TPTP Problem File: SLH0886^1.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 : Median_Method/0000_Median/prob_00142_005185__14673318_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1417 ( 479 unt; 146 typ; 0 def)
% Number of atoms : 3953 (1066 equ; 0 cnn)
% Maximal formula atoms : 26 ( 3 avg)
% Number of connectives : 11564 ( 431 ~; 135 |; 252 &;8853 @)
% ( 0 <=>;1893 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 621 ( 621 >; 0 *; 0 +; 0 <<)
% Number of symbols : 141 ( 138 usr; 15 con; 0-3 aty)
% Number of variables : 3887 ( 224 ^;3506 !; 157 ?;3887 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:43:43.660
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (138)
thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: real > int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Int__Oint,type,
condit7933062003635074389dd_int: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Nat__Onat,type,
condit7935552474144124665dd_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Real__Oreal,type,
condit1497324847667023189d_real: ( real > real > $o ) > ( real > real > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001tf__b,type,
condit4103000493307248662_bdd_b: ( b > b > $o ) > ( b > b > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Int__Oint,type,
condit4011256317322997495dd_int: ( int > int > $o ) > set_int > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Nat__Onat,type,
condit4013746787832047771dd_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Real__Oreal,type,
condit7290043087096337911d_real: ( real > real > $o ) > set_real > $o ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
sup_sup_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
sup_sup_set_real: set_real > set_real > set_real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
sup_sup_set_b: set_b > set_b > set_b ).
thf(sy_c_Median_Odown__ray_001t__Int__Oint,type,
down_ray_int: set_int > $o ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001t__Real__Oreal,type,
down_ray_real: set_real > $o ).
thf(sy_c_Median_Odown__ray_001tf__b,type,
down_ray_b: set_b > $o ).
thf(sy_c_Median_Ointerval_001t__Int__Oint,type,
interval_int: set_int > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001t__Real__Oreal,type,
interval_real: set_real > $o ).
thf(sy_c_Median_Ointerval_001tf__b,type,
interval_b: set_b > $o ).
thf(sy_c_Median_Osort__map_001tf__b,type,
sort_map_b: ( nat > b ) > nat > nat > b ).
thf(sy_c_Median_Oup__ray_001t__Int__Oint,type,
up_ray_int: set_int > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001t__Real__Oreal,type,
up_ray_real: set_real > $o ).
thf(sy_c_Median_Oup__ray_001tf__b,type,
up_ray_b: set_b > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__b_J,type,
ord_less_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__b,type,
ord_less_b: b > b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Int__Oint_J,type,
ord_less_eq_o_int: ( $o > int ) > ( $o > int ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J,type,
ord_less_eq_o_real: ( $o > real ) > ( $o > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__b_J,type,
ord_less_eq_o_b: ( $o > b ) > ( $o > b ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__b,type,
ord_less_eq_b: b > b > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__b,type,
order_Greatest_b: ( b > $o ) > b ).
thf(sy_c_Orderings_Oordering_001t__Int__Oint,type,
ordering_int: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
ordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001t__Real__Oreal,type,
ordering_real: ( real > real > $o ) > ( real > real > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001tf__b,type,
ordering_b: ( b > b > $o ) > ( b > b > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Int__Oint,type,
partia6820327588127286646ng_int: ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Nat__Onat,type,
partia6822818058636336922ng_nat: ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Real__Oreal,type,
partia7612893088228670710g_real: ( real > real > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001tf__b,type,
partia125584492769400373ring_b: ( b > b > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Int__Oint,type,
preordering_int: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Nat__Onat,type,
preordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Real__Oreal,type,
preordering_real: ( real > real > $o ) > ( real > real > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001tf__b,type,
preordering_b: ( b > b > $o ) > ( b > b > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__b,type,
set_or672772299803893940Most_b: b > b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__b,type,
set_or5139330845457685136Than_b: b > b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Int__Oint,type,
set_ord_atLeast_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001tf__b,type,
set_ord_atLeast_b: b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__b,type,
set_ord_atMost_b: b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Real__Oreal,type,
set_or2392270231875598684t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__b,type,
set_or4472690218693186639Most_b: b > b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001tf__b,type,
set_or5939364468397584555Than_b: b > b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
set_or1207661135979820486an_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001tf__b,type,
set_or8632414552788122085Than_b: b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001tf__b,type,
set_ord_lessThan_b: b > set_b ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_f,type,
f: nat > b ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1263)
thf(fact_0_order__refl,axiom,
! [X: b] : ( ord_less_eq_b @ X @ X ) ).
% order_refl
thf(fact_1_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_2_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_3_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_4_dual__order_Orefl,axiom,
! [A: b] : ( ord_less_eq_b @ A @ A ) ).
% dual_order.refl
thf(fact_5_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_6_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_8_minf_I8_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ~ ( ord_less_eq_b @ T @ X2 ) ) ).
% minf(8)
thf(fact_9_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% minf(8)
thf(fact_10_minf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_11_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_12_minf_I6_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( ord_less_eq_b @ X2 @ T ) ) ).
% minf(6)
thf(fact_13_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ T ) ) ).
% minf(6)
thf(fact_14_minf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_eq_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_15_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_16_pinf_I8_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( ord_less_eq_b @ T @ X2 ) ) ).
% pinf(8)
thf(fact_17_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ T @ X2 ) ) ).
% pinf(8)
thf(fact_18_pinf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_eq_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_19_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_20_pinf_I6_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ~ ( ord_less_eq_b @ X2 @ T ) ) ).
% pinf(6)
thf(fact_21_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_22_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_23_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_24_verit__comp__simplify1_I3_J,axiom,
! [B: b,A2: b] :
( ( ~ ( ord_less_eq_b @ B @ A2 ) )
= ( ord_less_b @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_25_verit__comp__simplify1_I3_J,axiom,
! [B: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B @ A2 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_26_verit__comp__simplify1_I3_J,axiom,
! [B: real,A2: real] :
( ( ~ ( ord_less_eq_real @ B @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_27_verit__comp__simplify1_I3_J,axiom,
! [B: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_28_leD,axiom,
! [Y: b,X: b] :
( ( ord_less_eq_b @ Y @ X )
=> ~ ( ord_less_b @ X @ Y ) ) ).
% leD
thf(fact_29_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_30_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_31_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_32_leI,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_b @ X @ Y )
=> ( ord_less_eq_b @ Y @ X ) ) ).
% leI
thf(fact_33_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_34_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_35_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_36_nless__le,axiom,
! [A: b,B2: b] :
( ( ~ ( ord_less_b @ A @ B2 ) )
= ( ~ ( ord_less_eq_b @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_37_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_38_nless__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_real @ A @ B2 ) )
= ( ~ ( ord_less_eq_real @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_39_nless__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_int @ A @ B2 ) )
= ( ~ ( ord_less_eq_int @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_40_antisym__conv1,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_b @ X @ Y )
=> ( ( ord_less_eq_b @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_41_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_42_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_43_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_44_antisym__conv2,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ~ ( ord_less_b @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_45_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_46_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_47_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_48_order__antisym__conv,axiom,
! [Y: b,X: b] :
( ( ord_less_eq_b @ Y @ X )
=> ( ( ord_less_eq_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_49_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_50_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_51_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_52_linorder__le__cases,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_eq_b @ X @ Y )
=> ( ord_less_eq_b @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_53_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_54_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_55_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_56_ord__le__eq__subst,axiom,
! [A: b,B2: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_57_ord__le__eq__subst,axiom,
! [A: b,B2: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_ord__le__eq__subst,axiom,
! [A: b,B2: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_59_ord__le__eq__subst,axiom,
! [A: b,B2: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_62_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_63_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_64_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_65_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_66_ord__eq__le__subst,axiom,
! [A: b,F: b > b,B2: b,C: b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_67_ord__eq__le__subst,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_68_ord__eq__le__subst,axiom,
! [A: real,F: b > real,B2: b,C: b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_69_ord__eq__le__subst,axiom,
! [A: int,F: b > int,B2: b,C: b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_70_ord__eq__le__subst,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_71_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_72_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_73_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_ord__eq__le__subst,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_75_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_76_linorder__linear,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
| ( ord_less_eq_b @ Y @ X ) ) ).
% linorder_linear
thf(fact_77_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_78_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_79_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_80_verit__la__disequality,axiom,
! [A: b,B2: b] :
( ( A = B2 )
| ~ ( ord_less_eq_b @ A @ B2 )
| ~ ( ord_less_eq_b @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_81_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_82_verit__la__disequality,axiom,
! [A: real,B2: real] :
( ( A = B2 )
| ~ ( ord_less_eq_real @ A @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_83_verit__la__disequality,axiom,
! [A: int,B2: int] :
( ( A = B2 )
| ~ ( ord_less_eq_int @ A @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_84_order__eq__refl,axiom,
! [X: b,Y: b] :
( ( X = Y )
=> ( ord_less_eq_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_85_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_86_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_87_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_88_order__subst2,axiom,
! [A: b,B2: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A: b,B2: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A: b,B2: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A: b,B2: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_95_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_96_order__subst2,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_97_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_98_order__subst1,axiom,
! [A: b,F: b > b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_99_order__subst1,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_100_order__subst1,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_order__subst1,axiom,
! [A: b,F: int > b,B2: int,C: int] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_102_order__subst1,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_103_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_105_order__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_106_order__subst1,axiom,
! [A: real,F: b > real,B2: b,C: b] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_107_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_108_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: b,Z2: b] : ( Y3 = Z2 ) )
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_109_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_110_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_111_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_112_antisym,axiom,
! [A: b,B2: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_b @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_113_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_114_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_115_antisym,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_116_dual__order_Otrans,axiom,
! [B2: b,A: b,C: b] :
( ( ord_less_eq_b @ B2 @ A )
=> ( ( ord_less_eq_b @ C @ B2 )
=> ( ord_less_eq_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_117_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_118_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_119_dual__order_Otrans,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_120_dual__order_Oantisym,axiom,
! [B2: b,A: b] :
( ( ord_less_eq_b @ B2 @ A )
=> ( ( ord_less_eq_b @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_121_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_122_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_123_dual__order_Oantisym,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_124_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: b,Z2: b] : ( Y3 = Z2 ) )
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_125_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_126_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_127_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_128_linorder__wlog,axiom,
! [P: b > b > $o,A: b,B2: b] :
( ! [A4: b,B4: b] :
( ( ord_less_eq_b @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: b,B4: b] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_129_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_130_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_131_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_132_order__trans,axiom,
! [X: b,Y: b,Z3: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ Z3 )
=> ( ord_less_eq_b @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_133_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_134_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_135_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_136_order_Otrans,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% order.trans
thf(fact_137_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_138_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_139_order_Otrans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_140_order__antisym,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_141_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_142_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_143_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_144_ord__le__eq__trans,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_145_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_146_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_147_ord__le__eq__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_148_ord__eq__le__trans,axiom,
! [A: b,B2: b,C: b] :
( ( A = B2 )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_149_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_150_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_151_ord__eq__le__trans,axiom,
! [A: int,B2: int,C: int] :
( ( A = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_152_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: b,Z2: b] : ( Y3 = Z2 ) )
= ( ^ [X4: b,Y4: b] :
( ( ord_less_eq_b @ X4 @ Y4 )
& ( ord_less_eq_b @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_153_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_154_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_155_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_156_le__cases3,axiom,
! [X: b,Y: b,Z3: b] :
( ( ( ord_less_eq_b @ X @ Y )
=> ~ ( ord_less_eq_b @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_b @ Y @ X )
=> ~ ( ord_less_eq_b @ X @ Z3 ) )
=> ( ( ( ord_less_eq_b @ X @ Z3 )
=> ~ ( ord_less_eq_b @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_b @ Z3 @ Y )
=> ~ ( ord_less_eq_b @ Y @ X ) )
=> ( ( ( ord_less_eq_b @ Y @ Z3 )
=> ~ ( ord_less_eq_b @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_b @ Z3 @ X )
=> ~ ( ord_less_eq_b @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_157_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_158_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_159_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_160_nle__le,axiom,
! [A: b,B2: b] :
( ( ~ ( ord_less_eq_b @ A @ B2 ) )
= ( ( ord_less_eq_b @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_161_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_162_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_163_nle__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_164_verit__comp__simplify1_I2_J,axiom,
! [A: b] : ( ord_less_eq_b @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_165_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_166_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_167_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_168_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_169_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_170_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_171_order__less__imp__not__less,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ~ ( ord_less_b @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_172_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_173_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_174_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_175_order__less__imp__not__eq2,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_176_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_177_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_178_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_179_order__less__imp__not__eq,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_180_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_181_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_182_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_183_linorder__less__linear,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
| ( X = Y )
| ( ord_less_b @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_184_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_185_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_186_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_187_order__less__imp__triv,axiom,
! [X: b,Y: b,P: $o] :
( ( ord_less_b @ X @ Y )
=> ( ( ord_less_b @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_188_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_189_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_190_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_191_Collect__mem__eq,axiom,
! [A5: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_192_Collect__mem__eq,axiom,
! [A5: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_193_Collect__mem__eq,axiom,
! [A5: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_194_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_195_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_196_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_197_order__less__not__sym,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ~ ( ord_less_b @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_198_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_199_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_200_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_201_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_202_order__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_203_order__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_204_order__less__subst2,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_205_order__less__subst2,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_206_order__less__subst2,axiom,
! [A: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_207_order__less__subst2,axiom,
! [A: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_208_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_209_order__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_210_order__less__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_211_order__less__subst1,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_212_order__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_213_order__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_214_order__less__subst1,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_215_order__less__subst1,axiom,
! [A: real,F: b > real,B2: b,C: b] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_216_order__less__subst1,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_217_order__less__subst1,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_218_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_219_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_220_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_221_order__less__irrefl,axiom,
! [X: b] :
~ ( ord_less_b @ X @ X ) ).
% order_less_irrefl
thf(fact_222_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_223_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_224_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_225_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_226_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_227_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_228_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_229_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_230_ord__less__eq__subst,axiom,
! [A: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_231_ord__less__eq__subst,axiom,
! [A: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_232_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_233_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_234_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_235_ord__eq__less__subst,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_236_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_237_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_238_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_239_ord__eq__less__subst,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_240_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_241_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_242_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_243_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_244_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_245_order__less__trans,axiom,
! [X: b,Y: b,Z3: b] :
( ( ord_less_b @ X @ Y )
=> ( ( ord_less_b @ Y @ Z3 )
=> ( ord_less_b @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_246_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_247_order__less__asym_H,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_248_order__less__asym_H,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_249_order__less__asym_H,axiom,
! [A: b,B2: b] :
( ( ord_less_b @ A @ B2 )
=> ~ ( ord_less_b @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_250_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_251_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_252_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_253_linorder__neq__iff,axiom,
! [X: b,Y: b] :
( ( X != Y )
= ( ( ord_less_b @ X @ Y )
| ( ord_less_b @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_254_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_255_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_256_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_257_order__less__asym,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ~ ( ord_less_b @ Y @ X ) ) ).
% order_less_asym
thf(fact_258_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_259_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_260_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_261_linorder__neqE,axiom,
! [X: b,Y: b] :
( ( X != Y )
=> ( ~ ( ord_less_b @ X @ Y )
=> ( ord_less_b @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_262_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_263_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_264_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_265_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: b,A: b] :
( ( ord_less_b @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_266_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_267_order_Ostrict__implies__not__eq,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_268_order_Ostrict__implies__not__eq,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_269_order_Ostrict__implies__not__eq,axiom,
! [A: b,B2: b] :
( ( ord_less_b @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_270_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_271_dual__order_Ostrict__trans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_272_dual__order_Ostrict__trans,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_273_dual__order_Ostrict__trans,axiom,
! [B2: b,A: b,C: b] :
( ( ord_less_b @ B2 @ A )
=> ( ( ord_less_b @ C @ B2 )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_274_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_275_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_276_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_277_not__less__iff__gr__or__eq,axiom,
! [X: b,Y: b] :
( ( ~ ( ord_less_b @ X @ Y ) )
= ( ( ord_less_b @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_278_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_279_order_Ostrict__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_280_order_Ostrict__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_281_order_Ostrict__trans,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_b @ B2 @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_282_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_283_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_284_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_285_linorder__less__wlog,axiom,
! [P: b > b > $o,A: b,B2: b] :
( ! [A4: b,B4: b] :
( ( ord_less_b @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: b] : ( P @ A4 @ A4 )
=> ( ! [A4: b,B4: b] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_286_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_287_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_288_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_289_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_290_dual__order_Oirrefl,axiom,
! [A: b] :
~ ( ord_less_b @ A @ A ) ).
% dual_order.irrefl
thf(fact_291_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_292_dual__order_Oasym,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ~ ( ord_less_real @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_293_dual__order_Oasym,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ~ ( ord_less_int @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_294_dual__order_Oasym,axiom,
! [B2: b,A: b] :
( ( ord_less_b @ B2 @ A )
=> ~ ( ord_less_b @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_295_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_296_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_297_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_298_linorder__cases,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_b @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_b @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_299_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_300_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_301_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_302_antisym__conv3,axiom,
! [Y: b,X: b] :
( ~ ( ord_less_b @ Y @ X )
=> ( ( ~ ( ord_less_b @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_303_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_304_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_305_ord__less__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_306_ord__less__eq__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_307_ord__less__eq__trans,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_b @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_308_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_309_ord__eq__less__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_310_ord__eq__less__trans,axiom,
! [A: int,B2: int,C: int] :
( ( A = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_311_ord__eq__less__trans,axiom,
! [A: b,B2: b,C: b] :
( ( A = B2 )
=> ( ( ord_less_b @ B2 @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_312_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_313_order_Oasym,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order.asym
thf(fact_314_order_Oasym,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order.asym
thf(fact_315_order_Oasym,axiom,
! [A: b,B2: b] :
( ( ord_less_b @ A @ B2 )
=> ~ ( ord_less_b @ B2 @ A ) ) ).
% order.asym
thf(fact_316_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_317_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_318_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_319_less__imp__neq,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_320_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_321_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_322_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_323_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_324_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_325_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_326_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_327_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_328_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_329_verit__comp__simplify1_I1_J,axiom,
! [A: b] :
~ ( ord_less_b @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_330_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_331_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_332_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_333_pinf_I1_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_334_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_335_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_336_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_337_pinf_I2_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_338_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_339_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_340_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_341_pinf_I3_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_342_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_343_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_344_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_345_pinf_I4_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_346_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_347_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_348_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_349_pinf_I5_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ~ ( ord_less_b @ X2 @ T ) ) ).
% pinf(5)
thf(fact_350_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_351_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_352_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_353_pinf_I7_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( ord_less_b @ T @ X2 ) ) ).
% pinf(7)
thf(fact_354_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_355_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_356_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_357_minf_I1_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_358_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_359_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_360_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_361_minf_I2_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_362_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_363_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_364_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_365_minf_I3_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_366_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_367_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_368_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_369_minf_I4_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_370_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_371_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_372_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_373_minf_I5_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( ord_less_b @ X2 @ T ) ) ).
% minf(5)
thf(fact_374_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_375_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_376_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_377_minf_I7_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ~ ( ord_less_b @ T @ X2 ) ) ).
% minf(7)
thf(fact_378_order__le__imp__less__or__eq,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_b @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_379_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_380_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_381_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_382_linorder__le__less__linear,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
| ( ord_less_b @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_383_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_384_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_385_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_386_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_387_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_388_order__less__le__subst2,axiom,
! [A: int,B2: int,F: int > b,C: b] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_389_order__less__le__subst2,axiom,
! [A: b,B2: b,F: b > b,C: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_390_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_391_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_392_order__less__le__subst2,axiom,
! [A: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_393_order__less__le__subst2,axiom,
! [A: b,B2: b,F: b > nat,C: nat] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_394_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_395_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_396_order__less__le__subst1,axiom,
! [A: b,F: b > b,B2: b,C: b] :
( ( ord_less_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_397_order__less__le__subst1,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_398_order__less__le__subst1,axiom,
! [A: real,F: b > real,B2: b,C: b] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_399_order__less__le__subst1,axiom,
! [A: int,F: b > int,B2: b,C: b] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_400_order__less__le__subst1,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( ord_less_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_401_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_402_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_403_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_404_order__less__le__subst1,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( ord_less_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_405_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_406_order__le__less__subst2,axiom,
! [A: b,B2: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_407_order__le__less__subst2,axiom,
! [A: b,B2: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_408_order__le__less__subst2,axiom,
! [A: b,B2: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_409_order__le__less__subst2,axiom,
! [A: b,B2: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_410_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_411_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_412_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_413_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_414_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_415_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_416_order__le__less__subst1,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_417_order__le__less__subst1,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_418_order__le__less__subst1,axiom,
! [A: b,F: int > b,B2: int,C: int] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_419_order__le__less__subst1,axiom,
! [A: b,F: b > b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_420_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_421_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_422_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_423_order__le__less__subst1,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_424_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_425_order__le__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_426_order__less__le__trans,axiom,
! [X: b,Y: b,Z3: b] :
( ( ord_less_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ Z3 )
=> ( ord_less_b @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_427_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_428_order__less__le__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_429_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_430_order__le__less__trans,axiom,
! [X: b,Y: b,Z3: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_b @ Y @ Z3 )
=> ( ord_less_b @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_431_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_432_order__le__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_433_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_434_order__neq__le__trans,axiom,
! [A: b,B2: b] :
( ( A != B2 )
=> ( ( ord_less_eq_b @ A @ B2 )
=> ( ord_less_b @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_435_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_436_order__neq__le__trans,axiom,
! [A: real,B2: real] :
( ( A != B2 )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_437_order__neq__le__trans,axiom,
! [A: int,B2: int] :
( ( A != B2 )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_438_order__le__neq__trans,axiom,
! [A: b,B2: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_b @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_439_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_440_order__le__neq__trans,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_441_order__le__neq__trans,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_442_order__less__imp__le,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ( ord_less_eq_b @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_443_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_444_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_445_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_446_linorder__not__less,axiom,
! [X: b,Y: b] :
( ( ~ ( ord_less_b @ X @ Y ) )
= ( ord_less_eq_b @ Y @ X ) ) ).
% linorder_not_less
thf(fact_447_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_448_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_449_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_450_linorder__not__le,axiom,
! [X: b,Y: b] :
( ( ~ ( ord_less_eq_b @ X @ Y ) )
= ( ord_less_b @ Y @ X ) ) ).
% linorder_not_le
thf(fact_451_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_452_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_453_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_454_order__less__le,axiom,
( ord_less_b
= ( ^ [X4: b,Y4: b] :
( ( ord_less_eq_b @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_455_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_456_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_457_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_458_order__le__less,axiom,
( ord_less_eq_b
= ( ^ [X4: b,Y4: b] :
( ( ord_less_b @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_459_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_460_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_461_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_462_dual__order_Ostrict__implies__order,axiom,
! [B2: b,A: b] :
( ( ord_less_b @ B2 @ A )
=> ( ord_less_eq_b @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_463_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_464_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_eq_real @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_465_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( ord_less_eq_int @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_466_order_Ostrict__implies__order,axiom,
! [A: b,B2: b] :
( ( ord_less_b @ A @ B2 )
=> ( ord_less_eq_b @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_467_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_468_order_Ostrict__implies__order,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_469_order_Ostrict__implies__order,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_470_dual__order_Ostrict__iff__not,axiom,
( ord_less_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ~ ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_471_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_472_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_473_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_474_dual__order_Ostrict__trans2,axiom,
! [B2: b,A: b,C: b] :
( ( ord_less_b @ B2 @ A )
=> ( ( ord_less_eq_b @ C @ B2 )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_475_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_476_dual__order_Ostrict__trans2,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_477_dual__order_Ostrict__trans2,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_478_dual__order_Ostrict__trans1,axiom,
! [B2: b,A: b,C: b] :
( ( ord_less_eq_b @ B2 @ A )
=> ( ( ord_less_b @ C @ B2 )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_479_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_480_dual__order_Ostrict__trans1,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_481_dual__order_Ostrict__trans1,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_482_dual__order_Ostrict__iff__order,axiom,
( ord_less_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_483_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_484_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_485_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_486_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_b @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_487_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_488_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_489_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_490_dense__le__bounded,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_491_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_492_order_Ostrict__iff__not,axiom,
( ord_less_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ~ ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_493_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_494_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_495_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_496_order_Ostrict__trans2,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_497_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_498_order_Ostrict__trans2,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_499_order_Ostrict__trans2,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_500_order_Ostrict__trans1,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_b @ B2 @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_501_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_502_order_Ostrict__trans1,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_503_order_Ostrict__trans1,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_504_order_Ostrict__iff__order,axiom,
( ord_less_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_505_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_506_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_507_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_508_order_Oorder__iff__strict,axiom,
( ord_less_eq_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_b @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_509_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_510_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_511_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_512_not__le__imp__less,axiom,
! [Y: b,X: b] :
( ~ ( ord_less_eq_b @ Y @ X )
=> ( ord_less_b @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_513_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_514_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_515_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_516_less__le__not__le,axiom,
( ord_less_b
= ( ^ [X4: b,Y4: b] :
( ( ord_less_eq_b @ X4 @ Y4 )
& ~ ( ord_less_eq_b @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_517_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_518_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_519_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_520_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_521_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_522_complete__interval,axiom,
! [A: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_523_complete__interval,axiom,
! [A: real,B2: real,P: real > $o] :
( ( ord_less_real @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B2 )
& ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_524_complete__interval,axiom,
! [A: int,B2: int,P: int > $o] :
( ( ord_less_int @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B2 )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_525_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_526_interval__def,axiom,
( interval_b
= ( ^ [I: set_b] :
! [X4: b,Y4: b,Z5: b] :
( ( member_b @ X4 @ I )
=> ( ( member_b @ Z5 @ I )
=> ( ( ord_less_eq_b @ X4 @ Y4 )
=> ( ( ord_less_eq_b @ Y4 @ Z5 )
=> ( member_b @ Y4 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_527_interval__def,axiom,
( interval_nat
= ( ^ [I: set_nat] :
! [X4: nat,Y4: nat,Z5: nat] :
( ( member_nat @ X4 @ I )
=> ( ( member_nat @ Z5 @ I )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z5 )
=> ( member_nat @ Y4 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_528_interval__def,axiom,
( interval_real
= ( ^ [I: set_real] :
! [X4: real,Y4: real,Z5: real] :
( ( member_real @ X4 @ I )
=> ( ( member_real @ Z5 @ I )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z5 )
=> ( member_real @ Y4 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_529_interval__def,axiom,
( interval_int
= ( ^ [I: set_int] :
! [X4: int,Y4: int,Z5: int] :
( ( member_int @ X4 @ I )
=> ( ( member_int @ Z5 @ I )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z5 )
=> ( member_int @ Y4 @ I ) ) ) ) ) ) ) ).
% interval_def
thf(fact_530_down__ray__def,axiom,
( down_ray_b
= ( ^ [I: set_b] :
! [X4: b,Y4: b] :
( ( member_b @ Y4 @ I )
=> ( ( ord_less_eq_b @ X4 @ Y4 )
=> ( member_b @ X4 @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_531_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I: set_nat] :
! [X4: nat,Y4: nat] :
( ( member_nat @ Y4 @ I )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( member_nat @ X4 @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_532_down__ray__def,axiom,
( down_ray_real
= ( ^ [I: set_real] :
! [X4: real,Y4: real] :
( ( member_real @ Y4 @ I )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
=> ( member_real @ X4 @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_533_down__ray__def,axiom,
( down_ray_int
= ( ^ [I: set_int] :
! [X4: int,Y4: int] :
( ( member_int @ Y4 @ I )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
=> ( member_int @ X4 @ I ) ) ) ) ) ).
% down_ray_def
thf(fact_534_up__ray__def,axiom,
( up_ray_b
= ( ^ [I: set_b] :
! [X4: b,Y4: b] :
( ( member_b @ X4 @ I )
=> ( ( ord_less_eq_b @ X4 @ Y4 )
=> ( member_b @ Y4 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_535_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I: set_nat] :
! [X4: nat,Y4: nat] :
( ( member_nat @ X4 @ I )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( member_nat @ Y4 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_536_up__ray__def,axiom,
( up_ray_real
= ( ^ [I: set_real] :
! [X4: real,Y4: real] :
( ( member_real @ X4 @ I )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
=> ( member_real @ Y4 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_537_up__ray__def,axiom,
( up_ray_int
= ( ^ [I: set_int] :
! [X4: int,Y4: int] :
( ( member_int @ X4 @ I )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
=> ( member_int @ Y4 @ I ) ) ) ) ) ).
% up_ray_def
thf(fact_538_Greatest__equality,axiom,
! [P: b > $o,X: b] :
( ( P @ X )
=> ( ! [Y2: b] :
( ( P @ Y2 )
=> ( ord_less_eq_b @ Y2 @ X ) )
=> ( ( order_Greatest_b @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_539_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_540_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_541_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_542_GreatestI2__order,axiom,
! [P: b > $o,X: b,Q: b > $o] :
( ( P @ X )
=> ( ! [Y2: b] :
( ( P @ Y2 )
=> ( ord_less_eq_b @ Y2 @ X ) )
=> ( ! [X3: b] :
( ( P @ X3 )
=> ( ! [Y5: b] :
( ( P @ Y5 )
=> ( ord_less_eq_b @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_543_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q: real > $o] :
( ( P @ X )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ! [X3: real] :
( ( P @ X3 )
=> ( ! [Y5: real] :
( ( P @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_544_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( ! [Y5: int] :
( ( P @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_545_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_546_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N2 @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K @ I2 )
=> ( P @ I2 ) )
=> ( P @ K ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_547_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_548_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_549_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_550_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_551_le__trans,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_552_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_553_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_554_GreatestI__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_555_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_556_Greatest__le__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_557_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_558_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_559_ex__gt__or__lt,axiom,
! [A: real] :
? [B4: real] :
( ( ord_less_real @ A @ B4 )
| ( ord_less_real @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_560_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_561_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_562_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_563_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_564_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_565_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_566_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_567_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_568_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_569_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_570_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_571_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_572_seq__mono__lemma,axiom,
! [M2: nat,D2: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D2 @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M2 ) ) )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D2 @ N4 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_573_linordered__field__no__lb,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% linordered_field_no_lb
thf(fact_574_linordered__field__no__ub,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_575_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_576_complete__real,axiom,
! [S2: set_real] :
( ? [X2: real] : ( member_real @ X2 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_577_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_578_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_579_bdd__above_Opreordering__bdd__axioms,axiom,
condit4103000493307248662_bdd_b @ ord_less_eq_b @ ord_less_b ).
% bdd_above.preordering_bdd_axioms
thf(fact_580_bdd__above_Opreordering__bdd__axioms,axiom,
condit7935552474144124665dd_nat @ ord_less_eq_nat @ ord_less_nat ).
% bdd_above.preordering_bdd_axioms
thf(fact_581_bdd__above_Opreordering__bdd__axioms,axiom,
condit1497324847667023189d_real @ ord_less_eq_real @ ord_less_real ).
% bdd_above.preordering_bdd_axioms
thf(fact_582_bdd__above_Opreordering__bdd__axioms,axiom,
condit7933062003635074389dd_int @ ord_less_eq_int @ ord_less_int ).
% bdd_above.preordering_bdd_axioms
thf(fact_583_le__left__mono,axiom,
! [X: b,Y: b,A: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ A )
=> ( ord_less_eq_b @ X @ A ) ) ) ).
% le_left_mono
thf(fact_584_le__left__mono,axiom,
! [X: nat,Y: nat,A: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ A )
=> ( ord_less_eq_nat @ X @ A ) ) ) ).
% le_left_mono
thf(fact_585_le__left__mono,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ord_less_eq_real @ X @ A ) ) ) ).
% le_left_mono
thf(fact_586_le__left__mono,axiom,
! [X: int,Y: int,A: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ord_less_eq_int @ X @ A ) ) ) ).
% le_left_mono
thf(fact_587_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_b
= ( ^ [X6: $o > b,Y6: $o > b] :
( ( ord_less_eq_b @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_b @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_588_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X6: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_589_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_real
= ( ^ [X6: $o > real,Y6: $o > real] :
( ( ord_less_eq_real @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_real @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_590_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_int
= ( ^ [X6: $o > int,Y6: $o > int] :
( ( ord_less_eq_int @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_int @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_591_order_Oordering__axioms,axiom,
ordering_b @ ord_less_eq_b @ ord_less_b ).
% order.ordering_axioms
thf(fact_592_order_Oordering__axioms,axiom,
ordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.ordering_axioms
thf(fact_593_order_Oordering__axioms,axiom,
ordering_real @ ord_less_eq_real @ ord_less_real ).
% order.ordering_axioms
thf(fact_594_order_Oordering__axioms,axiom,
ordering_int @ ord_less_eq_int @ ord_less_int ).
% order.ordering_axioms
thf(fact_595_order_Opreordering__axioms,axiom,
preordering_b @ ord_less_eq_b @ ord_less_b ).
% order.preordering_axioms
thf(fact_596_order_Opreordering__axioms,axiom,
preordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.preordering_axioms
thf(fact_597_order_Opreordering__axioms,axiom,
preordering_real @ ord_less_eq_real @ ord_less_real ).
% order.preordering_axioms
thf(fact_598_order_Opreordering__axioms,axiom,
preordering_int @ ord_less_eq_int @ ord_less_int ).
% order.preordering_axioms
thf(fact_599_greaterThanAtMost__iff,axiom,
! [I3: b,L: b,U: b] :
( ( member_b @ I3 @ ( set_or4472690218693186639Most_b @ L @ U ) )
= ( ( ord_less_b @ L @ I3 )
& ( ord_less_eq_b @ I3 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_600_greaterThanAtMost__iff,axiom,
! [I3: real,L: real,U: real] :
( ( member_real @ I3 @ ( set_or2392270231875598684t_real @ L @ U ) )
= ( ( ord_less_real @ L @ I3 )
& ( ord_less_eq_real @ I3 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_601_greaterThanAtMost__iff,axiom,
! [I3: nat,L: nat,U: nat] :
( ( member_nat @ I3 @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I3 )
& ( ord_less_eq_nat @ I3 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_602_greaterThanAtMost__iff,axiom,
! [I3: int,L: int,U: int] :
( ( member_int @ I3 @ ( set_or6656581121297822940st_int @ L @ U ) )
= ( ( ord_less_int @ L @ I3 )
& ( ord_less_eq_int @ I3 @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_603_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_604_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_605_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_606_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_607_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_608_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_609_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_610_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_611_Ioc__inj,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ( set_or4472690218693186639Most_b @ A @ B2 )
= ( set_or4472690218693186639Most_b @ C @ D2 ) )
= ( ( ( ord_less_eq_b @ B2 @ A )
& ( ord_less_eq_b @ D2 @ C ) )
| ( ( A = C )
& ( B2 = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_612_Ioc__inj,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ( set_or2392270231875598684t_real @ A @ B2 )
= ( set_or2392270231875598684t_real @ C @ D2 ) )
= ( ( ( ord_less_eq_real @ B2 @ A )
& ( ord_less_eq_real @ D2 @ C ) )
| ( ( A = C )
& ( B2 = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_613_Ioc__inj,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or6659071591806873216st_nat @ A @ B2 )
= ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ D2 @ C ) )
| ( ( A = C )
& ( B2 = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_614_Ioc__inj,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ( set_or6656581121297822940st_int @ A @ B2 )
= ( set_or6656581121297822940st_int @ C @ D2 ) )
= ( ( ( ord_less_eq_int @ B2 @ A )
& ( ord_less_eq_int @ D2 @ C ) )
| ( ( A = C )
& ( B2 = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_615_Ioc__subset__iff,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ord_less_eq_set_b @ ( set_or4472690218693186639Most_b @ A @ B2 ) @ ( set_or4472690218693186639Most_b @ C @ D2 ) )
= ( ( ord_less_eq_b @ B2 @ A )
| ( ( ord_less_eq_b @ C @ A )
& ( ord_less_eq_b @ B2 @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_616_Ioc__subset__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A @ B2 ) @ ( set_or2392270231875598684t_real @ C @ D2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_617_Ioc__subset__iff,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A @ B2 ) @ ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_618_Ioc__subset__iff,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_set_int @ ( set_or6656581121297822940st_int @ A @ B2 ) @ ( set_or6656581121297822940st_int @ C @ D2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B2 @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_619_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_620_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_621_of__nat__mono,axiom,
! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_622_of__nat__mono,axiom,
! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I3 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_623_of__nat__mono,axiom,
! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_624_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_625_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_626_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_627_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_628_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_629_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_630_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_631_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_632_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_633_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B2 ) @ ( set_or2392270231875598684t_real @ C @ D2 ) )
= ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_634_greaterThanLessThan__iff,axiom,
! [I3: b,L: b,U: b] :
( ( member_b @ I3 @ ( set_or5939364468397584555Than_b @ L @ U ) )
= ( ( ord_less_b @ L @ I3 )
& ( ord_less_b @ I3 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_635_greaterThanLessThan__iff,axiom,
! [I3: real,L: real,U: real] :
( ( member_real @ I3 @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( ( ord_less_real @ L @ I3 )
& ( ord_less_real @ I3 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_636_greaterThanLessThan__iff,axiom,
! [I3: nat,L: nat,U: nat] :
( ( member_nat @ I3 @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I3 )
& ( ord_less_nat @ I3 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_637_greaterThanLessThan__iff,axiom,
! [I3: int,L: int,U: int] :
( ( member_int @ I3 @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( ( ord_less_int @ L @ I3 )
& ( ord_less_int @ I3 @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_638_int__int__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% int_int_eq
thf(fact_639_int__if,axiom,
! [P: $o,A: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_640_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_641_order_Opartial__preordering__axioms,axiom,
partia125584492769400373ring_b @ ord_less_eq_b ).
% order.partial_preordering_axioms
thf(fact_642_order_Opartial__preordering__axioms,axiom,
partia6822818058636336922ng_nat @ ord_less_eq_nat ).
% order.partial_preordering_axioms
thf(fact_643_order_Opartial__preordering__axioms,axiom,
partia7612893088228670710g_real @ ord_less_eq_real ).
% order.partial_preordering_axioms
thf(fact_644_order_Opartial__preordering__axioms,axiom,
partia6820327588127286646ng_int @ ord_less_eq_int ).
% order.partial_preordering_axioms
thf(fact_645_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B2 ) @ ( set_or1633881224788618240n_real @ C @ D2 ) )
= ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_646_ivl__disj__un__two_I2_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_eq_b @ L @ M2 )
=> ( ( ord_less_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or4472690218693186639Most_b @ L @ M2 ) @ ( set_or5939364468397584555Than_b @ M2 @ U ) )
= ( set_or5939364468397584555Than_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_647_ivl__disj__un__two_I2_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M2 ) @ ( set_or1633881224788618240n_real @ M2 @ U ) )
= ( set_or1633881224788618240n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_648_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or5834768355832116004an_nat @ M2 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_649_ivl__disj__un__two_I2_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M2 ) @ ( set_or5832277885323065728an_int @ M2 @ U ) )
= ( set_or5832277885323065728an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_650_round__of__nat,axiom,
! [N2: nat] :
( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( semiri1314217659103216013at_int @ N2 ) ) ).
% round_of_nat
thf(fact_651_of__nat__less__of__int__iff,axiom,
! [N2: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_652_of__nat__less__of__int__iff,axiom,
! [N2: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_653_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A @ B2 ) @ ( set_or66887138388493659n_real @ C @ D2 ) )
= ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_real @ B2 @ D2 ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_654_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D2 ) )
= ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_655_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B2 ) @ ( set_or66887138388493659n_real @ C @ D2 ) )
= ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_656_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or1633881224788618240n_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D2 ) )
= ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_657_Icc__eq__Icc,axiom,
! [L: b,H: b,L2: b,H2: b] :
( ( ( set_or672772299803893940Most_b @ L @ H )
= ( set_or672772299803893940Most_b @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_b @ L @ H )
& ~ ( ord_less_eq_b @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_658_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_659_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_660_Icc__eq__Icc,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or1266510415728281911st_int @ L @ H )
= ( set_or1266510415728281911st_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_int @ L @ H )
& ~ ( ord_less_eq_int @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_661_atLeastAtMost__iff,axiom,
! [I3: b,L: b,U: b] :
( ( member_b @ I3 @ ( set_or672772299803893940Most_b @ L @ U ) )
= ( ( ord_less_eq_b @ L @ I3 )
& ( ord_less_eq_b @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_662_atLeastAtMost__iff,axiom,
! [I3: real,L: real,U: real] :
( ( member_real @ I3 @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I3 )
& ( ord_less_eq_real @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_663_atLeastAtMost__iff,axiom,
! [I3: nat,L: nat,U: nat] :
( ( member_nat @ I3 @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I3 )
& ( ord_less_eq_nat @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_664_atLeastAtMost__iff,axiom,
! [I3: int,L: int,U: int] :
( ( member_int @ I3 @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I3 )
& ( ord_less_eq_int @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_665_atLeastatMost__subset__iff,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ A @ B2 ) @ ( set_or672772299803893940Most_b @ C @ D2 ) )
= ( ~ ( ord_less_eq_b @ A @ B2 )
| ( ( ord_less_eq_b @ C @ A )
& ( ord_less_eq_b @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_666_atLeastatMost__subset__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D2 ) )
= ( ~ ( ord_less_eq_real @ A @ B2 )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_667_atLeastatMost__subset__iff,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_668_atLeastatMost__subset__iff,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D2 ) )
= ( ~ ( ord_less_eq_int @ A @ B2 )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_669_atLeastLessThan__iff,axiom,
! [I3: b,L: b,U: b] :
( ( member_b @ I3 @ ( set_or5139330845457685136Than_b @ L @ U ) )
= ( ( ord_less_eq_b @ L @ I3 )
& ( ord_less_b @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_670_atLeastLessThan__iff,axiom,
! [I3: real,L: real,U: real] :
( ( member_real @ I3 @ ( set_or66887138388493659n_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I3 )
& ( ord_less_real @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_671_atLeastLessThan__iff,axiom,
! [I3: nat,L: nat,U: nat] :
( ( member_nat @ I3 @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I3 )
& ( ord_less_nat @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_672_atLeastLessThan__iff,axiom,
! [I3: int,L: int,U: int] :
( ( member_int @ I3 @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I3 )
& ( ord_less_int @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_673_ivl__subset,axiom,
! [I3: b,J: b,M2: b,N2: b] :
( ( ord_less_eq_set_b @ ( set_or5139330845457685136Than_b @ I3 @ J ) @ ( set_or5139330845457685136Than_b @ M2 @ N2 ) )
= ( ( ord_less_eq_b @ J @ I3 )
| ( ( ord_less_eq_b @ M2 @ I3 )
& ( ord_less_eq_b @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_674_ivl__subset,axiom,
! [I3: real,J: real,M2: real,N2: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I3 @ J ) @ ( set_or66887138388493659n_real @ M2 @ N2 ) )
= ( ( ord_less_eq_real @ J @ I3 )
| ( ( ord_less_eq_real @ M2 @ I3 )
& ( ord_less_eq_real @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_675_ivl__subset,axiom,
! [I3: nat,J: nat,M2: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ J ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ J @ I3 )
| ( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_eq_nat @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_676_ivl__subset,axiom,
! [I3: int,J: int,M2: int,N2: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I3 @ J ) @ ( set_or4662586982721622107an_int @ M2 @ N2 ) )
= ( ( ord_less_eq_int @ J @ I3 )
| ( ( ord_less_eq_int @ M2 @ I3 )
& ( ord_less_eq_int @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_677_of__int__le__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% of_int_le_iff
thf(fact_678_of__int__le__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% of_int_le_iff
thf(fact_679_of__int__less__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% of_int_less_iff
thf(fact_680_of__int__less__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% of_int_less_iff
thf(fact_681_of__int__of__nat__eq,axiom,
! [N2: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_int_of_nat_eq
thf(fact_682_of__int__of__nat__eq,axiom,
! [N2: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_int_of_nat_eq
thf(fact_683_ivl__disj__un__two__touch_I2_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_eq_b @ L @ M2 )
=> ( ( ord_less_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or672772299803893940Most_b @ L @ M2 ) @ ( set_or5139330845457685136Than_b @ M2 @ U ) )
= ( set_or5139330845457685136Than_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_684_ivl__disj__un__two__touch_I2_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M2 ) @ ( set_or66887138388493659n_real @ M2 @ U ) )
= ( set_or66887138388493659n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_685_ivl__disj__un__two__touch_I2_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_686_ivl__disj__un__two__touch_I2_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or4662586982721622107an_int @ M2 @ U ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_687_ivl__disj__un__two__touch_I4_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_eq_b @ L @ M2 )
=> ( ( ord_less_eq_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or672772299803893940Most_b @ L @ M2 ) @ ( set_or672772299803893940Most_b @ M2 @ U ) )
= ( set_or672772299803893940Most_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_688_ivl__disj__un__two__touch_I4_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M2 ) @ ( set_or1222579329274155063t_real @ M2 @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_689_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_690_ivl__disj__un__two__touch_I4_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_691_ivl__disj__un__two_I3_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_eq_b @ L @ M2 )
=> ( ( ord_less_eq_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or5139330845457685136Than_b @ L @ M2 ) @ ( set_or5139330845457685136Than_b @ M2 @ U ) )
= ( set_or5139330845457685136Than_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_692_ivl__disj__un__two_I3_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or66887138388493659n_real @ L @ M2 ) @ ( set_or66887138388493659n_real @ M2 @ U ) )
= ( set_or66887138388493659n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_693_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_694_ivl__disj__un__two_I3_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M2 ) @ ( set_or4662586982721622107an_int @ M2 @ U ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_695_ivl__disj__un__two_I7_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_eq_b @ L @ M2 )
=> ( ( ord_less_eq_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or5139330845457685136Than_b @ L @ M2 ) @ ( set_or672772299803893940Most_b @ M2 @ U ) )
= ( set_or672772299803893940Most_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_696_ivl__disj__un__two_I7_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or66887138388493659n_real @ L @ M2 ) @ ( set_or1222579329274155063t_real @ M2 @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_697_ivl__disj__un__two_I7_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_698_ivl__disj__un__two_I7_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_699_ivl__disj__un__two_I4_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_eq_b @ L @ M2 )
=> ( ( ord_less_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or672772299803893940Most_b @ L @ M2 ) @ ( set_or5939364468397584555Than_b @ M2 @ U ) )
= ( set_or5139330845457685136Than_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_700_ivl__disj__un__two_I4_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M2 ) @ ( set_or1633881224788618240n_real @ M2 @ U ) )
= ( set_or66887138388493659n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_701_ivl__disj__un__two_I4_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or5834768355832116004an_nat @ M2 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_702_ivl__disj__un__two_I4_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or5832277885323065728an_int @ M2 @ U ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_703_ivl__disj__un__two_I8_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_eq_b @ L @ M2 )
=> ( ( ord_less_eq_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or672772299803893940Most_b @ L @ M2 ) @ ( set_or4472690218693186639Most_b @ M2 @ U ) )
= ( set_or672772299803893940Most_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_704_ivl__disj__un__two_I8_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M2 ) @ ( set_or2392270231875598684t_real @ M2 @ U ) )
= ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_705_ivl__disj__un__two_I8_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or6659071591806873216st_nat @ M2 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_706_ivl__disj__un__two_I8_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or6656581121297822940st_int @ M2 @ U ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_707_atLeastLessThan__inj_I2_J,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ( set_or66887138388493659n_real @ A @ B2 )
= ( set_or66887138388493659n_real @ C @ D2 ) )
=> ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D2 )
=> ( B2 = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_708_atLeastLessThan__inj_I2_J,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ( set_or5139330845457685136Than_b @ A @ B2 )
= ( set_or5139330845457685136Than_b @ C @ D2 ) )
=> ( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_b @ C @ D2 )
=> ( B2 = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_709_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( B2 = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_710_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ( set_or4662586982721622107an_int @ A @ B2 )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( B2 = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_711_atLeastLessThan__inj_I1_J,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ( set_or66887138388493659n_real @ A @ B2 )
= ( set_or66887138388493659n_real @ C @ D2 ) )
=> ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_712_atLeastLessThan__inj_I1_J,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ( set_or5139330845457685136Than_b @ A @ B2 )
= ( set_or5139330845457685136Than_b @ C @ D2 ) )
=> ( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_b @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_713_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_714_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ( set_or4662586982721622107an_int @ A @ B2 )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_715_Ico__eq__Ico,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or66887138388493659n_real @ L @ H )
= ( set_or66887138388493659n_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_real @ L @ H )
& ~ ( ord_less_real @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_716_Ico__eq__Ico,axiom,
! [L: b,H: b,L2: b,H2: b] :
( ( ( set_or5139330845457685136Than_b @ L @ H )
= ( set_or5139330845457685136Than_b @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_b @ L @ H )
& ~ ( ord_less_b @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_717_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_718_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_719_atLeastLessThan__eq__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D2 )
=> ( ( ( set_or66887138388493659n_real @ A @ B2 )
= ( set_or66887138388493659n_real @ C @ D2 ) )
= ( ( A = C )
& ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_720_atLeastLessThan__eq__iff,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_b @ C @ D2 )
=> ( ( ( set_or5139330845457685136Than_b @ A @ B2 )
= ( set_or5139330845457685136Than_b @ C @ D2 ) )
= ( ( A = C )
& ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_721_atLeastLessThan__eq__iff,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( A = C )
& ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_722_atLeastLessThan__eq__iff,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ( set_or4662586982721622107an_int @ A @ B2 )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
= ( ( A = C )
& ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_723_ex__le__of__int,axiom,
! [X: real] :
? [Z: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ).
% ex_le_of_int
thf(fact_724_ex__of__int__less,axiom,
! [X: real] :
? [Z: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ).
% ex_of_int_less
thf(fact_725_ex__less__of__int,axiom,
! [X: real] :
? [Z: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ).
% ex_less_of_int
thf(fact_726_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ A @ B2 ) @ ( set_or5139330845457685136Than_b @ C @ D2 ) )
= ( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_b @ C @ A )
& ( ord_less_b @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_727_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or66887138388493659n_real @ C @ D2 ) )
= ( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_real @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_728_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_nat @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_729_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
= ( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ A )
& ( ord_less_int @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_730_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D2 ) )
= ( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_731_ivl__disj__un__two_I1_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_b @ L @ M2 )
=> ( ( ord_less_eq_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or5939364468397584555Than_b @ L @ M2 ) @ ( set_or5139330845457685136Than_b @ M2 @ U ) )
= ( set_or5939364468397584555Than_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_732_ivl__disj__un__two_I1_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1633881224788618240n_real @ L @ M2 ) @ ( set_or66887138388493659n_real @ M2 @ U ) )
= ( set_or1633881224788618240n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_733_ivl__disj__un__two_I1_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_734_ivl__disj__un__two_I1_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or5832277885323065728an_int @ L @ M2 ) @ ( set_or4662586982721622107an_int @ M2 @ U ) )
= ( set_or5832277885323065728an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_735_ivl__disj__un__two__touch_I3_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_b @ L @ M2 )
=> ( ( ord_less_eq_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or4472690218693186639Most_b @ L @ M2 ) @ ( set_or672772299803893940Most_b @ M2 @ U ) )
= ( set_or4472690218693186639Most_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_736_ivl__disj__un__two__touch_I3_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M2 ) @ ( set_or1222579329274155063t_real @ M2 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_737_ivl__disj__un__two__touch_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_738_ivl__disj__un__two__touch_I3_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
= ( set_or6656581121297822940st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_739_ivl__disj__un__two__touch_I1_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_b @ L @ M2 )
=> ( ( ord_less_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or4472690218693186639Most_b @ L @ M2 ) @ ( set_or5139330845457685136Than_b @ M2 @ U ) )
= ( set_or5939364468397584555Than_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_740_ivl__disj__un__two__touch_I1_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_real @ L @ M2 )
=> ( ( ord_less_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M2 ) @ ( set_or66887138388493659n_real @ M2 @ U ) )
= ( set_or1633881224788618240n_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_741_ivl__disj__un__two__touch_I1_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_742_ivl__disj__un__two__touch_I1_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_int @ L @ M2 )
=> ( ( ord_less_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M2 ) @ ( set_or4662586982721622107an_int @ M2 @ U ) )
= ( set_or5832277885323065728an_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_743_atLeastatMost__psubset__iff,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ord_less_set_b @ ( set_or672772299803893940Most_b @ A @ B2 ) @ ( set_or672772299803893940Most_b @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_b @ A @ B2 )
| ( ( ord_less_eq_b @ C @ A )
& ( ord_less_eq_b @ B2 @ D2 )
& ( ( ord_less_b @ C @ A )
| ( ord_less_b @ B2 @ D2 ) ) ) )
& ( ord_less_eq_b @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_744_atLeastatMost__psubset__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_real @ A @ B2 )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B2 @ D2 ) ) ) )
& ( ord_less_eq_real @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_745_atLeastatMost__psubset__iff,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B2 @ D2 )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B2 @ D2 ) ) ) )
& ( ord_less_eq_nat @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_746_atLeastatMost__psubset__iff,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_int @ A @ B2 )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B2 @ D2 )
& ( ( ord_less_int @ C @ A )
| ( ord_less_int @ B2 @ D2 ) ) ) )
& ( ord_less_eq_int @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_747_atLeastLessThan__subset__iff,axiom,
! [A: b,B2: b,C: b,D2: b] :
( ( ord_less_eq_set_b @ ( set_or5139330845457685136Than_b @ A @ B2 ) @ ( set_or5139330845457685136Than_b @ C @ D2 ) )
=> ( ( ord_less_eq_b @ B2 @ A )
| ( ( ord_less_eq_b @ C @ A )
& ( ord_less_eq_b @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_748_atLeastLessThan__subset__iff,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B2 ) @ ( set_or66887138388493659n_real @ C @ D2 ) )
=> ( ( ord_less_eq_real @ B2 @ A )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_749_atLeastLessThan__subset__iff,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_eq_nat @ B2 @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_750_atLeastLessThan__subset__iff,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B2 ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ B2 @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_751_ivl__disj__un__two_I5_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_b @ L @ M2 )
=> ( ( ord_less_eq_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or5939364468397584555Than_b @ L @ M2 ) @ ( set_or672772299803893940Most_b @ M2 @ U ) )
= ( set_or4472690218693186639Most_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_752_ivl__disj__un__two_I5_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or1633881224788618240n_real @ L @ M2 ) @ ( set_or1222579329274155063t_real @ M2 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_753_ivl__disj__un__two_I5_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_754_ivl__disj__un__two_I5_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or5832277885323065728an_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
= ( set_or6656581121297822940st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_755_ivl__disj__un__two_I6_J,axiom,
! [L: b,M2: b,U: b] :
( ( ord_less_eq_b @ L @ M2 )
=> ( ( ord_less_eq_b @ M2 @ U )
=> ( ( sup_sup_set_b @ ( set_or4472690218693186639Most_b @ L @ M2 ) @ ( set_or4472690218693186639Most_b @ M2 @ U ) )
= ( set_or4472690218693186639Most_b @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_756_ivl__disj__un__two_I6_J,axiom,
! [L: real,M2: real,U: real] :
( ( ord_less_eq_real @ L @ M2 )
=> ( ( ord_less_eq_real @ M2 @ U )
=> ( ( sup_sup_set_real @ ( set_or2392270231875598684t_real @ L @ M2 ) @ ( set_or2392270231875598684t_real @ M2 @ U ) )
= ( set_or2392270231875598684t_real @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_757_ivl__disj__un__two_I6_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M2 ) @ ( set_or6659071591806873216st_nat @ M2 @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_758_ivl__disj__un__two_I6_J,axiom,
! [L: int,M2: int,U: int] :
( ( ord_less_eq_int @ L @ M2 )
=> ( ( ord_less_eq_int @ M2 @ U )
=> ( ( sup_sup_set_int @ ( set_or6656581121297822940st_int @ L @ M2 ) @ ( set_or6656581121297822940st_int @ M2 @ U ) )
= ( set_or6656581121297822940st_int @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_759_round__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim8280529875227126926d_real @ Y ) ) ) ).
% round_mono
thf(fact_760_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_set_nat @ X @ Z3 )
& ( ord_less_eq_set_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_761_le__sup__iff,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X @ Z3 )
& ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_762_le__sup__iff,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_real @ X @ Z3 )
& ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_763_le__sup__iff,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_int @ X @ Z3 )
& ( ord_less_eq_int @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_764_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B2 @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_765_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_766_sup_Obounded__iff,axiom,
! [B2: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C ) @ A )
= ( ( ord_less_eq_real @ B2 @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_767_sup_Obounded__iff,axiom,
! [B2: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A )
= ( ( ord_less_eq_int @ B2 @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_768_less__supI1,axiom,
! [X: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ X @ A )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_769_less__supI1,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_770_less__supI1,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_real @ X @ A )
=> ( ord_less_real @ X @ ( sup_sup_real @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_771_less__supI1,axiom,
! [X: int,A: int,B2: int] :
( ( ord_less_int @ X @ A )
=> ( ord_less_int @ X @ ( sup_sup_int @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_772_less__supI2,axiom,
! [X: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X @ B2 )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_773_less__supI2,axiom,
! [X: nat,B2: nat,A: nat] :
( ( ord_less_nat @ X @ B2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_774_less__supI2,axiom,
! [X: real,B2: real,A: real] :
( ( ord_less_real @ X @ B2 )
=> ( ord_less_real @ X @ ( sup_sup_real @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_775_less__supI2,axiom,
! [X: int,B2: int,A: int] :
( ( ord_less_int @ X @ B2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_776_sup_Oabsorb3,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_777_sup_Oabsorb3,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_778_sup_Oabsorb3,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( sup_sup_real @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_779_sup_Oabsorb3,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( sup_sup_int @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_780_sup_Oabsorb4,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_781_sup_Oabsorb4,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_782_sup_Oabsorb4,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( sup_sup_real @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_783_sup_Oabsorb4,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( sup_sup_int @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_784_sup_Ostrict__boundedE,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B2 @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_785_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_786_sup_Ostrict__boundedE,axiom,
! [B2: real,C: real,A: real] :
( ( ord_less_real @ ( sup_sup_real @ B2 @ C ) @ A )
=> ~ ( ( ord_less_real @ B2 @ A )
=> ~ ( ord_less_real @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_787_sup_Ostrict__boundedE,axiom,
! [B2: int,C: int,A: int] :
( ( ord_less_int @ ( sup_sup_int @ B2 @ C ) @ A )
=> ~ ( ( ord_less_int @ B2 @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_788_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_789_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_790_inf__sup__ord_I4_J,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_791_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_792_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_793_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_794_inf__sup__ord_I3_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_795_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_796_le__supE,axiom,
! [A: set_nat,B2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A @ X )
=> ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_797_le__supE,axiom,
! [A: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_798_le__supE,axiom,
! [A: real,B2: real,X: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_real @ A @ X )
=> ~ ( ord_less_eq_real @ B2 @ X ) ) ) ).
% le_supE
thf(fact_799_le__supE,axiom,
! [A: int,B2: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_int @ A @ X )
=> ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).
% le_supE
thf(fact_800_le__supI,axiom,
! [A: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_801_le__supI,axiom,
! [A: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_802_le__supI,axiom,
! [A: real,X: real,B2: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ B2 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_803_le__supI,axiom,
! [A: int,X: int,B2: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ( ord_less_eq_int @ B2 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_804_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_805_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_806_sup__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_807_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_808_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_809_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_810_sup__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_811_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_812_le__supI1,axiom,
! [X: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_813_le__supI1,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_814_le__supI1,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_815_le__supI1,axiom,
! [X: int,A: int,B2: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_816_le__supI2,axiom,
! [X: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_817_le__supI2,axiom,
! [X: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_818_le__supI2,axiom,
! [X: real,B2: real,A: real] :
( ( ord_less_eq_real @ X @ B2 )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_819_le__supI2,axiom,
! [X: int,B2: int,A: int] :
( ( ord_less_eq_int @ X @ B2 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_820_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_821_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_822_sup_Omono,axiom,
! [C: real,A: real,D2: real,B2: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ( ord_less_eq_real @ D2 @ B2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ C @ D2 ) @ ( sup_sup_real @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_823_sup_Omono,axiom,
! [C: int,A: int,D2: int,B2: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D2 @ B2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D2 ) @ ( sup_sup_int @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_824_sup__mono,axiom,
! [A: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_825_sup__mono,axiom,
! [A: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_826_sup__mono,axiom,
! [A: real,C: real,B2: real,D2: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B2 @ D2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B2 ) @ ( sup_sup_real @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_827_sup__mono,axiom,
! [A: int,C: int,B2: int,D2: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B2 @ D2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B2 ) @ ( sup_sup_int @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_828_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z3 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_829_sup__least,axiom,
! [Y: nat,X: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_830_sup__least,axiom,
! [Y: real,X: real,Z3: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ Z3 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_831_sup__least,axiom,
! [Y: int,X: int,Z3: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z3 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_832_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_833_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( sup_sup_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_834_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( sup_sup_real @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_835_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( sup_sup_int @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_836_sup_OorderE,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_837_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_838_sup_OorderE,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( A
= ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_839_sup_OorderE,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( A
= ( sup_sup_int @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_840_sup_OorderI,axiom,
! [A: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_841_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_842_sup_OorderI,axiom,
! [A: real,B2: real] :
( ( A
= ( sup_sup_real @ A @ B2 ) )
=> ( ord_less_eq_real @ B2 @ A ) ) ).
% sup.orderI
thf(fact_843_sup_OorderI,axiom,
! [A: int,B2: int] :
( ( A
= ( sup_sup_int @ A @ B2 ) )
=> ( ord_less_eq_int @ B2 @ A ) ) ).
% sup.orderI
thf(fact_844_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_845_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_846_sup__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X3: real,Y2: real] : ( ord_less_eq_real @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: real,Y2: real] : ( ord_less_eq_real @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: real,Y2: real,Z: real] :
( ( ord_less_eq_real @ Y2 @ X3 )
=> ( ( ord_less_eq_real @ Z @ X3 )
=> ( ord_less_eq_real @ ( F @ Y2 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_847_sup__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X3: int,Y2: int] : ( ord_less_eq_int @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int,Z: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ( ( ord_less_eq_int @ Z @ X3 )
=> ( ord_less_eq_int @ ( F @ Y2 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_848_sup_Oabsorb1,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_849_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_850_sup_Oabsorb1,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( sup_sup_real @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_851_sup_Oabsorb1,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( sup_sup_int @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_852_sup_Oabsorb2,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_853_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_854_sup_Oabsorb2,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( sup_sup_real @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_855_sup_Oabsorb2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( sup_sup_int @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_856_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_857_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_858_sup__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( sup_sup_real @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_859_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_860_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_861_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_862_sup__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( sup_sup_real @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_863_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_864_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_865_sup_OboundedE,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_866_sup_OboundedE,axiom,
! [B2: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_real @ B2 @ A )
=> ~ ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_867_sup_OboundedE,axiom,
! [B2: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B2 @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_868_sup_OboundedI,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_869_sup_OboundedI,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_870_sup_OboundedI,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_871_sup_OboundedI,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_872_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_873_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_874_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( A3
= ( sup_sup_real @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_875_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( A3
= ( sup_sup_int @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_876_sup_Ocobounded1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_877_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_878_sup_Ocobounded1,axiom,
! [A: real,B2: real] : ( ord_less_eq_real @ A @ ( sup_sup_real @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_879_sup_Ocobounded1,axiom,
! [A: int,B2: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_880_sup_Ocobounded2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_881_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_882_sup_Ocobounded2,axiom,
! [B2: real,A: real] : ( ord_less_eq_real @ B2 @ ( sup_sup_real @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_883_sup_Ocobounded2,axiom,
! [B2: int,A: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_884_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_885_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_886_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( sup_sup_real @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_887_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( sup_sup_int @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_888_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_889_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_890_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( sup_sup_real @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_891_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( sup_sup_int @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_892_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_893_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_894_sup_OcoboundedI1,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_895_sup_OcoboundedI1,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_896_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_897_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_898_sup_OcoboundedI2,axiom,
! [C: real,B2: real,A: real] :
( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_899_sup_OcoboundedI2,axiom,
! [C: int,B2: int,A: int] :
( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_900_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_901_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_902_sup_Ostrict__coboundedI2,axiom,
! [C: real,B2: real,A: real] :
( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_903_sup_Ostrict__coboundedI2,axiom,
! [C: int,B2: int,A: int] :
( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_904_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_905_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_906_sup_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ C @ A )
=> ( ord_less_real @ C @ ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_907_sup_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( sup_sup_int @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_908_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_909_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_910_sup_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( A3
= ( sup_sup_real @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_911_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( A3
= ( sup_sup_int @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_912_Icc__subset__Ici__iff,axiom,
! [L: b,H: b,L2: b] :
( ( ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ L @ H ) @ ( set_ord_atLeast_b @ L2 ) )
= ( ~ ( ord_less_eq_b @ L @ H )
| ( ord_less_eq_b @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_913_Icc__subset__Ici__iff,axiom,
! [L: real,H: real,L2: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atLeast_real @ L2 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_914_Icc__subset__Ici__iff,axiom,
! [L: nat,H: nat,L2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atLeast_nat @ L2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_915_Icc__subset__Ici__iff,axiom,
! [L: int,H: int,L2: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atLeast_int @ L2 ) )
= ( ~ ( ord_less_eq_int @ L @ H )
| ( ord_less_eq_int @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_916_Icc__subset__Iic__iff,axiom,
! [L: b,H: b,H2: b] :
( ( ord_less_eq_set_b @ ( set_or672772299803893940Most_b @ L @ H ) @ ( set_ord_atMost_b @ H2 ) )
= ( ~ ( ord_less_eq_b @ L @ H )
| ( ord_less_eq_b @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_917_Icc__subset__Iic__iff,axiom,
! [L: real,H: real,H2: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atMost_real @ H2 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_918_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_919_Icc__subset__Iic__iff,axiom,
! [L: int,H: int,H2: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atMost_int @ H2 ) )
= ( ~ ( ord_less_eq_int @ L @ H )
| ( ord_less_eq_int @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_920_greaterThanLessThan__eq__iff,axiom,
! [R: real,S: real,T: real,U: real] :
( ( ( set_or1633881224788618240n_real @ R @ S )
= ( set_or1633881224788618240n_real @ T @ U ) )
= ( ( ( ord_less_eq_real @ S @ R )
& ( ord_less_eq_real @ U @ T ) )
| ( ( R = T )
& ( S = U ) ) ) ) ).
% greaterThanLessThan_eq_iff
thf(fact_921_ivl__disj__un__one_I3_J,axiom,
! [L: b,U: b] :
( ( ord_less_eq_b @ L @ U )
=> ( ( sup_sup_set_b @ ( set_ord_atMost_b @ L ) @ ( set_or4472690218693186639Most_b @ L @ U ) )
= ( set_ord_atMost_b @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_922_ivl__disj__un__one_I3_J,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sup_sup_set_real @ ( set_ord_atMost_real @ L ) @ ( set_or2392270231875598684t_real @ L @ U ) )
= ( set_ord_atMost_real @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_923_ivl__disj__un__one_I3_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( set_ord_atMost_nat @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_924_ivl__disj__un__one_I3_J,axiom,
! [L: int,U: int] :
( ( ord_less_eq_int @ L @ U )
=> ( ( sup_sup_set_int @ ( set_ord_atMost_int @ L ) @ ( set_or6656581121297822940st_int @ L @ U ) )
= ( set_ord_atMost_int @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_925_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_926_atLeast__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atLeast_nat @ X )
= ( set_ord_atLeast_nat @ Y ) )
= ( X = Y ) ) ).
% atLeast_eq_iff
thf(fact_927_atMost__iff,axiom,
! [I3: b,K2: b] :
( ( member_b @ I3 @ ( set_ord_atMost_b @ K2 ) )
= ( ord_less_eq_b @ I3 @ K2 ) ) ).
% atMost_iff
thf(fact_928_atMost__iff,axiom,
! [I3: real,K2: real] :
( ( member_real @ I3 @ ( set_ord_atMost_real @ K2 ) )
= ( ord_less_eq_real @ I3 @ K2 ) ) ).
% atMost_iff
thf(fact_929_atMost__iff,axiom,
! [I3: int,K2: int] :
( ( member_int @ I3 @ ( set_ord_atMost_int @ K2 ) )
= ( ord_less_eq_int @ I3 @ K2 ) ) ).
% atMost_iff
thf(fact_930_atMost__iff,axiom,
! [I3: nat,K2: nat] :
( ( member_nat @ I3 @ ( set_ord_atMost_nat @ K2 ) )
= ( ord_less_eq_nat @ I3 @ K2 ) ) ).
% atMost_iff
thf(fact_931_atLeast__iff,axiom,
! [I3: b,K2: b] :
( ( member_b @ I3 @ ( set_ord_atLeast_b @ K2 ) )
= ( ord_less_eq_b @ K2 @ I3 ) ) ).
% atLeast_iff
thf(fact_932_atLeast__iff,axiom,
! [I3: real,K2: real] :
( ( member_real @ I3 @ ( set_ord_atLeast_real @ K2 ) )
= ( ord_less_eq_real @ K2 @ I3 ) ) ).
% atLeast_iff
thf(fact_933_atLeast__iff,axiom,
! [I3: int,K2: int] :
( ( member_int @ I3 @ ( set_ord_atLeast_int @ K2 ) )
= ( ord_less_eq_int @ K2 @ I3 ) ) ).
% atLeast_iff
thf(fact_934_atLeast__iff,axiom,
! [I3: nat,K2: nat] :
( ( member_nat @ I3 @ ( set_ord_atLeast_nat @ K2 ) )
= ( ord_less_eq_nat @ K2 @ I3 ) ) ).
% atLeast_iff
thf(fact_935_atMost__subset__iff,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_set_b @ ( set_ord_atMost_b @ X ) @ ( set_ord_atMost_b @ Y ) )
= ( ord_less_eq_b @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_936_atMost__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_937_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_938_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_939_atLeast__subset__iff,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_set_b @ ( set_ord_atLeast_b @ X ) @ ( set_ord_atLeast_b @ Y ) )
= ( ord_less_eq_b @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_940_atLeast__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ X ) @ ( set_ord_atLeast_real @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_941_atLeast__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atLeast_int @ X ) @ ( set_ord_atLeast_int @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_942_atLeast__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X ) @ ( set_ord_atLeast_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_943_not__Ici__le__Iic,axiom,
! [L: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L ) @ ( set_ord_atMost_nat @ H2 ) ) ).
% not_Ici_le_Iic
thf(fact_944_not__Iic__eq__Ici,axiom,
! [H: nat,L2: nat] :
( ( set_ord_atMost_nat @ H )
!= ( set_ord_atLeast_nat @ L2 ) ) ).
% not_Iic_eq_Ici
thf(fact_945_not__Iic__eq__Icc,axiom,
! [H2: int,L: int,H: int] :
( ( set_ord_atMost_int @ H2 )
!= ( set_or1266510415728281911st_int @ L @ H ) ) ).
% not_Iic_eq_Icc
thf(fact_946_not__Ici__eq__Icc,axiom,
! [L2: nat,L: nat,H: nat] :
( ( set_ord_atLeast_nat @ L2 )
!= ( set_or1269000886237332187st_nat @ L @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_947_not__Ici__eq__Icc,axiom,
! [L2: int,L: int,H: int] :
( ( set_ord_atLeast_int @ L2 )
!= ( set_or1266510415728281911st_int @ L @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_948_not__Iic__le__Icc,axiom,
! [H: int,L2: int,H2: int] :
~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H ) @ ( set_or1266510415728281911st_int @ L2 @ H2 ) ) ).
% not_Iic_le_Icc
thf(fact_949_not__Ici__le__Icc,axiom,
! [L: nat,L2: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L ) @ ( set_or1269000886237332187st_nat @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_950_not__Ici__le__Icc,axiom,
! [L: int,L2: int,H2: int] :
~ ( ord_less_eq_set_int @ ( set_ord_atLeast_int @ L ) @ ( set_or1266510415728281911st_int @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_951_preordering__bdd_OI,axiom,
! [Less_eq: real > real > $o,Less: real > real > $o,A5: set_real,M4: real] :
( ( condit1497324847667023189d_real @ Less_eq @ Less )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A5 )
=> ( Less_eq @ X3 @ M4 ) )
=> ( condit7290043087096337911d_real @ Less_eq @ A5 ) ) ) ).
% preordering_bdd.I
thf(fact_952_preordering__bdd_OI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,A5: set_nat,M4: nat] :
( ( condit7935552474144124665dd_nat @ Less_eq @ Less )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( Less_eq @ X3 @ M4 ) )
=> ( condit4013746787832047771dd_nat @ Less_eq @ A5 ) ) ) ).
% preordering_bdd.I
thf(fact_953_preordering__bdd_OI,axiom,
! [Less_eq: int > int > $o,Less: int > int > $o,A5: set_int,M4: int] :
( ( condit7933062003635074389dd_int @ Less_eq @ Less )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A5 )
=> ( Less_eq @ X3 @ M4 ) )
=> ( condit4011256317322997495dd_int @ Less_eq @ A5 ) ) ) ).
% preordering_bdd.I
thf(fact_954_preordering__bdd_OE,axiom,
! [Less_eq: real > real > $o,Less: real > real > $o,A5: set_real] :
( ( condit1497324847667023189d_real @ Less_eq @ Less )
=> ( ( condit7290043087096337911d_real @ Less_eq @ A5 )
=> ~ ! [M6: real] :
~ ! [X2: real] :
( ( member_real @ X2 @ A5 )
=> ( Less_eq @ X2 @ M6 ) ) ) ) ).
% preordering_bdd.E
thf(fact_955_preordering__bdd_OE,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,A5: set_nat] :
( ( condit7935552474144124665dd_nat @ Less_eq @ Less )
=> ( ( condit4013746787832047771dd_nat @ Less_eq @ A5 )
=> ~ ! [M6: nat] :
~ ! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( Less_eq @ X2 @ M6 ) ) ) ) ).
% preordering_bdd.E
thf(fact_956_preordering__bdd_OE,axiom,
! [Less_eq: int > int > $o,Less: int > int > $o,A5: set_int] :
( ( condit7933062003635074389dd_int @ Less_eq @ Less )
=> ( ( condit4011256317322997495dd_int @ Less_eq @ A5 )
=> ~ ! [M6: int] :
~ ! [X2: int] :
( ( member_int @ X2 @ A5 )
=> ( Less_eq @ X2 @ M6 ) ) ) ) ).
% preordering_bdd.E
thf(fact_957_ivl__disj__un__one_I8_J,axiom,
! [L: b,U: b] :
( ( ord_less_eq_b @ L @ U )
=> ( ( sup_sup_set_b @ ( set_or5139330845457685136Than_b @ L @ U ) @ ( set_ord_atLeast_b @ U ) )
= ( set_ord_atLeast_b @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_958_ivl__disj__un__one_I8_J,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sup_sup_set_real @ ( set_or66887138388493659n_real @ L @ U ) @ ( set_ord_atLeast_real @ U ) )
= ( set_ord_atLeast_real @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_959_ivl__disj__un__one_I8_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( set_ord_atLeast_nat @ U ) )
= ( set_ord_atLeast_nat @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_960_ivl__disj__un__one_I8_J,axiom,
! [L: int,U: int] :
( ( ord_less_eq_int @ L @ U )
=> ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ U ) @ ( set_ord_atLeast_int @ U ) )
= ( set_ord_atLeast_int @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_961_ivl__disj__un__one_I6_J,axiom,
! [L: b,U: b] :
( ( ord_less_b @ L @ U )
=> ( ( sup_sup_set_b @ ( set_or5939364468397584555Than_b @ L @ U ) @ ( set_ord_atLeast_b @ U ) )
= ( set_or8632414552788122085Than_b @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_962_ivl__disj__un__one_I6_J,axiom,
! [L: real,U: real] :
( ( ord_less_real @ L @ U )
=> ( ( sup_sup_set_real @ ( set_or1633881224788618240n_real @ L @ U ) @ ( set_ord_atLeast_real @ U ) )
= ( set_or5849166863359141190n_real @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_963_ivl__disj__un__one_I6_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U ) @ ( set_ord_atLeast_nat @ U ) )
= ( set_or1210151606488870762an_nat @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_964_ivl__disj__un__one_I6_J,axiom,
! [L: int,U: int] :
( ( ord_less_int @ L @ U )
=> ( ( sup_sup_set_int @ ( set_or5832277885323065728an_int @ L @ U ) @ ( set_ord_atLeast_int @ U ) )
= ( set_or1207661135979820486an_int @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_965_ivl__disj__un__one_I1_J,axiom,
! [L: b,U: b] :
( ( ord_less_b @ L @ U )
=> ( ( sup_sup_set_b @ ( set_ord_atMost_b @ L ) @ ( set_or5939364468397584555Than_b @ L @ U ) )
= ( set_ord_lessThan_b @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_966_ivl__disj__un__one_I1_J,axiom,
! [L: real,U: real] :
( ( ord_less_real @ L @ U )
=> ( ( sup_sup_set_real @ ( set_ord_atMost_real @ L ) @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( set_or5984915006950818249n_real @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_967_ivl__disj__un__one_I1_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_968_ivl__disj__un__one_I1_J,axiom,
! [L: int,U: int] :
( ( ord_less_int @ L @ U )
=> ( ( sup_sup_set_int @ ( set_ord_atMost_int @ L ) @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( set_ord_lessThan_int @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_969_ivl__disj__un__one_I7_J,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ U ) @ ( set_or5849166863359141190n_real @ U ) )
= ( set_ord_atLeast_real @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_970_ivl__disj__un__one_I7_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ U ) @ ( set_or1210151606488870762an_nat @ U ) )
= ( set_ord_atLeast_nat @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_971_ivl__disj__un__one_I7_J,axiom,
! [L: int,U: int] :
( ( ord_less_eq_int @ L @ U )
=> ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ U ) @ ( set_or1207661135979820486an_int @ U ) )
= ( set_ord_atLeast_int @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_972_diff__diff__cancel,axiom,
! [I3: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_973_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M5: nat,N3: nat] :
( Z3
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_974_diff__commute,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_975_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_976_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_977_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_978_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_979_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_980_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_981_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_982_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_983_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_984_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_985_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_986_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_987_int__ops_I6_J,axiom,
! [A: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_988_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_989_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_990_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_991_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_992_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_993_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_994_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_995_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_996_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_997_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_998_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_999_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1000_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1001_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1002_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1003_nat__int,axiom,
! [N2: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
= N2 ) ).
% nat_int
thf(fact_1004_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1005_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1006_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1007_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_1008_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1009_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_1010_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1011_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1012_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1013_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1014_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1015_split__nat,axiom,
! [P: nat > $o,I3: int] :
( ( P @ ( nat2 @ I3 ) )
= ( ! [N: nat] :
( ( I3
= ( semiri1314217659103216013at_int @ N ) )
=> ( P @ N ) )
& ( ( ord_less_int @ I3 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1016_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1017_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1018_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1019_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1020_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1021_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1022_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1023_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1024_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1025_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1026_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1027_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1028_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1029_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1030_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1031_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1032_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1033_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_1034_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1035_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1036_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_1037_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1038_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z3: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1039_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_1040_int__eq__iff,axiom,
! [M2: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z3 )
= ( ( M2
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_1041_int__minus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1042_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1043_le__nat__iff,axiom,
! [K2: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_1044_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1045_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1046_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1047_ex__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ( P @ M ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1048_all__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( P @ M ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1049_all__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( P @ M ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_1050_Inf__nat__def1,axiom,
! [K3: set_nat] :
( ( K3 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K3 ) @ K3 ) ) ).
% Inf_nat_def1
thf(fact_1051_ex__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( P @ M ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_1052_Bolzano,axiom,
! [A: real,B2: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ! [A4: real,B4: real,C2: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C2 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( P @ A4 @ C2 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B2 )
=> ? [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X3 )
& ( ord_less_eq_real @ X3 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1053_atLeast1__atMost__eq__remove0,axiom,
! [N2: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_1054_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1055_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1056_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_1057_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1058_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1059_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_1060_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1061_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1062_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1063_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1064_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1065_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
= ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_1066_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_1067_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_1068_atLeast__Suc__greaterThan,axiom,
! [K2: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
= ( set_or1210151606488870762an_nat @ K2 ) ) ).
% atLeast_Suc_greaterThan
thf(fact_1069_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1070_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1071_Nat_OlessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( K2
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1072_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_1073_Suc__lessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1074_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_1075_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_1076_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1077_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N2 ) )
& ( P @ I5 ) ) )
= ( ( P @ N2 )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
& ( P @ I5 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1078_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_1079_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1080_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N2 ) )
=> ( P @ I5 ) ) )
= ( ( P @ N2 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( P @ I5 ) ) ) ) ).
% All_less_Suc
thf(fact_1081_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1082_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_1083_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_1084_less__trans__Suc,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I3 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1085_less__Suc__induct,axiom,
! [I3: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I4 @ K ) ) ) ) )
=> ( P @ I3 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1086_strict__inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_1087_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_1088_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_1089_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1090_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1091_Suc__le__D,axiom,
! [N2: nat,M8: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M8 )
=> ? [M5: nat] :
( M8
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1092_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1093_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1094_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1095_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1096_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1097_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1098_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I3: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1099_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1100_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1101_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1102_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1103_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1104_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1105_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1106_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1107_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1108_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1109_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1110_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1111_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N2 ) )
& ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
& ( P @ ( suc @ I5 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1112_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1113_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N2 ) )
=> ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( P @ ( suc @ I5 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1114_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1115_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_1116_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1117_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1118_dec__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ I3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1119_inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_1120_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1121_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1122_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1123_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1124_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1125_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1126_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1127_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1128_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_1129_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( set_ord_atMost_nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_1130_lessThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( insert_nat @ K2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% lessThan_Suc
thf(fact_1131_atMost__Suc,axiom,
! [K2: nat] :
( ( set_ord_atMost_nat @ ( suc @ K2 ) )
= ( insert_nat @ ( suc @ K2 ) @ ( set_ord_atMost_nat @ K2 ) ) ) ).
% atMost_Suc
thf(fact_1132_greaterThan__Suc,axiom,
! [K2: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K2 ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K2 ) @ ( insert_nat @ ( suc @ K2 ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_1133_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N2 )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1134_diff__Suc__less,axiom,
! [N2: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1135_atLeast0__atMost__Suc,axiom,
! [N2: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
= ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_1136_atLeast0__lessThan__Suc,axiom,
! [N2: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
= ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_1137_atLeastAtMost__insertL,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
= ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1138_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) )
= ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1139_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( set_or1269000886237332187st_nat @ M2 @ N2 )
= ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1140_atLeast__Suc,axiom,
! [K2: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K2 ) @ ( insert_nat @ K2 @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_1141_atLeastLessThanSuc,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) )
= ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_1142_atLeast1__lessThan__eq__remove0,axiom,
! [N2: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_1143_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1144_nat__ivt__aux,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
& ( ( F @ I4 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1145_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1146_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1147_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1148_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1149_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1150_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1151_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1152_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1153_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1154_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1155_nat__abs__int__diff,axiom,
! [A: nat,B2: nat] :
( ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ B2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ A @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1156_nat__intermed__int__val,axiom,
! [M2: nat,N2: nat,F: nat > int,K2: int] :
( ! [I4: nat] :
( ( ( ord_less_eq_nat @ M2 @ I4 )
& ( ord_less_nat @ I4 @ N2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ M2 @ I4 )
& ( ord_less_eq_nat @ I4 @ N2 )
& ( ( F @ I4 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1157_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I4: nat] :
( ( Q @ I4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I4 ) )
& ( ord_less_eq_real @ ( X3 @ I4 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X2: nat > real,I2: nat] : ( ord_less_eq_nat @ ( L3 @ X2 @ I2 ) @ one_one_nat )
& ! [X2: nat > real,I2: nat] :
( ( ( P @ X2 )
& ( Q @ I2 )
& ( ( X2 @ I2 )
= zero_zero_real ) )
=> ( ( L3 @ X2 @ I2 )
= zero_zero_nat ) )
& ! [X2: nat > real,I2: nat] :
( ( ( P @ X2 )
& ( Q @ I2 )
& ( ( X2 @ I2 )
= one_one_real ) )
=> ( ( L3 @ X2 @ I2 )
= one_one_nat ) )
& ! [X2: nat > real,I2: nat] :
( ( ( P @ X2 )
& ( Q @ I2 )
& ( ( L3 @ X2 @ I2 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X2 @ I2 ) @ ( F @ X2 @ I2 ) ) )
& ! [X2: nat > real,I2: nat] :
( ( ( P @ X2 )
& ( Q @ I2 )
& ( ( L3 @ X2 @ I2 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X2 @ I2 ) @ ( X2 @ I2 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1158_real__of__nat__ge__one__iff,axiom,
! [N2: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1159_lemma__interval,axiom,
! [A: real,X: real,B2: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
=> ( ( ord_less_eq_real @ A @ Y5 )
& ( ord_less_eq_real @ Y5 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1160_lemma__interval__lt,axiom,
! [A: real,X: real,B2: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
=> ( ( ord_less_real @ A @ Y5 )
& ( ord_less_real @ Y5 @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1161_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K2: int] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N2 ) )
=> ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N2 )
& ( ( F @ I4 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1162_aset_I8_J,axiom,
! [D4: int,A5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A5 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X2 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_1163_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1164_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_1165_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_1166_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1167_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1168_diff__diff__left,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K2 )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1169_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_1170_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1171_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1172_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1173_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1174_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1175_nat__abs__triangle__ineq,axiom,
! [K2: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K2 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1176_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_1177_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1178_nat__arith_Osuc1,axiom,
! [A5: nat,K2: nat,A: nat] :
( ( A5
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1179_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1180_diff__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1181_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1182_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1183_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_1184_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1185_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K4: nat] :
( N
= ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1186_trans__le__add2,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1187_trans__le__add1,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1188_add__le__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1189_add__le__mono,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1190_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1191_add__leD2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_1192_add__leD1,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1193_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1194_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1195_add__leE,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_1196_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1197_trans__less__add2,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1198_trans__less__add1,axiom,
! [I3: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1199_add__less__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1200_not__add__less2,axiom,
! [J: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1201_not__add__less1,axiom,
! [I3: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).
% not_add_less1
thf(fact_1202_add__less__mono,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1203_add__lessD1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K2 )
=> ( ord_less_nat @ I3 @ K2 ) ) ).
% add_lessD1
thf(fact_1204_atLeastLessThan__add__Un,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( set_or4665077453230672383an_nat @ I3 @ ( plus_plus_nat @ J @ K2 ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1205_less__diff__conv,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1206_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1207_le__diff__conv,axiom,
! [J: nat,K2: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1208_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1209_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K2 )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1210_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1211_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1212_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1213_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1214_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1215_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1216_less__add__Suc2,axiom,
! [I3: nat,M2: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M2 @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_1217_less__add__Suc1,axiom,
! [I3: nat,M2: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1218_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1219_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I3 @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1220_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_1221_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_1222_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z5: int] :
? [N: nat] :
( Z5
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1223_zadd__int__left,axiom,
! [M2: nat,N2: nat,Z3: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_1224_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_1225_int__ops_I5_J,axiom,
! [A: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_1226_nat__int__add,axiom,
! [A: nat,B2: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
= ( plus_plus_nat @ A @ B2 ) ) ).
% nat_int_add
thf(fact_1227_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or5832277885323065728an_int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_1228_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_1229_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( set_or6656581121297822940st_int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_1230_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1231_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1232_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1233_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ( ( ord_less_nat @ A @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1234_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1235_less__diff__conv2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1236_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1237_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1238_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z5: int] :
? [N: nat] :
( Z5
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1239_atLeastAtMostPlus1__int__conv,axiom,
! [M2: int,N2: int] :
( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N2 ) )
=> ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N2 ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M2 @ N2 ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_1240_simp__from__to,axiom,
( set_or1266510415728281911st_int
= ( ^ [I5: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I5 ) @ bot_bot_set_int @ ( insert_int @ I5 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_1241_bset_I1_J,axiom,
! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
=> ( ( P @ ( minus_minus_int @ X2 @ D4 ) )
& ( Q @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_1242_bset_I2_J,axiom,
! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
=> ( ( P @ ( minus_minus_int @ X2 @ D4 ) )
| ( Q @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_1243_aset_I1_J,axiom,
! [D4: int,A5: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A5 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
=> ( ( P @ ( plus_plus_int @ X2 @ D4 ) )
& ( Q @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_1244_aset_I2_J,axiom,
! [D4: int,A5: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A5 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
=> ( ( P @ ( plus_plus_int @ X2 @ D4 ) )
| ( Q @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_1245_kuhn__lemma,axiom,
! [P5: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P5 )
=> ( ! [X3: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I2 ) @ P5 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ( ( Label @ X3 @ I4 )
= zero_zero_nat )
| ( ( Label @ X3 @ I4 )
= one_one_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I2 ) @ P5 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ( ( X3 @ I4 )
= zero_zero_nat )
=> ( ( Label @ X3 @ I4 )
= zero_zero_nat ) ) ) )
=> ( ! [X3: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_nat @ ( X3 @ I2 ) @ P5 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ( ( X3 @ I4 )
= P5 )
=> ( ( Label @ X3 @ I4 )
= one_one_nat ) ) ) )
=> ~ ! [Q3: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_nat @ ( Q3 @ I2 ) @ P5 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ? [R3: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J4 ) @ ( R3 @ J4 ) )
& ( ord_less_eq_nat @ ( R3 @ J4 ) @ ( plus_plus_nat @ ( Q3 @ J4 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q3 @ J4 ) @ ( S3 @ J4 ) )
& ( ord_less_eq_nat @ ( S3 @ J4 ) @ ( plus_plus_nat @ ( Q3 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I2 )
!= ( Label @ S3 @ I2 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1246_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1247_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1248_bset_I3_J,axiom,
! [D4: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X2 = T )
=> ( ( minus_minus_int @ X2 @ D4 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_1249_bset_I4_J,axiom,
! [D4: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ B5 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X2 != T )
=> ( ( minus_minus_int @ X2 @ D4 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_1250_bset_I5_J,axiom,
! [D4: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X2 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X2 @ D4 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_1251_bset_I7_J,axiom,
! [D4: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ B5 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X2 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_1252_aset_I3_J,axiom,
! [D4: int,T: int,A5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A5 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A5 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X2 = T )
=> ( ( plus_plus_int @ X2 @ D4 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_1253_aset_I4_J,axiom,
! [D4: int,T: int,A5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ A5 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A5 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X2 != T )
=> ( ( plus_plus_int @ X2 @ D4 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_1254_aset_I5_J,axiom,
! [D4: int,T: int,A5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ A5 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A5 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X2 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_1255_aset_I7_J,axiom,
! [D4: int,A5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A5 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X2 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_1256_bset_I6_J,axiom,
! [D4: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X2 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D4 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_1257_bset_I8_J,axiom,
! [D4: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X2 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_1258_aset_I6_J,axiom,
! [D4: int,T: int,A5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A5 )
=> ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A5 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X2 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D4 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_1259_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N: int,M: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M ) ) ) ) ).
% int_less_real_le
thf(fact_1260_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N: int,M: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1261_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N: nat,M: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% nat_less_real_le
thf(fact_1262_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N: nat,M: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
% Helper facts (5)
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_3_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $true @ X @ Y )
= X ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
ord_less_nat @ j @ n ).
thf(conj_1,hypothesis,
ord_less_nat @ i @ j ).
thf(conj_2,conjecture,
ord_less_eq_b @ ( sort_map_b @ f @ n @ i ) @ ( sort_map_b @ f @ n @ j ) ).
%------------------------------------------------------------------------------