TPTP Problem File: SLH0376^1.p
View Solutions
- 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 : Dedekind_Real/0000_Dedekind_Real/prob_00313_008438__5618342_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1370 ( 609 unt; 94 typ; 0 def)
% Number of atoms : 3749 (1228 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10199 ( 349 ~; 98 |; 232 &;8084 @)
% ( 0 <=>;1436 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 369 ( 369 >; 0 *; 0 +; 0 <<)
% Number of symbols : 90 ( 89 usr; 12 con; 0-4 aty)
% Number of variables : 3383 ( 202 ^;3131 !; 50 ?;3383 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:20:21.935
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
set_set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Rat__Orat_M_Eo_J_J,type,
set_rat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
% Explicit typings (89)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Rat__Orat_M_Eo_J,type,
comple2477142665972227838_rat_o: set_rat_o > rat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Rat__Orat_J,type,
comple4298007329820168263et_rat: set_set_rat > set_rat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Rat__Orat,type,
condit1103211067700513672ow_rat: set_rat > $o ).
thf(sy_c_Dedekind__Real_Oadd__set,type,
dedekind_add_set: set_rat > set_rat > set_rat ).
thf(sy_c_Dedekind__Real_Ocut,type,
dedekind_cut: set_rat > $o ).
thf(sy_c_Dedekind__Real_Omult__set,type,
dedekind_mult_set: set_rat > set_rat > set_rat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
inverse_inverse_rat: rat > rat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
minus_minus_rat: rat > rat > rat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Rat__Orat_J,type,
minus_minus_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
one_one_rat: rat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Rat__Orat_M_Eo_J,type,
inf_inf_rat_o: ( rat > $o ) > ( rat > $o ) > rat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Rat__Orat,type,
inf_inf_rat: rat > rat > rat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Rat__Orat_J,type,
inf_inf_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
inf_inf_set_set_rat: set_set_rat > set_set_rat > set_set_rat ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Rat__Orat_J,type,
semila7382412365081457248et_rat: ( set_rat > set_rat > set_rat ) > set_rat > ( set_rat > set_rat > $o ) > ( set_rat > set_rat > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Rat__Orat_M_Eo_J,type,
sup_sup_rat_o: ( rat > $o ) > ( rat > $o ) > rat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
sup_sup_rat: rat > rat > rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
sup_sup_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
sup_sup_set_set_rat: set_set_rat > set_set_rat > set_set_rat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Rat__Orat_M_Eo_J,type,
bot_bot_rat_o: rat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
bot_bot_set_rat_o: set_rat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
bot_bot_set_rat: set_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
bot_bot_set_set_rat: set_set_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
bot_bo6619408370577057422et_rat: set_set_set_rat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Rat__Orat_M_Eo_J,type,
ord_less_rat_o: ( rat > $o ) > ( rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
ord_less_set_set_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
ord_le8797576461720604238et_rat: set_set_set_rat > set_set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Rat__Orat_M_Eo_J,type,
ord_less_eq_rat_o: ( rat > $o ) > ( rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
ord_le513522071413781156et_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
ord_le8552383839478139994et_rat: set_set_set_rat > set_set_set_rat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Rat__Orat,type,
order_Greatest_rat: ( rat > $o ) > rat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Rat__Orat_J,type,
order_2216579580035808117et_rat: ( set_rat > $o ) > set_rat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
order_4122807098444226603et_rat: ( set_set_rat > $o ) > set_set_rat ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Rat__Orat_J,type,
ordering_top_set_rat: ( set_rat > set_rat > $o ) > ( set_rat > set_rat > $o ) > set_rat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Rat__Orat_M_Eo_J,type,
top_top_rat_o: rat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Rat__Orat_J,type,
top_top_set_rat: set_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
top_top_set_set_rat: set_set_rat ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Rat__Orat,type,
field_2639924705303425560at_rat: rat > rat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
divide_divide_rat: rat > rat > rat ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Rat__Orat_J,type,
collect_set_rat: ( set_rat > $o ) > set_set_rat ).
thf(sy_c_Set_Oimage_001_062_It__Rat__Orat_M_Eo_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_rat_o_set_rat: ( ( rat > $o ) > set_rat ) > set_rat_o > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Rat__Orat,type,
image_rat_rat: ( rat > rat ) > set_rat > set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_3939399684171694371et_rat: ( set_rat > set_rat ) > set_set_rat > set_set_rat ).
thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
insert_rat: rat > set_rat > set_rat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Rat__Orat_J,type,
insert_set_rat: set_rat > set_set_rat > set_set_rat ).
thf(sy_c_Set_Ois__empty_001t__Rat__Orat,type,
is_empty_rat: set_rat > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__Rat__Orat_J,type,
is_empty_set_rat: set_set_rat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Rat__Orat,type,
is_singleton_rat: set_rat > $o ).
thf(sy_c_Set_Opairwise_001t__Rat__Orat,type,
pairwise_rat: ( rat > rat > $o ) > set_rat > $o ).
thf(sy_c_Set_Oremove_001t__Rat__Orat,type,
remove_rat: rat > set_rat > set_rat ).
thf(sy_c_Set_Othe__elem_001t__Rat__Orat,type,
the_elem_rat: set_rat > rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Rat__Orat_J,type,
set_or1040488700251649177et_rat: set_rat > set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_or2757889799628458319et_rat: set_set_rat > set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
set_or4029947393144176647an_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or32047845639629757et_rat: set_rat > set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_or8253465997870395507et_rat: set_set_rat > set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Rat__Orat,type,
set_ord_atLeast_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Rat__Orat_J,type,
set_or7446828528931440131et_rat: set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_or4639696602114667193et_rat: set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
set_ord_atMost_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Rat__Orat_J,type,
set_or728397472755099399et_rat: set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Rat__Orat,type,
set_or6023941531720377480st_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Rat__Orat_J,type,
set_or3565782072395811902et_rat: set_rat > set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_or1109970963052301556et_rat: set_set_rat > set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Rat__Orat,type,
set_or5199638295745620268an_rat: rat > rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or5117453967338258658et_rat: set_rat > set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_or7639671272556130712et_rat: set_set_rat > set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Rat__Orat,type,
set_or575021546402375026an_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or6174011595382531368et_rat: set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_or6674600949247491550et_rat: set_set_rat > set_set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or6605270734133118763et_rat: set_rat > set_set_rat ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Rat__Orat_J,type,
member_set_rat: set_rat > set_set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
member_set_set_rat: set_set_rat > set_set_set_rat > $o ).
thf(sy_v_A,type,
a: set_rat ).
thf(sy_v_B,type,
b: set_rat ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_thesisa____,type,
thesisa: $o ).
% Relevant facts (1274)
thf(fact_0_preal__exists__bound,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
& ~ ( member_rat @ X @ A ) ) ) ).
% preal_exists_bound
thf(fact_1_assms_I2_J,axiom,
dedekind_cut @ b ).
% assms(2)
thf(fact_2_assms_I1_J,axiom,
dedekind_cut @ a ).
% assms(1)
thf(fact_3_that,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X2 )
=> ( ~ ( member_rat @ X2 @ a )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ~ ( member_rat @ Y @ b )
=> thesisa ) ) ) ) ).
% that
thf(fact_4__092_060open_062_092_060And_062q_O_A_092_060lbrakk_0620_A_060_Aq_059_Aq_A_092_060notin_062_Amult__set_AA_AB_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_____092_060close_062,axiom,
! [Q: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q )
=> ( ~ ( member_rat @ Q @ ( dedekind_mult_set @ a @ b ) )
=> thesis ) ) ).
% \<open>\<And>q. \<lbrakk>0 < q; q \<notin> mult_set A B\<rbrakk> \<Longrightarrow> thesis__\<close>
thf(fact_5_preal__imp__pos,axiom,
! [A: set_rat,R: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ R @ A )
=> ( ord_less_rat @ zero_zero_rat @ R ) ) ) ).
% preal_imp_pos
thf(fact_6_preal__nonempty,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] :
( ( member_rat @ X @ A )
& ( ord_less_rat @ zero_zero_rat @ X ) ) ) ).
% preal_nonempty
thf(fact_7_not__in__preal__ub,axiom,
! [A: set_rat,X2: rat,Y: rat] :
( ( dedekind_cut @ A )
=> ( ~ ( member_rat @ X2 @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ X2 )
=> ( ord_less_rat @ Y @ X2 ) ) ) ) ) ).
% not_in_preal_ub
thf(fact_8_preal__downwards__closed,axiom,
! [A: set_rat,Y: rat,Z: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ Z @ Y )
=> ( member_rat @ Z @ A ) ) ) ) ) ).
% preal_downwards_closed
thf(fact_9_preal__exists__greater,axiom,
! [A: set_rat,Y: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y @ A )
=> ? [X: rat] :
( ( member_rat @ X @ A )
& ( ord_less_rat @ Y @ X ) ) ) ) ).
% preal_exists_greater
thf(fact_10_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_11_preal__Ex__mem,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] : ( member_rat @ X @ A ) ) ).
% preal_Ex_mem
thf(fact_12_zero__reorient,axiom,
! [X2: rat] :
( ( zero_zero_rat = X2 )
= ( X2 = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_13_calculation,axiom,
ord_less_set_rat @ bot_bot_set_rat @ ( dedekind_mult_set @ a @ b ) ).
% calculation
thf(fact_14_field__lbound__gt__zero,axiom,
! [D1: rat,D2: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D2 )
=> ? [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
& ( ord_less_rat @ E @ D1 )
& ( ord_less_rat @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_15_mem__add__set,axiom,
! [A: set_rat,B: set_rat] :
( ( dedekind_cut @ A )
=> ( ( dedekind_cut @ B )
=> ( dedekind_cut @ ( dedekind_add_set @ A @ B ) ) ) ) ).
% mem_add_set
thf(fact_16_preal__downwards__closed_H,axiom,
! [A: set_rat,Y: rat,Z: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ Z @ Y )
=> ( member_rat @ Z @ A ) ) ) ) ) ).
% preal_downwards_closed'
thf(fact_17_zero__less__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( field_2639924705303425560at_rat @ R ) )
= ( ord_less_rat @ zero_zero_rat @ R ) ) ).
% zero_less_of_rat_iff
thf(fact_18_of__rat__less__0__iff,axiom,
! [R: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ zero_zero_rat )
= ( ord_less_rat @ R @ zero_zero_rat ) ) ).
% of_rat_less_0_iff
thf(fact_19_order__less__imp__not__less,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_set_rat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_20_order__less__imp__not__less,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ~ ( ord_less_rat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_21_order__less__imp__not__less,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_rat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_22_order__less__imp__not__eq2,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_23_order__less__imp__not__eq2,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_24_order__less__imp__not__eq2,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_25_order__less__imp__not__eq,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_26_order__less__imp__not__eq,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_27_order__less__imp__not__eq,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_28_linorder__less__linear,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_rat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_29_order__less__imp__triv,axiom,
! [X2: set_set_rat,Y: set_set_rat,P: $o] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ( ( ord_less_set_set_rat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_30_order__less__imp__triv,axiom,
! [X2: rat,Y: rat,P: $o] :
( ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_rat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_31_order__less__imp__triv,axiom,
! [X2: set_rat,Y: set_rat,P: $o] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ( ord_less_set_rat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_32_order__less__not__sym,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_set_rat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_33_order__less__not__sym,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ~ ( ord_less_rat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_34_order__less__not__sym,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_rat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_35_dual__order_Orefl,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_36_dual__order_Orefl,axiom,
! [A2: set_set_rat] : ( ord_le513522071413781156et_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_37_dual__order_Orefl,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_38_order__refl,axiom,
! [X2: set_rat] : ( ord_less_eq_set_rat @ X2 @ X2 ) ).
% order_refl
thf(fact_39_order__refl,axiom,
! [X2: set_set_rat] : ( ord_le513522071413781156et_rat @ X2 @ X2 ) ).
% order_refl
thf(fact_40_order__refl,axiom,
! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).
% order_refl
thf(fact_41_bot__apply,axiom,
( bot_bot_rat_o
= ( ^ [X3: rat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_42_of__rat__eq__iff,axiom,
! [A2: rat,B2: rat] :
( ( ( field_2639924705303425560at_rat @ A2 )
= ( field_2639924705303425560at_rat @ B2 ) )
= ( A2 = B2 ) ) ).
% of_rat_eq_iff
thf(fact_43_of__rat__0,axiom,
( ( field_2639924705303425560at_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% of_rat_0
thf(fact_44_of__rat__eq__0__iff,axiom,
! [A2: rat] :
( ( ( field_2639924705303425560at_rat @ A2 )
= zero_zero_rat )
= ( A2 = zero_zero_rat ) ) ).
% of_rat_eq_0_iff
thf(fact_45_zero__eq__of__rat__iff,axiom,
! [A2: rat] :
( ( zero_zero_rat
= ( field_2639924705303425560at_rat @ A2 ) )
= ( zero_zero_rat = A2 ) ) ).
% zero_eq_of_rat_iff
thf(fact_46_of__rat__le__0__iff,axiom,
! [R: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ zero_zero_rat )
= ( ord_less_eq_rat @ R @ zero_zero_rat ) ) ).
% of_rat_le_0_iff
thf(fact_47_zero__le__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( field_2639924705303425560at_rat @ R ) )
= ( ord_less_eq_rat @ zero_zero_rat @ R ) ) ).
% zero_le_of_rat_iff
thf(fact_48_order__antisym__conv,axiom,
! [Y: set_rat,X2: set_rat] :
( ( ord_less_eq_set_rat @ Y @ X2 )
=> ( ( ord_less_eq_set_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_49_order__antisym__conv,axiom,
! [Y: set_set_rat,X2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ Y @ X2 )
=> ( ( ord_le513522071413781156et_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_50_order__antisym__conv,axiom,
! [Y: rat,X2: rat] :
( ( ord_less_eq_rat @ Y @ X2 )
=> ( ( ord_less_eq_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_51_linorder__le__cases,axiom,
! [X2: rat,Y: rat] :
( ~ ( ord_less_eq_rat @ X2 @ Y )
=> ( ord_less_eq_rat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_52_ord__le__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > set_rat,C: set_rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_53_ord__le__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > set_set_rat,C: set_set_rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_54_ord__le__eq__subst,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > rat,C: rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_55_ord__le__eq__subst,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_56_ord__le__eq__subst,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_57_ord__le__eq__subst,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > rat,C: rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_ord__le__eq__subst,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_rat,C: set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_59_ord__le__eq__subst,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_ord__le__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_ord__eq__le__subst,axiom,
! [A2: set_rat,F: rat > set_rat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_62_ord__eq__le__subst,axiom,
! [A2: set_set_rat,F: rat > set_set_rat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_63_ord__eq__le__subst,axiom,
! [A2: rat,F: set_rat > rat,B2: set_rat,C: set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_64_ord__eq__le__subst,axiom,
! [A2: set_rat,F: set_rat > set_rat,B2: set_rat,C: set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_65_ord__eq__le__subst,axiom,
! [A2: set_set_rat,F: set_rat > set_set_rat,B2: set_rat,C: set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_66_ord__eq__le__subst,axiom,
! [A2: rat,F: set_set_rat > rat,B2: set_set_rat,C: set_set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_67_ord__eq__le__subst,axiom,
! [A2: set_rat,F: set_set_rat > set_rat,B2: set_set_rat,C: set_set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_68_ord__eq__le__subst,axiom,
! [A2: set_set_rat,F: set_set_rat > set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_69_ord__eq__le__subst,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_70_mem__Collect__eq,axiom,
! [A2: set_rat,P: set_rat > $o] :
( ( member_set_rat @ A2 @ ( collect_set_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A2: rat,P: rat > $o] :
( ( member_rat @ A2 @ ( collect_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_set_rat] :
( ( collect_set_rat
@ ^ [X3: set_rat] : ( member_set_rat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_rat] :
( ( collect_rat
@ ^ [X3: rat] : ( member_rat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_Collect__cong,axiom,
! [P: rat > $o,Q2: rat > $o] :
( ! [X: rat] :
( ( P @ X )
= ( Q2 @ X ) )
=> ( ( collect_rat @ P )
= ( collect_rat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_75_linorder__linear,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
| ( ord_less_eq_rat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_76_order__eq__refl,axiom,
! [X2: set_rat,Y: set_rat] :
( ( X2 = Y )
=> ( ord_less_eq_set_rat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_77_order__eq__refl,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( X2 = Y )
=> ( ord_le513522071413781156et_rat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_78_order__eq__refl,axiom,
! [X2: rat,Y: rat] :
( ( X2 = Y )
=> ( ord_less_eq_rat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_79_order__subst2,axiom,
! [A2: rat,B2: rat,F: rat > set_rat,C: set_rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_80_order__subst2,axiom,
! [A2: rat,B2: rat,F: rat > set_set_rat,C: set_set_rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_81_order__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > rat,C: rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_82_order__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_83_order__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_84_order__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > rat,C: rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_85_order__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_rat,C: set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_86_order__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst2,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst1,axiom,
! [A2: rat,F: set_rat > rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_89_order__subst1,axiom,
! [A2: rat,F: set_set_rat > rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_90_order__subst1,axiom,
! [A2: set_rat,F: rat > set_rat,B2: rat,C: rat] :
( ( ord_less_eq_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_91_order__subst1,axiom,
! [A2: set_rat,F: set_rat > set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_92_order__subst1,axiom,
! [A2: set_rat,F: set_set_rat > set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_eq_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_93_order__subst1,axiom,
! [A2: set_set_rat,F: rat > set_set_rat,B2: rat,C: rat] :
( ( ord_le513522071413781156et_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_94_order__subst1,axiom,
! [A2: set_set_rat,F: set_rat > set_set_rat,B2: set_rat,C: set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_95_order__subst1,axiom,
! [A2: set_set_rat,F: set_set_rat > set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le513522071413781156et_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_96_order__subst1,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_97_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_rat,Z2: set_rat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B3 )
& ( ord_less_eq_set_rat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_98_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_rat,Z2: set_set_rat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_rat,B3: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A3 @ B3 )
& ( ord_le513522071413781156et_rat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_99_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
& ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_100_bot__fun__def,axiom,
( bot_bot_rat_o
= ( ^ [X3: rat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_101_of__rat__less__eq,axiom,
! [R: rat,S: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ ( field_2639924705303425560at_rat @ S ) )
= ( ord_less_eq_rat @ R @ S ) ) ).
% of_rat_less_eq
thf(fact_102_antisym,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_103_antisym,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_104_antisym,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_105_bot_Oextremum__uniqueI,axiom,
! [A2: rat > $o] :
( ( ord_less_eq_rat_o @ A2 @ bot_bot_rat_o )
=> ( A2 = bot_bot_rat_o ) ) ).
% bot.extremum_uniqueI
thf(fact_106_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ bot_bot_set_set_rat )
=> ( A2 = bot_bot_set_set_rat ) ) ).
% bot.extremum_uniqueI
thf(fact_107_bot_Oextremum__uniqueI,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ bot_bot_set_rat )
=> ( A2 = bot_bot_set_rat ) ) ).
% bot.extremum_uniqueI
thf(fact_108_bot_Oextremum__unique,axiom,
! [A2: rat > $o] :
( ( ord_less_eq_rat_o @ A2 @ bot_bot_rat_o )
= ( A2 = bot_bot_rat_o ) ) ).
% bot.extremum_unique
thf(fact_109_bot_Oextremum__unique,axiom,
! [A2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ bot_bot_set_set_rat )
= ( A2 = bot_bot_set_set_rat ) ) ).
% bot.extremum_unique
thf(fact_110_bot_Oextremum__unique,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ bot_bot_set_rat )
= ( A2 = bot_bot_set_rat ) ) ).
% bot.extremum_unique
thf(fact_111_dual__order_Otrans,axiom,
! [B2: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ B2 @ A2 )
=> ( ( ord_less_eq_set_rat @ C @ B2 )
=> ( ord_less_eq_set_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_112_dual__order_Otrans,axiom,
! [B2: set_set_rat,A2: set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B2 @ A2 )
=> ( ( ord_le513522071413781156et_rat @ C @ B2 )
=> ( ord_le513522071413781156et_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_113_dual__order_Otrans,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ B2 )
=> ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_114_dual__order_Oantisym,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_eq_set_rat @ B2 @ A2 )
=> ( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_115_dual__order_Oantisym,axiom,
! [B2: set_set_rat,A2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B2 @ A2 )
=> ( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_116_dual__order_Oantisym,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_117_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_rat,Z2: set_rat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ B3 @ A3 )
& ( ord_less_eq_set_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_118_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_rat,Z2: set_set_rat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_rat,B3: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B3 @ A3 )
& ( ord_le513522071413781156et_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_119_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
& ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_120_bot_Oextremum,axiom,
! [A2: rat > $o] : ( ord_less_eq_rat_o @ bot_bot_rat_o @ A2 ) ).
% bot.extremum
thf(fact_121_bot_Oextremum,axiom,
! [A2: set_set_rat] : ( ord_le513522071413781156et_rat @ bot_bot_set_set_rat @ A2 ) ).
% bot.extremum
thf(fact_122_bot_Oextremum,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ bot_bot_set_rat @ A2 ) ).
% bot.extremum
thf(fact_123_linorder__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B2: rat] :
( ! [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: rat,B4: rat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_124_order__trans,axiom,
! [X2: set_rat,Y: set_rat,Z: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ord_less_eq_set_rat @ Y @ Z )
=> ( ord_less_eq_set_rat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_125_order__trans,axiom,
! [X2: set_set_rat,Y: set_set_rat,Z: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X2 @ Y )
=> ( ( ord_le513522071413781156et_rat @ Y @ Z )
=> ( ord_le513522071413781156et_rat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_126_order__trans,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_127_order_Otrans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_128_order_Otrans,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ord_le513522071413781156et_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_129_order_Otrans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_130_order__antisym,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ord_less_eq_set_rat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_131_order__antisym,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X2 @ Y )
=> ( ( ord_le513522071413781156et_rat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_132_order__antisym,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_133_ord__le__eq__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_134_ord__le__eq__trans,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le513522071413781156et_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_135_ord__le__eq__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_136_ord__eq__le__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_137_ord__eq__le__trans,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( A2 = B2 )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ord_le513522071413781156et_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_138_ord__eq__le__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_139_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_rat,Z2: set_rat] : ( Y3 = Z2 ) )
= ( ^ [X3: set_rat,Y4: set_rat] :
( ( ord_less_eq_set_rat @ X3 @ Y4 )
& ( ord_less_eq_set_rat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_140_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_rat,Z2: set_set_rat] : ( Y3 = Z2 ) )
= ( ^ [X3: set_set_rat,Y4: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X3 @ Y4 )
& ( ord_le513522071413781156et_rat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_141_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
= ( ^ [X3: rat,Y4: rat] :
( ( ord_less_eq_rat @ X3 @ Y4 )
& ( ord_less_eq_rat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_142_le__cases3,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X2 @ Y )
=> ~ ( ord_less_eq_rat @ Y @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y @ X2 )
=> ~ ( ord_less_eq_rat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_rat @ X2 @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y )
=> ~ ( ord_less_eq_rat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_rat @ Y @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X2 )
=> ~ ( ord_less_eq_rat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_143_nle__le,axiom,
! [A2: rat,B2: rat] :
( ( ~ ( ord_less_eq_rat @ A2 @ B2 ) )
= ( ( ord_less_eq_rat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_144_bot_Oextremum__strict,axiom,
! [A2: rat > $o] :
~ ( ord_less_rat_o @ A2 @ bot_bot_rat_o ) ).
% bot.extremum_strict
thf(fact_145_bot_Oextremum__strict,axiom,
! [A2: set_set_rat] :
~ ( ord_less_set_set_rat @ A2 @ bot_bot_set_set_rat ) ).
% bot.extremum_strict
thf(fact_146_bot_Oextremum__strict,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ bot_bot_set_rat ) ).
% bot.extremum_strict
thf(fact_147_bot_Onot__eq__extremum,axiom,
! [A2: rat > $o] :
( ( A2 != bot_bot_rat_o )
= ( ord_less_rat_o @ bot_bot_rat_o @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_148_bot_Onot__eq__extremum,axiom,
! [A2: set_set_rat] :
( ( A2 != bot_bot_set_set_rat )
= ( ord_less_set_set_rat @ bot_bot_set_set_rat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_149_bot_Onot__eq__extremum,axiom,
! [A2: set_rat] :
( ( A2 != bot_bot_set_rat )
= ( ord_less_set_rat @ bot_bot_set_rat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_150_leD,axiom,
! [Y: set_set_rat,X2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ Y @ X2 )
=> ~ ( ord_less_set_set_rat @ X2 @ Y ) ) ).
% leD
thf(fact_151_leD,axiom,
! [Y: set_rat,X2: set_rat] :
( ( ord_less_eq_set_rat @ Y @ X2 )
=> ~ ( ord_less_set_rat @ X2 @ Y ) ) ).
% leD
thf(fact_152_leD,axiom,
! [Y: rat,X2: rat] :
( ( ord_less_eq_rat @ Y @ X2 )
=> ~ ( ord_less_rat @ X2 @ Y ) ) ).
% leD
thf(fact_153_leI,axiom,
! [X2: rat,Y: rat] :
( ~ ( ord_less_rat @ X2 @ Y )
=> ( ord_less_eq_rat @ Y @ X2 ) ) ).
% leI
thf(fact_154_nless__le,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ~ ( ord_less_set_set_rat @ A2 @ B2 ) )
= ( ~ ( ord_le513522071413781156et_rat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_155_nless__le,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ~ ( ord_less_set_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_rat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_156_nless__le,axiom,
! [A2: rat,B2: rat] :
( ( ~ ( ord_less_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_rat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_157_antisym__conv1,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ~ ( ord_less_set_set_rat @ X2 @ Y )
=> ( ( ord_le513522071413781156et_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_158_antisym__conv1,axiom,
! [X2: set_rat,Y: set_rat] :
( ~ ( ord_less_set_rat @ X2 @ Y )
=> ( ( ord_less_eq_set_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_159_antisym__conv1,axiom,
! [X2: rat,Y: rat] :
( ~ ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_160_antisym__conv2,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_set_rat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_161_antisym__conv2,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_rat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_162_antisym__conv2,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ~ ( ord_less_rat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_163_dense__ge,axiom,
! [Z: rat,Y: rat] :
( ! [X: rat] :
( ( ord_less_rat @ Z @ X )
=> ( ord_less_eq_rat @ Y @ X ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_ge
thf(fact_164_dense__le,axiom,
! [Y: rat,Z: rat] :
( ! [X: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ X @ Z ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_le
thf(fact_165_less__le__not__le,axiom,
( ord_less_set_set_rat
= ( ^ [X3: set_set_rat,Y4: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X3 @ Y4 )
& ~ ( ord_le513522071413781156et_rat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_166_less__le__not__le,axiom,
( ord_less_set_rat
= ( ^ [X3: set_rat,Y4: set_rat] :
( ( ord_less_eq_set_rat @ X3 @ Y4 )
& ~ ( ord_less_eq_set_rat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_167_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y4: rat] :
( ( ord_less_eq_rat @ X3 @ Y4 )
& ~ ( ord_less_eq_rat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_168_not__le__imp__less,axiom,
! [Y: rat,X2: rat] :
( ~ ( ord_less_eq_rat @ Y @ X2 )
=> ( ord_less_rat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_169_order_Oorder__iff__strict,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A3: set_set_rat,B3: set_set_rat] :
( ( ord_less_set_set_rat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_170_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_rat
= ( ^ [A3: set_rat,B3: set_rat] :
( ( ord_less_set_rat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_171_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_rat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_172_order_Ostrict__iff__order,axiom,
( ord_less_set_set_rat
= ( ^ [A3: set_set_rat,B3: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_173_order_Ostrict__iff__order,axiom,
( ord_less_set_rat
= ( ^ [A3: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_174_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_175_order_Ostrict__trans1,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ord_less_set_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_176_order_Ostrict__trans1,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_177_order_Ostrict__trans1,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_178_order_Ostrict__trans2,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ord_less_set_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_179_order_Ostrict__trans2,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_180_order_Ostrict__trans2,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_181_order_Ostrict__iff__not,axiom,
( ord_less_set_set_rat
= ( ^ [A3: set_set_rat,B3: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A3 @ B3 )
& ~ ( ord_le513522071413781156et_rat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_182_order_Ostrict__iff__not,axiom,
( ord_less_set_rat
= ( ^ [A3: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B3 )
& ~ ( ord_less_eq_set_rat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_183_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] :
( ( ord_less_eq_rat @ A3 @ B3 )
& ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_184_dense__ge__bounded,axiom,
! [Z: rat,X2: rat,Y: rat] :
( ( ord_less_rat @ Z @ X2 )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z @ W )
=> ( ( ord_less_rat @ W @ X2 )
=> ( ord_less_eq_rat @ Y @ W ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_185_dense__le__bounded,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ! [W: rat] :
( ( ord_less_rat @ X2 @ W )
=> ( ( ord_less_rat @ W @ Y )
=> ( ord_less_eq_rat @ W @ Z ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_186_dual__order_Oorder__iff__strict,axiom,
( ord_le513522071413781156et_rat
= ( ^ [B3: set_set_rat,A3: set_set_rat] :
( ( ord_less_set_set_rat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_187_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_rat
= ( ^ [B3: set_rat,A3: set_rat] :
( ( ord_less_set_rat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_188_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_rat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_189_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_rat
= ( ^ [B3: set_set_rat,A3: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_190_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_rat
= ( ^ [B3: set_rat,A3: set_rat] :
( ( ord_less_eq_set_rat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_191_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_192_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_rat,A2: set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B2 @ A2 )
=> ( ( ord_less_set_set_rat @ C @ B2 )
=> ( ord_less_set_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_193_dual__order_Ostrict__trans1,axiom,
! [B2: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ B2 @ A2 )
=> ( ( ord_less_set_rat @ C @ B2 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_194_dual__order_Ostrict__trans1,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ B2 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_195_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_rat,A2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ B2 @ A2 )
=> ( ( ord_le513522071413781156et_rat @ C @ B2 )
=> ( ord_less_set_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_196_dual__order_Ostrict__trans2,axiom,
! [B2: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ( ord_less_eq_set_rat @ C @ B2 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_197_dual__order_Ostrict__trans2,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ B2 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_198_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_rat
= ( ^ [B3: set_set_rat,A3: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B3 @ A3 )
& ~ ( ord_le513522071413781156et_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_199_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_rat
= ( ^ [B3: set_rat,A3: set_rat] :
( ( ord_less_eq_set_rat @ B3 @ A3 )
& ~ ( ord_less_eq_set_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_200_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B3: rat,A3: rat] :
( ( ord_less_eq_rat @ B3 @ A3 )
& ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_201_order_Ostrict__implies__order,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ord_le513522071413781156et_rat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_202_order_Ostrict__implies__order,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ord_less_eq_set_rat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_203_order_Ostrict__implies__order,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ord_less_eq_rat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_204_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_rat,A2: set_set_rat] :
( ( ord_less_set_set_rat @ B2 @ A2 )
=> ( ord_le513522071413781156et_rat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_205_dual__order_Ostrict__implies__order,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ord_less_eq_set_rat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_206_dual__order_Ostrict__implies__order,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_207_order__le__less,axiom,
( ord_le513522071413781156et_rat
= ( ^ [X3: set_set_rat,Y4: set_set_rat] :
( ( ord_less_set_set_rat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_208_order__le__less,axiom,
( ord_less_eq_set_rat
= ( ^ [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_209_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_210_order__less__le,axiom,
( ord_less_set_set_rat
= ( ^ [X3: set_set_rat,Y4: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_211_order__less__le,axiom,
( ord_less_set_rat
= ( ^ [X3: set_rat,Y4: set_rat] :
( ( ord_less_eq_set_rat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_212_order__less__le,axiom,
( ord_less_rat
= ( ^ [X3: rat,Y4: rat] :
( ( ord_less_eq_rat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_213_linorder__not__le,axiom,
! [X2: rat,Y: rat] :
( ( ~ ( ord_less_eq_rat @ X2 @ Y ) )
= ( ord_less_rat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_214_linorder__not__less,axiom,
! [X2: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X2 @ Y ) )
= ( ord_less_eq_rat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_215_order__less__imp__le,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ( ord_le513522071413781156et_rat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_216_order__less__imp__le,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ord_less_eq_set_rat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_217_order__less__imp__le,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ord_less_eq_rat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_218_order__le__neq__trans,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_rat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_219_order__le__neq__trans,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_rat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_220_order__le__neq__trans,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_rat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_221_order__neq__le__trans,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( A2 != B2 )
=> ( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ord_less_set_set_rat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_222_order__neq__le__trans,axiom,
! [A2: set_rat,B2: set_rat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ord_less_set_rat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_223_order__neq__le__trans,axiom,
! [A2: rat,B2: rat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ord_less_rat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_224_order__le__less__trans,axiom,
! [X2: set_set_rat,Y: set_set_rat,Z: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X2 @ Y )
=> ( ( ord_less_set_set_rat @ Y @ Z )
=> ( ord_less_set_set_rat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_225_order__le__less__trans,axiom,
! [X2: set_rat,Y: set_rat,Z: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ord_less_set_rat @ Y @ Z )
=> ( ord_less_set_rat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_226_order__le__less__trans,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_227_order__less__le__trans,axiom,
! [X2: set_set_rat,Y: set_set_rat,Z: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ( ( ord_le513522071413781156et_rat @ Y @ Z )
=> ( ord_less_set_set_rat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_228_order__less__le__trans,axiom,
! [X2: set_rat,Y: set_rat,Z: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ( ord_less_eq_set_rat @ Y @ Z )
=> ( ord_less_set_rat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_229_order__less__le__trans,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_rat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_230_order__le__less__subst1,axiom,
! [A2: rat,F: set_set_rat > rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_231_order__le__less__subst1,axiom,
! [A2: set_rat,F: set_set_rat > set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_eq_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_232_order__le__less__subst1,axiom,
! [A2: set_set_rat,F: rat > set_set_rat,B2: rat,C: rat] :
( ( ord_le513522071413781156et_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_233_order__le__less__subst1,axiom,
! [A2: set_set_rat,F: set_rat > set_set_rat,B2: set_rat,C: set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_234_order__le__less__subst1,axiom,
! [A2: set_set_rat,F: set_set_rat > set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_235_order__le__less__subst1,axiom,
! [A2: set_rat,F: rat > set_rat,B2: rat,C: rat] :
( ( ord_less_eq_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_236_order__le__less__subst1,axiom,
! [A2: set_rat,F: set_rat > set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_237_order__le__less__subst1,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_238_order__le__less__subst1,axiom,
! [A2: rat,F: set_rat > rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_239_order__le__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > set_set_rat,C: set_set_rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_set_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_240_order__le__less__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > rat,C: rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_241_order__le__less__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_242_order__le__less__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_243_order__le__less__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > rat,C: rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_244_order__le__less__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_rat,C: set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_245_order__le__less__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_less_set_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_246_order__le__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > set_rat,C: set_rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_247_order__le__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_248_order__less__le__subst1,axiom,
! [A2: set_set_rat,F: rat > set_set_rat,B2: rat,C: rat] :
( ( ord_less_set_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_249_order__less__le__subst1,axiom,
! [A2: rat,F: set_rat > rat,B2: set_rat,C: set_rat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_250_order__less__le__subst1,axiom,
! [A2: set_rat,F: set_rat > set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_251_order__less__le__subst1,axiom,
! [A2: set_set_rat,F: set_rat > set_set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_252_order__less__le__subst1,axiom,
! [A2: rat,F: set_set_rat > rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_253_order__less__le__subst1,axiom,
! [A2: set_rat,F: set_set_rat > set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_254_order__less__le__subst1,axiom,
! [A2: set_set_rat,F: set_set_rat > set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le513522071413781156et_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X @ Y2 )
=> ( ord_le513522071413781156et_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_255_order__less__le__subst1,axiom,
! [A2: set_rat,F: rat > set_rat,B2: rat,C: rat] :
( ( ord_less_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_256_order__less__le__subst1,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_257_order__less__le__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > rat,C: rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_258_order__less__le__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_rat,C: set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_259_order__less__le__subst2,axiom,
! [A2: rat,B2: rat,F: rat > set_set_rat,C: set_set_rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_260_order__less__le__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_261_order__less__le__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_262_order__less__le__subst2,axiom,
! [A2: rat,B2: rat,F: rat > set_rat,C: set_rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_263_order__less__le__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_264_order__less__le__subst2,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_265_order__less__le__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > rat,C: rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_266_linorder__le__less__linear,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
| ( ord_less_rat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_267_order__le__imp__less__or__eq,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X2 @ Y )
=> ( ( ord_less_set_set_rat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_268_order__le__imp__less__or__eq,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ Y )
=> ( ( ord_less_set_rat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_269_order__le__imp__less__or__eq,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_rat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_270_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% less_eq_rat_def
thf(fact_271_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_272_of__rat__less,axiom,
! [R: rat,S: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ ( field_2639924705303425560at_rat @ S ) )
= ( ord_less_rat @ R @ S ) ) ).
% of_rat_less
thf(fact_273_lt__ex,axiom,
! [X2: rat] :
? [Y2: rat] : ( ord_less_rat @ Y2 @ X2 ) ).
% lt_ex
thf(fact_274_gt__ex,axiom,
! [X2: rat] :
? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_275_dense,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ? [Z3: rat] :
( ( ord_less_rat @ X2 @ Z3 )
& ( ord_less_rat @ Z3 @ Y ) ) ) ).
% dense
thf(fact_276_less__imp__neq,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_277_less__imp__neq,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_278_less__imp__neq,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_279_order_Oasym,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ~ ( ord_less_set_set_rat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_280_order_Oasym,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ~ ( ord_less_rat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_281_order_Oasym,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ~ ( ord_less_set_rat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_282_ord__eq__less__trans,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( A2 = B2 )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ord_less_set_set_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_283_ord__eq__less__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( A2 = B2 )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_284_ord__eq__less__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( A2 = B2 )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_285_ord__less__eq__trans,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_set_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_286_ord__less__eq__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_287_ord__less__eq__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_288_antisym__conv3,axiom,
! [Y: rat,X2: rat] :
( ~ ( ord_less_rat @ Y @ X2 )
=> ( ( ~ ( ord_less_rat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_289_linorder__cases,axiom,
! [X2: rat,Y: rat] :
( ~ ( ord_less_rat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_rat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_290_dual__order_Oasym,axiom,
! [B2: set_set_rat,A2: set_set_rat] :
( ( ord_less_set_set_rat @ B2 @ A2 )
=> ~ ( ord_less_set_set_rat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_291_dual__order_Oasym,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ~ ( ord_less_rat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_292_dual__order_Oasym,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ~ ( ord_less_set_rat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_293_dual__order_Oirrefl,axiom,
! [A2: set_set_rat] :
~ ( ord_less_set_set_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_294_dual__order_Oirrefl,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_295_dual__order_Oirrefl,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_296_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B2: rat] :
( ! [A4: rat,B4: rat] :
( ( ord_less_rat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: rat] : ( P @ A4 @ A4 )
=> ( ! [A4: rat,B4: rat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_297_order_Ostrict__trans,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ord_less_set_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_298_order_Ostrict__trans,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_299_order_Ostrict__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_300_not__less__iff__gr__or__eq,axiom,
! [X2: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X2 @ Y ) )
= ( ( ord_less_rat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_301_dual__order_Ostrict__trans,axiom,
! [B2: set_set_rat,A2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ B2 @ A2 )
=> ( ( ord_less_set_set_rat @ C @ B2 )
=> ( ord_less_set_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_302_dual__order_Ostrict__trans,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ B2 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_303_dual__order_Ostrict__trans,axiom,
! [B2: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ( ord_less_set_rat @ C @ B2 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_304_order_Ostrict__implies__not__eq,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_305_order_Ostrict__implies__not__eq,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_306_order_Ostrict__implies__not__eq,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_307_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_set_rat,A2: set_set_rat] :
( ( ord_less_set_set_rat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_308_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_309_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_310_linorder__neqE,axiom,
! [X2: rat,Y: rat] :
( ( X2 != Y )
=> ( ~ ( ord_less_rat @ X2 @ Y )
=> ( ord_less_rat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_311_order__less__asym,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_set_rat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_312_order__less__asym,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ~ ( ord_less_rat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_313_order__less__asym,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ~ ( ord_less_set_rat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_314_linorder__neq__iff,axiom,
! [X2: rat,Y: rat] :
( ( X2 != Y )
= ( ( ord_less_rat @ X2 @ Y )
| ( ord_less_rat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_315_order__less__asym_H,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ~ ( ord_less_set_set_rat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_316_order__less__asym_H,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ~ ( ord_less_rat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_317_order__less__asym_H,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ~ ( ord_less_set_rat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_318_order__less__trans,axiom,
! [X2: set_set_rat,Y: set_set_rat,Z: set_set_rat] :
( ( ord_less_set_set_rat @ X2 @ Y )
=> ( ( ord_less_set_set_rat @ Y @ Z )
=> ( ord_less_set_set_rat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_319_order__less__trans,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_320_order__less__trans,axiom,
! [X2: set_rat,Y: set_rat,Z: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ( ord_less_set_rat @ Y @ Z )
=> ( ord_less_set_rat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_321_ord__eq__less__subst,axiom,
! [A2: set_set_rat,F: rat > set_set_rat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_322_ord__eq__less__subst,axiom,
! [A2: set_set_rat,F: set_rat > set_set_rat,B2: set_rat,C: set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_323_ord__eq__less__subst,axiom,
! [A2: rat,F: set_set_rat > rat,B2: set_set_rat,C: set_set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_324_ord__eq__less__subst,axiom,
! [A2: set_rat,F: set_set_rat > set_rat,B2: set_set_rat,C: set_set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_325_ord__eq__less__subst,axiom,
! [A2: set_set_rat,F: set_set_rat > set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_326_ord__eq__less__subst,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_327_ord__eq__less__subst,axiom,
! [A2: set_rat,F: rat > set_rat,B2: rat,C: rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_328_ord__eq__less__subst,axiom,
! [A2: rat,F: set_rat > rat,B2: set_rat,C: set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_329_ord__eq__less__subst,axiom,
! [A2: set_rat,F: set_rat > set_rat,B2: set_rat,C: set_rat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_330_ord__less__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > set_set_rat,C: set_set_rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_331_ord__less__eq__subst,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_332_ord__less__eq__subst,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > rat,C: rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_333_ord__less__eq__subst,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_rat,C: set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_334_ord__less__eq__subst,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_335_ord__less__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_336_ord__less__eq__subst,axiom,
! [A2: rat,B2: rat,F: rat > set_rat,C: set_rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_337_ord__less__eq__subst,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > rat,C: rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_338_ord__less__eq__subst,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_339_order__less__irrefl,axiom,
! [X2: set_set_rat] :
~ ( ord_less_set_set_rat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_340_order__less__irrefl,axiom,
! [X2: rat] :
~ ( ord_less_rat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_341_order__less__irrefl,axiom,
! [X2: set_rat] :
~ ( ord_less_set_rat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_342_order__less__subst1,axiom,
! [A2: rat,F: set_set_rat > rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_343_order__less__subst1,axiom,
! [A2: set_rat,F: set_set_rat > set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_344_order__less__subst1,axiom,
! [A2: set_set_rat,F: rat > set_set_rat,B2: rat,C: rat] :
( ( ord_less_set_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_345_order__less__subst1,axiom,
! [A2: set_set_rat,F: set_rat > set_set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_346_order__less__subst1,axiom,
! [A2: set_set_rat,F: set_set_rat > set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_set_rat @ B2 @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_347_order__less__subst1,axiom,
! [A2: rat,F: rat > rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_348_order__less__subst1,axiom,
! [A2: rat,F: set_rat > rat,B2: set_rat,C: set_rat] :
( ( ord_less_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_349_order__less__subst1,axiom,
! [A2: set_rat,F: rat > set_rat,B2: rat,C: rat] :
( ( ord_less_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_rat @ B2 @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_350_order__less__subst1,axiom,
! [A2: set_rat,F: set_rat > set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_351_order__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > set_set_rat,C: set_set_rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_set_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_352_order__less__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_353_order__less__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > rat,C: rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_354_order__less__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_rat,C: set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_355_order__less__subst2,axiom,
! [A2: set_set_rat,B2: set_set_rat,F: set_set_rat > set_set_rat,C: set_set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_set_rat,Y2: set_set_rat] :
( ( ord_less_set_set_rat @ X @ Y2 )
=> ( ord_less_set_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_356_order__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_357_order__less__subst2,axiom,
! [A2: rat,B2: rat,F: rat > set_rat,C: set_rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: rat,Y2: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_358_order__less__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > rat,C: rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_359_order__less__subst2,axiom,
! [A2: set_rat,B2: set_rat,F: set_rat > set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ ( F @ B2 ) @ C )
=> ( ! [X: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X @ Y2 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_360_empty__Collect__eq,axiom,
! [P: set_rat > $o] :
( ( bot_bot_set_set_rat
= ( collect_set_rat @ P ) )
= ( ! [X3: set_rat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_361_empty__Collect__eq,axiom,
! [P: rat > $o] :
( ( bot_bot_set_rat
= ( collect_rat @ P ) )
= ( ! [X3: rat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_362_Collect__empty__eq,axiom,
! [P: set_rat > $o] :
( ( ( collect_set_rat @ P )
= bot_bot_set_set_rat )
= ( ! [X3: set_rat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_363_Collect__empty__eq,axiom,
! [P: rat > $o] :
( ( ( collect_rat @ P )
= bot_bot_set_rat )
= ( ! [X3: rat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_364_all__not__in__conv,axiom,
! [A: set_set_rat] :
( ( ! [X3: set_rat] :
~ ( member_set_rat @ X3 @ A ) )
= ( A = bot_bot_set_set_rat ) ) ).
% all_not_in_conv
thf(fact_365_all__not__in__conv,axiom,
! [A: set_rat] :
( ( ! [X3: rat] :
~ ( member_rat @ X3 @ A ) )
= ( A = bot_bot_set_rat ) ) ).
% all_not_in_conv
thf(fact_366_empty__iff,axiom,
! [C: set_rat] :
~ ( member_set_rat @ C @ bot_bot_set_set_rat ) ).
% empty_iff
thf(fact_367_empty__iff,axiom,
! [C: rat] :
~ ( member_rat @ C @ bot_bot_set_rat ) ).
% empty_iff
thf(fact_368_Dedekind__Real_Ocut__def,axiom,
( dedekind_cut
= ( ^ [A5: set_rat] :
( ( ord_less_set_rat @ bot_bot_set_rat @ A5 )
& ( ord_less_set_rat @ A5 @ ( set_or575021546402375026an_rat @ zero_zero_rat ) )
& ! [X3: rat] :
( ( member_rat @ X3 @ A5 )
=> ( ! [Z4: rat] :
( ( ( ord_less_rat @ zero_zero_rat @ Z4 )
& ( ord_less_rat @ Z4 @ X3 ) )
=> ( member_rat @ Z4 @ A5 ) )
& ? [Y4: rat] :
( ( member_rat @ Y4 @ A5 )
& ( ord_less_rat @ X3 @ Y4 ) ) ) ) ) ) ) ).
% Dedekind_Real.cut_def
thf(fact_369_not__psubset__empty,axiom,
! [A: set_set_rat] :
~ ( ord_less_set_set_rat @ A @ bot_bot_set_set_rat ) ).
% not_psubset_empty
thf(fact_370_not__psubset__empty,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ A @ bot_bot_set_rat ) ).
% not_psubset_empty
thf(fact_371_minf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z3 )
=> ~ ( ord_less_eq_rat @ T @ X4 ) ) ).
% minf(8)
thf(fact_372_minf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z3 )
=> ( ord_less_eq_rat @ X4 @ T ) ) ).
% minf(6)
thf(fact_373_pinf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z3 @ X4 )
=> ( ord_less_eq_rat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_374_pinf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z3 @ X4 )
=> ~ ( ord_less_eq_rat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_375_verit__comp__simplify1_I3_J,axiom,
! [B5: rat,A6: rat] :
( ( ~ ( ord_less_eq_rat @ B5 @ A6 ) )
= ( ord_less_rat @ A6 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_376_subset__empty,axiom,
! [A: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ bot_bot_set_set_rat )
= ( A = bot_bot_set_set_rat ) ) ).
% subset_empty
thf(fact_377_subset__empty,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ A @ bot_bot_set_rat )
= ( A = bot_bot_set_rat ) ) ).
% subset_empty
thf(fact_378_empty__subsetI,axiom,
! [A: set_set_rat] : ( ord_le513522071413781156et_rat @ bot_bot_set_set_rat @ A ) ).
% empty_subsetI
thf(fact_379_empty__subsetI,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ bot_bot_set_rat @ A ) ).
% empty_subsetI
thf(fact_380_psubsetI,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_rat @ A @ B ) ) ) ).
% psubsetI
thf(fact_381_psubsetI,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_rat @ A @ B ) ) ) ).
% psubsetI
thf(fact_382_bot__set__def,axiom,
( bot_bot_set_set_rat
= ( collect_set_rat @ bot_bot_set_rat_o ) ) ).
% bot_set_def
thf(fact_383_bot__set__def,axiom,
( bot_bot_set_rat
= ( collect_rat @ bot_bot_rat_o ) ) ).
% bot_set_def
thf(fact_384_subset__iff__psubset__eq,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A5: set_set_rat,B6: set_set_rat] :
( ( ord_less_set_set_rat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_385_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B6: set_rat] :
( ( ord_less_set_rat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_386_subset__psubset__trans,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( ord_less_set_set_rat @ B @ C2 )
=> ( ord_less_set_set_rat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_387_subset__psubset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_388_subset__not__subset__eq,axiom,
( ord_less_set_set_rat
= ( ^ [A5: set_set_rat,B6: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A5 @ B6 )
& ~ ( ord_le513522071413781156et_rat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_389_subset__not__subset__eq,axiom,
( ord_less_set_rat
= ( ^ [A5: set_rat,B6: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B6 )
& ~ ( ord_less_eq_set_rat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_390_psubset__subset__trans,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( ord_less_set_set_rat @ A @ B )
=> ( ( ord_le513522071413781156et_rat @ B @ C2 )
=> ( ord_less_set_set_rat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_391_psubset__subset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_392_psubset__imp__subset,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ord_less_set_set_rat @ A @ B )
=> ( ord_le513522071413781156et_rat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_393_psubset__imp__subset,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_394_psubset__eq,axiom,
( ord_less_set_set_rat
= ( ^ [A5: set_set_rat,B6: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_395_psubset__eq,axiom,
( ord_less_set_rat
= ( ^ [A5: set_rat,B6: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_396_psubsetE,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ord_less_set_set_rat @ A @ B )
=> ~ ( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ord_le513522071413781156et_rat @ B @ A ) ) ) ).
% psubsetE
thf(fact_397_psubsetE,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ~ ( ( ord_less_eq_set_rat @ A @ B )
=> ( ord_less_eq_set_rat @ B @ A ) ) ) ).
% psubsetE
thf(fact_398_verit__comp__simplify1_I2_J,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_399_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_rat] : ( ord_le513522071413781156et_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_400_verit__comp__simplify1_I2_J,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_401_verit__la__disequality,axiom,
! [A2: rat,B2: rat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_rat @ A2 @ B2 )
| ~ ( ord_less_eq_rat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_402_verit__comp__simplify1_I1_J,axiom,
! [A2: set_set_rat] :
~ ( ord_less_set_set_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_403_verit__comp__simplify1_I1_J,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_404_verit__comp__simplify1_I1_J,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_405_pinf_I1_J,axiom,
! [P: rat > $o,P2: rat > $o,Q2: rat > $o,Q3: rat > $o] :
( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ Z5 @ X )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q2 @ X4 ) )
= ( ( P2 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_406_pinf_I2_J,axiom,
! [P: rat > $o,P2: rat > $o,Q2: rat > $o,Q3: rat > $o] :
( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ Z5 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ Z5 @ X )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q2 @ X4 ) )
= ( ( P2 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_407_pinf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_408_pinf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_409_pinf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z3 @ X4 )
=> ~ ( ord_less_rat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_410_pinf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ Z3 @ X4 )
=> ( ord_less_rat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_411_minf_I1_J,axiom,
! [P: rat > $o,P2: rat > $o,Q2: rat > $o,Q3: rat > $o] :
( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z5 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z5 )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q2 @ X4 ) )
= ( ( P2 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_412_minf_I2_J,axiom,
! [P: rat > $o,P2: rat > $o,Q2: rat > $o,Q3: rat > $o] :
( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z5 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z5: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z5 )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q2 @ X4 ) )
= ( ( P2 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_413_minf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_414_minf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_415_minf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z3 )
=> ( ord_less_rat @ X4 @ T ) ) ).
% minf(5)
thf(fact_416_minf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( ( ord_less_rat @ X4 @ Z3 )
=> ~ ( ord_less_rat @ T @ X4 ) ) ).
% minf(7)
thf(fact_417_emptyE,axiom,
! [A2: set_rat] :
~ ( member_set_rat @ A2 @ bot_bot_set_set_rat ) ).
% emptyE
thf(fact_418_emptyE,axiom,
! [A2: rat] :
~ ( member_rat @ A2 @ bot_bot_set_rat ) ).
% emptyE
thf(fact_419_equals0D,axiom,
! [A: set_set_rat,A2: set_rat] :
( ( A = bot_bot_set_set_rat )
=> ~ ( member_set_rat @ A2 @ A ) ) ).
% equals0D
thf(fact_420_equals0D,axiom,
! [A: set_rat,A2: rat] :
( ( A = bot_bot_set_rat )
=> ~ ( member_rat @ A2 @ A ) ) ).
% equals0D
thf(fact_421_equals0I,axiom,
! [A: set_set_rat] :
( ! [Y2: set_rat] :
~ ( member_set_rat @ Y2 @ A )
=> ( A = bot_bot_set_set_rat ) ) ).
% equals0I
thf(fact_422_equals0I,axiom,
! [A: set_rat] :
( ! [Y2: rat] :
~ ( member_rat @ Y2 @ A )
=> ( A = bot_bot_set_rat ) ) ).
% equals0I
thf(fact_423_ex__in__conv,axiom,
! [A: set_set_rat] :
( ( ? [X3: set_rat] : ( member_set_rat @ X3 @ A ) )
= ( A != bot_bot_set_set_rat ) ) ).
% ex_in_conv
thf(fact_424_ex__in__conv,axiom,
! [A: set_rat] :
( ( ? [X3: rat] : ( member_rat @ X3 @ A ) )
= ( A != bot_bot_set_rat ) ) ).
% ex_in_conv
thf(fact_425_psubset__trans,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( ord_less_set_set_rat @ A @ B )
=> ( ( ord_less_set_set_rat @ B @ C2 )
=> ( ord_less_set_set_rat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_426_psubset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_427_psubsetD,axiom,
! [A: set_set_rat,B: set_set_rat,C: set_rat] :
( ( ord_less_set_set_rat @ A @ B )
=> ( ( member_set_rat @ C @ A )
=> ( member_set_rat @ C @ B ) ) ) ).
% psubsetD
thf(fact_428_psubsetD,axiom,
! [A: set_rat,B: set_rat,C: rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B ) ) ) ).
% psubsetD
thf(fact_429_greaterThan__iff,axiom,
! [I: set_set_rat,K: set_set_rat] :
( ( member_set_set_rat @ I @ ( set_or6674600949247491550et_rat @ K ) )
= ( ord_less_set_set_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_430_greaterThan__iff,axiom,
! [I: set_rat,K: set_rat] :
( ( member_set_rat @ I @ ( set_or6174011595382531368et_rat @ K ) )
= ( ord_less_set_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_431_greaterThan__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_or575021546402375026an_rat @ K ) )
= ( ord_less_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_432_greaterThan__subset__iff,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_or575021546402375026an_rat @ X2 ) @ ( set_or575021546402375026an_rat @ Y ) )
= ( ord_less_eq_rat @ Y @ X2 ) ) ).
% greaterThan_subset_iff
thf(fact_433_greaterThan__eq__iff,axiom,
! [X2: rat,Y: rat] :
( ( ( set_or575021546402375026an_rat @ X2 )
= ( set_or575021546402375026an_rat @ Y ) )
= ( X2 = Y ) ) ).
% greaterThan_eq_iff
thf(fact_434_greaterThan__non__empty,axiom,
! [X2: rat] :
( ( set_or575021546402375026an_rat @ X2 )
!= bot_bot_set_rat ) ).
% greaterThan_non_empty
thf(fact_435_Set_Ois__empty__def,axiom,
( is_empty_set_rat
= ( ^ [A5: set_set_rat] : ( A5 = bot_bot_set_set_rat ) ) ) ).
% Set.is_empty_def
thf(fact_436_Set_Ois__empty__def,axiom,
( is_empty_rat
= ( ^ [A5: set_rat] : ( A5 = bot_bot_set_rat ) ) ) ).
% Set.is_empty_def
thf(fact_437_GreatestI2__order,axiom,
! [P: set_rat > $o,X2: set_rat,Q2: set_rat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_rat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_rat @ Y2 @ X2 ) )
=> ( ! [X: set_rat] :
( ( P @ X )
=> ( ! [Y5: set_rat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_rat @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_2216579580035808117et_rat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_438_GreatestI2__order,axiom,
! [P: set_set_rat > $o,X2: set_set_rat,Q2: set_set_rat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_set_rat] :
( ( P @ Y2 )
=> ( ord_le513522071413781156et_rat @ Y2 @ X2 ) )
=> ( ! [X: set_set_rat] :
( ( P @ X )
=> ( ! [Y5: set_set_rat] :
( ( P @ Y5 )
=> ( ord_le513522071413781156et_rat @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_4122807098444226603et_rat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_439_GreatestI2__order,axiom,
! [P: rat > $o,X2: rat,Q2: rat > $o] :
( ( P @ X2 )
=> ( ! [Y2: rat] :
( ( P @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X2 ) )
=> ( ! [X: rat] :
( ( P @ X )
=> ( ! [Y5: rat] :
( ( P @ Y5 )
=> ( ord_less_eq_rat @ Y5 @ X ) )
=> ( Q2 @ X ) ) )
=> ( Q2 @ ( order_Greatest_rat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_440_Greatest__equality,axiom,
! [P: set_rat > $o,X2: set_rat] :
( ( P @ X2 )
=> ( ! [Y2: set_rat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_rat @ Y2 @ X2 ) )
=> ( ( order_2216579580035808117et_rat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_441_Greatest__equality,axiom,
! [P: set_set_rat > $o,X2: set_set_rat] :
( ( P @ X2 )
=> ( ! [Y2: set_set_rat] :
( ( P @ Y2 )
=> ( ord_le513522071413781156et_rat @ Y2 @ X2 ) )
=> ( ( order_4122807098444226603et_rat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_442_Greatest__equality,axiom,
! [P: rat > $o,X2: rat] :
( ( P @ X2 )
=> ( ! [Y2: rat] :
( ( P @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X2 ) )
=> ( ( order_Greatest_rat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_443_subset__antisym,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_444_subset__antisym,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( ord_le513522071413781156et_rat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_445_subsetI,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( member_set_rat @ X @ B ) )
=> ( ord_le513522071413781156et_rat @ A @ B ) ) ).
% subsetI
thf(fact_446_subsetI,axiom,
! [A: set_rat,B: set_rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ X @ B ) )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% subsetI
thf(fact_447_Collect__mono__iff,axiom,
! [P: rat > $o,Q2: rat > $o] :
( ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q2 ) )
= ( ! [X3: rat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_448_Collect__mono__iff,axiom,
! [P: set_rat > $o,Q2: set_rat > $o] :
( ( ord_le513522071413781156et_rat @ ( collect_set_rat @ P ) @ ( collect_set_rat @ Q2 ) )
= ( ! [X3: set_rat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_449_set__eq__subset,axiom,
( ( ^ [Y3: set_rat,Z2: set_rat] : ( Y3 = Z2 ) )
= ( ^ [A5: set_rat,B6: set_rat] :
( ( ord_less_eq_set_rat @ A5 @ B6 )
& ( ord_less_eq_set_rat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_450_set__eq__subset,axiom,
( ( ^ [Y3: set_set_rat,Z2: set_set_rat] : ( Y3 = Z2 ) )
= ( ^ [A5: set_set_rat,B6: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A5 @ B6 )
& ( ord_le513522071413781156et_rat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_451_subset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ C2 )
=> ( ord_less_eq_set_rat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_452_subset__trans,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( ord_le513522071413781156et_rat @ B @ C2 )
=> ( ord_le513522071413781156et_rat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_453_Collect__mono,axiom,
! [P: rat > $o,Q2: rat > $o] :
( ! [X: rat] :
( ( P @ X )
=> ( Q2 @ X ) )
=> ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_454_Collect__mono,axiom,
! [P: set_rat > $o,Q2: set_rat > $o] :
( ! [X: set_rat] :
( ( P @ X )
=> ( Q2 @ X ) )
=> ( ord_le513522071413781156et_rat @ ( collect_set_rat @ P ) @ ( collect_set_rat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_455_subset__refl,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ A ) ).
% subset_refl
thf(fact_456_subset__refl,axiom,
! [A: set_set_rat] : ( ord_le513522071413781156et_rat @ A @ A ) ).
% subset_refl
thf(fact_457_subset__iff,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A5: set_set_rat,B6: set_set_rat] :
! [T2: set_rat] :
( ( member_set_rat @ T2 @ A5 )
=> ( member_set_rat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_458_subset__iff,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B6: set_rat] :
! [T2: rat] :
( ( member_rat @ T2 @ A5 )
=> ( member_rat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_459_equalityD2,axiom,
! [A: set_rat,B: set_rat] :
( ( A = B )
=> ( ord_less_eq_set_rat @ B @ A ) ) ).
% equalityD2
thf(fact_460_equalityD2,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( A = B )
=> ( ord_le513522071413781156et_rat @ B @ A ) ) ).
% equalityD2
thf(fact_461_equalityD1,axiom,
! [A: set_rat,B: set_rat] :
( ( A = B )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% equalityD1
thf(fact_462_equalityD1,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( A = B )
=> ( ord_le513522071413781156et_rat @ A @ B ) ) ).
% equalityD1
thf(fact_463_subset__eq,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A5: set_set_rat,B6: set_set_rat] :
! [X3: set_rat] :
( ( member_set_rat @ X3 @ A5 )
=> ( member_set_rat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_464_subset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B6: set_rat] :
! [X3: rat] :
( ( member_rat @ X3 @ A5 )
=> ( member_rat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_465_equalityE,axiom,
! [A: set_rat,B: set_rat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_rat @ A @ B )
=> ~ ( ord_less_eq_set_rat @ B @ A ) ) ) ).
% equalityE
thf(fact_466_equalityE,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( A = B )
=> ~ ( ( ord_le513522071413781156et_rat @ A @ B )
=> ~ ( ord_le513522071413781156et_rat @ B @ A ) ) ) ).
% equalityE
thf(fact_467_subsetD,axiom,
! [A: set_set_rat,B: set_set_rat,C: set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( member_set_rat @ C @ A )
=> ( member_set_rat @ C @ B ) ) ) ).
% subsetD
thf(fact_468_subsetD,axiom,
! [A: set_rat,B: set_rat,C: rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B ) ) ) ).
% subsetD
thf(fact_469_in__mono,axiom,
! [A: set_set_rat,B: set_set_rat,X2: set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( member_set_rat @ X2 @ A )
=> ( member_set_rat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_470_in__mono,axiom,
! [A: set_rat,B: set_rat,X2: rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( member_rat @ X2 @ A )
=> ( member_rat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_471_Collect__empty__eq__bot,axiom,
! [P: set_rat > $o] :
( ( ( collect_set_rat @ P )
= bot_bot_set_set_rat )
= ( P = bot_bot_set_rat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_472_Collect__empty__eq__bot,axiom,
! [P: rat > $o] :
( ( ( collect_rat @ P )
= bot_bot_set_rat )
= ( P = bot_bot_rat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_473_bot__empty__eq,axiom,
( bot_bot_set_rat_o
= ( ^ [X3: set_rat] : ( member_set_rat @ X3 @ bot_bot_set_set_rat ) ) ) ).
% bot_empty_eq
thf(fact_474_bot__empty__eq,axiom,
( bot_bot_rat_o
= ( ^ [X3: rat] : ( member_rat @ X3 @ bot_bot_set_rat ) ) ) ).
% bot_empty_eq
thf(fact_475_subset__emptyI,axiom,
! [A: set_set_rat] :
( ! [X: set_rat] :
~ ( member_set_rat @ X @ A )
=> ( ord_le513522071413781156et_rat @ A @ bot_bot_set_set_rat ) ) ).
% subset_emptyI
thf(fact_476_subset__emptyI,axiom,
! [A: set_rat] :
( ! [X: rat] :
~ ( member_rat @ X @ A )
=> ( ord_less_eq_set_rat @ A @ bot_bot_set_rat ) ) ).
% subset_emptyI
thf(fact_477_Ici__subset__Ioi__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atLeast_rat @ A2 ) @ ( set_or575021546402375026an_rat @ B2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ).
% Ici_subset_Ioi_iff
thf(fact_478_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A2 @ B2 ) @ ( set_or5199638295745620268an_rat @ C @ D ) )
= ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_479_linordered__field__no__lb,axiom,
! [X4: rat] :
? [Y2: rat] : ( ord_less_rat @ Y2 @ X4 ) ).
% linordered_field_no_lb
thf(fact_480_linordered__field__no__ub,axiom,
! [X4: rat] :
? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_481_atLeast__eq__iff,axiom,
! [X2: rat,Y: rat] :
( ( ( set_ord_atLeast_rat @ X2 )
= ( set_ord_atLeast_rat @ Y ) )
= ( X2 = Y ) ) ).
% atLeast_eq_iff
thf(fact_482_atLeast__eq__iff,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ( set_or7446828528931440131et_rat @ X2 )
= ( set_or7446828528931440131et_rat @ Y ) )
= ( X2 = Y ) ) ).
% atLeast_eq_iff
thf(fact_483_greaterThanLessThan__iff,axiom,
! [I: set_set_rat,L: set_set_rat,U: set_set_rat] :
( ( member_set_set_rat @ I @ ( set_or7639671272556130712et_rat @ L @ U ) )
= ( ( ord_less_set_set_rat @ L @ I )
& ( ord_less_set_set_rat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_484_greaterThanLessThan__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or5199638295745620268an_rat @ L @ U ) )
= ( ( ord_less_rat @ L @ I )
& ( ord_less_rat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_485_greaterThanLessThan__iff,axiom,
! [I: set_rat,L: set_rat,U: set_rat] :
( ( member_set_rat @ I @ ( set_or5117453967338258658et_rat @ L @ U ) )
= ( ( ord_less_set_rat @ L @ I )
& ( ord_less_set_rat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_486_atLeast__iff,axiom,
! [I: set_set_rat,K: set_set_rat] :
( ( member_set_set_rat @ I @ ( set_or4639696602114667193et_rat @ K ) )
= ( ord_le513522071413781156et_rat @ K @ I ) ) ).
% atLeast_iff
thf(fact_487_atLeast__iff,axiom,
! [I: set_rat,K: set_rat] :
( ( member_set_rat @ I @ ( set_or7446828528931440131et_rat @ K ) )
= ( ord_less_eq_set_rat @ K @ I ) ) ).
% atLeast_iff
thf(fact_488_atLeast__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_atLeast_rat @ K ) )
= ( ord_less_eq_rat @ K @ I ) ) ).
% atLeast_iff
thf(fact_489_greaterThanLessThan__empty,axiom,
! [L: set_set_rat,K: set_set_rat] :
( ( ord_le513522071413781156et_rat @ L @ K )
=> ( ( set_or7639671272556130712et_rat @ K @ L )
= bot_bo6619408370577057422et_rat ) ) ).
% greaterThanLessThan_empty
thf(fact_490_greaterThanLessThan__empty,axiom,
! [L: set_rat,K: set_rat] :
( ( ord_less_eq_set_rat @ L @ K )
=> ( ( set_or5117453967338258658et_rat @ K @ L )
= bot_bot_set_set_rat ) ) ).
% greaterThanLessThan_empty
thf(fact_491_greaterThanLessThan__empty,axiom,
! [L: rat,K: rat] :
( ( ord_less_eq_rat @ L @ K )
=> ( ( set_or5199638295745620268an_rat @ K @ L )
= bot_bot_set_rat ) ) ).
% greaterThanLessThan_empty
thf(fact_492_greaterThanLessThan__empty__iff,axiom,
! [A2: rat,B2: rat] :
( ( ( set_or5199638295745620268an_rat @ A2 @ B2 )
= bot_bot_set_rat )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_493_greaterThanLessThan__empty__iff2,axiom,
! [A2: rat,B2: rat] :
( ( bot_bot_set_rat
= ( set_or5199638295745620268an_rat @ A2 @ B2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_494_atLeast__subset__iff,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ord_le8552383839478139994et_rat @ ( set_or4639696602114667193et_rat @ X2 ) @ ( set_or4639696602114667193et_rat @ Y ) )
= ( ord_le513522071413781156et_rat @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_495_atLeast__subset__iff,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_le513522071413781156et_rat @ ( set_or7446828528931440131et_rat @ X2 ) @ ( set_or7446828528931440131et_rat @ Y ) )
= ( ord_less_eq_set_rat @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_496_atLeast__subset__iff,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atLeast_rat @ X2 ) @ ( set_ord_atLeast_rat @ Y ) )
= ( ord_less_eq_rat @ Y @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_497_not__empty__eq__Ici__eq__empty,axiom,
! [L: set_rat] :
( bot_bot_set_set_rat
!= ( set_or7446828528931440131et_rat @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_498_not__empty__eq__Ici__eq__empty,axiom,
! [L: rat] :
( bot_bot_set_rat
!= ( set_ord_atLeast_rat @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_499_Ioi__le__Ico,axiom,
! [A2: set_rat] : ( ord_le513522071413781156et_rat @ ( set_or6174011595382531368et_rat @ A2 ) @ ( set_or7446828528931440131et_rat @ A2 ) ) ).
% Ioi_le_Ico
thf(fact_500_Ioi__le__Ico,axiom,
! [A2: rat] : ( ord_less_eq_set_rat @ ( set_or575021546402375026an_rat @ A2 ) @ ( set_ord_atLeast_rat @ A2 ) ) ).
% Ioi_le_Ico
thf(fact_501_ivl__disj__un__one_I6_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_or5199638295745620268an_rat @ L @ U ) @ ( set_ord_atLeast_rat @ U ) )
= ( set_or575021546402375026an_rat @ L ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_502_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A2 @ B2 ) @ ( set_or6023941531720377480st_rat @ C @ D ) )
= ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_503_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A2 @ B2 ) @ ( set_or4029947393144176647an_rat @ C @ D ) )
= ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_504_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A2 @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_505_Icc__subset__Ici__iff,axiom,
! [L: set_set_rat,H: set_set_rat,L2: set_set_rat] :
( ( ord_le8552383839478139994et_rat @ ( set_or2757889799628458319et_rat @ L @ H ) @ ( set_or4639696602114667193et_rat @ L2 ) )
= ( ~ ( ord_le513522071413781156et_rat @ L @ H )
| ( ord_le513522071413781156et_rat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_506_Icc__subset__Ici__iff,axiom,
! [L: set_rat,H: set_rat,L2: set_rat] :
( ( ord_le513522071413781156et_rat @ ( set_or1040488700251649177et_rat @ L @ H ) @ ( set_or7446828528931440131et_rat @ L2 ) )
= ( ~ ( ord_less_eq_set_rat @ L @ H )
| ( ord_less_eq_set_rat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_507_Icc__subset__Ici__iff,axiom,
! [L: rat,H: rat,L2: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H ) @ ( set_ord_atLeast_rat @ L2 ) )
= ( ~ ( ord_less_eq_rat @ L @ H )
| ( ord_less_eq_rat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_508_ivl__disj__int__one_I6_J,axiom,
! [L: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or5117453967338258658et_rat @ L @ U ) @ ( set_or7446828528931440131et_rat @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_one(6)
thf(fact_509_ivl__disj__int__one_I6_J,axiom,
! [L: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or5199638295745620268an_rat @ L @ U ) @ ( set_ord_atLeast_rat @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_one(6)
thf(fact_510_inverse__le__iff__le__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_511_IntI,axiom,
! [C: set_rat,A: set_set_rat,B: set_set_rat] :
( ( member_set_rat @ C @ A )
=> ( ( member_set_rat @ C @ B )
=> ( member_set_rat @ C @ ( inf_inf_set_set_rat @ A @ B ) ) ) ) ).
% IntI
thf(fact_512_IntI,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ A )
=> ( ( member_rat @ C @ B )
=> ( member_rat @ C @ ( inf_inf_set_rat @ A @ B ) ) ) ) ).
% IntI
thf(fact_513_Int__iff,axiom,
! [C: set_rat,A: set_set_rat,B: set_set_rat] :
( ( member_set_rat @ C @ ( inf_inf_set_set_rat @ A @ B ) )
= ( ( member_set_rat @ C @ A )
& ( member_set_rat @ C @ B ) ) ) ).
% Int_iff
thf(fact_514_Int__iff,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( inf_inf_set_rat @ A @ B ) )
= ( ( member_rat @ C @ A )
& ( member_rat @ C @ B ) ) ) ).
% Int_iff
thf(fact_515_UnCI,axiom,
! [C: set_rat,B: set_set_rat,A: set_set_rat] :
( ( ~ ( member_set_rat @ C @ B )
=> ( member_set_rat @ C @ A ) )
=> ( member_set_rat @ C @ ( sup_sup_set_set_rat @ A @ B ) ) ) ).
% UnCI
thf(fact_516_UnCI,axiom,
! [C: rat,B: set_rat,A: set_rat] :
( ( ~ ( member_rat @ C @ B )
=> ( member_rat @ C @ A ) )
=> ( member_rat @ C @ ( sup_sup_set_rat @ A @ B ) ) ) ).
% UnCI
thf(fact_517_Un__iff,axiom,
! [C: set_rat,A: set_set_rat,B: set_set_rat] :
( ( member_set_rat @ C @ ( sup_sup_set_set_rat @ A @ B ) )
= ( ( member_set_rat @ C @ A )
| ( member_set_rat @ C @ B ) ) ) ).
% Un_iff
thf(fact_518_Un__iff,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( sup_sup_set_rat @ A @ B ) )
= ( ( member_rat @ C @ A )
| ( member_rat @ C @ B ) ) ) ).
% Un_iff
thf(fact_519_inverse__eq__iff__eq,axiom,
! [A2: rat,B2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= ( inverse_inverse_rat @ B2 ) )
= ( A2 = B2 ) ) ).
% inverse_eq_iff_eq
thf(fact_520_inverse__inverse__eq,axiom,
! [A2: rat] :
( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A2 ) )
= A2 ) ).
% inverse_inverse_eq
thf(fact_521_Icc__eq__Icc,axiom,
! [L: set_set_rat,H: set_set_rat,L2: set_set_rat,H2: set_set_rat] :
( ( ( set_or2757889799628458319et_rat @ L @ H )
= ( set_or2757889799628458319et_rat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le513522071413781156et_rat @ L @ H )
& ~ ( ord_le513522071413781156et_rat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_522_Icc__eq__Icc,axiom,
! [L: set_rat,H: set_rat,L2: set_rat,H2: set_rat] :
( ( ( set_or1040488700251649177et_rat @ L @ H )
= ( set_or1040488700251649177et_rat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_rat @ L @ H )
& ~ ( ord_less_eq_set_rat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_523_Icc__eq__Icc,axiom,
! [L: rat,H: rat,L2: rat,H2: rat] :
( ( ( set_or633870826150836451st_rat @ L @ H )
= ( set_or633870826150836451st_rat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_rat @ L @ H )
& ~ ( ord_less_eq_rat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_524_atLeastAtMost__iff,axiom,
! [I: set_set_rat,L: set_set_rat,U: set_set_rat] :
( ( member_set_set_rat @ I @ ( set_or2757889799628458319et_rat @ L @ U ) )
= ( ( ord_le513522071413781156et_rat @ L @ I )
& ( ord_le513522071413781156et_rat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_525_atLeastAtMost__iff,axiom,
! [I: set_rat,L: set_rat,U: set_rat] :
( ( member_set_rat @ I @ ( set_or1040488700251649177et_rat @ L @ U ) )
= ( ( ord_less_eq_set_rat @ L @ I )
& ( ord_less_eq_set_rat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_526_atLeastAtMost__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
= ( ( ord_less_eq_rat @ L @ I )
& ( ord_less_eq_rat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_527_inverse__nonzero__iff__nonzero,axiom,
! [A2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= zero_zero_rat )
= ( A2 = zero_zero_rat ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_528_inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% inverse_zero
thf(fact_529_Int__subset__iff,axiom,
! [C2: set_rat,A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ C2 @ ( inf_inf_set_rat @ A @ B ) )
= ( ( ord_less_eq_set_rat @ C2 @ A )
& ( ord_less_eq_set_rat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_530_Int__subset__iff,axiom,
! [C2: set_set_rat,A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ C2 @ ( inf_inf_set_set_rat @ A @ B ) )
= ( ( ord_le513522071413781156et_rat @ C2 @ A )
& ( ord_le513522071413781156et_rat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_531_Un__empty,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ( sup_sup_set_set_rat @ A @ B )
= bot_bot_set_set_rat )
= ( ( A = bot_bot_set_set_rat )
& ( B = bot_bot_set_set_rat ) ) ) ).
% Un_empty
thf(fact_532_Un__empty,axiom,
! [A: set_rat,B: set_rat] :
( ( ( sup_sup_set_rat @ A @ B )
= bot_bot_set_rat )
= ( ( A = bot_bot_set_rat )
& ( B = bot_bot_set_rat ) ) ) ).
% Un_empty
thf(fact_533_Un__subset__iff,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ ( sup_sup_set_rat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_rat @ A @ C2 )
& ( ord_less_eq_set_rat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_534_Un__subset__iff,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ ( sup_sup_set_set_rat @ A @ B ) @ C2 )
= ( ( ord_le513522071413781156et_rat @ A @ C2 )
& ( ord_le513522071413781156et_rat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_535_Int__Un__eq_I4_J,axiom,
! [T3: set_set_rat,S2: set_set_rat] :
( ( sup_sup_set_set_rat @ T3 @ ( inf_inf_set_set_rat @ S2 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_536_Int__Un__eq_I4_J,axiom,
! [T3: set_rat,S2: set_rat] :
( ( sup_sup_set_rat @ T3 @ ( inf_inf_set_rat @ S2 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_537_Int__Un__eq_I3_J,axiom,
! [S2: set_set_rat,T3: set_set_rat] :
( ( sup_sup_set_set_rat @ S2 @ ( inf_inf_set_set_rat @ S2 @ T3 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_538_Int__Un__eq_I3_J,axiom,
! [S2: set_rat,T3: set_rat] :
( ( sup_sup_set_rat @ S2 @ ( inf_inf_set_rat @ S2 @ T3 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_539_Int__Un__eq_I2_J,axiom,
! [S2: set_set_rat,T3: set_set_rat] :
( ( sup_sup_set_set_rat @ ( inf_inf_set_set_rat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_540_Int__Un__eq_I2_J,axiom,
! [S2: set_rat,T3: set_rat] :
( ( sup_sup_set_rat @ ( inf_inf_set_rat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_541_Int__Un__eq_I1_J,axiom,
! [S2: set_set_rat,T3: set_set_rat] :
( ( sup_sup_set_set_rat @ ( inf_inf_set_set_rat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_542_Int__Un__eq_I1_J,axiom,
! [S2: set_rat,T3: set_rat] :
( ( sup_sup_set_rat @ ( inf_inf_set_rat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_543_Un__Int__eq_I4_J,axiom,
! [T3: set_set_rat,S2: set_set_rat] :
( ( inf_inf_set_set_rat @ T3 @ ( sup_sup_set_set_rat @ S2 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_544_Un__Int__eq_I4_J,axiom,
! [T3: set_rat,S2: set_rat] :
( ( inf_inf_set_rat @ T3 @ ( sup_sup_set_rat @ S2 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_545_Un__Int__eq_I3_J,axiom,
! [S2: set_set_rat,T3: set_set_rat] :
( ( inf_inf_set_set_rat @ S2 @ ( sup_sup_set_set_rat @ S2 @ T3 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_546_Un__Int__eq_I3_J,axiom,
! [S2: set_rat,T3: set_rat] :
( ( inf_inf_set_rat @ S2 @ ( sup_sup_set_rat @ S2 @ T3 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_547_Un__Int__eq_I2_J,axiom,
! [S2: set_set_rat,T3: set_set_rat] :
( ( inf_inf_set_set_rat @ ( sup_sup_set_set_rat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_548_Un__Int__eq_I2_J,axiom,
! [S2: set_rat,T3: set_rat] :
( ( inf_inf_set_rat @ ( sup_sup_set_rat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_549_Un__Int__eq_I1_J,axiom,
! [S2: set_set_rat,T3: set_set_rat] :
( ( inf_inf_set_set_rat @ ( sup_sup_set_set_rat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_550_Un__Int__eq_I1_J,axiom,
! [S2: set_rat,T3: set_rat] :
( ( inf_inf_set_rat @ ( sup_sup_set_rat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_551_inverse__nonnegative__iff__nonnegative,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_552_inverse__nonpositive__iff__nonpositive,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_553_inverse__less__iff__less,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% inverse_less_iff_less
thf(fact_554_inverse__less__iff__less__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_555_inverse__negative__iff__negative,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% inverse_negative_iff_negative
thf(fact_556_inverse__positive__iff__positive,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A2 ) )
= ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% inverse_positive_iff_positive
thf(fact_557_atLeastLessThan__iff,axiom,
! [I: set_set_rat,L: set_set_rat,U: set_set_rat] :
( ( member_set_set_rat @ I @ ( set_or8253465997870395507et_rat @ L @ U ) )
= ( ( ord_le513522071413781156et_rat @ L @ I )
& ( ord_less_set_set_rat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_558_atLeastLessThan__iff,axiom,
! [I: set_rat,L: set_rat,U: set_rat] :
( ( member_set_rat @ I @ ( set_or32047845639629757et_rat @ L @ U ) )
= ( ( ord_less_eq_set_rat @ L @ I )
& ( ord_less_set_rat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_559_atLeastLessThan__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or4029947393144176647an_rat @ L @ U ) )
= ( ( ord_less_eq_rat @ L @ I )
& ( ord_less_rat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_560_atLeastatMost__empty__iff2,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( bot_bo6619408370577057422et_rat
= ( set_or2757889799628458319et_rat @ A2 @ B2 ) )
= ( ~ ( ord_le513522071413781156et_rat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_561_atLeastatMost__empty__iff2,axiom,
! [A2: set_rat,B2: set_rat] :
( ( bot_bot_set_set_rat
= ( set_or1040488700251649177et_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_rat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_562_atLeastatMost__empty__iff2,axiom,
! [A2: rat,B2: rat] :
( ( bot_bot_set_rat
= ( set_or633870826150836451st_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_563_atLeastatMost__empty__iff,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ( set_or2757889799628458319et_rat @ A2 @ B2 )
= bot_bo6619408370577057422et_rat )
= ( ~ ( ord_le513522071413781156et_rat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_564_atLeastatMost__empty__iff,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ( set_or1040488700251649177et_rat @ A2 @ B2 )
= bot_bot_set_set_rat )
= ( ~ ( ord_less_eq_set_rat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_565_atLeastatMost__empty__iff,axiom,
! [A2: rat,B2: rat] :
( ( ( set_or633870826150836451st_rat @ A2 @ B2 )
= bot_bot_set_rat )
= ( ~ ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_566_atLeastatMost__subset__iff,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat,D: set_set_rat] :
( ( ord_le8552383839478139994et_rat @ ( set_or2757889799628458319et_rat @ A2 @ B2 ) @ ( set_or2757889799628458319et_rat @ C @ D ) )
= ( ~ ( ord_le513522071413781156et_rat @ A2 @ B2 )
| ( ( ord_le513522071413781156et_rat @ C @ A2 )
& ( ord_le513522071413781156et_rat @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_567_atLeastatMost__subset__iff,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat,D: set_rat] :
( ( ord_le513522071413781156et_rat @ ( set_or1040488700251649177et_rat @ A2 @ B2 ) @ ( set_or1040488700251649177et_rat @ C @ D ) )
= ( ~ ( ord_less_eq_set_rat @ A2 @ B2 )
| ( ( ord_less_eq_set_rat @ C @ A2 )
& ( ord_less_eq_set_rat @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_568_atLeastatMost__subset__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ~ ( ord_less_eq_rat @ A2 @ B2 )
| ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_569_atLeastatMost__empty,axiom,
! [B2: set_set_rat,A2: set_set_rat] :
( ( ord_less_set_set_rat @ B2 @ A2 )
=> ( ( set_or2757889799628458319et_rat @ A2 @ B2 )
= bot_bo6619408370577057422et_rat ) ) ).
% atLeastatMost_empty
thf(fact_570_atLeastatMost__empty,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( set_or633870826150836451st_rat @ A2 @ B2 )
= bot_bot_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_571_atLeastatMost__empty,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ( set_or1040488700251649177et_rat @ A2 @ B2 )
= bot_bot_set_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_572_atLeastLessThan__empty,axiom,
! [B2: set_set_rat,A2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B2 @ A2 )
=> ( ( set_or8253465997870395507et_rat @ A2 @ B2 )
= bot_bo6619408370577057422et_rat ) ) ).
% atLeastLessThan_empty
thf(fact_573_atLeastLessThan__empty,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_eq_set_rat @ B2 @ A2 )
=> ( ( set_or32047845639629757et_rat @ A2 @ B2 )
= bot_bot_set_set_rat ) ) ).
% atLeastLessThan_empty
thf(fact_574_atLeastLessThan__empty,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
= bot_bot_set_rat ) ) ).
% atLeastLessThan_empty
thf(fact_575_ivl__subset,axiom,
! [I: rat,J: rat,M: rat,N: rat] :
( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ I @ J ) @ ( set_or4029947393144176647an_rat @ M @ N ) )
= ( ( ord_less_eq_rat @ J @ I )
| ( ( ord_less_eq_rat @ M @ I )
& ( ord_less_eq_rat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_576_atLeastLessThan__empty__iff,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ( set_or8253465997870395507et_rat @ A2 @ B2 )
= bot_bo6619408370577057422et_rat )
= ( ~ ( ord_less_set_set_rat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_577_atLeastLessThan__empty__iff,axiom,
! [A2: rat,B2: rat] :
( ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
= bot_bot_set_rat )
= ( ~ ( ord_less_rat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_578_atLeastLessThan__empty__iff,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ( set_or32047845639629757et_rat @ A2 @ B2 )
= bot_bot_set_set_rat )
= ( ~ ( ord_less_set_rat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_579_atLeastLessThan__empty__iff2,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( bot_bo6619408370577057422et_rat
= ( set_or8253465997870395507et_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_set_set_rat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_580_atLeastLessThan__empty__iff2,axiom,
! [A2: rat,B2: rat] :
( ( bot_bot_set_rat
= ( set_or4029947393144176647an_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_rat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_581_atLeastLessThan__empty__iff2,axiom,
! [A2: set_rat,B2: set_rat] :
( ( bot_bot_set_set_rat
= ( set_or32047845639629757et_rat @ A2 @ B2 ) )
= ( ~ ( ord_less_set_rat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_582_greaterThanAtMost__iff,axiom,
! [I: set_set_rat,L: set_set_rat,U: set_set_rat] :
( ( member_set_set_rat @ I @ ( set_or1109970963052301556et_rat @ L @ U ) )
= ( ( ord_less_set_set_rat @ L @ I )
& ( ord_le513522071413781156et_rat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_583_greaterThanAtMost__iff,axiom,
! [I: set_rat,L: set_rat,U: set_rat] :
( ( member_set_rat @ I @ ( set_or3565782072395811902et_rat @ L @ U ) )
= ( ( ord_less_set_rat @ L @ I )
& ( ord_less_eq_set_rat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_584_greaterThanAtMost__iff,axiom,
! [I: rat,L: rat,U: rat] :
( ( member_rat @ I @ ( set_or6023941531720377480st_rat @ L @ U ) )
= ( ( ord_less_rat @ L @ I )
& ( ord_less_eq_rat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_585_greaterThanAtMost__empty,axiom,
! [L: set_set_rat,K: set_set_rat] :
( ( ord_le513522071413781156et_rat @ L @ K )
=> ( ( set_or1109970963052301556et_rat @ K @ L )
= bot_bo6619408370577057422et_rat ) ) ).
% greaterThanAtMost_empty
thf(fact_586_greaterThanAtMost__empty,axiom,
! [L: set_rat,K: set_rat] :
( ( ord_less_eq_set_rat @ L @ K )
=> ( ( set_or3565782072395811902et_rat @ K @ L )
= bot_bot_set_set_rat ) ) ).
% greaterThanAtMost_empty
thf(fact_587_greaterThanAtMost__empty,axiom,
! [L: rat,K: rat] :
( ( ord_less_eq_rat @ L @ K )
=> ( ( set_or6023941531720377480st_rat @ K @ L )
= bot_bot_set_rat ) ) ).
% greaterThanAtMost_empty
thf(fact_588_greaterThanAtMost__empty__iff,axiom,
! [K: set_set_rat,L: set_set_rat] :
( ( ( set_or1109970963052301556et_rat @ K @ L )
= bot_bo6619408370577057422et_rat )
= ( ~ ( ord_less_set_set_rat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_589_greaterThanAtMost__empty__iff,axiom,
! [K: rat,L: rat] :
( ( ( set_or6023941531720377480st_rat @ K @ L )
= bot_bot_set_rat )
= ( ~ ( ord_less_rat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_590_greaterThanAtMost__empty__iff,axiom,
! [K: set_rat,L: set_rat] :
( ( ( set_or3565782072395811902et_rat @ K @ L )
= bot_bot_set_set_rat )
= ( ~ ( ord_less_set_rat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_591_greaterThanAtMost__empty__iff2,axiom,
! [K: set_set_rat,L: set_set_rat] :
( ( bot_bo6619408370577057422et_rat
= ( set_or1109970963052301556et_rat @ K @ L ) )
= ( ~ ( ord_less_set_set_rat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_592_greaterThanAtMost__empty__iff2,axiom,
! [K: rat,L: rat] :
( ( bot_bot_set_rat
= ( set_or6023941531720377480st_rat @ K @ L ) )
= ( ~ ( ord_less_rat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_593_greaterThanAtMost__empty__iff2,axiom,
! [K: set_rat,L: set_rat] :
( ( bot_bot_set_set_rat
= ( set_or3565782072395811902et_rat @ K @ L ) )
= ( ~ ( ord_less_set_rat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_594_inverse__le__iff__le,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% inverse_le_iff_le
thf(fact_595_ivl__disj__un__two__touch_I1_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_rat @ L @ M )
=> ( ( ord_less_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or6023941531720377480st_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
= ( set_or5199638295745620268an_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_596_inverse__eq__imp__eq,axiom,
! [A2: rat,B2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= ( inverse_inverse_rat @ B2 ) )
=> ( A2 = B2 ) ) ).
% inverse_eq_imp_eq
thf(fact_597_UnE,axiom,
! [C: set_rat,A: set_set_rat,B: set_set_rat] :
( ( member_set_rat @ C @ ( sup_sup_set_set_rat @ A @ B ) )
=> ( ~ ( member_set_rat @ C @ A )
=> ( member_set_rat @ C @ B ) ) ) ).
% UnE
thf(fact_598_UnE,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( sup_sup_set_rat @ A @ B ) )
=> ( ~ ( member_rat @ C @ A )
=> ( member_rat @ C @ B ) ) ) ).
% UnE
thf(fact_599_IntE,axiom,
! [C: set_rat,A: set_set_rat,B: set_set_rat] :
( ( member_set_rat @ C @ ( inf_inf_set_set_rat @ A @ B ) )
=> ~ ( ( member_set_rat @ C @ A )
=> ~ ( member_set_rat @ C @ B ) ) ) ).
% IntE
thf(fact_600_IntE,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( inf_inf_set_rat @ A @ B ) )
=> ~ ( ( member_rat @ C @ A )
=> ~ ( member_rat @ C @ B ) ) ) ).
% IntE
thf(fact_601_UnI1,axiom,
! [C: set_rat,A: set_set_rat,B: set_set_rat] :
( ( member_set_rat @ C @ A )
=> ( member_set_rat @ C @ ( sup_sup_set_set_rat @ A @ B ) ) ) ).
% UnI1
thf(fact_602_UnI1,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ A )
=> ( member_rat @ C @ ( sup_sup_set_rat @ A @ B ) ) ) ).
% UnI1
thf(fact_603_UnI2,axiom,
! [C: set_rat,B: set_set_rat,A: set_set_rat] :
( ( member_set_rat @ C @ B )
=> ( member_set_rat @ C @ ( sup_sup_set_set_rat @ A @ B ) ) ) ).
% UnI2
thf(fact_604_UnI2,axiom,
! [C: rat,B: set_rat,A: set_rat] :
( ( member_rat @ C @ B )
=> ( member_rat @ C @ ( sup_sup_set_rat @ A @ B ) ) ) ).
% UnI2
thf(fact_605_IntD1,axiom,
! [C: set_rat,A: set_set_rat,B: set_set_rat] :
( ( member_set_rat @ C @ ( inf_inf_set_set_rat @ A @ B ) )
=> ( member_set_rat @ C @ A ) ) ).
% IntD1
thf(fact_606_IntD1,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( inf_inf_set_rat @ A @ B ) )
=> ( member_rat @ C @ A ) ) ).
% IntD1
thf(fact_607_IntD2,axiom,
! [C: set_rat,A: set_set_rat,B: set_set_rat] :
( ( member_set_rat @ C @ ( inf_inf_set_set_rat @ A @ B ) )
=> ( member_set_rat @ C @ B ) ) ).
% IntD2
thf(fact_608_IntD2,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( inf_inf_set_rat @ A @ B ) )
=> ( member_rat @ C @ B ) ) ).
% IntD2
thf(fact_609_bex__Un,axiom,
! [A: set_rat,B: set_rat,P: rat > $o] :
( ( ? [X3: rat] :
( ( member_rat @ X3 @ ( sup_sup_set_rat @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: rat] :
( ( member_rat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: rat] :
( ( member_rat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_610_ball__Un,axiom,
! [A: set_rat,B: set_rat,P: rat > $o] :
( ( ! [X3: rat] :
( ( member_rat @ X3 @ ( sup_sup_set_rat @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: rat] :
( ( member_rat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: rat] :
( ( member_rat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_611_Un__assoc,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( sup_sup_set_rat @ ( sup_sup_set_rat @ A @ B ) @ C2 )
= ( sup_sup_set_rat @ A @ ( sup_sup_set_rat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_612_Int__assoc,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( inf_inf_set_rat @ ( inf_inf_set_rat @ A @ B ) @ C2 )
= ( inf_inf_set_rat @ A @ ( inf_inf_set_rat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_613_Int__assoc,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( inf_inf_set_set_rat @ ( inf_inf_set_set_rat @ A @ B ) @ C2 )
= ( inf_inf_set_set_rat @ A @ ( inf_inf_set_set_rat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_614_Un__absorb,axiom,
! [A: set_rat] :
( ( sup_sup_set_rat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_615_Int__absorb,axiom,
! [A: set_rat] :
( ( inf_inf_set_rat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_616_Int__absorb,axiom,
! [A: set_set_rat] :
( ( inf_inf_set_set_rat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_617_Un__commute,axiom,
( sup_sup_set_rat
= ( ^ [A5: set_rat,B6: set_rat] : ( sup_sup_set_rat @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_618_Int__commute,axiom,
( inf_inf_set_rat
= ( ^ [A5: set_rat,B6: set_rat] : ( inf_inf_set_rat @ B6 @ A5 ) ) ) ).
% Int_commute
thf(fact_619_Int__commute,axiom,
( inf_inf_set_set_rat
= ( ^ [A5: set_set_rat,B6: set_set_rat] : ( inf_inf_set_set_rat @ B6 @ A5 ) ) ) ).
% Int_commute
thf(fact_620_Un__Int__crazy,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( sup_sup_set_set_rat @ ( sup_sup_set_set_rat @ ( inf_inf_set_set_rat @ A @ B ) @ ( inf_inf_set_set_rat @ B @ C2 ) ) @ ( inf_inf_set_set_rat @ C2 @ A ) )
= ( inf_inf_set_set_rat @ ( inf_inf_set_set_rat @ ( sup_sup_set_set_rat @ A @ B ) @ ( sup_sup_set_set_rat @ B @ C2 ) ) @ ( sup_sup_set_set_rat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_621_Un__Int__crazy,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( sup_sup_set_rat @ ( sup_sup_set_rat @ ( inf_inf_set_rat @ A @ B ) @ ( inf_inf_set_rat @ B @ C2 ) ) @ ( inf_inf_set_rat @ C2 @ A ) )
= ( inf_inf_set_rat @ ( inf_inf_set_rat @ ( sup_sup_set_rat @ A @ B ) @ ( sup_sup_set_rat @ B @ C2 ) ) @ ( sup_sup_set_rat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_622_Int__Un__distrib,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( inf_inf_set_set_rat @ A @ ( sup_sup_set_set_rat @ B @ C2 ) )
= ( sup_sup_set_set_rat @ ( inf_inf_set_set_rat @ A @ B ) @ ( inf_inf_set_set_rat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_623_Int__Un__distrib,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( inf_inf_set_rat @ A @ ( sup_sup_set_rat @ B @ C2 ) )
= ( sup_sup_set_rat @ ( inf_inf_set_rat @ A @ B ) @ ( inf_inf_set_rat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_624_Un__Int__distrib,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( sup_sup_set_set_rat @ A @ ( inf_inf_set_set_rat @ B @ C2 ) )
= ( inf_inf_set_set_rat @ ( sup_sup_set_set_rat @ A @ B ) @ ( sup_sup_set_set_rat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_625_Un__Int__distrib,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( sup_sup_set_rat @ A @ ( inf_inf_set_rat @ B @ C2 ) )
= ( inf_inf_set_rat @ ( sup_sup_set_rat @ A @ B ) @ ( sup_sup_set_rat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_626_Un__left__absorb,axiom,
! [A: set_rat,B: set_rat] :
( ( sup_sup_set_rat @ A @ ( sup_sup_set_rat @ A @ B ) )
= ( sup_sup_set_rat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_627_Int__Un__distrib2,axiom,
! [B: set_set_rat,C2: set_set_rat,A: set_set_rat] :
( ( inf_inf_set_set_rat @ ( sup_sup_set_set_rat @ B @ C2 ) @ A )
= ( sup_sup_set_set_rat @ ( inf_inf_set_set_rat @ B @ A ) @ ( inf_inf_set_set_rat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_628_Int__Un__distrib2,axiom,
! [B: set_rat,C2: set_rat,A: set_rat] :
( ( inf_inf_set_rat @ ( sup_sup_set_rat @ B @ C2 ) @ A )
= ( sup_sup_set_rat @ ( inf_inf_set_rat @ B @ A ) @ ( inf_inf_set_rat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_629_Int__left__absorb,axiom,
! [A: set_rat,B: set_rat] :
( ( inf_inf_set_rat @ A @ ( inf_inf_set_rat @ A @ B ) )
= ( inf_inf_set_rat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_630_Int__left__absorb,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( inf_inf_set_set_rat @ A @ ( inf_inf_set_set_rat @ A @ B ) )
= ( inf_inf_set_set_rat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_631_Un__Int__distrib2,axiom,
! [B: set_set_rat,C2: set_set_rat,A: set_set_rat] :
( ( sup_sup_set_set_rat @ ( inf_inf_set_set_rat @ B @ C2 ) @ A )
= ( inf_inf_set_set_rat @ ( sup_sup_set_set_rat @ B @ A ) @ ( sup_sup_set_set_rat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_632_Un__Int__distrib2,axiom,
! [B: set_rat,C2: set_rat,A: set_rat] :
( ( sup_sup_set_rat @ ( inf_inf_set_rat @ B @ C2 ) @ A )
= ( inf_inf_set_rat @ ( sup_sup_set_rat @ B @ A ) @ ( sup_sup_set_rat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_633_Un__left__commute,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( sup_sup_set_rat @ A @ ( sup_sup_set_rat @ B @ C2 ) )
= ( sup_sup_set_rat @ B @ ( sup_sup_set_rat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_634_Int__left__commute,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( inf_inf_set_rat @ A @ ( inf_inf_set_rat @ B @ C2 ) )
= ( inf_inf_set_rat @ B @ ( inf_inf_set_rat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_635_Int__left__commute,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( inf_inf_set_set_rat @ A @ ( inf_inf_set_set_rat @ B @ C2 ) )
= ( inf_inf_set_set_rat @ B @ ( inf_inf_set_set_rat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_636_of__rat__inverse,axiom,
! [A2: rat] :
( ( field_2639924705303425560at_rat @ ( inverse_inverse_rat @ A2 ) )
= ( inverse_inverse_rat @ ( field_2639924705303425560at_rat @ A2 ) ) ) ).
% of_rat_inverse
thf(fact_637_ivl__disj__un__two_I8_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or6023941531720377480st_rat @ M @ U ) )
= ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_638_ivl__disj__un__two_I7_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
= ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_639_ivl__disj__un__two_I6_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or6023941531720377480st_rat @ L @ M ) @ ( set_or6023941531720377480st_rat @ M @ U ) )
= ( set_or6023941531720377480st_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_640_ivl__disj__un__two_I3_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
= ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_641_ivl__disj__int__two_I8_J,axiom,
! [L: set_rat,M: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or1040488700251649177et_rat @ L @ M ) @ ( set_or3565782072395811902et_rat @ M @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_two(8)
thf(fact_642_ivl__disj__int__two_I8_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or6023941531720377480st_rat @ M @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_two(8)
thf(fact_643_ivl__disj__int__two_I7_J,axiom,
! [L: set_rat,M: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or32047845639629757et_rat @ L @ M ) @ ( set_or1040488700251649177et_rat @ M @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_two(7)
thf(fact_644_ivl__disj__int__two_I7_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_two(7)
thf(fact_645_ivl__disj__int__two_I6_J,axiom,
! [L: set_rat,M: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or3565782072395811902et_rat @ L @ M ) @ ( set_or3565782072395811902et_rat @ M @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_two(6)
thf(fact_646_ivl__disj__int__two_I6_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or6023941531720377480st_rat @ L @ M ) @ ( set_or6023941531720377480st_rat @ M @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_two(6)
thf(fact_647_ivl__disj__int__two_I3_J,axiom,
! [L: set_rat,M: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or32047845639629757et_rat @ L @ M ) @ ( set_or32047845639629757et_rat @ M @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_two(3)
thf(fact_648_ivl__disj__int__two_I3_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or4029947393144176647an_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_two(3)
thf(fact_649_ivl__disj__un__two__touch_I4_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
= ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_650_Un__Int__assoc__eq,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ( sup_sup_set_rat @ ( inf_inf_set_rat @ A @ B ) @ C2 )
= ( inf_inf_set_rat @ A @ ( sup_sup_set_rat @ B @ C2 ) ) )
= ( ord_less_eq_set_rat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_651_Un__Int__assoc__eq,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_set_rat] :
( ( ( sup_sup_set_set_rat @ ( inf_inf_set_set_rat @ A @ B ) @ C2 )
= ( inf_inf_set_set_rat @ A @ ( sup_sup_set_set_rat @ B @ C2 ) ) )
= ( ord_le513522071413781156et_rat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_652_ivl__disj__un__two__touch_I2_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
= ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_653_ivl__disj__un__two__touch_I3_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or6023941531720377480st_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
= ( set_or6023941531720377480st_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_654_ivl__disj__un__two_I4_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or5199638295745620268an_rat @ M @ U ) )
= ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_655_ivl__disj__un__two_I5_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or5199638295745620268an_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
= ( set_or6023941531720377480st_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_656_ivl__disj__int__one_I7_J,axiom,
! [L: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or1040488700251649177et_rat @ L @ U ) @ ( set_or6174011595382531368et_rat @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_one(7)
thf(fact_657_ivl__disj__int__one_I7_J,axiom,
! [L: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or633870826150836451st_rat @ L @ U ) @ ( set_or575021546402375026an_rat @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_one(7)
thf(fact_658_Ioc__disjoint,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ( inf_inf_set_rat @ ( set_or6023941531720377480st_rat @ A2 @ B2 ) @ ( set_or6023941531720377480st_rat @ C @ D ) )
= bot_bot_set_rat )
= ( ( ord_less_eq_rat @ B2 @ A2 )
| ( ord_less_eq_rat @ D @ C )
| ( ord_less_eq_rat @ B2 @ C )
| ( ord_less_eq_rat @ D @ A2 ) ) ) ).
% Ioc_disjoint
thf(fact_659_ivl__disj__int__two_I5_J,axiom,
! [L: set_rat,M: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or5117453967338258658et_rat @ L @ M ) @ ( set_or1040488700251649177et_rat @ M @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_two(5)
thf(fact_660_ivl__disj__int__two_I5_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or5199638295745620268an_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_two(5)
thf(fact_661_ivl__disj__int__two_I4_J,axiom,
! [L: set_rat,M: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or1040488700251649177et_rat @ L @ M ) @ ( set_or5117453967338258658et_rat @ M @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_two(4)
thf(fact_662_ivl__disj__int__two_I4_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or5199638295745620268an_rat @ M @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_two(4)
thf(fact_663_ivl__disj__un__one_I8_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ U ) @ ( set_ord_atLeast_rat @ U ) )
= ( set_ord_atLeast_rat @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_664_ivl__disj__int__two_I1_J,axiom,
! [L: set_rat,M: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or5117453967338258658et_rat @ L @ M ) @ ( set_or32047845639629757et_rat @ M @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_two(1)
thf(fact_665_ivl__disj__int__two_I1_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or5199638295745620268an_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_two(1)
thf(fact_666_ivl__disj__un__one_I5_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_or6023941531720377480st_rat @ L @ U ) @ ( set_or575021546402375026an_rat @ U ) )
= ( set_or575021546402375026an_rat @ L ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_667_ivl__disj__int__one_I8_J,axiom,
! [L: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or32047845639629757et_rat @ L @ U ) @ ( set_or7446828528931440131et_rat @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_one(8)
thf(fact_668_ivl__disj__int__one_I8_J,axiom,
! [L: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or4029947393144176647an_rat @ L @ U ) @ ( set_ord_atLeast_rat @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_one(8)
thf(fact_669_nonzero__of__rat__inverse,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( field_2639924705303425560at_rat @ ( inverse_inverse_rat @ A2 ) )
= ( inverse_inverse_rat @ ( field_2639924705303425560at_rat @ A2 ) ) ) ) ).
% nonzero_of_rat_inverse
thf(fact_670_ivl__disj__int__one_I5_J,axiom,
! [L: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or3565782072395811902et_rat @ L @ U ) @ ( set_or6174011595382531368et_rat @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_one(5)
thf(fact_671_ivl__disj__int__one_I5_J,axiom,
! [L: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or6023941531720377480st_rat @ L @ U ) @ ( set_or575021546402375026an_rat @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_one(5)
thf(fact_672_nonzero__imp__inverse__nonzero,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A2 )
!= zero_zero_rat ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_673_nonzero__inverse__inverse__eq,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A2 ) )
= A2 ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_674_nonzero__inverse__eq__imp__eq,axiom,
! [A2: rat,B2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= ( inverse_inverse_rat @ B2 ) )
=> ( ( A2 != zero_zero_rat )
=> ( ( B2 != zero_zero_rat )
=> ( A2 = B2 ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_675_inverse__zero__imp__zero,axiom,
! [A2: rat] :
( ( ( inverse_inverse_rat @ A2 )
= zero_zero_rat )
=> ( A2 = zero_zero_rat ) ) ).
% inverse_zero_imp_zero
thf(fact_676_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% field_class.field_inverse_zero
thf(fact_677_ivl__disj__int__two_I2_J,axiom,
! [L: set_rat,M: set_rat,U: set_rat] :
( ( inf_inf_set_set_rat @ ( set_or3565782072395811902et_rat @ L @ M ) @ ( set_or5117453967338258658et_rat @ M @ U ) )
= bot_bot_set_set_rat ) ).
% ivl_disj_int_two(2)
thf(fact_678_ivl__disj__int__two_I2_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_or6023941531720377480st_rat @ L @ M ) @ ( set_or5199638295745620268an_rat @ M @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_two(2)
thf(fact_679_atLeastLessThan__eq__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
= ( set_or4029947393144176647an_rat @ C @ D ) )
= ( ( A2 = C )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_680_Ico__eq__Ico,axiom,
! [L: rat,H: rat,L2: rat,H2: rat] :
( ( ( set_or4029947393144176647an_rat @ L @ H )
= ( set_or4029947393144176647an_rat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_rat @ L @ H )
& ~ ( ord_less_rat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_681_atLeastLessThan__inj_I1_J,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
= ( set_or4029947393144176647an_rat @ C @ D ) )
=> ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ C @ D )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_682_atLeastLessThan__inj_I2_J,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
= ( set_or4029947393144176647an_rat @ C @ D ) )
=> ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ C @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_683_Un__empty__right,axiom,
! [A: set_set_rat] :
( ( sup_sup_set_set_rat @ A @ bot_bot_set_set_rat )
= A ) ).
% Un_empty_right
thf(fact_684_Un__empty__right,axiom,
! [A: set_rat] :
( ( sup_sup_set_rat @ A @ bot_bot_set_rat )
= A ) ).
% Un_empty_right
thf(fact_685_Un__empty__left,axiom,
! [B: set_set_rat] :
( ( sup_sup_set_set_rat @ bot_bot_set_set_rat @ B )
= B ) ).
% Un_empty_left
thf(fact_686_Un__empty__left,axiom,
! [B: set_rat] :
( ( sup_sup_set_rat @ bot_bot_set_rat @ B )
= B ) ).
% Un_empty_left
thf(fact_687_subset__Un__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A5: set_rat,B6: set_rat] :
( ( sup_sup_set_rat @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_688_subset__Un__eq,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A5: set_set_rat,B6: set_set_rat] :
( ( sup_sup_set_set_rat @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_689_subset__UnE,axiom,
! [C2: set_rat,A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ C2 @ ( sup_sup_set_rat @ A @ B ) )
=> ~ ! [A7: set_rat] :
( ( ord_less_eq_set_rat @ A7 @ A )
=> ! [B7: set_rat] :
( ( ord_less_eq_set_rat @ B7 @ B )
=> ( C2
!= ( sup_sup_set_rat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_690_subset__UnE,axiom,
! [C2: set_set_rat,A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ C2 @ ( sup_sup_set_set_rat @ A @ B ) )
=> ~ ! [A7: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A7 @ A )
=> ! [B7: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B7 @ B )
=> ( C2
!= ( sup_sup_set_set_rat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_691_Un__absorb2,axiom,
! [B: set_rat,A: set_rat] :
( ( ord_less_eq_set_rat @ B @ A )
=> ( ( sup_sup_set_rat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_692_Un__absorb2,axiom,
! [B: set_set_rat,A: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B @ A )
=> ( ( sup_sup_set_set_rat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_693_Un__absorb1,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( sup_sup_set_rat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_694_Un__absorb1,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( sup_sup_set_set_rat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_695_Un__upper2,axiom,
! [B: set_rat,A: set_rat] : ( ord_less_eq_set_rat @ B @ ( sup_sup_set_rat @ A @ B ) ) ).
% Un_upper2
thf(fact_696_Un__upper2,axiom,
! [B: set_set_rat,A: set_set_rat] : ( ord_le513522071413781156et_rat @ B @ ( sup_sup_set_set_rat @ A @ B ) ) ).
% Un_upper2
thf(fact_697_Un__upper1,axiom,
! [A: set_rat,B: set_rat] : ( ord_less_eq_set_rat @ A @ ( sup_sup_set_rat @ A @ B ) ) ).
% Un_upper1
thf(fact_698_Un__upper1,axiom,
! [A: set_set_rat,B: set_set_rat] : ( ord_le513522071413781156et_rat @ A @ ( sup_sup_set_set_rat @ A @ B ) ) ).
% Un_upper1
thf(fact_699_Un__least,axiom,
! [A: set_rat,C2: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ C2 )
=> ( ( ord_less_eq_set_rat @ B @ C2 )
=> ( ord_less_eq_set_rat @ ( sup_sup_set_rat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_700_Un__least,axiom,
! [A: set_set_rat,C2: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ C2 )
=> ( ( ord_le513522071413781156et_rat @ B @ C2 )
=> ( ord_le513522071413781156et_rat @ ( sup_sup_set_set_rat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_701_Un__mono,axiom,
! [A: set_rat,C2: set_rat,B: set_rat,D3: set_rat] :
( ( ord_less_eq_set_rat @ A @ C2 )
=> ( ( ord_less_eq_set_rat @ B @ D3 )
=> ( ord_less_eq_set_rat @ ( sup_sup_set_rat @ A @ B ) @ ( sup_sup_set_rat @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_702_Un__mono,axiom,
! [A: set_set_rat,C2: set_set_rat,B: set_set_rat,D3: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ C2 )
=> ( ( ord_le513522071413781156et_rat @ B @ D3 )
=> ( ord_le513522071413781156et_rat @ ( sup_sup_set_set_rat @ A @ B ) @ ( sup_sup_set_set_rat @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_703_disjoint__iff__not__equal,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ( inf_inf_set_set_rat @ A @ B )
= bot_bot_set_set_rat )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ! [Y4: set_rat] :
( ( member_set_rat @ Y4 @ B )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_704_disjoint__iff__not__equal,axiom,
! [A: set_rat,B: set_rat] :
( ( ( inf_inf_set_rat @ A @ B )
= bot_bot_set_rat )
= ( ! [X3: rat] :
( ( member_rat @ X3 @ A )
=> ! [Y4: rat] :
( ( member_rat @ Y4 @ B )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_705_Int__empty__right,axiom,
! [A: set_set_rat] :
( ( inf_inf_set_set_rat @ A @ bot_bot_set_set_rat )
= bot_bot_set_set_rat ) ).
% Int_empty_right
thf(fact_706_Int__empty__right,axiom,
! [A: set_rat] :
( ( inf_inf_set_rat @ A @ bot_bot_set_rat )
= bot_bot_set_rat ) ).
% Int_empty_right
thf(fact_707_Int__empty__left,axiom,
! [B: set_set_rat] :
( ( inf_inf_set_set_rat @ bot_bot_set_set_rat @ B )
= bot_bot_set_set_rat ) ).
% Int_empty_left
thf(fact_708_Int__empty__left,axiom,
! [B: set_rat] :
( ( inf_inf_set_rat @ bot_bot_set_rat @ B )
= bot_bot_set_rat ) ).
% Int_empty_left
thf(fact_709_disjoint__iff,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ( inf_inf_set_set_rat @ A @ B )
= bot_bot_set_set_rat )
= ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ~ ( member_set_rat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_710_disjoint__iff,axiom,
! [A: set_rat,B: set_rat] :
( ( ( inf_inf_set_rat @ A @ B )
= bot_bot_set_rat )
= ( ! [X3: rat] :
( ( member_rat @ X3 @ A )
=> ~ ( member_rat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_711_Int__emptyI,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ~ ( member_set_rat @ X @ B ) )
=> ( ( inf_inf_set_set_rat @ A @ B )
= bot_bot_set_set_rat ) ) ).
% Int_emptyI
thf(fact_712_Int__emptyI,axiom,
! [A: set_rat,B: set_rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ~ ( member_rat @ X @ B ) )
=> ( ( inf_inf_set_rat @ A @ B )
= bot_bot_set_rat ) ) ).
% Int_emptyI
thf(fact_713_Int__Collect__mono,axiom,
! [A: set_set_rat,B: set_set_rat,P: set_rat > $o,Q2: set_rat > $o] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ( P @ X )
=> ( Q2 @ X ) ) )
=> ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ A @ ( collect_set_rat @ P ) ) @ ( inf_inf_set_set_rat @ B @ ( collect_set_rat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_714_Int__Collect__mono,axiom,
! [A: set_rat,B: set_rat,P: rat > $o,Q2: rat > $o] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ( P @ X )
=> ( Q2 @ X ) ) )
=> ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ A @ ( collect_rat @ P ) ) @ ( inf_inf_set_rat @ B @ ( collect_rat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_715_Int__greatest,axiom,
! [C2: set_rat,A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ C2 @ A )
=> ( ( ord_less_eq_set_rat @ C2 @ B )
=> ( ord_less_eq_set_rat @ C2 @ ( inf_inf_set_rat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_716_Int__greatest,axiom,
! [C2: set_set_rat,A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ C2 @ A )
=> ( ( ord_le513522071413781156et_rat @ C2 @ B )
=> ( ord_le513522071413781156et_rat @ C2 @ ( inf_inf_set_set_rat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_717_Int__absorb2,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( inf_inf_set_rat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_718_Int__absorb2,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( inf_inf_set_set_rat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_719_Int__absorb1,axiom,
! [B: set_rat,A: set_rat] :
( ( ord_less_eq_set_rat @ B @ A )
=> ( ( inf_inf_set_rat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_720_Int__absorb1,axiom,
! [B: set_set_rat,A: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B @ A )
=> ( ( inf_inf_set_set_rat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_721_Int__lower2,axiom,
! [A: set_rat,B: set_rat] : ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_722_Int__lower2,axiom,
! [A: set_set_rat,B: set_set_rat] : ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_723_Int__lower1,axiom,
! [A: set_rat,B: set_rat] : ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_724_Int__lower1,axiom,
! [A: set_set_rat,B: set_set_rat] : ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_725_Int__mono,axiom,
! [A: set_rat,C2: set_rat,B: set_rat,D3: set_rat] :
( ( ord_less_eq_set_rat @ A @ C2 )
=> ( ( ord_less_eq_set_rat @ B @ D3 )
=> ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ A @ B ) @ ( inf_inf_set_rat @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_726_Int__mono,axiom,
! [A: set_set_rat,C2: set_set_rat,B: set_set_rat,D3: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ C2 )
=> ( ( ord_le513522071413781156et_rat @ B @ D3 )
=> ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ A @ B ) @ ( inf_inf_set_set_rat @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_727_Ioc__inj,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ( set_or6023941531720377480st_rat @ A2 @ B2 )
= ( set_or6023941531720377480st_rat @ C @ D ) )
= ( ( ( ord_less_eq_rat @ B2 @ A2 )
& ( ord_less_eq_rat @ D @ C ) )
| ( ( A2 = C )
& ( B2 = D ) ) ) ) ).
% Ioc_inj
thf(fact_728_not__Ici__eq__Icc,axiom,
! [L2: rat,L: rat,H: rat] :
( ( set_ord_atLeast_rat @ L2 )
!= ( set_or633870826150836451st_rat @ L @ H ) ) ).
% not_Ici_eq_Icc
thf(fact_729_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A2 @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_730_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat,D: set_set_rat] :
( ( ord_le8552383839478139994et_rat @ ( set_or2757889799628458319et_rat @ A2 @ B2 ) @ ( set_or8253465997870395507et_rat @ C @ D ) )
= ( ( ord_le513522071413781156et_rat @ A2 @ B2 )
=> ( ( ord_le513522071413781156et_rat @ C @ A2 )
& ( ord_less_set_set_rat @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_731_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat,D: set_rat] :
( ( ord_le513522071413781156et_rat @ ( set_or1040488700251649177et_rat @ A2 @ B2 ) @ ( set_or32047845639629757et_rat @ C @ D ) )
= ( ( ord_less_eq_set_rat @ A2 @ B2 )
=> ( ( ord_less_eq_set_rat @ C @ A2 )
& ( ord_less_set_rat @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_732_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) @ ( set_or4029947393144176647an_rat @ C @ D ) )
= ( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_rat @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_733_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or6023941531720377480st_rat @ A2 @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_734_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or6023941531720377480st_rat @ A2 @ B2 ) @ ( set_or4029947393144176647an_rat @ C @ D ) )
= ( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_rat @ B2 @ D ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_735_ivl__disj__un__two_I1_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_rat @ L @ M )
=> ( ( ord_less_eq_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or5199638295745620268an_rat @ L @ M ) @ ( set_or4029947393144176647an_rat @ M @ U ) )
= ( set_or5199638295745620268an_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_736_ivl__disj__un__one_I7_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ U ) @ ( set_or575021546402375026an_rat @ U ) )
= ( set_ord_atLeast_rat @ L ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_737_ivl__disj__un__two_I2_J,axiom,
! [L: rat,M: rat,U: rat] :
( ( ord_less_eq_rat @ L @ M )
=> ( ( ord_less_rat @ M @ U )
=> ( ( sup_sup_set_rat @ ( set_or6023941531720377480st_rat @ L @ M ) @ ( set_or5199638295745620268an_rat @ M @ U ) )
= ( set_or5199638295745620268an_rat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_738_inverse__less__imp__less,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ B2 @ A2 ) ) ) ).
% inverse_less_imp_less
thf(fact_739_less__imp__inverse__less,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% less_imp_inverse_less
thf(fact_740_inverse__less__imp__less__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ B2 @ A2 ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_741_less__imp__inverse__less__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_742_inverse__negative__imp__negative,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ zero_zero_rat )
=> ( ( A2 != zero_zero_rat )
=> ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).
% inverse_negative_imp_negative
thf(fact_743_inverse__positive__imp__positive,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A2 ) )
=> ( ( A2 != zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ).
% inverse_positive_imp_positive
thf(fact_744_negative__imp__inverse__negative,axiom,
! [A2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ord_less_rat @ ( inverse_inverse_rat @ A2 ) @ zero_zero_rat ) ) ).
% negative_imp_inverse_negative
thf(fact_745_positive__imp__inverse__positive,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A2 ) ) ) ).
% positive_imp_inverse_positive
thf(fact_746_atLeastLessThan__subset__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A2 @ B2 ) @ ( set_or4029947393144176647an_rat @ C @ D ) )
=> ( ( ord_less_eq_rat @ B2 @ A2 )
| ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_747_not__Ici__le__Icc,axiom,
! [L: rat,L2: rat,H2: rat] :
~ ( ord_less_eq_set_rat @ ( set_ord_atLeast_rat @ L ) @ ( set_or633870826150836451st_rat @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_748_Ioc__subset__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or6023941531720377480st_rat @ A2 @ B2 ) @ ( set_or6023941531720377480st_rat @ C @ D ) )
= ( ( ord_less_eq_rat @ B2 @ A2 )
| ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_749_inverse__le__imp__le,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).
% inverse_le_imp_le
thf(fact_750_le__imp__inverse__le,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% le_imp_inverse_le
thf(fact_751_inverse__le__imp__le__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_752_le__imp__inverse__le__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_753_atLeastatMost__psubset__iff,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat,D: set_set_rat] :
( ( ord_le8797576461720604238et_rat @ ( set_or2757889799628458319et_rat @ A2 @ B2 ) @ ( set_or2757889799628458319et_rat @ C @ D ) )
= ( ( ~ ( ord_le513522071413781156et_rat @ A2 @ B2 )
| ( ( ord_le513522071413781156et_rat @ C @ A2 )
& ( ord_le513522071413781156et_rat @ B2 @ D )
& ( ( ord_less_set_set_rat @ C @ A2 )
| ( ord_less_set_set_rat @ B2 @ D ) ) ) )
& ( ord_le513522071413781156et_rat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_754_atLeastatMost__psubset__iff,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat,D: set_rat] :
( ( ord_less_set_set_rat @ ( set_or1040488700251649177et_rat @ A2 @ B2 ) @ ( set_or1040488700251649177et_rat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_rat @ A2 @ B2 )
| ( ( ord_less_eq_set_rat @ C @ A2 )
& ( ord_less_eq_set_rat @ B2 @ D )
& ( ( ord_less_set_rat @ C @ A2 )
| ( ord_less_set_rat @ B2 @ D ) ) ) )
& ( ord_less_eq_set_rat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_755_atLeastatMost__psubset__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ( ~ ( ord_less_eq_rat @ A2 @ B2 )
| ( ( ord_less_eq_rat @ C @ A2 )
& ( ord_less_eq_rat @ B2 @ D )
& ( ( ord_less_rat @ C @ A2 )
| ( ord_less_rat @ B2 @ D ) ) ) )
& ( ord_less_eq_rat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_756_sup__bot__left,axiom,
! [X2: rat > $o] :
( ( sup_sup_rat_o @ bot_bot_rat_o @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_757_sup__bot__left,axiom,
! [X2: set_set_rat] :
( ( sup_sup_set_set_rat @ bot_bot_set_set_rat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_758_sup__bot__left,axiom,
! [X2: set_rat] :
( ( sup_sup_set_rat @ bot_bot_set_rat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_759_sup__bot__right,axiom,
! [X2: rat > $o] :
( ( sup_sup_rat_o @ X2 @ bot_bot_rat_o )
= X2 ) ).
% sup_bot_right
thf(fact_760_sup__bot__right,axiom,
! [X2: set_set_rat] :
( ( sup_sup_set_set_rat @ X2 @ bot_bot_set_set_rat )
= X2 ) ).
% sup_bot_right
thf(fact_761_sup__bot__right,axiom,
! [X2: set_rat] :
( ( sup_sup_set_rat @ X2 @ bot_bot_set_rat )
= X2 ) ).
% sup_bot_right
thf(fact_762_bot__eq__sup__iff,axiom,
! [X2: rat > $o,Y: rat > $o] :
( ( bot_bot_rat_o
= ( sup_sup_rat_o @ X2 @ Y ) )
= ( ( X2 = bot_bot_rat_o )
& ( Y = bot_bot_rat_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_763_bot__eq__sup__iff,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( bot_bot_set_set_rat
= ( sup_sup_set_set_rat @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_set_rat )
& ( Y = bot_bot_set_set_rat ) ) ) ).
% bot_eq_sup_iff
thf(fact_764_bot__eq__sup__iff,axiom,
! [X2: set_rat,Y: set_rat] :
( ( bot_bot_set_rat
= ( sup_sup_set_rat @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_rat )
& ( Y = bot_bot_set_rat ) ) ) ).
% bot_eq_sup_iff
thf(fact_765_sup__eq__bot__iff,axiom,
! [X2: rat > $o,Y: rat > $o] :
( ( ( sup_sup_rat_o @ X2 @ Y )
= bot_bot_rat_o )
= ( ( X2 = bot_bot_rat_o )
& ( Y = bot_bot_rat_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_766_sup__eq__bot__iff,axiom,
! [X2: set_set_rat,Y: set_set_rat] :
( ( ( sup_sup_set_set_rat @ X2 @ Y )
= bot_bot_set_set_rat )
= ( ( X2 = bot_bot_set_set_rat )
& ( Y = bot_bot_set_set_rat ) ) ) ).
% sup_eq_bot_iff
thf(fact_767_sup__eq__bot__iff,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ( sup_sup_set_rat @ X2 @ Y )
= bot_bot_set_rat )
= ( ( X2 = bot_bot_set_rat )
& ( Y = bot_bot_set_rat ) ) ) ).
% sup_eq_bot_iff
thf(fact_768_sup__bot_Oeq__neutr__iff,axiom,
! [A2: rat > $o,B2: rat > $o] :
( ( ( sup_sup_rat_o @ A2 @ B2 )
= bot_bot_rat_o )
= ( ( A2 = bot_bot_rat_o )
& ( B2 = bot_bot_rat_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_769_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( ( sup_sup_set_set_rat @ A2 @ B2 )
= bot_bot_set_set_rat )
= ( ( A2 = bot_bot_set_set_rat )
& ( B2 = bot_bot_set_set_rat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_770_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ( sup_sup_set_rat @ A2 @ B2 )
= bot_bot_set_rat )
= ( ( A2 = bot_bot_set_rat )
& ( B2 = bot_bot_set_rat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_771_sup__bot_Oleft__neutral,axiom,
! [A2: rat > $o] :
( ( sup_sup_rat_o @ bot_bot_rat_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_772_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_rat] :
( ( sup_sup_set_set_rat @ bot_bot_set_set_rat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_773_sup__bot_Oleft__neutral,axiom,
! [A2: set_rat] :
( ( sup_sup_set_rat @ bot_bot_set_rat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_774_sup__bot_Oneutr__eq__iff,axiom,
! [A2: rat > $o,B2: rat > $o] :
( ( bot_bot_rat_o
= ( sup_sup_rat_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_rat_o )
& ( B2 = bot_bot_rat_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_775_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( bot_bot_set_set_rat
= ( sup_sup_set_set_rat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_rat )
& ( B2 = bot_bot_set_set_rat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_776_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_rat,B2: set_rat] :
( ( bot_bot_set_rat
= ( sup_sup_set_rat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_rat )
& ( B2 = bot_bot_set_rat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_777_le__inf__iff,axiom,
! [X2: set_rat,Y: set_rat,Z: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ ( inf_inf_set_rat @ Y @ Z ) )
= ( ( ord_less_eq_set_rat @ X2 @ Y )
& ( ord_less_eq_set_rat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_778_le__inf__iff,axiom,
! [X2: set_set_rat,Y: set_set_rat,Z: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X2 @ ( inf_inf_set_set_rat @ Y @ Z ) )
= ( ( ord_le513522071413781156et_rat @ X2 @ Y )
& ( ord_le513522071413781156et_rat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_779_le__inf__iff,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ Y @ Z ) )
= ( ( ord_less_eq_rat @ X2 @ Y )
& ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_780_inf_Obounded__iff,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ ( inf_inf_set_rat @ B2 @ C ) )
= ( ( ord_less_eq_set_rat @ A2 @ B2 )
& ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_781_inf_Obounded__iff,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ ( inf_inf_set_set_rat @ B2 @ C ) )
= ( ( ord_le513522071413781156et_rat @ A2 @ B2 )
& ( ord_le513522071413781156et_rat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_782_inf_Obounded__iff,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B2 @ C ) )
= ( ( ord_less_eq_rat @ A2 @ B2 )
& ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_783_le__sup__iff,axiom,
! [X2: set_rat,Y: set_rat,Z: set_rat] :
( ( ord_less_eq_set_rat @ ( sup_sup_set_rat @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_set_rat @ X2 @ Z )
& ( ord_less_eq_set_rat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_784_le__sup__iff,axiom,
! [X2: set_set_rat,Y: set_set_rat,Z: set_set_rat] :
( ( ord_le513522071413781156et_rat @ ( sup_sup_set_set_rat @ X2 @ Y ) @ Z )
= ( ( ord_le513522071413781156et_rat @ X2 @ Z )
& ( ord_le513522071413781156et_rat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_785_le__sup__iff,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ ( sup_sup_rat @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_rat @ X2 @ Z )
& ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_786_sup_Obounded__iff,axiom,
! [B2: set_rat,C: set_rat,A2: set_rat] :
( ( ord_less_eq_set_rat @ ( sup_sup_set_rat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_rat @ B2 @ A2 )
& ( ord_less_eq_set_rat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_787_sup_Obounded__iff,axiom,
! [B2: set_set_rat,C: set_set_rat,A2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ ( sup_sup_set_set_rat @ B2 @ C ) @ A2 )
= ( ( ord_le513522071413781156et_rat @ B2 @ A2 )
& ( ord_le513522071413781156et_rat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_788_sup_Obounded__iff,axiom,
! [B2: rat,C: rat,A2: rat] :
( ( ord_less_eq_rat @ ( sup_sup_rat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_rat @ B2 @ A2 )
& ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_789_inf__bot__right,axiom,
! [X2: rat > $o] :
( ( inf_inf_rat_o @ X2 @ bot_bot_rat_o )
= bot_bot_rat_o ) ).
% inf_bot_right
thf(fact_790_inf__bot__right,axiom,
! [X2: set_set_rat] :
( ( inf_inf_set_set_rat @ X2 @ bot_bot_set_set_rat )
= bot_bot_set_set_rat ) ).
% inf_bot_right
thf(fact_791_inf__bot__right,axiom,
! [X2: set_rat] :
( ( inf_inf_set_rat @ X2 @ bot_bot_set_rat )
= bot_bot_set_rat ) ).
% inf_bot_right
thf(fact_792_inf__bot__left,axiom,
! [X2: rat > $o] :
( ( inf_inf_rat_o @ bot_bot_rat_o @ X2 )
= bot_bot_rat_o ) ).
% inf_bot_left
thf(fact_793_inf__bot__left,axiom,
! [X2: set_set_rat] :
( ( inf_inf_set_set_rat @ bot_bot_set_set_rat @ X2 )
= bot_bot_set_set_rat ) ).
% inf_bot_left
thf(fact_794_inf__bot__left,axiom,
! [X2: set_rat] :
( ( inf_inf_set_rat @ bot_bot_set_rat @ X2 )
= bot_bot_set_rat ) ).
% inf_bot_left
thf(fact_795_sup__bot_Oright__neutral,axiom,
! [A2: rat > $o] :
( ( sup_sup_rat_o @ A2 @ bot_bot_rat_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_796_sup__bot_Oright__neutral,axiom,
! [A2: set_set_rat] :
( ( sup_sup_set_set_rat @ A2 @ bot_bot_set_set_rat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_797_sup__bot_Oright__neutral,axiom,
! [A2: set_rat] :
( ( sup_sup_set_rat @ A2 @ bot_bot_set_rat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_798_inf__sup__ord_I2_J,axiom,
! [X2: set_rat,Y: set_rat] : ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_799_inf__sup__ord_I2_J,axiom,
! [X2: set_set_rat,Y: set_set_rat] : ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_800_inf__sup__ord_I2_J,axiom,
! [X2: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_801_inf__sup__ord_I1_J,axiom,
! [X2: set_rat,Y: set_rat] : ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_802_inf__sup__ord_I1_J,axiom,
! [X2: set_set_rat,Y: set_set_rat] : ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_803_inf__sup__ord_I1_J,axiom,
! [X2: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_804_inf__le1,axiom,
! [X2: set_rat,Y: set_rat] : ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_805_inf__le1,axiom,
! [X2: set_set_rat,Y: set_set_rat] : ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_806_inf__le1,axiom,
! [X2: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_807_inf__le2,axiom,
! [X2: set_rat,Y: set_rat] : ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_808_inf__le2,axiom,
! [X2: set_set_rat,Y: set_set_rat] : ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_809_inf__le2,axiom,
! [X2: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_810_le__infE,axiom,
! [X2: set_rat,A2: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ ( inf_inf_set_rat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_rat @ X2 @ A2 )
=> ~ ( ord_less_eq_set_rat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_811_le__infE,axiom,
! [X2: set_set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X2 @ ( inf_inf_set_set_rat @ A2 @ B2 ) )
=> ~ ( ( ord_le513522071413781156et_rat @ X2 @ A2 )
=> ~ ( ord_le513522071413781156et_rat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_812_le__infE,axiom,
! [X2: rat,A2: rat,B2: rat] :
( ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_rat @ X2 @ A2 )
=> ~ ( ord_less_eq_rat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_813_le__infI,axiom,
! [X2: set_rat,A2: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ X2 @ A2 )
=> ( ( ord_less_eq_set_rat @ X2 @ B2 )
=> ( ord_less_eq_set_rat @ X2 @ ( inf_inf_set_rat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_814_le__infI,axiom,
! [X2: set_set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ X2 @ A2 )
=> ( ( ord_le513522071413781156et_rat @ X2 @ B2 )
=> ( ord_le513522071413781156et_rat @ X2 @ ( inf_inf_set_set_rat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_815_le__infI,axiom,
! [X2: rat,A2: rat,B2: rat] :
( ( ord_less_eq_rat @ X2 @ A2 )
=> ( ( ord_less_eq_rat @ X2 @ B2 )
=> ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_816_inf__mono,axiom,
! [A2: set_rat,C: set_rat,B2: set_rat,D: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ C )
=> ( ( ord_less_eq_set_rat @ B2 @ D )
=> ( ord_less_eq_set_rat @ ( inf_inf_set_rat @ A2 @ B2 ) @ ( inf_inf_set_rat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_817_inf__mono,axiom,
! [A2: set_set_rat,C: set_set_rat,B2: set_set_rat,D: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ C )
=> ( ( ord_le513522071413781156et_rat @ B2 @ D )
=> ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ A2 @ B2 ) @ ( inf_inf_set_set_rat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_818_inf__mono,axiom,
! [A2: rat,C: rat,B2: rat,D: rat] :
( ( ord_less_eq_rat @ A2 @ C )
=> ( ( ord_less_eq_rat @ B2 @ D )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B2 ) @ ( inf_inf_rat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_819_le__infI1,axiom,
! [A2: set_set_rat,X2: set_set_rat,B2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A2 @ X2 )
=> ( ord_le513522071413781156et_rat @ ( inf_inf_set_set_rat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_820_le__infI1,axiom,
! [A2: rat,X2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ X2 )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_821_le__infI2,axiom,
! [B2: rat,X2: rat,A2: rat] :
( ( ord_less_eq_rat @ B2 @ X2 )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_822_inf_OorderE,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( A2
= ( inf_inf_rat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_823_inf_OorderI,axiom,
! [A2: rat,B2: rat] :
( ( A2
= ( inf_inf_rat @ A2 @ B2 ) )
=> ( ord_less_eq_rat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_824_inf__unique,axiom,
! [F: rat > rat > rat,X2: rat,Y: rat] :
( ! [X: rat,Y2: rat] : ( ord_less_eq_rat @ ( F @ X @ Y2 ) @ X )
=> ( ! [X: rat,Y2: rat] : ( ord_less_eq_rat @ ( F @ X @ Y2 ) @ Y2 )
=> ( ! [X: rat,Y2: rat,Z3: rat] :
( ( ord_less_eq_rat @ X @ Y2 )
=> ( ( ord_less_eq_rat @ X @ Z3 )
=> ( ord_less_eq_rat @ X @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_rat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_825_le__iff__inf,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y4: rat] :
( ( inf_inf_rat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_826_inf_Oabsorb1,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( inf_inf_rat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_827_inf_Oabsorb2,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( inf_inf_rat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_828_inf__absorb1,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( inf_inf_rat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_829_inf__absorb2,axiom,
! [Y: rat,X2: rat] :
( ( ord_less_eq_rat @ Y @ X2 )
=> ( ( inf_inf_rat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_830_inf_OboundedE,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_rat @ A2 @ B2 )
=> ~ ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_831_inf_OboundedI,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ A2 @ C )
=> ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_832_inf__greatest,axiom,
! [X2: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_eq_rat @ X2 @ Z )
=> ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_833_inf_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( A3
= ( inf_inf_rat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_834_inf_Ocobounded1,axiom,
! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_835_inf_Ocobounded2,axiom,
! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_836_inf_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( ( inf_inf_rat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_837_inf_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( ( inf_inf_rat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_838_inf_OcoboundedI1,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ C )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_839_inf_OcoboundedI2,axiom,
! [B2: rat,C: rat,A2: rat] :
( ( ord_less_eq_rat @ B2 @ C )
=> ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_840_inf__sup__ord_I4_J,axiom,
! [Y: rat,X2: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_841_inf__sup__ord_I3_J,axiom,
! [X2: rat,Y: rat] : ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_842_le__supE,axiom,
! [A2: rat,B2: rat,X2: rat] :
( ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_rat @ A2 @ X2 )
=> ~ ( ord_less_eq_rat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_843_le__supI,axiom,
! [A2: rat,X2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ X2 )
=> ( ( ord_less_eq_rat @ B2 @ X2 )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_844_sup__ge1,axiom,
! [X2: rat,Y: rat] : ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_845_sup__ge2,axiom,
! [Y: rat,X2: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_846_le__supI1,axiom,
! [X2: rat,A2: rat,B2: rat] :
( ( ord_less_eq_rat @ X2 @ A2 )
=> ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_847_le__supI2,axiom,
! [X2: rat,B2: rat,A2: rat] :
( ( ord_less_eq_rat @ X2 @ B2 )
=> ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_848_sup_Omono,axiom,
! [C: rat,A2: rat,D: rat,B2: rat] :
( ( ord_less_eq_rat @ C @ A2 )
=> ( ( ord_less_eq_rat @ D @ B2 )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D ) @ ( sup_sup_rat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_849_sup__mono,axiom,
! [A2: rat,C: rat,B2: rat,D: rat] :
( ( ord_less_eq_rat @ A2 @ C )
=> ( ( ord_less_eq_rat @ B2 @ D )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B2 ) @ ( sup_sup_rat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_850_sup__least,axiom,
! [Y: rat,X2: rat,Z: rat] :
( ( ord_less_eq_rat @ Y @ X2 )
=> ( ( ord_less_eq_rat @ Z @ X2 )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_851_le__iff__sup,axiom,
( ord_less_eq_rat
= ( ^ [X3: rat,Y4: rat] :
( ( sup_sup_rat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_852_sup_OorderE,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( A2
= ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_853_sup_OorderI,axiom,
! [A2: rat,B2: rat] :
( ( A2
= ( sup_sup_rat @ A2 @ B2 ) )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_854_sup__unique,axiom,
! [F: rat > rat > rat,X2: rat,Y: rat] :
( ! [X: rat,Y2: rat] : ( ord_less_eq_rat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: rat,Y2: rat] : ( ord_less_eq_rat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: rat,Y2: rat,Z3: rat] :
( ( ord_less_eq_rat @ Y2 @ X )
=> ( ( ord_less_eq_rat @ Z3 @ X )
=> ( ord_less_eq_rat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_rat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_855_sup_Oabsorb1,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( sup_sup_rat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_856_sup_Oabsorb2,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( sup_sup_rat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_857_sup__absorb1,axiom,
! [Y: rat,X2: rat] :
( ( ord_less_eq_rat @ Y @ X2 )
=> ( ( sup_sup_rat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_858_sup__absorb2,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( sup_sup_rat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_859_sup_OboundedE,axiom,
! [B2: rat,C: rat,A2: rat] :
( ( ord_less_eq_rat @ ( sup_sup_rat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_rat @ B2 @ A2 )
=> ~ ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_860_sup_OboundedI,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ A2 )
=> ( ord_less_eq_rat @ ( sup_sup_rat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_861_sup_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( A3
= ( sup_sup_rat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_862_sup_Ocobounded1,axiom,
! [A2: rat,B2: rat] : ( ord_less_eq_rat @ A2 @ ( sup_sup_rat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_863_sup_Ocobounded2,axiom,
! [B2: rat,A2: rat] : ( ord_less_eq_rat @ B2 @ ( sup_sup_rat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_864_sup_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A3: rat] :
( ( sup_sup_rat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_865_sup_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] :
( ( sup_sup_rat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_866_sup_OcoboundedI1,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_eq_rat @ C @ A2 )
=> ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_867_sup_OcoboundedI2,axiom,
! [C: rat,B2: rat,A2: rat] :
( ( ord_less_eq_rat @ C @ B2 )
=> ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_868_inf_Ostrict__coboundedI2,axiom,
! [B2: rat,C: rat,A2: rat] :
( ( ord_less_rat @ B2 @ C )
=> ( ord_less_rat @ ( inf_inf_rat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_869_inf_Ostrict__coboundedI2,axiom,
! [B2: set_rat,C: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ ( inf_inf_set_rat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_870_inf_Ostrict__coboundedI1,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_rat @ A2 @ C )
=> ( ord_less_rat @ ( inf_inf_rat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_871_inf_Ostrict__coboundedI1,axiom,
! [A2: set_rat,C: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ C )
=> ( ord_less_set_rat @ ( inf_inf_set_rat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_872_inf_Ostrict__order__iff,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] :
( ( A3
= ( inf_inf_rat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_873_inf_Ostrict__order__iff,axiom,
( ord_less_set_rat
= ( ^ [A3: set_rat,B3: set_rat] :
( ( A3
= ( inf_inf_set_rat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_874_inf_Ostrict__boundedE,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( inf_inf_rat @ B2 @ C ) )
=> ~ ( ( ord_less_rat @ A2 @ B2 )
=> ~ ( ord_less_rat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_875_inf_Ostrict__boundedE,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ ( inf_inf_set_rat @ B2 @ C ) )
=> ~ ( ( ord_less_set_rat @ A2 @ B2 )
=> ~ ( ord_less_set_rat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_876_inf_Oabsorb4,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( inf_inf_rat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_877_inf_Oabsorb4,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ( inf_inf_set_rat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_878_inf_Oabsorb3,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( inf_inf_rat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_879_inf_Oabsorb3,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( inf_inf_set_rat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_880_less__infI2,axiom,
! [B2: rat,X2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ X2 )
=> ( ord_less_rat @ ( inf_inf_rat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_881_less__infI2,axiom,
! [B2: set_rat,X2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ X2 )
=> ( ord_less_set_rat @ ( inf_inf_set_rat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_882_less__infI1,axiom,
! [A2: rat,X2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ X2 )
=> ( ord_less_rat @ ( inf_inf_rat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_883_less__infI1,axiom,
! [A2: set_rat,X2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ X2 )
=> ( ord_less_set_rat @ ( inf_inf_set_rat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_884_sup_Ostrict__coboundedI2,axiom,
! [C: rat,B2: rat,A2: rat] :
( ( ord_less_rat @ C @ B2 )
=> ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_885_sup_Ostrict__coboundedI2,axiom,
! [C: set_rat,B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ C @ B2 )
=> ( ord_less_set_rat @ C @ ( sup_sup_set_rat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_886_sup_Ostrict__coboundedI1,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ C @ A2 )
=> ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_887_sup_Ostrict__coboundedI1,axiom,
! [C: set_rat,A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ C @ A2 )
=> ( ord_less_set_rat @ C @ ( sup_sup_set_rat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_888_sup_Ostrict__order__iff,axiom,
( ord_less_rat
= ( ^ [B3: rat,A3: rat] :
( ( A3
= ( sup_sup_rat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_889_sup_Ostrict__order__iff,axiom,
( ord_less_set_rat
= ( ^ [B3: set_rat,A3: set_rat] :
( ( A3
= ( sup_sup_set_rat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_890_sup_Ostrict__boundedE,axiom,
! [B2: rat,C: rat,A2: rat] :
( ( ord_less_rat @ ( sup_sup_rat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_rat @ B2 @ A2 )
=> ~ ( ord_less_rat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_891_sup_Ostrict__boundedE,axiom,
! [B2: set_rat,C: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ ( sup_sup_set_rat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_rat @ B2 @ A2 )
=> ~ ( ord_less_set_rat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_892_sup_Oabsorb4,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( sup_sup_rat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_893_sup_Oabsorb4,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( sup_sup_set_rat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_894_sup_Oabsorb3,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( sup_sup_rat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_895_sup_Oabsorb3,axiom,
! [B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B2 @ A2 )
=> ( ( sup_sup_set_rat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_896_less__supI2,axiom,
! [X2: rat,B2: rat,A2: rat] :
( ( ord_less_rat @ X2 @ B2 )
=> ( ord_less_rat @ X2 @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_897_less__supI2,axiom,
! [X2: set_rat,B2: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ X2 @ B2 )
=> ( ord_less_set_rat @ X2 @ ( sup_sup_set_rat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_898_less__supI1,axiom,
! [X2: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ X2 @ A2 )
=> ( ord_less_rat @ X2 @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_899_less__supI1,axiom,
! [X2: set_rat,A2: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ X2 @ A2 )
=> ( ord_less_set_rat @ X2 @ ( sup_sup_set_rat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_900_distrib__sup__le,axiom,
! [X2: rat,Y: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X2 @ ( inf_inf_rat @ Y @ Z ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X2 @ Y ) @ ( sup_sup_rat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_901_distrib__inf__le,axiom,
! [X2: rat,Y: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X2 @ Y ) @ ( inf_inf_rat @ X2 @ Z ) ) @ ( inf_inf_rat @ X2 @ ( sup_sup_rat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_902_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_rat] :
( ( inf_inf_set_rat @ X2 @ bot_bot_set_rat )
= bot_bot_set_rat ) ).
% boolean_algebra.conj_zero_right
thf(fact_903_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_rat] :
( ( inf_inf_set_rat @ bot_bot_set_rat @ X2 )
= bot_bot_set_rat ) ).
% boolean_algebra.conj_zero_left
thf(fact_904_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_rat] :
( ( sup_sup_set_rat @ X2 @ bot_bot_set_rat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_905_ivl__disj__un__singleton_I4_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_or5199638295745620268an_rat @ L @ U ) @ ( insert_rat @ U @ bot_bot_set_rat ) )
= ( set_or6023941531720377480st_rat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_906_ivl__disj__un__singleton_I3_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( insert_rat @ L @ bot_bot_set_rat ) @ ( set_or5199638295745620268an_rat @ L @ U ) )
= ( set_or4029947393144176647an_rat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_907_insertCI,axiom,
! [A2: rat,B: set_rat,B2: rat] :
( ( ~ ( member_rat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_rat @ A2 @ ( insert_rat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_908_insert__iff,axiom,
! [A2: rat,B2: rat,A: set_rat] :
( ( member_rat @ A2 @ ( insert_rat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_rat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_909_singletonI,axiom,
! [A2: rat] : ( member_rat @ A2 @ ( insert_rat @ A2 @ bot_bot_set_rat ) ) ).
% singletonI
thf(fact_910_insert__subset,axiom,
! [X2: rat,A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ ( insert_rat @ X2 @ A ) @ B )
= ( ( member_rat @ X2 @ B )
& ( ord_less_eq_set_rat @ A @ B ) ) ) ).
% insert_subset
thf(fact_911_Int__insert__left__if0,axiom,
! [A2: rat,C2: set_rat,B: set_rat] :
( ~ ( member_rat @ A2 @ C2 )
=> ( ( inf_inf_set_rat @ ( insert_rat @ A2 @ B ) @ C2 )
= ( inf_inf_set_rat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_912_Int__insert__left__if1,axiom,
! [A2: rat,C2: set_rat,B: set_rat] :
( ( member_rat @ A2 @ C2 )
=> ( ( inf_inf_set_rat @ ( insert_rat @ A2 @ B ) @ C2 )
= ( insert_rat @ A2 @ ( inf_inf_set_rat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_913_Int__insert__right__if0,axiom,
! [A2: rat,A: set_rat,B: set_rat] :
( ~ ( member_rat @ A2 @ A )
=> ( ( inf_inf_set_rat @ A @ ( insert_rat @ A2 @ B ) )
= ( inf_inf_set_rat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_914_Int__insert__right__if1,axiom,
! [A2: rat,A: set_rat,B: set_rat] :
( ( member_rat @ A2 @ A )
=> ( ( inf_inf_set_rat @ A @ ( insert_rat @ A2 @ B ) )
= ( insert_rat @ A2 @ ( inf_inf_set_rat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_915_singleton__insert__inj__eq,axiom,
! [B2: rat,A2: rat,A: set_rat] :
( ( ( insert_rat @ B2 @ bot_bot_set_rat )
= ( insert_rat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_rat @ A @ ( insert_rat @ B2 @ bot_bot_set_rat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_916_singleton__insert__inj__eq_H,axiom,
! [A2: rat,A: set_rat,B2: rat] :
( ( ( insert_rat @ A2 @ A )
= ( insert_rat @ B2 @ bot_bot_set_rat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_rat @ A @ ( insert_rat @ B2 @ bot_bot_set_rat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_917_disjoint__insert_I2_J,axiom,
! [A: set_rat,B2: rat,B: set_rat] :
( ( bot_bot_set_rat
= ( inf_inf_set_rat @ A @ ( insert_rat @ B2 @ B ) ) )
= ( ~ ( member_rat @ B2 @ A )
& ( bot_bot_set_rat
= ( inf_inf_set_rat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_918_disjoint__insert_I1_J,axiom,
! [B: set_rat,A2: rat,A: set_rat] :
( ( ( inf_inf_set_rat @ B @ ( insert_rat @ A2 @ A ) )
= bot_bot_set_rat )
= ( ~ ( member_rat @ A2 @ B )
& ( ( inf_inf_set_rat @ B @ A )
= bot_bot_set_rat ) ) ) ).
% disjoint_insert(1)
thf(fact_919_insert__disjoint_I2_J,axiom,
! [A2: rat,A: set_rat,B: set_rat] :
( ( bot_bot_set_rat
= ( inf_inf_set_rat @ ( insert_rat @ A2 @ A ) @ B ) )
= ( ~ ( member_rat @ A2 @ B )
& ( bot_bot_set_rat
= ( inf_inf_set_rat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_920_insert__disjoint_I1_J,axiom,
! [A2: rat,A: set_rat,B: set_rat] :
( ( ( inf_inf_set_rat @ ( insert_rat @ A2 @ A ) @ B )
= bot_bot_set_rat )
= ( ~ ( member_rat @ A2 @ B )
& ( ( inf_inf_set_rat @ A @ B )
= bot_bot_set_rat ) ) ) ).
% insert_disjoint(1)
thf(fact_921_atLeastAtMost__singleton,axiom,
! [A2: rat] :
( ( set_or633870826150836451st_rat @ A2 @ A2 )
= ( insert_rat @ A2 @ bot_bot_set_rat ) ) ).
% atLeastAtMost_singleton
thf(fact_922_atLeastAtMost__singleton__iff,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ( set_or633870826150836451st_rat @ A2 @ B2 )
= ( insert_rat @ C @ bot_bot_set_rat ) )
= ( ( A2 = B2 )
& ( B2 = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_923_singletonD,axiom,
! [B2: rat,A2: rat] :
( ( member_rat @ B2 @ ( insert_rat @ A2 @ bot_bot_set_rat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_924_singleton__iff,axiom,
! [B2: rat,A2: rat] :
( ( member_rat @ B2 @ ( insert_rat @ A2 @ bot_bot_set_rat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_925_doubleton__eq__iff,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ( insert_rat @ A2 @ ( insert_rat @ B2 @ bot_bot_set_rat ) )
= ( insert_rat @ C @ ( insert_rat @ D @ bot_bot_set_rat ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_926_insert__not__empty,axiom,
! [A2: rat,A: set_rat] :
( ( insert_rat @ A2 @ A )
!= bot_bot_set_rat ) ).
% insert_not_empty
thf(fact_927_singleton__inject,axiom,
! [A2: rat,B2: rat] :
( ( ( insert_rat @ A2 @ bot_bot_set_rat )
= ( insert_rat @ B2 @ bot_bot_set_rat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_928_insert__subsetI,axiom,
! [X2: rat,A: set_rat,X5: set_rat] :
( ( member_rat @ X2 @ A )
=> ( ( ord_less_eq_set_rat @ X5 @ A )
=> ( ord_less_eq_set_rat @ ( insert_rat @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_929_subset__insert,axiom,
! [X2: rat,A: set_rat,B: set_rat] :
( ~ ( member_rat @ X2 @ A )
=> ( ( ord_less_eq_set_rat @ A @ ( insert_rat @ X2 @ B ) )
= ( ord_less_eq_set_rat @ A @ B ) ) ) ).
% subset_insert
thf(fact_930_insertE,axiom,
! [A2: rat,B2: rat,A: set_rat] :
( ( member_rat @ A2 @ ( insert_rat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_rat @ A2 @ A ) ) ) ).
% insertE
thf(fact_931_insertI1,axiom,
! [A2: rat,B: set_rat] : ( member_rat @ A2 @ ( insert_rat @ A2 @ B ) ) ).
% insertI1
thf(fact_932_insertI2,axiom,
! [A2: rat,B: set_rat,B2: rat] :
( ( member_rat @ A2 @ B )
=> ( member_rat @ A2 @ ( insert_rat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_933_Set_Oset__insert,axiom,
! [X2: rat,A: set_rat] :
( ( member_rat @ X2 @ A )
=> ~ ! [B8: set_rat] :
( ( A
= ( insert_rat @ X2 @ B8 ) )
=> ( member_rat @ X2 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_934_insert__ident,axiom,
! [X2: rat,A: set_rat,B: set_rat] :
( ~ ( member_rat @ X2 @ A )
=> ( ~ ( member_rat @ X2 @ B )
=> ( ( ( insert_rat @ X2 @ A )
= ( insert_rat @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_935_insert__absorb,axiom,
! [A2: rat,A: set_rat] :
( ( member_rat @ A2 @ A )
=> ( ( insert_rat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_936_insert__eq__iff,axiom,
! [A2: rat,A: set_rat,B2: rat,B: set_rat] :
( ~ ( member_rat @ A2 @ A )
=> ( ~ ( member_rat @ B2 @ B )
=> ( ( ( insert_rat @ A2 @ A )
= ( insert_rat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_rat] :
( ( A
= ( insert_rat @ B2 @ C3 ) )
& ~ ( member_rat @ B2 @ C3 )
& ( B
= ( insert_rat @ A2 @ C3 ) )
& ~ ( member_rat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_937_mk__disjoint__insert,axiom,
! [A2: rat,A: set_rat] :
( ( member_rat @ A2 @ A )
=> ? [B8: set_rat] :
( ( A
= ( insert_rat @ A2 @ B8 ) )
& ~ ( member_rat @ A2 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_938_Int__insert__left,axiom,
! [A2: rat,C2: set_rat,B: set_rat] :
( ( ( member_rat @ A2 @ C2 )
=> ( ( inf_inf_set_rat @ ( insert_rat @ A2 @ B ) @ C2 )
= ( insert_rat @ A2 @ ( inf_inf_set_rat @ B @ C2 ) ) ) )
& ( ~ ( member_rat @ A2 @ C2 )
=> ( ( inf_inf_set_rat @ ( insert_rat @ A2 @ B ) @ C2 )
= ( inf_inf_set_rat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_939_Int__insert__right,axiom,
! [A2: rat,A: set_rat,B: set_rat] :
( ( ( member_rat @ A2 @ A )
=> ( ( inf_inf_set_rat @ A @ ( insert_rat @ A2 @ B ) )
= ( insert_rat @ A2 @ ( inf_inf_set_rat @ A @ B ) ) ) )
& ( ~ ( member_rat @ A2 @ A )
=> ( ( inf_inf_set_rat @ A @ ( insert_rat @ A2 @ B ) )
= ( inf_inf_set_rat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_940_subset__singletonD,axiom,
! [A: set_rat,X2: rat] :
( ( ord_less_eq_set_rat @ A @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
=> ( ( A = bot_bot_set_rat )
| ( A
= ( insert_rat @ X2 @ bot_bot_set_rat ) ) ) ) ).
% subset_singletonD
thf(fact_941_subset__singleton__iff,axiom,
! [X5: set_rat,A2: rat] :
( ( ord_less_eq_set_rat @ X5 @ ( insert_rat @ A2 @ bot_bot_set_rat ) )
= ( ( X5 = bot_bot_set_rat )
| ( X5
= ( insert_rat @ A2 @ bot_bot_set_rat ) ) ) ) ).
% subset_singleton_iff
thf(fact_942_atLeastAtMost__singleton_H,axiom,
! [A2: rat,B2: rat] :
( ( A2 = B2 )
=> ( ( set_or633870826150836451st_rat @ A2 @ B2 )
= ( insert_rat @ A2 @ bot_bot_set_rat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_943_insert__is__Un,axiom,
( insert_rat
= ( ^ [A3: rat] : ( sup_sup_set_rat @ ( insert_rat @ A3 @ bot_bot_set_rat ) ) ) ) ).
% insert_is_Un
thf(fact_944_Un__singleton__iff,axiom,
! [A: set_rat,B: set_rat,X2: rat] :
( ( ( sup_sup_set_rat @ A @ B )
= ( insert_rat @ X2 @ bot_bot_set_rat ) )
= ( ( ( A = bot_bot_set_rat )
& ( B
= ( insert_rat @ X2 @ bot_bot_set_rat ) ) )
| ( ( A
= ( insert_rat @ X2 @ bot_bot_set_rat ) )
& ( B = bot_bot_set_rat ) )
| ( ( A
= ( insert_rat @ X2 @ bot_bot_set_rat ) )
& ( B
= ( insert_rat @ X2 @ bot_bot_set_rat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_945_singleton__Un__iff,axiom,
! [X2: rat,A: set_rat,B: set_rat] :
( ( ( insert_rat @ X2 @ bot_bot_set_rat )
= ( sup_sup_set_rat @ A @ B ) )
= ( ( ( A = bot_bot_set_rat )
& ( B
= ( insert_rat @ X2 @ bot_bot_set_rat ) ) )
| ( ( A
= ( insert_rat @ X2 @ bot_bot_set_rat ) )
& ( B = bot_bot_set_rat ) )
| ( ( A
= ( insert_rat @ X2 @ bot_bot_set_rat ) )
& ( B
= ( insert_rat @ X2 @ bot_bot_set_rat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_946_ivl__disj__un__singleton_I1_J,axiom,
! [L: rat] :
( ( sup_sup_set_rat @ ( insert_rat @ L @ bot_bot_set_rat ) @ ( set_or575021546402375026an_rat @ L ) )
= ( set_ord_atLeast_rat @ L ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_947_ivl__disj__un__singleton_I6_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ U ) @ ( insert_rat @ U @ bot_bot_set_rat ) )
= ( set_or633870826150836451st_rat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_948_ivl__disj__un__singleton_I5_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( insert_rat @ L @ bot_bot_set_rat ) @ ( set_or6023941531720377480st_rat @ L @ U ) )
= ( set_or633870826150836451st_rat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_949_the__elem__eq,axiom,
! [X2: rat] :
( ( the_elem_rat @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
= X2 ) ).
% the_elem_eq
thf(fact_950_is__singletonI,axiom,
! [X2: rat] : ( is_singleton_rat @ ( insert_rat @ X2 @ bot_bot_set_rat ) ) ).
% is_singletonI
thf(fact_951_is__singleton__def,axiom,
( is_singleton_rat
= ( ^ [A5: set_rat] :
? [X3: rat] :
( A5
= ( insert_rat @ X3 @ bot_bot_set_rat ) ) ) ) ).
% is_singleton_def
thf(fact_952_is__singletonE,axiom,
! [A: set_rat] :
( ( is_singleton_rat @ A )
=> ~ ! [X: rat] :
( A
!= ( insert_rat @ X @ bot_bot_set_rat ) ) ) ).
% is_singletonE
thf(fact_953_is__singleton__the__elem,axiom,
( is_singleton_rat
= ( ^ [A5: set_rat] :
( A5
= ( insert_rat @ ( the_elem_rat @ A5 ) @ bot_bot_set_rat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_954_is__singletonI_H,axiom,
! [A: set_rat] :
( ( A != bot_bot_set_rat )
=> ( ! [X: rat,Y2: rat] :
( ( member_rat @ X @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_singleton_rat @ A ) ) ) ).
% is_singletonI'
thf(fact_955_atMost__Int__atLeast,axiom,
! [N: rat] :
( ( inf_inf_set_rat @ ( set_ord_atMost_rat @ N ) @ ( set_ord_atLeast_rat @ N ) )
= ( insert_rat @ N @ bot_bot_set_rat ) ) ).
% atMost_Int_atLeast
thf(fact_956_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or6023941531720377480st_rat
= ( ^ [A3: rat,B3: rat] : ( minus_minus_set_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) @ ( insert_rat @ A3 @ bot_bot_set_rat ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_957_atLeastAtMost__diff__ends,axiom,
! [A2: rat,B2: rat] :
( ( minus_minus_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) @ ( insert_rat @ A2 @ ( insert_rat @ B2 @ bot_bot_set_rat ) ) )
= ( set_or5199638295745620268an_rat @ A2 @ B2 ) ) ).
% atLeastAtMost_diff_ends
thf(fact_958_one__le__inverse,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ one_one_rat )
=> ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% one_le_inverse
thf(fact_959_DiffI,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ A )
=> ( ~ ( member_rat @ C @ B )
=> ( member_rat @ C @ ( minus_minus_set_rat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_960_Diff__iff,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( minus_minus_set_rat @ A @ B ) )
= ( ( member_rat @ C @ A )
& ~ ( member_rat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_961_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: rat] :
( ( minus_minus_rat @ A2 @ A2 )
= zero_zero_rat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_962_diff__zero,axiom,
! [A2: rat] :
( ( minus_minus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% diff_zero
thf(fact_963_diff__0__right,axiom,
! [A2: rat] :
( ( minus_minus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% diff_0_right
thf(fact_964_diff__self,axiom,
! [A2: rat] :
( ( minus_minus_rat @ A2 @ A2 )
= zero_zero_rat ) ).
% diff_self
thf(fact_965_inverse__eq__1__iff,axiom,
! [X2: rat] :
( ( ( inverse_inverse_rat @ X2 )
= one_one_rat )
= ( X2 = one_one_rat ) ) ).
% inverse_eq_1_iff
thf(fact_966_inverse__1,axiom,
( ( inverse_inverse_rat @ one_one_rat )
= one_one_rat ) ).
% inverse_1
thf(fact_967_atMost__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
= ( ord_less_eq_rat @ I @ K ) ) ).
% atMost_iff
thf(fact_968_Diff__empty,axiom,
! [A: set_rat] :
( ( minus_minus_set_rat @ A @ bot_bot_set_rat )
= A ) ).
% Diff_empty
thf(fact_969_empty__Diff,axiom,
! [A: set_rat] :
( ( minus_minus_set_rat @ bot_bot_set_rat @ A )
= bot_bot_set_rat ) ).
% empty_Diff
thf(fact_970_Diff__cancel,axiom,
! [A: set_rat] :
( ( minus_minus_set_rat @ A @ A )
= bot_bot_set_rat ) ).
% Diff_cancel
thf(fact_971_insert__Diff1,axiom,
! [X2: rat,B: set_rat,A: set_rat] :
( ( member_rat @ X2 @ B )
=> ( ( minus_minus_set_rat @ ( insert_rat @ X2 @ A ) @ B )
= ( minus_minus_set_rat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_972_Diff__insert0,axiom,
! [X2: rat,A: set_rat,B: set_rat] :
( ~ ( member_rat @ X2 @ A )
=> ( ( minus_minus_set_rat @ A @ ( insert_rat @ X2 @ B ) )
= ( minus_minus_set_rat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_973_diff__ge__0__iff__ge,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_974_diff__gt__0__iff__gt,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_975_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_976_atMost__subset__iff,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X2 ) @ ( set_ord_atMost_rat @ Y ) )
= ( ord_less_eq_rat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_977_Diff__eq__empty__iff,axiom,
! [A: set_rat,B: set_rat] :
( ( ( minus_minus_set_rat @ A @ B )
= bot_bot_set_rat )
= ( ord_less_eq_set_rat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_978_insert__Diff__single,axiom,
! [A2: rat,A: set_rat] :
( ( insert_rat @ A2 @ ( minus_minus_set_rat @ A @ ( insert_rat @ A2 @ bot_bot_set_rat ) ) )
= ( insert_rat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_979_ivl__diff,axiom,
! [I: rat,N: rat,M: rat] :
( ( ord_less_eq_rat @ I @ N )
=> ( ( minus_minus_set_rat @ ( set_or4029947393144176647an_rat @ I @ M ) @ ( set_or4029947393144176647an_rat @ I @ N ) )
= ( set_or4029947393144176647an_rat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_980_Diff__disjoint,axiom,
! [A: set_rat,B: set_rat] :
( ( inf_inf_set_rat @ A @ ( minus_minus_set_rat @ B @ A ) )
= bot_bot_set_rat ) ).
% Diff_disjoint
thf(fact_981_Icc__subset__Iic__iff,axiom,
! [L: rat,H: rat,H2: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H ) @ ( set_ord_atMost_rat @ H2 ) )
= ( ~ ( ord_less_eq_rat @ L @ H )
| ( ord_less_eq_rat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_982_of__rat__less__1__iff,axiom,
! [R: rat] :
( ( ord_less_rat @ ( field_2639924705303425560at_rat @ R ) @ one_one_rat )
= ( ord_less_rat @ R @ one_one_rat ) ) ).
% of_rat_less_1_iff
thf(fact_983_one__less__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R ) )
= ( ord_less_rat @ one_one_rat @ R ) ) ).
% one_less_of_rat_iff
thf(fact_984_one__le__of__rat__iff,axiom,
! [R: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( field_2639924705303425560at_rat @ R ) )
= ( ord_less_eq_rat @ one_one_rat @ R ) ) ).
% one_le_of_rat_iff
thf(fact_985_of__rat__le__1__iff,axiom,
! [R: rat] :
( ( ord_less_eq_rat @ ( field_2639924705303425560at_rat @ R ) @ one_one_rat )
= ( ord_less_eq_rat @ R @ one_one_rat ) ) ).
% of_rat_le_1_iff
thf(fact_986_diff__eq__diff__less__eq,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A2 @ B2 )
= ( minus_minus_rat @ C @ D ) )
=> ( ( ord_less_eq_rat @ A2 @ B2 )
= ( ord_less_eq_rat @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_987_diff__right__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_988_diff__left__mono,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_989_diff__mono,axiom,
! [A2: rat,B2: rat,D: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ D @ C )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_990_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
= ( ^ [A3: rat,B3: rat] :
( ( minus_minus_rat @ A3 @ B3 )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_991_diff__strict__right__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_992_diff__strict__left__mono,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_993_diff__eq__diff__less,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A2 @ B2 )
= ( minus_minus_rat @ C @ D ) )
=> ( ( ord_less_rat @ A2 @ B2 )
= ( ord_less_rat @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_994_diff__strict__mono,axiom,
! [A2: rat,B2: rat,D: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ D @ C )
=> ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_995_DiffE,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( minus_minus_set_rat @ A @ B ) )
=> ~ ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B ) ) ) ).
% DiffE
thf(fact_996_DiffD1,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( minus_minus_set_rat @ A @ B ) )
=> ( member_rat @ C @ A ) ) ).
% DiffD1
thf(fact_997_DiffD2,axiom,
! [C: rat,A: set_rat,B: set_rat] :
( ( member_rat @ C @ ( minus_minus_set_rat @ A @ B ) )
=> ~ ( member_rat @ C @ B ) ) ).
% DiffD2
thf(fact_998_psubset__imp__ex__mem,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ? [B4: rat] : ( member_rat @ B4 @ ( minus_minus_set_rat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_999_not__empty__eq__Iic__eq__empty,axiom,
! [H: rat] :
( bot_bot_set_rat
!= ( set_ord_atMost_rat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_1000_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_1001_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_1002_insert__Diff__if,axiom,
! [X2: rat,B: set_rat,A: set_rat] :
( ( ( member_rat @ X2 @ B )
=> ( ( minus_minus_set_rat @ ( insert_rat @ X2 @ A ) @ B )
= ( minus_minus_set_rat @ A @ B ) ) )
& ( ~ ( member_rat @ X2 @ B )
=> ( ( minus_minus_set_rat @ ( insert_rat @ X2 @ A ) @ B )
= ( insert_rat @ X2 @ ( minus_minus_set_rat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1003_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_1004_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_1005_diff__shunt__var,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ( minus_minus_set_rat @ X2 @ Y )
= bot_bot_set_rat )
= ( ord_less_eq_set_rat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_1006_Diff__insert,axiom,
! [A: set_rat,A2: rat,B: set_rat] :
( ( minus_minus_set_rat @ A @ ( insert_rat @ A2 @ B ) )
= ( minus_minus_set_rat @ ( minus_minus_set_rat @ A @ B ) @ ( insert_rat @ A2 @ bot_bot_set_rat ) ) ) ).
% Diff_insert
thf(fact_1007_insert__Diff,axiom,
! [A2: rat,A: set_rat] :
( ( member_rat @ A2 @ A )
=> ( ( insert_rat @ A2 @ ( minus_minus_set_rat @ A @ ( insert_rat @ A2 @ bot_bot_set_rat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1008_Diff__insert2,axiom,
! [A: set_rat,A2: rat,B: set_rat] :
( ( minus_minus_set_rat @ A @ ( insert_rat @ A2 @ B ) )
= ( minus_minus_set_rat @ ( minus_minus_set_rat @ A @ ( insert_rat @ A2 @ bot_bot_set_rat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1009_Diff__insert__absorb,axiom,
! [X2: rat,A: set_rat] :
( ~ ( member_rat @ X2 @ A )
=> ( ( minus_minus_set_rat @ ( insert_rat @ X2 @ A ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1010_subset__Diff__insert,axiom,
! [A: set_rat,B: set_rat,X2: rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A @ ( minus_minus_set_rat @ B @ ( insert_rat @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_rat @ A @ ( minus_minus_set_rat @ B @ C2 ) )
& ~ ( member_rat @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1011_Diff__triv,axiom,
! [A: set_rat,B: set_rat] :
( ( ( inf_inf_set_rat @ A @ B )
= bot_bot_set_rat )
=> ( ( minus_minus_set_rat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1012_Int__Diff__disjoint,axiom,
! [A: set_rat,B: set_rat] :
( ( inf_inf_set_rat @ ( inf_inf_set_rat @ A @ B ) @ ( minus_minus_set_rat @ A @ B ) )
= bot_bot_set_rat ) ).
% Int_Diff_disjoint
thf(fact_1013_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_1014_subset__insert__iff,axiom,
! [A: set_rat,X2: rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ ( insert_rat @ X2 @ B ) )
= ( ( ( member_rat @ X2 @ A )
=> ( ord_less_eq_set_rat @ ( minus_minus_set_rat @ A @ ( insert_rat @ X2 @ bot_bot_set_rat ) ) @ B ) )
& ( ~ ( member_rat @ X2 @ A )
=> ( ord_less_eq_set_rat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1015_Diff__single__insert,axiom,
! [A: set_rat,X2: rat,B: set_rat] :
( ( ord_less_eq_set_rat @ ( minus_minus_set_rat @ A @ ( insert_rat @ X2 @ bot_bot_set_rat ) ) @ B )
=> ( ord_less_eq_set_rat @ A @ ( insert_rat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1016_inverse__le__1__iff,axiom,
! [X2: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X2 ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
| ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).
% inverse_le_1_iff
thf(fact_1017_one__less__inverse,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ A2 @ one_one_rat )
=> ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A2 ) ) ) ) ).
% one_less_inverse
thf(fact_1018_one__less__inverse__iff,axiom,
! [X2: rat] :
( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X2 ) )
= ( ( ord_less_rat @ zero_zero_rat @ X2 )
& ( ord_less_rat @ X2 @ one_one_rat ) ) ) ).
% one_less_inverse_iff
thf(fact_1019_ivl__disj__un__one_I3_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_ord_atMost_rat @ L ) @ ( set_or6023941531720377480st_rat @ L @ U ) )
= ( set_ord_atMost_rat @ U ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_1020_ivl__disj__int__one_I1_J,axiom,
! [L: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_ord_atMost_rat @ L ) @ ( set_or5199638295745620268an_rat @ L @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_one(1)
thf(fact_1021_ivl__disj__int__one_I3_J,axiom,
! [L: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_ord_atMost_rat @ L ) @ ( set_or6023941531720377480st_rat @ L @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_one(3)
thf(fact_1022_greaterThanAtMost__def,axiom,
( set_or6023941531720377480st_rat
= ( ^ [L3: rat,U2: rat] : ( inf_inf_set_rat @ ( set_or575021546402375026an_rat @ L3 ) @ ( set_ord_atMost_rat @ U2 ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_1023_psubset__insert__iff,axiom,
! [A: set_rat,X2: rat,B: set_rat] :
( ( ord_less_set_rat @ A @ ( insert_rat @ X2 @ B ) )
= ( ( ( member_rat @ X2 @ B )
=> ( ord_less_set_rat @ A @ B ) )
& ( ~ ( member_rat @ X2 @ B )
=> ( ( ( member_rat @ X2 @ A )
=> ( ord_less_set_rat @ ( minus_minus_set_rat @ A @ ( insert_rat @ X2 @ bot_bot_set_rat ) ) @ B ) )
& ( ~ ( member_rat @ X2 @ A )
=> ( ord_less_eq_set_rat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1024_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4029947393144176647an_rat
= ( ^ [A3: rat,B3: rat] : ( minus_minus_set_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) @ ( insert_rat @ B3 @ bot_bot_set_rat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1025_one__le__inverse__iff,axiom,
! [X2: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X2 ) )
= ( ( ord_less_rat @ zero_zero_rat @ X2 )
& ( ord_less_eq_rat @ X2 @ one_one_rat ) ) ) ).
% one_le_inverse_iff
thf(fact_1026_inverse__less__1__iff,axiom,
! [X2: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ X2 ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
| ( ord_less_rat @ one_one_rat @ X2 ) ) ) ).
% inverse_less_1_iff
thf(fact_1027_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_1028_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_1029_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_1030_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1031_linorder__neqE__linordered__idom,axiom,
! [X2: rat,Y: rat] :
( ( X2 != Y )
=> ( ~ ( ord_less_rat @ X2 @ Y )
=> ( ord_less_rat @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1032_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_1033_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_1034_remove__def,axiom,
( remove_rat
= ( ^ [X3: rat,A5: set_rat] : ( minus_minus_set_rat @ A5 @ ( insert_rat @ X3 @ bot_bot_set_rat ) ) ) ) ).
% remove_def
thf(fact_1035_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_1036_member__remove,axiom,
! [X2: rat,Y: rat,A: set_rat] :
( ( member_rat @ X2 @ ( remove_rat @ Y @ A ) )
= ( ( member_rat @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_1037_le__divide__eq__1__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
= ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1038_le__divide__eq__1__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1039_UNIV__I,axiom,
! [X2: rat] : ( member_rat @ X2 @ top_top_set_rat ) ).
% UNIV_I
thf(fact_1040_atLeast__empty__triv,axiom,
( ( set_or7446828528931440131et_rat @ bot_bot_set_rat )
= top_top_set_set_rat ) ).
% atLeast_empty_triv
thf(fact_1041_div__0,axiom,
! [A2: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A2 )
= zero_zero_rat ) ).
% div_0
thf(fact_1042_div__by__0,axiom,
! [A2: rat] :
( ( divide_divide_rat @ A2 @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_1043_divide__eq__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ( divide_divide_rat @ A2 @ B2 )
= zero_zero_rat )
= ( ( A2 = zero_zero_rat )
| ( B2 = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_1044_divide__cancel__left,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ( divide_divide_rat @ C @ A2 )
= ( divide_divide_rat @ C @ B2 ) )
= ( ( C = zero_zero_rat )
| ( A2 = B2 ) ) ) ).
% divide_cancel_left
thf(fact_1045_divide__cancel__right,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ( divide_divide_rat @ A2 @ C )
= ( divide_divide_rat @ B2 @ C ) )
= ( ( C = zero_zero_rat )
| ( A2 = B2 ) ) ) ).
% divide_cancel_right
thf(fact_1046_division__ring__divide__zero,axiom,
! [A2: rat] :
( ( divide_divide_rat @ A2 @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_1047_lessThan__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
= ( ord_less_rat @ I @ K ) ) ).
% lessThan_iff
thf(fact_1048_lessThan__iff,axiom,
! [I: set_rat,K: set_rat] :
( ( member_set_rat @ I @ ( set_or6605270734133118763et_rat @ K ) )
= ( ord_less_set_rat @ I @ K ) ) ).
% lessThan_iff
thf(fact_1049_inverse__divide,axiom,
! [A2: rat,B2: rat] :
( ( inverse_inverse_rat @ ( divide_divide_rat @ A2 @ B2 ) )
= ( divide_divide_rat @ B2 @ A2 ) ) ).
% inverse_divide
thf(fact_1050_zero__eq__1__divide__iff,axiom,
! [A2: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A2 ) )
= ( A2 = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_1051_one__divide__eq__0__iff,axiom,
! [A2: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A2 )
= zero_zero_rat )
= ( A2 = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_1052_eq__divide__eq__1,axiom,
! [B2: rat,A2: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B2 @ A2 ) )
= ( ( A2 != zero_zero_rat )
& ( A2 = B2 ) ) ) ).
% eq_divide_eq_1
thf(fact_1053_divide__eq__eq__1,axiom,
! [B2: rat,A2: rat] :
( ( ( divide_divide_rat @ B2 @ A2 )
= one_one_rat )
= ( ( A2 != zero_zero_rat )
& ( A2 = B2 ) ) ) ).
% divide_eq_eq_1
thf(fact_1054_divide__self__if,axiom,
! [A2: rat] :
( ( ( A2 = zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ A2 )
= zero_zero_rat ) )
& ( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ A2 )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_1055_divide__self,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ A2 )
= one_one_rat ) ) ).
% divide_self
thf(fact_1056_one__eq__divide__iff,axiom,
! [A2: rat,B2: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A2 @ B2 ) )
= ( ( B2 != zero_zero_rat )
& ( A2 = B2 ) ) ) ).
% one_eq_divide_iff
thf(fact_1057_divide__eq__1__iff,axiom,
! [A2: rat,B2: rat] :
( ( ( divide_divide_rat @ A2 @ B2 )
= one_one_rat )
= ( ( B2 != zero_zero_rat )
& ( A2 = B2 ) ) ) ).
% divide_eq_1_iff
thf(fact_1058_div__self,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ A2 )
= one_one_rat ) ) ).
% div_self
thf(fact_1059_lessThan__subset__iff,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X2 ) @ ( set_ord_lessThan_rat @ Y ) )
= ( ord_less_eq_rat @ X2 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_1060_Diff__UNIV,axiom,
! [A: set_rat] :
( ( minus_minus_set_rat @ A @ top_top_set_rat )
= bot_bot_set_rat ) ).
% Diff_UNIV
thf(fact_1061_divide__le__0__1__iff,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_1062_zero__le__divide__1__iff,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% zero_le_divide_1_iff
thf(fact_1063_divide__less__0__1__iff,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_1064_divide__less__eq__1__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
= ( ord_less_rat @ A2 @ B2 ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1065_divide__less__eq__1__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
= ( ord_less_rat @ B2 @ A2 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1066_less__divide__eq__1__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1067_less__divide__eq__1__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
= ( ord_less_rat @ A2 @ B2 ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1068_zero__less__divide__1__iff,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
= ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% zero_less_divide_1_iff
thf(fact_1069_single__Diff__lessThan,axiom,
! [K: rat] :
( ( minus_minus_set_rat @ ( insert_rat @ K @ bot_bot_set_rat ) @ ( set_ord_lessThan_rat @ K ) )
= ( insert_rat @ K @ bot_bot_set_rat ) ) ).
% single_Diff_lessThan
thf(fact_1070_divide__le__eq__1__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
= ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1071_divide__le__eq__1__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1072_diff__divide__distrib,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( divide_divide_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
= ( minus_minus_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% diff_divide_distrib
thf(fact_1073_atLeastAtMost__eq__UNIV__iff,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ( set_or1040488700251649177et_rat @ X2 @ Y )
= top_top_set_set_rat )
= ( ( X2 = bot_bot_set_rat )
& ( Y = top_top_set_rat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_1074_top_Onot__eq__extremum,axiom,
! [A2: set_rat] :
( ( A2 != top_top_set_rat )
= ( ord_less_set_rat @ A2 @ top_top_set_rat ) ) ).
% top.not_eq_extremum
thf(fact_1075_top_Oextremum__strict,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ top_top_set_rat @ A2 ) ).
% top.extremum_strict
thf(fact_1076_nonzero__of__rat__divide,axiom,
! [B2: rat,A2: rat] :
( ( B2 != zero_zero_rat )
=> ( ( field_2639924705303425560at_rat @ ( divide_divide_rat @ A2 @ B2 ) )
= ( divide_divide_rat @ ( field_2639924705303425560at_rat @ A2 ) @ ( field_2639924705303425560at_rat @ B2 ) ) ) ) ).
% nonzero_of_rat_divide
thf(fact_1077_lessThan__non__empty,axiom,
! [X2: rat] :
( ( set_ord_lessThan_rat @ X2 )
!= bot_bot_set_rat ) ).
% lessThan_non_empty
thf(fact_1078_UNIV__eq__I,axiom,
! [A: set_rat] :
( ! [X: rat] : ( member_rat @ X @ A )
=> ( top_top_set_rat = A ) ) ).
% UNIV_eq_I
thf(fact_1079_UNIV__witness,axiom,
? [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% UNIV_witness
thf(fact_1080_of__rat__divide,axiom,
! [A2: rat,B2: rat] :
( ( field_2639924705303425560at_rat @ ( divide_divide_rat @ A2 @ B2 ) )
= ( divide_divide_rat @ ( field_2639924705303425560at_rat @ A2 ) @ ( field_2639924705303425560at_rat @ B2 ) ) ) ).
% of_rat_divide
thf(fact_1081_empty__not__UNIV,axiom,
bot_bot_set_rat != top_top_set_rat ).
% empty_not_UNIV
thf(fact_1082_divide__le__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ B2 ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% divide_le_0_iff
thf(fact_1083_divide__right__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% divide_right_mono
thf(fact_1084_zero__le__divide__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B2 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1085_divide__nonneg__nonneg,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1086_divide__nonneg__nonpos,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1087_divide__nonpos__nonneg,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1088_divide__nonpos__nonpos,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1089_divide__right__mono__neg,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( divide_divide_rat @ A2 @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1090_divide__neg__neg,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_1091_divide__neg__pos,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_neg_pos
thf(fact_1092_divide__pos__neg,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_pos_neg
thf(fact_1093_divide__pos__pos,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_1094_divide__less__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A2 @ B2 ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ B2 @ zero_zero_rat ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% divide_less_0_iff
thf(fact_1095_divide__less__cancel,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B2 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B2 @ A2 ) )
& ( C != zero_zero_rat ) ) ) ).
% divide_less_cancel
thf(fact_1096_zero__less__divide__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ zero_zero_rat @ B2 ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1097_divide__strict__right__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1098_divide__strict__right__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1099_right__inverse__eq,axiom,
! [B2: rat,A2: rat] :
( ( B2 != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A2 @ B2 )
= one_one_rat )
= ( A2 = B2 ) ) ) ).
% right_inverse_eq
thf(fact_1100_lessThan__strict__subset__iff,axiom,
! [M: rat,N: rat] :
( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
= ( ord_less_rat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_1101_inverse__eq__divide,axiom,
( inverse_inverse_rat
= ( divide_divide_rat @ one_one_rat ) ) ).
% inverse_eq_divide
thf(fact_1102_atLeast__eq__UNIV__iff,axiom,
! [X2: set_rat] :
( ( ( set_or7446828528931440131et_rat @ X2 )
= top_top_set_set_rat )
= ( X2 = bot_bot_set_rat ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1103_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_rat,X2: set_rat,Y: set_rat] :
( ( ( inf_inf_set_rat @ A2 @ X2 )
= bot_bot_set_rat )
=> ( ( ( sup_sup_set_rat @ A2 @ X2 )
= top_top_set_rat )
=> ( ( ( inf_inf_set_rat @ A2 @ Y )
= bot_bot_set_rat )
=> ( ( ( sup_sup_set_rat @ A2 @ Y )
= top_top_set_rat )
=> ( X2 = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1104_frac__le,axiom,
! [Y: rat,X2: rat,W2: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_1105_frac__less,axiom,
! [X2: rat,Y: rat,W2: rat,Z: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_rat @ X2 @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_1106_frac__less2,axiom,
! [X2: rat,Y: rat,W2: rat,Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_rat @ W2 @ Z )
=> ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_1107_divide__le__cancel,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B2 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% divide_le_cancel
thf(fact_1108_divide__nonneg__neg,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_neg
thf(fact_1109_divide__nonneg__pos,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1110_divide__nonpos__neg,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_rat @ Y @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1111_divide__nonpos__pos,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_pos
thf(fact_1112_divide__less__eq__1,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ B2 @ A2 ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ A2 @ B2 ) )
| ( A2 = zero_zero_rat ) ) ) ).
% divide_less_eq_1
thf(fact_1113_less__divide__eq__1,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ A2 @ B2 ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% less_divide_eq_1
thf(fact_1114_Iic__subset__Iio__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A2 ) @ ( set_ord_lessThan_rat @ B2 ) )
= ( ord_less_rat @ A2 @ B2 ) ) ).
% Iic_subset_Iio_iff
thf(fact_1115_ivl__disj__un__one_I2_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or4029947393144176647an_rat @ L @ U ) )
= ( set_ord_lessThan_rat @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_1116_ivl__disj__int__one_I4_J,axiom,
! [L: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or633870826150836451st_rat @ L @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_one(4)
thf(fact_1117_nonzero__inverse__eq__divide,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A2 )
= ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_1118_ivl__disj__int__one_I2_J,axiom,
! [L: rat,U: rat] :
( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or4029947393144176647an_rat @ L @ U ) )
= bot_bot_set_rat ) ).
% ivl_disj_int_one(2)
thf(fact_1119_greaterThanLessThan__def,axiom,
( set_or5199638295745620268an_rat
= ( ^ [L3: rat,U2: rat] : ( inf_inf_set_rat @ ( set_or575021546402375026an_rat @ L3 ) @ ( set_ord_lessThan_rat @ U2 ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_1120_greaterThanLessThan__eq,axiom,
( set_or5199638295745620268an_rat
= ( ^ [A3: rat,B3: rat] : ( inf_inf_set_rat @ ( set_or575021546402375026an_rat @ A3 ) @ ( set_ord_lessThan_rat @ B3 ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_1121_le__divide__eq__1,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ A2 @ B2 ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1
thf(fact_1122_divide__le__eq__1,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ B2 @ A2 ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ A2 @ B2 ) )
| ( A2 = zero_zero_rat ) ) ) ).
% divide_le_eq_1
thf(fact_1123_Iio__Int__singleton,axiom,
! [X2: rat,K: rat] :
( ( ( ord_less_rat @ X2 @ K )
=> ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
= ( insert_rat @ X2 @ bot_bot_set_rat ) ) )
& ( ~ ( ord_less_rat @ X2 @ K )
=> ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
= bot_bot_set_rat ) ) ) ).
% Iio_Int_singleton
thf(fact_1124_Iio__Int__singleton,axiom,
! [X2: set_rat,K: set_rat] :
( ( ( ord_less_set_rat @ X2 @ K )
=> ( ( inf_inf_set_set_rat @ ( set_or6605270734133118763et_rat @ K ) @ ( insert_set_rat @ X2 @ bot_bot_set_set_rat ) )
= ( insert_set_rat @ X2 @ bot_bot_set_set_rat ) ) )
& ( ~ ( ord_less_set_rat @ X2 @ K )
=> ( ( inf_inf_set_set_rat @ ( set_or6605270734133118763et_rat @ K ) @ ( insert_set_rat @ X2 @ bot_bot_set_set_rat ) )
= bot_bot_set_set_rat ) ) ) ).
% Iio_Int_singleton
thf(fact_1125_ivl__disj__un__one_I4_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_eq_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or633870826150836451st_rat @ L @ U ) )
= ( set_ord_atMost_rat @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_1126_ivl__disj__un__singleton_I2_J,axiom,
! [U: rat] :
( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ U ) @ ( insert_rat @ U @ bot_bot_set_rat ) )
= ( set_ord_atMost_rat @ U ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_1127_ivl__disj__un__one_I1_J,axiom,
! [L: rat,U: rat] :
( ( ord_less_rat @ L @ U )
=> ( ( sup_sup_set_rat @ ( set_ord_atMost_rat @ L ) @ ( set_or5199638295745620268an_rat @ L @ U ) )
= ( set_ord_lessThan_rat @ U ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_1128_less__separate,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ? [A4: set_rat,B4: set_rat] :
( ( member_set_rat @ X2 @ ( set_or6605270734133118763et_rat @ A4 ) )
& ( member_set_rat @ Y @ ( set_or6174011595382531368et_rat @ B4 ) )
& ( ( inf_inf_set_set_rat @ ( set_or6605270734133118763et_rat @ A4 ) @ ( set_or6174011595382531368et_rat @ B4 ) )
= bot_bot_set_set_rat ) ) ) ).
% less_separate
thf(fact_1129_less__separate,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
=> ? [A4: rat,B4: rat] :
( ( member_rat @ X2 @ ( set_ord_lessThan_rat @ A4 ) )
& ( member_rat @ Y @ ( set_or575021546402375026an_rat @ B4 ) )
& ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ A4 ) @ ( set_or575021546402375026an_rat @ B4 ) )
= bot_bot_set_rat ) ) ) ).
% less_separate
thf(fact_1130_top__empty__eq,axiom,
( top_top_rat_o
= ( ^ [X3: rat] : ( member_rat @ X3 @ top_top_set_rat ) ) ) ).
% top_empty_eq
thf(fact_1131_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila7382412365081457248et_rat @ inf_inf_set_rat @ top_top_set_rat @ ord_less_eq_set_rat @ ord_less_set_rat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1132_top_Oordering__top__axioms,axiom,
ordering_top_set_rat @ ord_less_eq_set_rat @ ord_less_set_rat @ top_top_set_rat ).
% top.ordering_top_axioms
thf(fact_1133_Inf__atMostLessThan,axiom,
! [X2: set_rat] :
( ( ord_less_set_rat @ top_top_set_rat @ X2 )
=> ( ( comple4298007329820168263et_rat @ ( set_or6605270734133118763et_rat @ X2 ) )
= bot_bot_set_rat ) ) ).
% Inf_atMostLessThan
thf(fact_1134_pairwise__alt,axiom,
( pairwise_rat
= ( ^ [R2: rat > rat > $o,S3: set_rat] :
! [X3: rat] :
( ( member_rat @ X3 @ S3 )
=> ! [Y4: rat] :
( ( member_rat @ Y4 @ ( minus_minus_set_rat @ S3 @ ( insert_rat @ X3 @ bot_bot_set_rat ) ) )
=> ( R2 @ X3 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1135_Inf__atLeastLessThan,axiom,
! [X2: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X2 @ Y )
=> ( ( comple4298007329820168263et_rat @ ( set_or32047845639629757et_rat @ X2 @ Y ) )
= X2 ) ) ).
% Inf_atLeastLessThan
thf(fact_1136_cInf__atLeastLessThan,axiom,
! [Y: set_rat,X2: set_rat] :
( ( ord_less_set_rat @ Y @ X2 )
=> ( ( comple4298007329820168263et_rat @ ( set_or32047845639629757et_rat @ Y @ X2 ) )
= Y ) ) ).
% cInf_atLeastLessThan
thf(fact_1137_Inf__atMost,axiom,
! [X2: set_rat] :
( ( comple4298007329820168263et_rat @ ( set_or728397472755099399et_rat @ X2 ) )
= bot_bot_set_rat ) ).
% Inf_atMost
thf(fact_1138_pairwiseI,axiom,
! [S2: set_rat,R3: rat > rat > $o] :
( ! [X: rat,Y2: rat] :
( ( member_rat @ X @ S2 )
=> ( ( member_rat @ Y2 @ S2 )
=> ( ( X != Y2 )
=> ( R3 @ X @ Y2 ) ) ) )
=> ( pairwise_rat @ R3 @ S2 ) ) ).
% pairwiseI
thf(fact_1139_pairwiseD,axiom,
! [R3: rat > rat > $o,S2: set_rat,X2: rat,Y: rat] :
( ( pairwise_rat @ R3 @ S2 )
=> ( ( member_rat @ X2 @ S2 )
=> ( ( member_rat @ Y @ S2 )
=> ( ( X2 != Y )
=> ( R3 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1140_pairwise__insert,axiom,
! [R: rat > rat > $o,X2: rat,S: set_rat] :
( ( pairwise_rat @ R @ ( insert_rat @ X2 @ S ) )
= ( ! [Y4: rat] :
( ( ( member_rat @ Y4 @ S )
& ( Y4 != X2 ) )
=> ( ( R @ X2 @ Y4 )
& ( R @ Y4 @ X2 ) ) )
& ( pairwise_rat @ R @ S ) ) ) ).
% pairwise_insert
thf(fact_1141_pairwise__empty,axiom,
! [P: rat > rat > $o] : ( pairwise_rat @ P @ bot_bot_set_rat ) ).
% pairwise_empty
thf(fact_1142_pairwise__singleton,axiom,
! [P: rat > rat > $o,A: rat] : ( pairwise_rat @ P @ ( insert_rat @ A @ bot_bot_set_rat ) ) ).
% pairwise_singleton
thf(fact_1143_Inf__UNIV,axiom,
( ( comple4298007329820168263et_rat @ top_top_set_set_rat )
= bot_bot_set_rat ) ).
% Inf_UNIV
thf(fact_1144_Inter__UNIV,axiom,
( ( comple4298007329820168263et_rat @ top_top_set_set_rat )
= bot_bot_set_rat ) ).
% Inter_UNIV
thf(fact_1145_bdd__belowI,axiom,
! [A: set_rat,M: rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_rat @ M @ X ) )
=> ( condit1103211067700513672ow_rat @ A ) ) ).
% bdd_belowI
thf(fact_1146_bdd__below_OI,axiom,
! [A: set_rat,M2: rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_rat @ M2 @ X ) )
=> ( condit1103211067700513672ow_rat @ A ) ) ).
% bdd_below.I
thf(fact_1147_bdd__below__empty,axiom,
condit1103211067700513672ow_rat @ bot_bot_set_rat ).
% bdd_below_empty
thf(fact_1148_bdd__below__Ioi,axiom,
! [A2: rat] : ( condit1103211067700513672ow_rat @ ( set_or575021546402375026an_rat @ A2 ) ) ).
% bdd_below_Ioi
thf(fact_1149_bdd__below_OE,axiom,
! [A: set_rat] :
( ( condit1103211067700513672ow_rat @ A )
=> ~ ! [M3: rat] :
~ ! [X4: rat] :
( ( member_rat @ X4 @ A )
=> ( ord_less_eq_rat @ M3 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1150_bdd__below_Ounfold,axiom,
( condit1103211067700513672ow_rat
= ( ^ [A5: set_rat] :
? [M4: rat] :
! [X3: rat] :
( ( member_rat @ X3 @ A5 )
=> ( ord_less_eq_rat @ M4 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_1151_image__eqI,axiom,
! [B2: rat,F: rat > rat,X2: rat,A: set_rat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_rat @ X2 @ A )
=> ( member_rat @ B2 @ ( image_rat_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1152_image__is__empty,axiom,
! [F: rat > rat,A: set_rat] :
( ( ( image_rat_rat @ F @ A )
= bot_bot_set_rat )
= ( A = bot_bot_set_rat ) ) ).
% image_is_empty
thf(fact_1153_empty__is__image,axiom,
! [F: rat > rat,A: set_rat] :
( ( bot_bot_set_rat
= ( image_rat_rat @ F @ A ) )
= ( A = bot_bot_set_rat ) ) ).
% empty_is_image
thf(fact_1154_image__empty,axiom,
! [F: rat > rat] :
( ( image_rat_rat @ F @ bot_bot_set_rat )
= bot_bot_set_rat ) ).
% image_empty
thf(fact_1155_bdd__below_OI2,axiom,
! [A: set_rat,M2: rat,F: rat > rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_rat @ M2 @ ( F @ X ) ) )
=> ( condit1103211067700513672ow_rat @ ( image_rat_rat @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_1156_bdd__belowI2,axiom,
! [A: set_rat,M: rat,F: rat > rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_rat @ M @ ( F @ X ) ) )
=> ( condit1103211067700513672ow_rat @ ( image_rat_rat @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_1157_image__subsetI,axiom,
! [A: set_rat,F: rat > rat,B: set_rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_rat @ ( image_rat_rat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1158_Inf__INT__eq,axiom,
( comple2477142665972227838_rat_o
= ( ^ [S3: set_rat_o,X3: rat] : ( member_rat @ X3 @ ( comple4298007329820168263et_rat @ ( image_rat_o_set_rat @ collect_rat @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_1159_imageI,axiom,
! [X2: rat,A: set_rat,F: rat > rat] :
( ( member_rat @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ ( image_rat_rat @ F @ A ) ) ) ).
% imageI
thf(fact_1160_rev__image__eqI,axiom,
! [X2: rat,A: set_rat,B2: rat,F: rat > rat] :
( ( member_rat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_rat @ B2 @ ( image_rat_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1161_in__image__insert__iff,axiom,
! [B: set_set_rat,X2: rat,A: set_rat] :
( ! [C4: set_rat] :
( ( member_set_rat @ C4 @ B )
=> ~ ( member_rat @ X2 @ C4 ) )
=> ( ( member_set_rat @ A @ ( image_3939399684171694371et_rat @ ( insert_rat @ X2 ) @ B ) )
= ( ( member_rat @ X2 @ A )
& ( member_set_rat @ ( minus_minus_set_rat @ A @ ( insert_rat @ X2 @ bot_bot_set_rat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1162_image__mult__atLeastAtMost__if,axiom,
! [C: rat,X2: rat,Y: rat] :
( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X2 @ Y ) )
= ( set_or633870826150836451st_rat @ ( times_times_rat @ C @ X2 ) @ ( times_times_rat @ C @ Y ) ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_eq_rat @ X2 @ Y )
=> ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X2 @ Y ) )
= ( set_or633870826150836451st_rat @ ( times_times_rat @ C @ Y ) @ ( times_times_rat @ C @ X2 ) ) ) )
& ( ~ ( ord_less_eq_rat @ X2 @ Y )
=> ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X2 @ Y ) )
= bot_bot_set_rat ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_1163_mult__cancel__right,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ C )
= ( times_times_rat @ B2 @ C ) )
= ( ( C = zero_zero_rat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1164_mult__cancel__left,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ( times_times_rat @ C @ A2 )
= ( times_times_rat @ C @ B2 ) )
= ( ( C = zero_zero_rat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_1165_mult__eq__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ B2 )
= zero_zero_rat )
= ( ( A2 = zero_zero_rat )
| ( B2 = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_1166_mult__zero__right,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_1167_mult__zero__left,axiom,
! [A2: rat] :
( ( times_times_rat @ zero_zero_rat @ A2 )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_1168_mult__1,axiom,
! [A2: rat] :
( ( times_times_rat @ one_one_rat @ A2 )
= A2 ) ).
% mult_1
thf(fact_1169_mult_Oright__neutral,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ one_one_rat )
= A2 ) ).
% mult.right_neutral
thf(fact_1170_times__divide__eq__right,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( times_times_rat @ A2 @ ( divide_divide_rat @ B2 @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ C ) ) ).
% times_divide_eq_right
thf(fact_1171_divide__divide__eq__right,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( divide_divide_rat @ A2 @ ( divide_divide_rat @ B2 @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) ) ).
% divide_divide_eq_right
thf(fact_1172_divide__divide__eq__left,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A2 @ B2 ) @ C )
= ( divide_divide_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_1173_times__divide__eq__left,axiom,
! [B2: rat,C: rat,A2: rat] :
( ( times_times_rat @ ( divide_divide_rat @ B2 @ C ) @ A2 )
= ( divide_divide_rat @ ( times_times_rat @ B2 @ A2 ) @ C ) ) ).
% times_divide_eq_left
thf(fact_1174_inverse__mult__distrib,axiom,
! [A2: rat,B2: rat] :
( ( inverse_inverse_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ ( inverse_inverse_rat @ B2 ) ) ) ).
% inverse_mult_distrib
thf(fact_1175_mult__cancel__right2,axiom,
! [A2: rat,C: rat] :
( ( ( times_times_rat @ A2 @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A2 = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_1176_mult__cancel__right1,axiom,
! [C: rat,B2: rat] :
( ( C
= ( times_times_rat @ B2 @ C ) )
= ( ( C = zero_zero_rat )
| ( B2 = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_1177_mult__cancel__left2,axiom,
! [C: rat,A2: rat] :
( ( ( times_times_rat @ C @ A2 )
= C )
= ( ( C = zero_zero_rat )
| ( A2 = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_1178_mult__cancel__left1,axiom,
! [C: rat,B2: rat] :
( ( C
= ( times_times_rat @ C @ B2 ) )
= ( ( C = zero_zero_rat )
| ( B2 = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_1179_nonzero__mult__div__cancel__right,axiom,
! [B2: rat,A2: rat] :
( ( B2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1180_nonzero__mult__div__cancel__left,axiom,
! [A2: rat,B2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1181_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ C @ B2 ) )
= ( divide_divide_rat @ A2 @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1182_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
= ( divide_divide_rat @ A2 @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1183_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ B2 @ C ) )
= ( divide_divide_rat @ A2 @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1184_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( divide_divide_rat @ A2 @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1185_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ( C = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( divide_divide_rat @ A2 @ B2 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1186_nonzero__divide__mult__cancel__left,axiom,
! [A2: rat,B2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( divide_divide_rat @ A2 @ ( times_times_rat @ A2 @ B2 ) )
= ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1187_nonzero__divide__mult__cancel__right,axiom,
! [B2: rat,A2: rat] :
( ( B2 != zero_zero_rat )
=> ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A2 @ B2 ) )
= ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1188_right__inverse,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( times_times_rat @ A2 @ ( inverse_inverse_rat @ A2 ) )
= one_one_rat ) ) ).
% right_inverse
thf(fact_1189_left__inverse,axiom,
! [A2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A2 ) @ A2 )
= one_one_rat ) ) ).
% left_inverse
thf(fact_1190_image__mult__atLeastAtMost,axiom,
! [D: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ D )
=> ( ( image_rat_rat @ ( times_times_rat @ D ) @ ( set_or633870826150836451st_rat @ A2 @ B2 ) )
= ( set_or633870826150836451st_rat @ ( times_times_rat @ D @ A2 ) @ ( times_times_rat @ D @ B2 ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_1191_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: rat,X2: rat] :
( ( ( times_times_rat @ Y @ X2 )
= ( times_times_rat @ X2 @ Y ) )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ Y ) @ X2 )
= ( times_times_rat @ X2 @ ( inverse_inverse_rat @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_1192_mult_Oleft__commute,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( times_times_rat @ B2 @ ( times_times_rat @ A2 @ C ) )
= ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_1193_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_1194_mult_Oassoc,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C )
= ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_1195_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C )
= ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1196_of__rat__mult,axiom,
! [A2: rat,B2: rat] :
( ( field_2639924705303425560at_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( times_times_rat @ ( field_2639924705303425560at_rat @ A2 ) @ ( field_2639924705303425560at_rat @ B2 ) ) ) ).
% of_rat_mult
thf(fact_1197_mult__right__cancel,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A2 @ C )
= ( times_times_rat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_1198_mult__left__cancel,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A2 )
= ( times_times_rat @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_1199_no__zero__divisors,axiom,
! [A2: rat,B2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( B2 != zero_zero_rat )
=> ( ( times_times_rat @ A2 @ B2 )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_1200_divisors__zero,axiom,
! [A2: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ B2 )
= zero_zero_rat )
=> ( ( A2 = zero_zero_rat )
| ( B2 = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_1201_mult__not__zero,axiom,
! [A2: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ B2 )
!= zero_zero_rat )
=> ( ( A2 != zero_zero_rat )
& ( B2 != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_1202_mult_Ocomm__neutral,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ one_one_rat )
= A2 ) ).
% mult.comm_neutral
thf(fact_1203_comm__monoid__mult__class_Omult__1,axiom,
! [A2: rat] :
( ( times_times_rat @ one_one_rat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1204_inf__period_I1_J,axiom,
! [P: rat > $o,D3: rat,Q2: rat > $o] :
( ! [X: rat,K2: rat] :
( ( P @ X )
= ( P @ ( minus_minus_rat @ X @ ( times_times_rat @ K2 @ D3 ) ) ) )
=> ( ! [X: rat,K2: rat] :
( ( Q2 @ X )
= ( Q2 @ ( minus_minus_rat @ X @ ( times_times_rat @ K2 @ D3 ) ) ) )
=> ! [X4: rat,K3: rat] :
( ( ( P @ X4 )
& ( Q2 @ X4 ) )
= ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) )
& ( Q2 @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1205_inf__period_I2_J,axiom,
! [P: rat > $o,D3: rat,Q2: rat > $o] :
( ! [X: rat,K2: rat] :
( ( P @ X )
= ( P @ ( minus_minus_rat @ X @ ( times_times_rat @ K2 @ D3 ) ) ) )
=> ( ! [X: rat,K2: rat] :
( ( Q2 @ X )
= ( Q2 @ ( minus_minus_rat @ X @ ( times_times_rat @ K2 @ D3 ) ) ) )
=> ! [X4: rat,K3: rat] :
( ( ( P @ X4 )
| ( Q2 @ X4 ) )
= ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) )
| ( Q2 @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1206_field__class_Ofield__divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_1207_divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ A3 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_1208_divide__inverse__commute,axiom,
( divide_divide_rat
= ( ^ [A3: rat,B3: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B3 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_1209_inverse__unique,axiom,
! [A2: rat,B2: rat] :
( ( ( times_times_rat @ A2 @ B2 )
= one_one_rat )
=> ( ( inverse_inverse_rat @ A2 )
= B2 ) ) ).
% inverse_unique
thf(fact_1210_nonzero__inverse__mult__distrib,axiom,
! [A2: rat,B2: rat] :
( ( A2 != zero_zero_rat )
=> ( ( B2 != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( times_times_rat @ ( inverse_inverse_rat @ B2 ) @ ( inverse_inverse_rat @ A2 ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_1211_nonzero__eq__divide__eq,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( A2
= ( divide_divide_rat @ B2 @ C ) )
= ( ( times_times_rat @ A2 @ C )
= B2 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1212_nonzero__divide__eq__eq,axiom,
! [C: rat,B2: rat,A2: rat] :
( ( C != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B2 @ C )
= A2 )
= ( B2
= ( times_times_rat @ A2 @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1213_eq__divide__imp,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A2 @ C )
= B2 )
=> ( A2
= ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_1214_divide__eq__imp,axiom,
! [C: rat,B2: rat,A2: rat] :
( ( C != zero_zero_rat )
=> ( ( B2
= ( times_times_rat @ A2 @ C ) )
=> ( ( divide_divide_rat @ B2 @ C )
= A2 ) ) ) ).
% divide_eq_imp
thf(fact_1215_eq__divide__eq,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( A2
= ( divide_divide_rat @ B2 @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A2 @ C )
= B2 ) )
& ( ( C = zero_zero_rat )
=> ( A2 = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_1216_divide__eq__eq,axiom,
! [B2: rat,C: rat,A2: rat] :
( ( ( divide_divide_rat @ B2 @ C )
= A2 )
= ( ( ( C != zero_zero_rat )
=> ( B2
= ( times_times_rat @ A2 @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A2 = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_1217_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X2: rat,W2: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X2 @ Y )
= ( divide_divide_rat @ W2 @ Z ) )
= ( ( times_times_rat @ X2 @ Z )
= ( times_times_rat @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1218_less__1__mult,axiom,
! [M: rat,N: rat] :
( ( ord_less_rat @ one_one_rat @ M )
=> ( ( ord_less_rat @ one_one_rat @ N )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1219_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1220_mult__less__cancel__right__disj,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A2 @ B2 ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1221_mult__strict__right__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1222_mult__strict__right__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1223_mult__less__cancel__left__disj,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A2 @ B2 ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1224_mult__strict__left__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1225_mult__strict__left__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B2 @ A2 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1226_mult__less__cancel__left__pos,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ord_less_rat @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1227_mult__less__cancel__left__neg,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ord_less_rat @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1228_zero__less__mult__pos2,axiom,
! [B2: rat,A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1229_zero__less__mult__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1230_zero__less__mult__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ zero_zero_rat @ B2 ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1231_mult__pos__neg2,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_1232_mult__pos__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1233_mult__pos__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_1234_mult__neg__pos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_1235_mult__less__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ B2 @ zero_zero_rat ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_1236_not__square__less__zero,axiom,
! [A2: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_1237_mult__neg__neg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B2 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_1238_divide__divide__eq__left_H,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A2 @ B2 ) @ C )
= ( divide_divide_rat @ A2 @ ( times_times_rat @ C @ B2 ) ) ) ).
% divide_divide_eq_left'
thf(fact_1239_divide__divide__times__eq,axiom,
! [X2: rat,Y: rat,Z: rat,W2: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ X2 @ Y ) @ ( divide_divide_rat @ Z @ W2 ) )
= ( divide_divide_rat @ ( times_times_rat @ X2 @ W2 ) @ ( times_times_rat @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_1240_times__divide__times__eq,axiom,
! [X2: rat,Y: rat,Z: rat,W2: rat] :
( ( times_times_rat @ ( divide_divide_rat @ X2 @ Y ) @ ( divide_divide_rat @ Z @ W2 ) )
= ( divide_divide_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_1241_mult__mono,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1242_mult__mono_H,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1243_zero__le__square,axiom,
! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_1244_split__mult__pos__le,axiom,
! [A2: rat,B2: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_1245_mult__left__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1246_mult__nonpos__nonpos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1247_mult__left__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1248_mult__right__mono__neg,axiom,
! [B2: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B2 @ A2 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1249_mult__right__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1250_mult__le__0__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_1251_split__mult__neg__le,axiom,
! [A2: rat,B2: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_1252_mult__nonneg__nonneg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1253_mult__nonneg__nonpos,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1254_mult__nonpos__nonneg,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1255_mult__nonneg__nonpos2,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1256_zero__le__mult__iff,axiom,
! [A2: rat,B2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1257_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: rat,B2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1258_divide__rat__def,axiom,
( divide_divide_rat
= ( ^ [Q4: rat,R4: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R4 ) ) ) ) ).
% divide_rat_def
thf(fact_1259_mult__less__le__imp__less,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1260_mult__le__less__imp__less,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1261_mult__right__le__imp__le,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1262_mult__left__le__imp__le,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1263_mult__le__cancel__left__pos,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1264_mult__le__cancel__left__neg,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1265_mult__less__cancel__right,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B2 ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1266_mult__strict__mono_H,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1267_mult__right__less__imp__less,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1268_mult__less__cancel__left,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B2 ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1269_mult__strict__mono,axiom,
! [A2: rat,B2: rat,C: rat,D: rat] :
( ( ord_less_rat @ A2 @ B2 )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ B2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1270_mult__left__less__imp__less,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1271_mult__le__cancel__right,axiom,
! [A2: rat,C: rat,B2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B2 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1272_mult__le__cancel__left,axiom,
! [C: rat,A2: rat,B2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B2 ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1273_mult__left__le__one__le,axiom,
! [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
% Conjectures (2)
thf(conj_0,hypothesis,
! [X4: rat,Y5: rat] :
( ( ord_less_rat @ zero_zero_rat @ X4 )
=> ( ~ ( member_rat @ X4 @ a )
=> ( ( ord_less_rat @ zero_zero_rat @ Y5 )
=> ( ~ ( member_rat @ Y5 @ b )
=> thesisa ) ) ) ) ).
thf(conj_1,conjecture,
thesisa ).
%------------------------------------------------------------------------------