(fact_523_Compl__UNIV__eq,axiom,
( ( uminus7637779981753519036l_num1 @ top_to3689904429138878997l_num1 )
= bot_bo5651135754355059505l_num1 ) ).
% Compl_UNIV_eq
thf(fact_524_Compl__UNIV__eq,axiom,
( ( uminus7637779977450290235l_num0 @ top_to3689904424835650196l_num0 )
= bot_bo5651135750051830704l_num0 ) ).
% Compl_UNIV_eq
thf(fact_525_Compl__UNIV__eq,axiom,
( ( uminus3698395583336102443iteral @ top_top_set_literal )
= bot_bot_set_literal ) ).
% Compl_UNIV_eq
thf(fact_526_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_527_equation__minus__iff,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_528_equation__minus__iff,axiom,
! [A2: real,B: real] :
( ( A2
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_529_minus__equation__iff,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( uminus_uminus_int @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_530_minus__equation__iff,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= B )
= ( ( uminus_uminus_real @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_531_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X2: product_unit] : ( member_Product_unit @ X2 @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_532_UNIV__eq__I,axiom,
! [A: set_Numeral_num1] :
( ! [X2: numeral_num1] : ( member_Numeral_num1 @ X2 @ A )
=> ( top_to3689904429138878997l_num1 = A ) ) ).
% UNIV_eq_I
thf(fact_533_UNIV__eq__I,axiom,
! [A: set_Numeral_num0] :
( ! [X2: numeral_num0] : ( member_Numeral_num0 @ X2 @ A )
=> ( top_to3689904424835650196l_num0 = A ) ) ).
% UNIV_eq_I
thf(fact_534_UNIV__eq__I,axiom,
! [A: set_literal] :
( ! [X2: literal] : ( member_literal @ X2 @ A )
=> ( top_top_set_literal = A ) ) ).
% UNIV_eq_I
thf(fact_535_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat2 @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_536_UNIV__witness,axiom,
? [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_537_UNIV__witness,axiom,
? [X2: numeral_num1] : ( member_Numeral_num1 @ X2 @ top_to3689904429138878997l_num1 ) ).
% UNIV_witness
thf(fact_538_UNIV__witness,axiom,
? [X2: numeral_num0] : ( member_Numeral_num0 @ X2 @ top_to3689904424835650196l_num0 ) ).
% UNIV_witness
thf(fact_539_UNIV__witness,axiom,
? [X2: literal] : ( member_literal @ X2 @ top_top_set_literal ) ).
% UNIV_witness
thf(fact_540_UNIV__witness,axiom,
? [X2: nat] : ( member_nat2 @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_541_Finite__Set_Ofinite__set,axiom,
( ( finite9047747110432174090at_nat @ top_to7629004291339433233at_nat )
= ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ).
% Finite_Set.finite_set
thf(fact_542_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_543_Finite__Set_Ofinite__set,axiom,
( ( finite3465821969784575470l_num1 @ top_to3504832992239360757l_num1 )
= ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ).
% Finite_Set.finite_set
thf(fact_544_Finite__Set_Ofinite__set,axiom,
( ( finite3465821965481346669l_num0 @ top_to3433798952193014772l_num0 )
= ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ).
% Finite_Set.finite_set
thf(fact_545_Finite__Set_Ofinite__set,axiom,
( ( finite2869373537460367197iteral @ top_to5694933271948605156iteral )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ).
% Finite_Set.finite_set
thf(fact_546_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_547_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_548_less__top,axiom,
! [A2: set_Product_unit] :
( ( A2 != top_to1996260823553986621t_unit )
= ( ord_le8056459307392131481t_unit @ A2 @ top_to1996260823553986621t_unit ) ) ).
% less_top
thf(fact_549_less__top,axiom,
! [A2: set_Numeral_num1] :
( ( A2 != top_to3689904429138878997l_num1 )
= ( ord_le526730876122248049l_num1 @ A2 @ top_to3689904429138878997l_num1 ) ) ).
% less_top
thf(fact_550_less__top,axiom,
! [A2: set_Numeral_num0] :
( ( A2 != top_to3689904424835650196l_num0 )
= ( ord_le526730871819019248l_num0 @ A2 @ top_to3689904424835650196l_num0 ) ) ).
% less_top
thf(fact_551_less__top,axiom,
! [A2: set_literal] :
( ( A2 != top_top_set_literal )
= ( ord_less_set_literal @ A2 @ top_top_set_literal ) ) ).
% less_top
thf(fact_552_less__top,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% less_top
thf(fact_553_less__top,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( A2 != top_to1496364449551166952nnreal )
= ( ord_le7381754540660121996nnreal @ A2 @ top_to1496364449551166952nnreal ) ) ).
% less_top
thf(fact_554_not__top__less,axiom,
! [A2: set_Product_unit] :
~ ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ A2 ) ).
% not_top_less
thf(fact_555_not__top__less,axiom,
! [A2: set_Numeral_num1] :
~ ( ord_le526730876122248049l_num1 @ top_to3689904429138878997l_num1 @ A2 ) ).
% not_top_less
thf(fact_556_not__top__less,axiom,
! [A2: set_Numeral_num0] :
~ ( ord_le526730871819019248l_num0 @ top_to3689904424835650196l_num0 @ A2 ) ).
% not_top_less
thf(fact_557_not__top__less,axiom,
! [A2: set_literal] :
~ ( ord_less_set_literal @ top_top_set_literal @ A2 ) ).
% not_top_less
thf(fact_558_not__top__less,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% not_top_less
thf(fact_559_not__top__less,axiom,
! [A2: extend8495563244428889912nnreal] :
~ ( ord_le7381754540660121996nnreal @ top_to1496364449551166952nnreal @ A2 ) ).
% not_top_less
thf(fact_560_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_561_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( finite8510363019766020200l_num1 @ top_to3573342476081039471l_num1 ) ) ) ).
% finite_Prod_UNIV
thf(fact_562_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( finite8510363015462791399l_num0 @ top_to3502308436034693486l_num0 ) ) ) ).
% finite_Prod_UNIV
thf(fact_563_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite885979093102766295iteral @ top_to1987565607987313502iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_564_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_565_finite__Prod__UNIV,axiom,
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite7151458192227337912t_unit @ top_to7024180616081714879t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_566_finite__Prod__UNIV,axiom,
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( finite8845101797812230288l_num1 @ top_to8761715943182210199l_num1 ) ) ) ).
% finite_Prod_UNIV
thf(fact_567_finite__Prod__UNIV,axiom,
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( finite8845101793509001487l_num0 @ top_to8690681903135864214l_num0 ) ) ) ).
% finite_Prod_UNIV
thf(fact_568_finite__Prod__UNIV,axiom,
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite861741415382844671iteral @ top_to8999954214351618950iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_569_finite__Prod__UNIV,axiom,
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5121599250977343483m1_nat @ top_to5458413198516828874m1_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_570_finite__prod,axiom,
( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_571_finite__prod,axiom,
( ( finite8510363019766020200l_num1 @ top_to3573342476081039471l_num1 )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_prod
thf(fact_572_finite__prod,axiom,
( ( finite8510363015462791399l_num0 @ top_to3502308436034693486l_num0 )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_prod
thf(fact_573_finite__prod,axiom,
( ( finite885979093102766295iteral @ top_to1987565607987313502iteral )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_574_finite__prod,axiom,
( ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_575_finite__prod,axiom,
( ( finite7151458192227337912t_unit @ top_to7024180616081714879t_unit )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_576_finite__prod,axiom,
( ( finite8845101797812230288l_num1 @ top_to8761715943182210199l_num1 )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_prod
thf(fact_577_finite__prod,axiom,
( ( finite8845101793509001487l_num0 @ top_to8690681903135864214l_num0 )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_prod
thf(fact_578_finite__prod,axiom,
( ( finite861741415382844671iteral @ top_to8999954214351618950iteral )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_579_finite__prod,axiom,
( ( finite5121599250977343483m1_nat @ top_to5458413198516828874m1_nat )
= ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_580_less__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_581_less__minus__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A2 ) ) ) ).
% less_minus_iff
thf(fact_582_minus__less__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_583_minus__less__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_584_verit__negate__coefficient_I2_J,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_585_verit__negate__coefficient_I2_J,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_586_int__cases2,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% int_cases2
thf(fact_587_rangeI,axiom,
! [F: product_unit > nat,X: product_unit] : ( member_nat2 @ ( F @ X ) @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_588_rangeI,axiom,
! [F: numeral_num1 > nat,X: numeral_num1] : ( member_nat2 @ ( F @ X ) @ ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 ) ) ).
% rangeI
thf(fact_589_rangeI,axiom,
! [F: numeral_num0 > nat,X: numeral_num0] : ( member_nat2 @ ( F @ X ) @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) ) ).
% rangeI
thf(fact_590_rangeI,axiom,
! [F: literal > nat,X: literal] : ( member_nat2 @ ( F @ X ) @ ( image_literal_nat @ F @ top_top_set_literal ) ) ).
% rangeI
thf(fact_591_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_592_range__eqI,axiom,
! [B: nat,F: product_unit > nat,X: product_unit] :
( ( B
= ( F @ X ) )
=> ( member_nat2 @ B @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_593_range__eqI,axiom,
! [B: nat,F: numeral_num1 > nat,X: numeral_num1] :
( ( B
= ( F @ X ) )
=> ( member_nat2 @ B @ ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 ) ) ) ).
% range_eqI
thf(fact_594_range__eqI,axiom,
! [B: nat,F: numeral_num0 > nat,X: numeral_num0] :
( ( B
= ( F @ X ) )
=> ( member_nat2 @ B @ ( image_8797574156932312687m0_nat @ F @ top_to3689904424835650196l_num0 ) ) ) ).
% range_eqI
thf(fact_595_range__eqI,axiom,
! [B: nat,F: literal > nat,X: literal] :
( ( B
= ( F @ X ) )
=> ( member_nat2 @ B @ ( image_literal_nat @ F @ top_top_set_literal ) ) ) ).
% range_eqI
thf(fact_596_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat2 @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_597_UNIV__not__empty,axiom,
top_to4669805908274784177at_nat != bot_bo2099793752762293965at_nat ).
% UNIV_not_empty
thf(fact_598_UNIV__not__empty,axiom,
top_to1996260823553986621t_unit != bot_bo3957492148770167129t_unit ).
% UNIV_not_empty
thf(fact_599_UNIV__not__empty,axiom,
top_to3689904429138878997l_num1 != bot_bo5651135754355059505l_num1 ).
% UNIV_not_empty
thf(fact_600_UNIV__not__empty,axiom,
top_to3689904424835650196l_num0 != bot_bo5651135750051830704l_num0 ).
% UNIV_not_empty
thf(fact_601_UNIV__not__empty,axiom,
top_top_set_literal != bot_bot_set_literal ).
% UNIV_not_empty
thf(fact_602_UNIV__not__empty,axiom,
top_top_set_nat != bot_bot_set_nat ).
% UNIV_not_empty
thf(fact_603_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_604_ex__new__if__finite,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
=> ( ( finite6177210948735845034at_nat @ A )
=> ? [A4: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_605_ex__new__if__finite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A )
=> ? [A4: product_unit] :
~ ( member_Product_unit @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_606_ex__new__if__finite,axiom,
! [A: set_Numeral_num1] :
( ~ ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( finite1111429032697314574l_num1 @ A )
=> ? [A4: numeral_num1] :
~ ( member_Numeral_num1 @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_607_ex__new__if__finite,axiom,
! [A: set_Numeral_num0] :
( ~ ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( finite1111429032697314573l_num0 @ A )
=> ? [A4: numeral_num0] :
~ ( member_Numeral_num0 @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_608_ex__new__if__finite,axiom,
! [A: set_literal] :
( ~ ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite5847741373460823677iteral @ A )
=> ? [A4: literal] :
~ ( member_literal @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_609_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A4: nat] :
~ ( member_nat2 @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_610_finite__class_Ofinite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_class.finite_UNIV
thf(fact_611_finite__class_Ofinite__UNIV,axiom,
finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ).
% finite_class.finite_UNIV
thf(fact_612_enumerate__in__set,axiom,
! [S2: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( member_nat2 @ ( infini8530281810654367211te_nat @ S2 @ N2 ) @ S2 ) ) ).
% enumerate_in_set
thf(fact_613_not__int__zless__negative,axiom,
! [N2: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_614_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
=> ( ( ( finite711546835091564841at_nat @ A )
= ( finite711546835091564841at_nat @ top_to4669805908274784177at_nat ) )
=> ( A = top_to4669805908274784177at_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_615_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
=> ( A = top_to1996260823553986621t_unit ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_616_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Numeral_num1] :
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ( ( finite6454714172617411597l_num1 @ A )
= ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) )
=> ( A = top_to3689904429138878997l_num1 ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_617_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Numeral_num0] :
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ( ( finite6454714172617411596l_num0 @ A )
= ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 ) )
=> ( A = top_to3689904424835650196l_num0 ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_618_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( ( finite_card_literal @ A )
= ( finite_card_literal @ top_top_set_literal ) )
=> ( A = top_top_set_literal ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_619_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_620_int__cases4,axiom,
! [M2: int] :
( ! [N4: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_621_enumerate__mono,axiom,
! [M2: nat,N2: nat,S2: set_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) ) ) ) ).
% enumerate_mono
thf(fact_622_finite__enumerate__in__set,axiom,
! [S2: set_Numeral_num1,N2: nat] :
( ( finite1111429032697314574l_num1 @ S2 )
=> ( ( ord_less_nat @ N2 @ ( finite6454714172617411597l_num1 @ S2 ) )
=> ( member_Numeral_num1 @ ( infini4751236147369608586l_num1 @ S2 @ N2 ) @ S2 ) ) ) ).
% finite_enumerate_in_set
thf(fact_623_finite__enumerate__in__set,axiom,
! [S2: set_nat,N2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ( member_nat2 @ ( infini8530281810654367211te_nat @ S2 @ N2 ) @ S2 ) ) ) ).
% finite_enumerate_in_set
thf(fact_624_finite__enumerate__Ex,axiom,
! [S2: set_Numeral_num1,S: numeral_num1] :
( ( finite1111429032697314574l_num1 @ S2 )
=> ( ( member_Numeral_num1 @ S @ S2 )
=> ? [N4: nat] :
( ( ord_less_nat @ N4 @ ( finite6454714172617411597l_num1 @ S2 ) )
& ( ( infini4751236147369608586l_num1 @ S2 @ N4 )
= S ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_625_finite__enumerate__Ex,axiom,
! [S2: set_nat,S: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( member_nat2 @ S @ S2 )
=> ? [N4: nat] :
( ( ord_less_nat @ N4 @ ( finite_card_nat @ S2 ) )
& ( ( infini8530281810654367211te_nat @ S2 @ N4 )
= S ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_626_finite__enum__ext,axiom,
! [X6: set_Numeral_num1,Y6: set_Numeral_num1] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( finite6454714172617411597l_num1 @ X6 ) )
=> ( ( infini4751236147369608586l_num1 @ X6 @ I )
= ( infini4751236147369608586l_num1 @ Y6 @ I ) ) )
=> ( ( finite1111429032697314574l_num1 @ X6 )
=> ( ( finite1111429032697314574l_num1 @ Y6 )
=> ( ( ( finite6454714172617411597l_num1 @ X6 )
= ( finite6454714172617411597l_num1 @ Y6 ) )
=> ( X6 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_627_finite__enum__ext,axiom,
! [X6: set_nat,Y6: set_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_nat @ X6 ) )
=> ( ( infini8530281810654367211te_nat @ X6 @ I )
= ( infini8530281810654367211te_nat @ Y6 @ I ) ) )
=> ( ( finite_finite_nat @ X6 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ( finite_card_nat @ X6 )
= ( finite_card_nat @ Y6 ) )
=> ( X6 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_628_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_629_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D3: real,E: real] :
( ( ord_less_real @ D3 @ E )
=> ( ( P @ D3 )
=> ( P @ E ) ) )
=> ( ! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_630_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% int_cases3
thf(fact_631_finite__UNIV__card__ge__0,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_632_finite__UNIV__card__ge__0,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_633_finite__UNIV__card__ge__0,axiom,
( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_634_finite__UNIV__card__ge__0,axiom,
( ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 )
=> ( ord_less_nat @ zero_zero_nat @ ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_635_finite__UNIV__card__ge__0,axiom,
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_literal @ top_top_set_literal ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_636_finite__UNIV__card__ge__0,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ top_top_set_nat ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_637_finite__enumerate__mono,axiom,
! [M2: nat,N2: nat,S2: set_Numeral_num1] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( finite1111429032697314574l_num1 @ S2 )
=> ( ( ord_less_nat @ N2 @ ( finite6454714172617411597l_num1 @ S2 ) )
=> ( ord_le6405328735288452753l_num1 @ ( infini4751236147369608586l_num1 @ S2 @ M2 ) @ ( infini4751236147369608586l_num1 @ S2 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_638_finite__enumerate__mono,axiom,
! [M2: nat,N2: nat,S2: set_nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N2 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_639_inverse__positive__iff__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% inverse_positive_iff_positive
thf(fact_640_inverse__negative__iff__negative,axiom,
! [A2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_641_inverse__less__iff__less__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A2 ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_642_inverse__less__iff__less,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A2 ) ) ) ) ).
% inverse_less_iff_less
thf(fact_643_surj__uminus,axiom,
( ( image_6783865936125884741l_num1 @ uminus8996842218212420316l_num1 @ top_to3689904429138878997l_num1 )
= top_to3689904429138878997l_num1 ) ).
% surj_uminus
thf(fact_644_surj__uminus,axiom,
( ( image_int_int @ uminus_uminus_int @ top_top_set_int )
= top_top_set_int ) ).
% surj_uminus
thf(fact_645_surj__uminus,axiom,
( ( image_real_real @ uminus_uminus_real @ top_top_set_real )
= top_top_set_real ) ).
% surj_uminus
thf(fact_646_compl__bot__eq,axiom,
( ( uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat )
= top_to4669805908274784177at_nat ) ).
% compl_bot_eq
thf(fact_647_compl__bot__eq,axiom,
( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
= top_to1996260823553986621t_unit ) ).
% compl_bot_eq
thf(fact_648_compl__bot__eq,axiom,
( ( uminus7637779981753519036l_num1 @ bot_bo5651135754355059505l_num1 )
= top_to3689904429138878997l_num1 ) ).
% compl_bot_eq
thf(fact_649_compl__bot__eq,axiom,
( ( uminus7637779977450290235l_num0 @ bot_bo5651135750051830704l_num0 )
= top_to3689904424835650196l_num0 ) ).
% compl_bot_eq
thf(fact_650_compl__bot__eq,axiom,
( ( uminus3698395583336102443iteral @ bot_bot_set_literal )
= top_top_set_literal ) ).
% compl_bot_eq
thf(fact_651_compl__bot__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% compl_bot_eq
thf(fact_652_compl__top__eq,axiom,
( ( uminus6524753893492686040at_nat @ top_to4669805908274784177at_nat )
= bot_bo2099793752762293965at_nat ) ).
% compl_top_eq
thf(fact_653_compl__top__eq,axiom,
( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% compl_top_eq
thf(fact_654_compl__top__eq,axiom,
( ( uminus7637779981753519036l_num1 @ top_to3689904429138878997l_num1 )
= bot_bo5651135754355059505l_num1 ) ).
% compl_top_eq
thf(fact_655_compl__top__eq,axiom,
( ( uminus7637779977450290235l_num0 @ top_to3689904424835650196l_num0 )
= bot_bo5651135750051830704l_num0 ) ).
% compl_top_eq
thf(fact_656_compl__top__eq,axiom,
( ( uminus3698395583336102443iteral @ top_top_set_literal )
= bot_bot_set_literal ) ).
% compl_top_eq
thf(fact_657_compl__top__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% compl_top_eq
thf(fact_658_mem__simps_I5_J,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat2 @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat2 @ C2 @ A ) ) ) ).
% mem_simps(5)
thf(fact_659_ComplI,axiom,
! [C2: nat,A: set_nat] :
( ~ ( member_nat2 @ C2 @ A )
=> ( member_nat2 @ C2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_660_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_661_inverse__nonzero__iff__nonzero,axiom,
! [A2: real] :
( ( ( inverse_inverse_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_662_ComplD,axiom,
! [C2: nat,A: set_nat] :
( ( member_nat2 @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat2 @ C2 @ A ) ) ).
% ComplD
thf(fact_663_top__empty__eq,axiom,
( top_to2465898995584390880unit_o
= ( ^ [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ) ) ).
% top_empty_eq
thf(fact_664_top__empty__eq,axiom,
( top_to1749082287617889032num1_o
= ( ^ [X3: numeral_num1] : ( member_Numeral_num1 @ X3 @ top_to3689904429138878997l_num1 ) ) ) ).
% top_empty_eq
thf(fact_665_top__empty__eq,axiom,
( top_to4648304687082283337num0_o
= ( ^ [X3: numeral_num0] : ( member_Numeral_num0 @ X3 @ top_to3689904424835650196l_num0 ) ) ) ).
% top_empty_eq
thf(fact_666_top__empty__eq,axiom,
( top_top_literal_o
= ( ^ [X3: literal] : ( member_literal @ X3 @ top_top_set_literal ) ) ) ).
% top_empty_eq
thf(fact_667_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_668_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_669_top__set__def,axiom,
( top_to3689904429138878997l_num1
= ( collect_Numeral_num1 @ top_to1749082287617889032num1_o ) ) ).
% top_set_def
thf(fact_670_top__set__def,axiom,
( top_to3689904424835650196l_num0
= ( collect_Numeral_num0 @ top_to4648304687082283337num0_o ) ) ).
% top_set_def
thf(fact_671_top__set__def,axiom,
( top_top_set_literal
= ( collect_literal @ top_top_literal_o ) ) ).
% top_set_def
thf(fact_672_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_673_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_674_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_675_surjD,axiom,
! [F: product_unit > product_unit,Y: product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_676_surjD,axiom,
! [F: product_unit > numeral_num1,Y: numeral_num1] :
( ( ( image_6449127158079674653l_num1 @ F @ top_to1996260823553986621t_unit )
= top_to3689904429138878997l_num1 )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_677_surjD,axiom,
! [F: product_unit > numeral_num0,Y: numeral_num0] :
( ( ( image_6449127158079674652l_num0 @ F @ top_to1996260823553986621t_unit )
= top_to3689904424835650196l_num0 )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_678_surjD,axiom,
! [F: product_unit > literal,Y: literal] :
( ( ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_679_surjD,axiom,
! [F: product_unit > nat,Y: nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_680_surjD,axiom,
! [F: numeral_num1 > product_unit,Y: product_unit] :
( ( ( image_739801482541841261t_unit @ F @ top_to3689904429138878997l_num1 )
= top_to1996260823553986621t_unit )
=> ? [X2: numeral_num1] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_681_surjD,axiom,
! [F: numeral_num1 > numeral_num1,Y: numeral_num1] :
( ( ( image_6783865936125884741l_num1 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904429138878997l_num1 )
=> ? [X2: numeral_num1] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_682_surjD,axiom,
! [F: numeral_num1 > numeral_num0,Y: numeral_num0] :
( ( ( image_6783865936125884740l_num0 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904424835650196l_num0 )
=> ? [X2: numeral_num1] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_683_surjD,axiom,
! [F: numeral_num1 > literal,Y: literal] :
( ( ( image_5852747068178070836iteral @ F @ top_to3689904429138878997l_num1 )
= top_top_set_literal )
=> ? [X2: numeral_num1] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_684_surjD,axiom,
! [F: numeral_num1 > nat,Y: nat] :
( ( ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 )
= top_top_set_nat )
=> ? [X2: numeral_num1] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_685_surjE,axiom,
! [F: product_unit > product_unit,Y: product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_686_surjE,axiom,
! [F: product_unit > numeral_num1,Y: numeral_num1] :
( ( ( image_6449127158079674653l_num1 @ F @ top_to1996260823553986621t_unit )
= top_to3689904429138878997l_num1 )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_687_surjE,axiom,
! [F: product_unit > numeral_num0,Y: numeral_num0] :
( ( ( image_6449127158079674652l_num0 @ F @ top_to1996260823553986621t_unit )
= top_to3689904424835650196l_num0 )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_688_surjE,axiom,
! [F: product_unit > literal,Y: literal] :
( ( ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_689_surjE,axiom,
! [F: product_unit > nat,Y: nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_690_surjE,axiom,
! [F: numeral_num1 > product_unit,Y: product_unit] :
( ( ( image_739801482541841261t_unit @ F @ top_to3689904429138878997l_num1 )
= top_to1996260823553986621t_unit )
=> ~ ! [X2: numeral_num1] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_691_surjE,axiom,
! [F: numeral_num1 > numeral_num1,Y: numeral_num1] :
( ( ( image_6783865936125884741l_num1 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904429138878997l_num1 )
=> ~ ! [X2: numeral_num1] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_692_surjE,axiom,
! [F: numeral_num1 > numeral_num0,Y: numeral_num0] :
( ( ( image_6783865936125884740l_num0 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904424835650196l_num0 )
=> ~ ! [X2: numeral_num1] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_693_surjE,axiom,
! [F: numeral_num1 > literal,Y: literal] :
( ( ( image_5852747068178070836iteral @ F @ top_to3689904429138878997l_num1 )
= top_top_set_literal )
=> ~ ! [X2: numeral_num1] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_694_surjE,axiom,
! [F: numeral_num1 > nat,Y: nat] :
( ( ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 )
= top_top_set_nat )
=> ~ ! [X2: numeral_num1] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_695_surjI,axiom,
! [G: product_unit > product_unit,F: product_unit > product_unit] :
( ! [X2: product_unit] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_696_surjI,axiom,
! [G: product_unit > numeral_num1,F: numeral_num1 > product_unit] :
( ! [X2: numeral_num1] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_6449127158079674653l_num1 @ G @ top_to1996260823553986621t_unit )
= top_to3689904429138878997l_num1 ) ) ).
% surjI
thf(fact_697_surjI,axiom,
! [G: product_unit > numeral_num0,F: numeral_num0 > product_unit] :
( ! [X2: numeral_num0] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_6449127158079674652l_num0 @ G @ top_to1996260823553986621t_unit )
= top_to3689904424835650196l_num0 ) ) ).
% surjI
thf(fact_698_surjI,axiom,
! [G: product_unit > literal,F: literal > product_unit] :
( ! [X2: literal] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_5876984745897992460iteral @ G @ top_to1996260823553986621t_unit )
= top_top_set_literal ) ) ).
% surjI
thf(fact_699_surjI,axiom,
! [G: product_unit > nat,F: nat > product_unit] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ) ).
% surjI
thf(fact_700_surjI,axiom,
! [G: numeral_num1 > product_unit,F: product_unit > numeral_num1] :
( ! [X2: product_unit] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_739801482541841261t_unit @ G @ top_to3689904429138878997l_num1 )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_701_surjI,axiom,
! [G: numeral_num1 > numeral_num1,F: numeral_num1 > numeral_num1] :
( ! [X2: numeral_num1] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_6783865936125884741l_num1 @ G @ top_to3689904429138878997l_num1 )
= top_to3689904429138878997l_num1 ) ) ).
% surjI
thf(fact_702_surjI,axiom,
! [G: numeral_num1 > numeral_num0,F: numeral_num0 > numeral_num1] :
( ! [X2: numeral_num0] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_6783865936125884740l_num0 @ G @ top_to3689904429138878997l_num1 )
= top_to3689904424835650196l_num0 ) ) ).
% surjI
thf(fact_703_surjI,axiom,
! [G: numeral_num1 > literal,F: literal > numeral_num1] :
( ! [X2: literal] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_5852747068178070836iteral @ G @ top_to3689904429138878997l_num1 )
= top_top_set_literal ) ) ).
% surjI
thf(fact_704_surjI,axiom,
! [G: numeral_num1 > nat,F: nat > numeral_num1] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_809646449033931376m1_nat @ G @ top_to3689904429138878997l_num1 )
= top_top_set_nat ) ) ).
% surjI
thf(fact_705_surj__def,axiom,
! [F: product_unit > product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
= ( ! [Y3: product_unit] :
? [X3: product_unit] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_706_surj__def,axiom,
! [F: product_unit > numeral_num1] :
( ( ( image_6449127158079674653l_num1 @ F @ top_to1996260823553986621t_unit )
= top_to3689904429138878997l_num1 )
= ( ! [Y3: numeral_num1] :
? [X3: product_unit] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_707_surj__def,axiom,
! [F: product_unit > numeral_num0] :
( ( ( image_6449127158079674652l_num0 @ F @ top_to1996260823553986621t_unit )
= top_to3689904424835650196l_num0 )
= ( ! [Y3: numeral_num0] :
? [X3: product_unit] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_708_surj__def,axiom,
! [F: product_unit > literal] :
( ( ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal )
= ( ! [Y3: literal] :
? [X3: product_unit] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_709_surj__def,axiom,
! [F: product_unit > nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X3: product_unit] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_710_surj__def,axiom,
! [F: numeral_num1 > product_unit] :
( ( ( image_739801482541841261t_unit @ F @ top_to3689904429138878997l_num1 )
= top_to1996260823553986621t_unit )
= ( ! [Y3: product_unit] :
? [X3: numeral_num1] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_711_surj__def,axiom,
! [F: numeral_num1 > numeral_num1] :
( ( ( image_6783865936125884741l_num1 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904429138878997l_num1 )
= ( ! [Y3: numeral_num1] :
? [X3: numeral_num1] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_712_surj__def,axiom,
! [F: numeral_num1 > numeral_num0] :
( ( ( image_6783865936125884740l_num0 @ F @ top_to3689904429138878997l_num1 )
= top_to3689904424835650196l_num0 )
= ( ! [Y3: numeral_num0] :
? [X3: numeral_num1] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_713_surj__def,axiom,
! [F: numeral_num1 > literal] :
( ( ( image_5852747068178070836iteral @ F @ top_to3689904429138878997l_num1 )
= top_top_set_literal )
= ( ! [Y3: literal] :
? [X3: numeral_num1] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_714_surj__def,axiom,
! [F: numeral_num1 > nat] :
( ( ( image_809646449033931376m1_nat @ F @ top_to3689904429138878997l_num1 )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X3: numeral_num1] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_715_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_716_inverse__zero__imp__zero,axiom,
! [A2: real] :
( ( ( inverse_inverse_real @ A2 )
= zero_zero_real )
=> ( A2 = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_717_nonzero__inverse__eq__imp__eq,axiom,
! [A2: real,B: real] :
( ( ( inverse_inverse_real @ A2 )
= ( inverse_inverse_real @ B ) )
=> ( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A2 = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_718_nonzero__inverse__inverse__eq,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A2 ) )
= A2 ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_719_nonzero__imp__inverse__nonzero,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ A2 )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_720_inverse__less__imp__less,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ B @ A2 ) ) ) ).
% inverse_less_imp_less
thf(fact_721_less__imp__inverse__less,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% less_imp_inverse_less
thf(fact_722_inverse__less__imp__less__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ B @ A2 ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_723_less__imp__inverse__less__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_724_inverse__negative__imp__negative,axiom,
! [A2: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real )
=> ( ( A2 != zero_zero_real )
=> ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_725_inverse__positive__imp__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) )
=> ( ( A2 != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A2 ) ) ) ).
% inverse_positive_imp_positive
thf(fact_726_negative__imp__inverse__negative,axiom,
! [A2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A2 ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_727_positive__imp__inverse__positive,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A2 ) ) ) ).
% positive_imp_inverse_positive
thf(fact_728_nonzero__inverse__minus__eq,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A2 ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_729_real__arch__invD,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N4: nat] :
( ( N4 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ).
% real_arch_invD
thf(fact_730_card__nat,axiom,
( ( finite_card_nat @ top_top_set_nat )
= zero_zero_nat ) ).
% card_nat
thf(fact_731_finite__option__UNIV,axiom,
( ( finite6732403688824079472at_nat @ top_to3251141154256563319at_nat )
= ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ).
% finite_option_UNIV
thf(fact_732_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_733_finite__option__UNIV,axiom,
( ( finite3139260975159805780l_num1 @ top_to4428395536652758875l_num1 )
= ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ).
% finite_option_UNIV
thf(fact_734_finite__option__UNIV,axiom,
( ( finite3139260970856576979l_num0 @ top_to4357361496606412890l_num0 )
= ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ).
% finite_option_UNIV
thf(fact_735_finite__option__UNIV,axiom,
( ( finite5071707688241699267iteral @ top_to8248435444729185354iteral )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ).
% finite_option_UNIV
thf(fact_736_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_737_zero__less__card__finite,axiom,
ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ).
% zero_less_card_finite
thf(fact_738_zero__less__card__finite,axiom,
ord_less_nat @ zero_zero_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ).
% zero_less_card_finite
thf(fact_739_card__literal,axiom,
( ( finite_card_literal @ top_top_set_literal )
= zero_zero_nat ) ).
% card_literal
thf(fact_740_is__list__UNIV__iff,axiom,
( is_lis495297779641082652t_unit
= ( ^ [Xs2: list_Product_unit] :
( ( set_Product_unit2 @ Xs2 )
= top_to1996260823553986621t_unit ) ) ) ).
% is_list_UNIV_iff
thf(fact_741_is__list__UNIV__iff,axiom,
( is_lis6539362233225126132l_num1
= ( ^ [Xs2: list_Numeral_num1] :
( ( set_Numeral_num12 @ Xs2 )
= top_to3689904429138878997l_num1 ) ) ) ).
% is_list_UNIV_iff
thf(fact_742_is__list__UNIV__iff,axiom,
( is_lis6539362233225126131l_num0
= ( ^ [Xs2: list_Numeral_num0] :
( ( set_Numeral_num02 @ Xs2 )
= top_to3689904424835650196l_num0 ) ) ) ).
% is_list_UNIV_iff
thf(fact_743_is__list__UNIV__iff,axiom,
( is_list_UNIV_literal
= ( ^ [Xs2: list_literal] :
( ( set_literal2 @ Xs2 )
= top_top_set_literal ) ) ) ).
% is_list_UNIV_iff
thf(fact_744_is__list__UNIV__iff,axiom,
( is_list_UNIV_nat
= ( ^ [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= top_top_set_nat ) ) ) ).
% is_list_UNIV_iff
thf(fact_745_card__num0,axiom,
( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
= zero_zero_nat ) ).
% card_num0
thf(fact_746_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ N )
=> ( ord_less_nat @ X2 @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_747_infinite__literal,axiom,
~ ( finite5847741373460823677iteral @ top_top_set_literal ) ).
% infinite_literal
thf(fact_748_infinite__num0,axiom,
~ ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ).
% infinite_num0
thf(fact_749_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_750_fps__inverse__zero_H,axiom,
( ( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ zero_z7760665558314615101s_real )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_zero'
thf(fact_751_finite__UNIV__fun,axiom,
( ( finite6665322292308856380t_unit @ top_to658657236369668235t_unit )
= ( ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) )
| ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_752_finite__UNIV__fun,axiom,
( ( finite8358965897893748756l_num1 @ top_to2396192563470163555l_num1 )
= ( ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) )
| ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_753_finite__UNIV__fun,axiom,
( ( finite8358965893590519955l_num0 @ top_to2325158523423817570l_num0 )
= ( ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) )
| ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_754_finite__UNIV__fun,axiom,
( ( finite5086960133205840515iteral @ top_to2397521590294773586iteral )
= ( ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite5847741373460823677iteral @ top_top_set_literal ) )
| ( ( finite_card_literal @ top_top_set_literal )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_755_finite__UNIV__fun,axiom,
( ( finite4332129999517832055it_nat @ top_to5871476398150932990it_nat )
= ( ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) )
| ( ( finite_card_nat @ top_top_set_nat )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_756_finite__UNIV__fun,axiom,
( ( finite7000061070355066468t_unit @ top_to5847030703470838963t_unit )
= ( ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) )
| ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_757_finite__UNIV__fun,axiom,
( ( finite8693704675939958844l_num1 @ top_to7584566030571334283l_num1 )
= ( ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) )
| ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_758_finite__UNIV__fun,axiom,
( ( finite8693704671636730043l_num0 @ top_to7513531990524988298l_num0 )
= ( ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) )
| ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_759_finite__UNIV__fun,axiom,
( ( finite5062722455485918891iteral @ top_to186538159804303226iteral )
= ( ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite5847741373460823677iteral @ top_top_set_literal ) )
| ( ( finite_card_literal @ top_top_set_literal )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_760_finite__UNIV__fun,axiom,
( ( finite4266206433997009231m1_nat @ top_to5355779118554991574m1_nat )
= ( ( ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 )
& ( finite_finite_nat @ top_top_set_nat ) )
| ( ( finite_card_nat @ top_top_set_nat )
= one_one_nat ) ) ) ).
% finite_UNIV_fun
thf(fact_761_card__prod,axiom,
( ( finite5501798543469036431t_unit @ top_to1835807148980544151t_unit )
= ( times_times_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% card_prod
thf(fact_762_card__prod,axiom,
( ( finite7195442149053928807l_num1 @ top_to3573342476081039471l_num1 )
= ( times_times_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% card_prod
thf(fact_763_card__prod,axiom,
( ( finite7195442144750700006l_num0 @ top_to3502308436034693486l_num0 )
= ( times_times_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% card_prod
thf(fact_764_card__prod,axiom,
( ( finite4150021457272484950iteral @ top_to1987565607987313502iteral )
= ( times_times_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite_card_literal @ top_top_set_literal ) ) ) ).
% card_prod
thf(fact_765_card__prod,axiom,
( ( finite8220318816156418148it_nat @ top_to5974110478112770290it_nat )
= ( times_times_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite_card_nat @ top_top_set_nat ) ) ) ).
% card_prod
thf(fact_766_card__prod,axiom,
( ( finite5836537321515246519t_unit @ top_to7024180616081714879t_unit )
= ( times_times_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% card_prod
thf(fact_767_card__prod,axiom,
( ( finite7530180927100138895l_num1 @ top_to8761715943182210199l_num1 )
= ( times_times_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% card_prod
thf(fact_768_card__prod,axiom,
( ( finite7530180922796910094l_num0 @ top_to8690681903135864214l_num0 )
= ( times_times_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) @ ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% card_prod
thf(fact_769_card__prod,axiom,
( ( finite4125783779552563326iteral @ top_to8999954214351618950iteral )
= ( times_times_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) @ ( finite_card_literal @ top_top_set_literal ) ) ) ).
% card_prod
thf(fact_770_card__prod,axiom,
( ( finite8154395250635595324m1_nat @ top_to5458413198516828874m1_nat )
= ( times_times_nat @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) @ ( finite_card_nat @ top_top_set_nat ) ) ) ).
% card_prod
thf(fact_771_finite__enumerate__step,axiom,
! [S2: set_Numeral_num1,N2: nat] :
( ( finite1111429032697314574l_num1 @ S2 )
=> ( ( ord_less_nat @ ( suc @ N2 ) @ ( finite6454714172617411597l_num1 @ S2 ) )
=> ( ord_le6405328735288452753l_num1 @ ( infini4751236147369608586l_num1 @ S2 @ N2 ) @ ( infini4751236147369608586l_num1 @ S2 @ ( suc @ N2 ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_772_finite__enumerate__step,axiom,
! [S2: set_nat,N2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ ( suc @ N2 ) @ ( finite_card_nat @ S2 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ N2 ) @ ( infini8530281810654367211te_nat @ S2 @ ( suc @ N2 ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_773_nat_Osimps_I1_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.simps(1)
thf(fact_774_old_Onat_Osimps_I1_J,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.simps(1)
thf(fact_775_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_776_card__num1,axiom,
( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
= one_one_nat ) ).
% card_num1
thf(fact_777_arithmetic__simps_I62_J,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% arithmetic_simps(62)
thf(fact_778_arithmetic__simps_I62_J,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% arithmetic_simps(62)
thf(fact_779_arithmetic__simps_I62_J,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% arithmetic_simps(62)
thf(fact_780_arithmetic__simps_I63_J,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% arithmetic_simps(63)
thf(fact_781_arithmetic__simps_I63_J,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% arithmetic_simps(63)
thf(fact_782_arithmetic__simps_I63_J,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% arithmetic_simps(63)
thf(fact_783_more__arith__simps_I6_J,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A2 @ one_on2969667320475766781nnreal )
= A2 ) ).
% more_arith_simps(6)
thf(fact_784_more__arith__simps_I6_J,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% more_arith_simps(6)
thf(fact_785_more__arith__simps_I6_J,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% more_arith_simps(6)
thf(fact_786_more__arith__simps_I6_J,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% more_arith_simps(6)
thf(fact_787_more__arith__simps_I5_J,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A2 )
= A2 ) ).
% more_arith_simps(5)
thf(fact_788_more__arith__simps_I5_J,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% more_arith_simps(5)
thf(fact_789_more__arith__simps_I5_J,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% more_arith_simps(5)
thf(fact_790_more__arith__simps_I5_J,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% more_arith_simps(5)
thf(fact_791_more__arith__simps_I8_J,axiom,
! [A2: int,B: int] :
( ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% more_arith_simps(8)
thf(fact_792_more__arith__simps_I8_J,axiom,
! [A2: real,B: real] :
( ( times_times_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ B ) ) ) ).
% more_arith_simps(8)
thf(fact_793_more__arith__simps_I7_J,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% more_arith_simps(7)
thf(fact_794_more__arith__simps_I7_J,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ B ) ) ) ).
% more_arith_simps(7)
thf(fact_795_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_796_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_797_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_798_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_799_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_800_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_801_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_802_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_803_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_804_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_805_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_806_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_807_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_808_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_809_of__nat__1,axiom,
( ( semiri6283507881447550617nnreal @ one_one_nat )
= one_on2969667320475766781nnreal ) ).
% of_nat_1
thf(fact_810_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_811_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_812_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_813_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_on2969667320475766781nnreal
= ( semiri6283507881447550617nnreal @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_814_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_815_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_816_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_817_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri6283507881447550617nnreal @ N2 )
= one_on2969667320475766781nnreal )
= ( N2 = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_818_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_819_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_820_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_821_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_822_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_823_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_824_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_825_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_826_mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_827_one__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_828_mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel1
thf(fact_829_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_830_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_831_right__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ A2 @ ( inverse_inverse_real @ A2 ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_832_left__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A2 ) @ A2 )
= one_one_real ) ) ).
% left_inverse
thf(fact_833_negative__zless,axiom,
! [N2: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_834_card__option,axiom,
( ( finite4356775664083950075t_unit @ top_to2690860209552263555t_unit )
= ( suc @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% card_option
thf(fact_835_card__option,axiom,
( ( finite6050419269668842451l_num1 @ top_to4428395536652758875l_num1 )
= ( suc @ ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% card_option
thf(fact_836_field__class_Ofield__inverse,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A2 ) @ A2 )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_837_n__less__n__mult__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_838_n__less__m__mult__n,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_839_one__less__mult,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_840_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_841_more__arith__simps_I11_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C2 )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C2 ) ) ) ).
% more_arith_simps(11)
thf(fact_842_more__arith__simps_I11_J,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C2 )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C2 ) ) ) ).
% more_arith_simps(11)
thf(fact_843_more__arith__simps_I11_J,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C2 )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C2 ) ) ) ).
% more_arith_simps(11)
thf(fact_844_Groups_Omult__ac_I3_J,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C2 ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C2 ) ) ) ).
% Groups.mult_ac(3)
thf(fact_845_Groups_Omult__ac_I3_J,axiom,
! [B: int,A2: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C2 ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C2 ) ) ) ).
% Groups.mult_ac(3)
thf(fact_846_Groups_Omult__ac_I3_J,axiom,
! [B: real,A2: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A2 @ C2 ) )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C2 ) ) ) ).
% Groups.mult_ac(3)
thf(fact_847_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A6: nat,B5: nat] : ( times_times_nat @ B5 @ A6 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_848_Groups_Omult__ac_I2_J,axiom,
( times_times_int
= ( ^ [A6: int,B5: int] : ( times_times_int @ B5 @ A6 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_849_Groups_Omult__ac_I2_J,axiom,
( times_times_real
= ( ^ [A6: real,B5: real] : ( times_times_real @ B5 @ A6 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_850_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C2 )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_851_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C2 )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_852_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C2 )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_853_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_854_comm__monoid__mult__class_Omult__1,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ one_on2969667320475766781nnreal @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_855_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_856_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_857_comm__monoid__mult__class_Omult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_858_mult_Ocomm__neutral,axiom,
! [A2: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A2 @ one_on2969667320475766781nnreal )
= A2 ) ).
% mult.comm_neutral
thf(fact_859_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_860_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_861_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_862_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_863_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_864_Suc__n__not__n,axiom,
! [N2: nat] :
( ( suc @ N2 )
!= N2 ) ).
% Suc_n_not_n
thf(fact_865_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_866_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_867_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_868_one__reorient,axiom,
! [X: extend8495563244428889912nnreal] :
( ( one_on2969667320475766781nnreal = X )
= ( X = one_on2969667320475766781nnreal ) ) ).
% one_reorient
thf(fact_869_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M2 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_870_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_871_mult__delta__right,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_872_mult__delta__right,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_873_mult__delta__right,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_874_mult__delta__left,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_875_mult__delta__left,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_876_mult__delta__left,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_877_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_878_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_879_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_880_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_881_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_882_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_883_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_884_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_885_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_886_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_887_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_888_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_889_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_890_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_891_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_892_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_893_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_894_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_895_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_896_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_897_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_898_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_899_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_900_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_901_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_902_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_903_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_904_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_905_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_906_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_907_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_908_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_909_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_910_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_911_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_912_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_913_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_914_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_915_eq__numeral__extra_I1_J,axiom,
zero_z7100319975126383169nnreal != one_on2969667320475766781nnreal ).
% eq_numeral_extra(1)
thf(fact_916_eq__numeral__extra_I1_J,axiom,
zero_zero_nat != one_one_nat ).
% eq_numeral_extra(1)
thf(fact_917_eq__numeral__extra_I1_J,axiom,
zero_zero_real != one_one_real ).
% eq_numeral_extra(1)
thf(fact_918_eq__numeral__extra_I1_J,axiom,
zero_zero_int != one_one_int ).
% eq_numeral_extra(1)
thf(fact_919_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_920_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_921_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_922_less__numeral__extra_I4_J,axiom,
~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ one_on2969667320475766781nnreal ) ).
% less_numeral_extra(4)
thf(fact_923_rel__simps_I91_J,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% rel_simps(91)
thf(fact_924_rel__simps_I91_J,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% rel_simps(91)
thf(fact_925_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_926_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_927_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_928_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_929_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_930_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_931_lift__Suc__mono__less,axiom,
! [F: nat > extend8495563244428889912nnreal,N2: nat,N7: nat] :
( ! [N4: nat] : ( ord_le7381754540660121996nnreal @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N7 )
=> ( ord_le7381754540660121996nnreal @ ( F @ N2 ) @ ( F @ N7 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_932_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_933_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_934_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_935_lift__Suc__mono__less__iff,axiom,
! [F: nat > extend8495563244428889912nnreal,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_le7381754540660121996nnreal @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_le7381754540660121996nnreal @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_936_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_937_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_938_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_939_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_940_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_941_nonzero__inverse__mult__distrib,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A2 @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A2 ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_942_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_943_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_944_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_945_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_946_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_947_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_948_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_949_less__numeral__extra_I1_J,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% less_numeral_extra(1)
thf(fact_950_less__numeral__extra_I2_J,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% less_numeral_extra(2)
thf(fact_951_less__numeral__extra_I2_J,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% less_numeral_extra(2)
thf(fact_952_less__numeral__extra_I2_J,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% less_numeral_extra(2)
thf(fact_953_less__numeral__extra_I2_J,axiom,
~ ( ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ zero_z7100319975126383169nnreal ) ).
% less_numeral_extra(2)
thf(fact_954_verit__comp__simplify_I28_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% verit_comp_simplify(28)
thf(fact_955_verit__comp__simplify_I28_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(28)
thf(fact_956_verit__comp__simplify_I28_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% verit_comp_simplify(28)
thf(fact_957_verit__comp__simplify_I28_J,axiom,
ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ one_on2969667320475766781nnreal ).
% verit_comp_simplify(28)
thf(fact_958_semiring__norm_I156_J,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% semiring_norm(156)
thf(fact_959_semiring__norm_I156_J,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% semiring_norm(156)
thf(fact_960_mult__inverse__of__nat__commute,axiom,
! [Xa2: nat,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa2 ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa2 ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_961_semiring__norm_I132_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% semiring_norm(132)
thf(fact_962_semiring__norm_I132_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% semiring_norm(132)
thf(fact_963_semiring__norm_I134_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% semiring_norm(134)
thf(fact_964_semiring__norm_I134_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% semiring_norm(134)
thf(fact_965_int__cases,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% int_cases
thf(fact_966_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
=> ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_967_CARD__1,axiom,
( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
= one_one_nat ) ).
% CARD_1
thf(fact_968_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_969_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N4: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_970_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_971_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_972_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_973_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_974_one__less__inverse__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_975_one__less__inverse,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ A2 @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A2 ) ) ) ) ).
% one_less_inverse
thf(fact_976_enumerate__step,axiom,
! [S2: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ N2 ) @ ( infini8530281810654367211te_nat @ S2 @ ( suc @ N2 ) ) ) ) ).
% enumerate_step
thf(fact_977_reals__Archimedean,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N4: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ X ) ) ).
% reals_Archimedean
thf(fact_978_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D3: real,E: real] :
( ( ord_less_real @ D3 @ E )
=> ( ( P @ D3 )
=> ( P @ E ) ) )
=> ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_979_negative__zless__0,axiom,
! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_980_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N4: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% negD
thf(fact_981_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= one_one_nat )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_982_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri6283507881447550617nnreal @ X )
= one_on2969667320475766781nnreal )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_983_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= one_one_real )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_984_Totient_Oof__nat__eq__1__iff,axiom,
! [X: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= one_one_int )
= ( X = one_one_nat ) ) ).
% Totient.of_nat_eq_1_iff
thf(fact_985_finite__fun__UNIVD1,axiom,
( ( finite6665322292308856380t_unit @ top_to658657236369668235t_unit )
=> ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
!= ( suc @ zero_zero_nat ) )
=> ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_fun_UNIVD1
thf(fact_986_finite__fun__UNIVD1,axiom,
( ( finite7000061070355066468t_unit @ top_to5847030703470838963t_unit )
=> ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
!= ( suc @ zero_zero_nat ) )
=> ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_fun_UNIVD1
thf(fact_987_finite__fun__UNIVD1,axiom,
( ( finite3049169434736477989t_unit @ top_to8932786945460833524t_unit )
=> ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
!= ( suc @ zero_zero_nat ) )
=> ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_fun_UNIVD1
thf(fact_988_finite__fun__UNIVD1,axiom,
( ( finite1482354282328208437t_unit @ top_to9006910351538956292t_unit )
=> ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
!= ( suc @ zero_zero_nat ) )
=> ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_fun_UNIVD1
thf(fact_989_finite__fun__UNIVD1,axiom,
( ( finite4257689694021357085t_unit @ top_to8442108875268333988t_unit )
=> ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_990_finite__fun__UNIVD1,axiom,
( ( finite8358965897893748756l_num1 @ top_to2396192563470163555l_num1 )
=> ( ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
!= ( suc @ zero_zero_nat ) )
=> ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_fun_UNIVD1
thf(fact_991_finite__fun__UNIVD1,axiom,
( ( finite8693704675939958844l_num1 @ top_to7584566030571334283l_num1 )
=> ( ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
!= ( suc @ zero_zero_nat ) )
=> ( finite1111429032697314574l_num1 @ top_to3689904429138878997l_num1 ) ) ) ).
% finite_fun_UNIVD1
thf(fact_992_finite__fun__UNIVD1,axiom,
( ( finite4742813040321370365l_num1 @ top_to1446950235706553036l_num1 )
=> ( ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
!= ( suc @ zero_zero_nat ) )
=> ( finite1111429032697314573l_num0 @ top_to3689904424835650196l_num0 ) ) ) ).
% finite_fun_UNIVD1
thf(fact_993_finite__fun__UNIVD1,axiom,
( ( finite3175997887913100813l_num1 @ top_to1521073641784675804l_num1 )
=> ( ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
!= ( suc @ zero_zero_nat ) )
=> ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_fun_UNIVD1
thf(fact_994_finite__fun__UNIVD1,axiom,
( ( finite5951333299606249461l_num1 @ top_to956272165514053500l_num1 )
=> ( ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_995_mult__cancel__right2,axiom,
! [A2: real,C2: real] :
( ( ( times_times_real @ A2 @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_996_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_997_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_998_int__ops_I7_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_999_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1000_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N4: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1001_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1002_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1003_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_1004_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1005_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1006_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1007_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1008_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1009_mult__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ M2 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1010_plus__nat_Osimps_I1_J,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.simps(1)
thf(fact_1011_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1012_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1013_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1014_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1015_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1016_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1017_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1018_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1019_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1020_trans__less__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1021_trans__less__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1022_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1023_nat__distrib_I1_J,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% nat_distrib(1)
thf(fact_1024_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1025_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1026_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1027_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1028_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1029_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1030_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1031_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1032_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q2: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1033_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1034_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1035_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1036_times__nat_Osimps_I2_J,axiom,
! [M2: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).
% times_nat.simps(2)
thf(fact_1037_real__add__minus__iff,axiom,
! [X: real,A2: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X = A2 ) ) ).
% real_add_minus_iff
thf(fact_1038_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1039_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1040_zadd__int__left,axiom,
! [M2: nat,N2: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1041_int__ops_I5_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1042_int__plus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1043_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 ) ) ) ).
% zless_add1_eq
thf(fact_1044_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( 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 @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1045_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1046_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1047_odd__less__0__iff,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 ) ) ).
% odd_less_0_iff
thf(fact_1048_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1049_fib_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [N4: nat] :
( X
!= ( suc @ ( suc @ N4 ) ) ) ) ) ).
% fib.cases
thf(fact_1050_square__bound__lemma,axiom,
! [X: real] : ( ord_less_real @ X @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X ) @ ( plus_plus_real @ one_one_real @ X ) ) ) ).
% square_bound_lemma
thf(fact_1051_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1052_encode__unary__nat_Ocases,axiom,
! [X: nat] :
( ! [L2: nat] :
( X
!= ( suc @ L2 ) )
=> ( X = zero_zero_nat ) ) ).
% encode_unary_nat.cases
thf(fact_1053_nth__item_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% nth_item.cases
thf(fact_1054_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power_nat @ X @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1055_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1056_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1057_power__top__ennreal,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_6007165696250533058nnreal @ top_to1496364449551166952nnreal @ N2 )
= top_to1496364449551166952nnreal ) ) ).
% power_top_ennreal
thf(fact_1058_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1059_power__less__top__ennreal,axiom,
! [X: extend8495563244428889912nnreal,N2: nat] :
( ( ord_le7381754540660121996nnreal @ ( power_6007165696250533058nnreal @ X @ N2 ) @ top_to1496364449551166952nnreal )
= ( ( ord_le7381754540660121996nnreal @ X @ top_to1496364449551166952nnreal )
| ( N2 = zero_zero_nat ) ) ) ).
% power_less_top_ennreal
thf(fact_1060_nat__power__less__imp__less,axiom,
! [I2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1061_power__eq__top__ennreal__iff,axiom,
! [X: extend8495563244428889912nnreal,N2: nat] :
( ( ( power_6007165696250533058nnreal @ X @ N2 )
= top_to1496364449551166952nnreal )
= ( ( X = top_to1496364449551166952nnreal )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_top_ennreal_iff
thf(fact_1062_int__ops_I8_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1063_power__eq__top__ennreal,axiom,
! [X: extend8495563244428889912nnreal,N2: nat] :
( ( ( power_6007165696250533058nnreal @ X @ N2 )
= top_to1496364449551166952nnreal )
= ( ( N2 != zero_zero_nat )
& ( X = top_to1496364449551166952nnreal ) ) ) ).
% power_eq_top_ennreal
thf(fact_1064_top__power__ennreal,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ top_to1496364449551166952nnreal @ N2 )
= one_on2969667320475766781nnreal ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_6007165696250533058nnreal @ top_to1496364449551166952nnreal @ N2 )
= top_to1496364449551166952nnreal ) ) ) ).
% top_power_ennreal
thf(fact_1065_reals__power__lt__ex,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K2 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1066_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N4: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N4 ) ) ) ).
% real_arch_pow
thf(fact_1067_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_1068_realpow__pos__nth2,axiom,
! [A2: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ ( suc @ N2 ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_1069_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1070_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_nat2 @ m @ ( image_nat_nat @ ( count_list_nat @ as ) @ ( set_nat2 @ as ) ) ).
%------------------------------------------------------------------------------