TPTP Problem File: SLH0472^1.p
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%------------------------------------------------------------------------------
% 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 : Quasi_Borel_Spaces/0001_QuasiBorel/prob_00562_018427__15144140_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1490 ( 556 unt; 211 typ; 0 def)
% Number of atoms : 3698 ( 982 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10566 ( 318 ~; 72 |; 184 &;8383 @)
% ( 0 <=>;1609 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 905 ( 905 >; 0 *; 0 +; 0 <<)
% Number of symbols : 191 ( 188 usr; 13 con; 0-5 aty)
% Number of variables : 3616 ( 383 ^;3178 !; 55 ?;3616 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:08:28.461
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_n_t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
produc149604174626918183orel_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J_J,type,
produc680249927945959346real_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
produc4991720479562799725orel_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc2269578075601916641_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
set_Pr5845495582615845127_set_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__Nat__Onat_J,type,
produc1856989317385805731_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
produc562953661399087177orel_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1703568184450464039_set_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
produc5346009699553737693_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
produc2875793189560775939_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
set_quasi_borel_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
set_real_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
quasi_borel_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (188)
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001tf__a,type,
bNF_We1162827675446709994_rel_a: set_Product_prod_a_a > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001tf__a,type,
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thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001tf__a,type,
bNF_We5615626441682584778inim_a: set_Product_prod_a_a > set_a > a ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc_001tf__a,type,
bNF_We6154283375207884895_suc_a: set_Product_prod_a_a > set_a > a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Complete__Partial__Order2_Occpo_Ocompact_001t__Set__Oset_Itf__a_J,type,
comple704071332753839977_set_a: ( set_set_a > set_a ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001t__Nat__Onat,type,
comple7016393980872852640in_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Complete__Partial__Order_Ochain_001t__Set__Oset_Itf__a_J,type,
comple4316259127148425102_set_a: ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
condit3516415644855960584orel_a: set_quasi_borel_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_Itf__a_J,type,
condit3373647341569784514_set_a: set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Nat__Onat,type,
condit7935552474144124665dd_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Set__Oset_Itf__a_J,type,
condit6315317455391067509_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001tf__a,type,
condit4103000493307248661_bdd_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Nat__Onat,type,
condit4013746787832047771dd_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
condit6163609524079627545orel_a: ( quasi_borel_a > quasi_borel_a > $o ) > set_quasi_borel_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Set__Oset_Itf__a_J,type,
condit4774827555938943059_set_a: ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001tf__a,type,
condit6541519642617408243_bdd_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Groups_Oabel__semigroup_001t__Nat__Onat,type,
abel_semigroup_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Groups_Oabel__semigroup_001t__Set__Oset_Itf__a_J,type,
abel_semigroup_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
minus_4365393887724441320at_nat: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
minus_6817036919807184750od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
if_quasi_borel_a: $o > quasi_borel_a > quasi_borel_a > quasi_borel_a ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
inf_in4240866628268492783at_nat: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
inf_inf_set_real_a: set_real_a > set_real_a > set_real_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice_001t__Nat__Onat,type,
semilattice_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Lattices_Osemilattice_001t__Set__Oset_Itf__a_J,type,
semilattice_set_a: ( set_a > set_a > set_a ) > $o ).
thf(sy_c_Lattices_Osemilattice__order_001t__Nat__Onat,type,
semila1248733672344298208er_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices_Osemilattice__order_001t__Set__Oset_Itf__a_J,type,
semila4706084620769370446_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_Mt__Nat__Onat_J,type,
sup_sup_o_nat: ( $o > nat ) > ( $o > nat ) > $o > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
sup_sup_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
sup_su4120719815643632853at_nat: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
sup_su7624641699453423151_set_a: produc2875793189560775939_set_a > produc2875793189560775939_set_a > produc2875793189560775939_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
sup_su871486172591609097_a_nat: produc5346009699553737693_a_nat > produc5346009699553737693_a_nat > produc5346009699553737693_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
sup_su2807591092881064699_set_a: produc1703568184450464039_set_a > produc1703568184450464039_set_a > produc1703568184450464039_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
sup_su6298519176299948920orel_a: quasi_borel_a > quasi_borel_a > quasi_borel_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
sup_sup_set_real_a: set_real_a > set_real_a > set_real_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
sup_su3048258781599657691od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_Itf__a_J,type,
lattic8986249270076014136_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Nat__Onat,type,
measur8151441426001876059_a_nat: set_set_a > ( set_a > nat ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
measur6337775561432609113orel_a: set_set_a > ( set_a > quasi_borel_a ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_Itf__a_J,type,
measur7842569353079325843_set_a: set_set_a > ( set_a > set_a ) > $o ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__Nat__Onat,type,
measur4601247141005857854at_nat: nat > nat > ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
measur6224197021335510454orel_a: nat > nat > ( nat > quasi_borel_a ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
measur5940501938642862960_set_a: nat > nat > ( nat > set_a ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__lexord_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Nat__Onat,type,
measur1617175427248866264_a_nat: quasi_borel_a > quasi_borel_a > ( quasi_borel_a > nat ) > quasi_borel_a > quasi_borel_a > quasi_borel_a ).
thf(sy_c_Measure__Space_Osup__lexord_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
measur7073608842307378588orel_a: quasi_borel_a > quasi_borel_a > ( quasi_borel_a > quasi_borel_a ) > quasi_borel_a > quasi_borel_a > quasi_borel_a ).
thf(sy_c_Measure__Space_Osup__lexord_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
measur7302512200507534934_set_a: quasi_borel_a > quasi_borel_a > ( quasi_borel_a > set_a ) > quasi_borel_a > quasi_borel_a > quasi_borel_a ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
measur783711913603534354_a_nat: set_a > set_a > ( set_a > nat ) > set_a > set_a > set_a ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_Itf__a_J_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
measur8298705319509490402orel_a: set_a > set_a > ( set_a > quasi_borel_a ) > set_a > set_a > set_a ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
measur758946168800011932_set_a: set_a > set_a > ( set_a > set_a ) > set_a > set_a > set_a ).
thf(sy_c_Order__Relation_OAboveS_001tf__a,type,
order_AboveS_a: set_Product_prod_a_a > set_a > set_a ).
thf(sy_c_Order__Relation_Olinear__order__on_001tf__a,type,
order_8768733634509060147r_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Order__Relation_Oofilter_001tf__a,type,
order_ofilter_a: set_Product_prod_a_a > set_a > $o ).
thf(sy_c_Order__Relation_OunderS_001tf__a,type,
order_underS_a: set_Product_prod_a_a > a > set_a ).
thf(sy_c_Order__Relation_Ounder_001tf__a,type,
order_under_a: set_Product_prod_a_a > a > set_a ).
thf(sy_c_Order__Relation_Owell__order__on_001tf__a,type,
order_6972113574731384220r_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
bot_bo450704735548830736orel_a: quasi_borel_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
bot_bot_set_real_a: set_real_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mt__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
ord_le2255205531489447621orel_a: ( $o > quasi_borel_a ) > ( $o > quasi_borel_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le1203424502768444845at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
ord_le423894246401745751_set_a: produc2875793189560775939_set_a > produc2875793189560775939_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
ord_le3788981553861827664orel_a: quasi_borel_a > quasi_borel_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
ord_less_set_real_a: set_real_a > set_real_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le6819997720685908915od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
ord_le1636368021143733457orel_a: ( $o > quasi_borel_a ) > ( $o > quasi_borel_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le8460144461188290721at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
ord_le185366790440233129orel_a: produc562953661399087177orel_a > produc562953661399087177orel_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
ord_le6607365053213775203_set_a: produc2875793189560775939_set_a > produc2875793189560775939_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__Nat__Onat_J,type,
ord_le1479402446426951683_a_nat: produc1856989317385805731_a_nat > produc1856989317385805731_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
ord_le5941759531670611655orel_a: produc149604174626918183orel_a > produc149604174626918183orel_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__QuasiBorel__Oquasi____borel_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_le5001701540438981249_set_a: produc2269578075601916641_set_a > produc2269578075601916641_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
ord_le9077581563206736957_a_nat: produc5346009699553737693_a_nat > produc5346009699553737693_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__QuasiBorel__Oquasi____borel_Itf__a_J_J,type,
ord_le7723843944399864333orel_a: produc4991720479562799725orel_a > produc4991720479562799725orel_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
ord_le7186201010124320967_set_a: produc1703568184450464039_set_a > produc1703568184450464039_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
ord_le1843388692487544644orel_a: quasi_borel_a > quasi_borel_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
ord_le5743406823621094409real_a: set_real_a > set_real_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le746702958409616551od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
ord_mi5312030764815082635orel_a: quasi_borel_a > quasi_borel_a > quasi_borel_a ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_Itf__a_J,type,
ord_min_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
order_2939245480137816317orel_a: ( quasi_borel_a > $o ) > quasi_borel_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
produc5438933203794904571orel_a: nat > quasi_borel_a > produc562953661399087177orel_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
produc4534374738092020277_set_a: nat > set_a > produc2875793189560775939_set_a ).
thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Nat__Onat,type,
produc831911609708260381_a_nat: quasi_borel_a > nat > produc1856989317385805731_a_nat ).
thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
produc6592225216727209751orel_a: quasi_borel_a > quasi_borel_a > produc149604174626918183orel_a ).
thf(sy_c_Product__Type_OPair_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc6811942899106390353_set_a: quasi_borel_a > set_a > produc2269578075601916641_set_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
produc8600956749907467479_a_nat: set_a > nat > produc5346009699553737693_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
produc7808136018108345821orel_a: set_a > quasi_borel_a > produc4991720479562799725orel_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
produc8198169892514940194real_a: set_a > set_real_a > produc680249927945959346real_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc9088192753505129239_set_a: set_a > set_a > produc1703568184450464039_set_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
produc9127234019004855055orel_a: produc562953661399087177orel_a > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
produc7883749236145314121_set_a: produc2875793189560775939_set_a > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Nat__Onat,type,
produc4520212424918210865_a_nat: produc1856989317385805731_a_nat > quasi_borel_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
produc2874185341511397635orel_a: produc149604174626918183orel_a > quasi_borel_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc5500841278744043837_set_a: produc2269578075601916641_set_a > quasi_borel_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
produc2726959211105985515_a_nat: produc5346009699553737693_a_nat > set_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_Itf__a_J_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
produc6497034397745999305orel_a: produc4991720479562799725orel_a > set_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_Itf__a_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
produc7517347076341637902real_a: produc680249927945959346real_a > set_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc9088895665703139587_set_a: produc1703568184450464039_set_a > set_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
produc570164119032387661orel_a: produc562953661399087177orel_a > quasi_borel_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
produc7069095437425259911_set_a: produc2875793189560775939_set_a > set_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Nat__Onat,type,
produc5186514561800519279_a_nat: produc1856989317385805731_a_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
produc7830989722923372101orel_a: produc149604174626918183orel_a > quasi_borel_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__QuasiBorel__Oquasi____borel_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc760345589714667391_set_a: produc2269578075601916641_set_a > set_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
produc1912305412385931305_a_nat: produc5346009699553737693_a_nat > nat ).
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produc1756538708716622859orel_a: produc4991720479562799725orel_a > quasi_borel_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc1983107199584856133_set_a: produc1703568184450464039_set_a > set_a ).
thf(sy_c_QuasiBorel_Oempty__quasi__borel_001tf__a,type,
empty_quasi_borel_a: quasi_borel_a ).
thf(sy_c_QuasiBorel_Ogenerating__Mx_001tf__a,type,
generating_Mx_a: set_a > set_real_a > set_real_a ).
thf(sy_c_QuasiBorel_Oinf__quasi__borel_001tf__a,type,
inf_quasi_borel_a: quasi_borel_a > quasi_borel_a > quasi_borel_a ).
thf(sy_c_QuasiBorel_Omap__qbs_001tf__a_001tf__a,type,
map_qbs_a_a: ( a > a ) > quasi_borel_a > quasi_borel_a ).
thf(sy_c_QuasiBorel_Omax__quasi__borel_001tf__a,type,
max_quasi_borel_a: set_a > quasi_borel_a ).
thf(sy_c_QuasiBorel_Oqbs__Mx_001tf__a,type,
qbs_Mx_a: quasi_borel_a > set_real_a ).
thf(sy_c_QuasiBorel_Oqbs__space_001tf__a,type,
qbs_space_a: quasi_borel_a > set_a ).
thf(sy_c_QuasiBorel_Oquasi__borel_OAbs__quasi__borel_001tf__a,type,
quasi_2002468295286565184orel_a: produc680249927945959346real_a > quasi_borel_a ).
thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001tf__a,type,
quasi_4958298574314430517orel_a: quasi_borel_a > produc680249927945959346real_a ).
thf(sy_c_QuasiBorel_Osub__qbs_001tf__a,type,
sub_qbs_a: quasi_borel_a > set_a > quasi_borel_a ).
thf(sy_c_Relation_ODomain_001tf__a_001tf__a,type,
domain_a_a: set_Product_prod_a_a > set_a ).
thf(sy_c_Relation_OField_001tf__a,type,
field_a: set_Product_prod_a_a > set_a ).
thf(sy_c_Relation_OId_001tf__a,type,
id_a: set_Product_prod_a_a ).
thf(sy_c_Relation_ORange_001tf__a_001tf__a,type,
range_a_a: set_Product_prod_a_a > set_a ).
thf(sy_c_Relation_Orefl__on_001tf__a,type,
refl_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Ototal__on_001tf__a,type,
total_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mtf__a_J,type,
collect_real_a: ( ( real > a ) > $o ) > set_real_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
insert4534936382041156343od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Wellfounded_Omax__ext_001tf__a,type,
max_ext_a: set_Product_prod_a_a > set_Pr5845495582615845127_set_a ).
thf(sy_c_Wellfounded_Owf_001tf__a,type,
wf_a: set_Product_prod_a_a > $o ).
thf(sy_c_Zorn_Ochain__subset_001tf__a,type,
chain_subset_a: set_set_a > $o ).
thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_member_001_062_It__Real__Oreal_Mtf__a_J,type,
member_real_a: ( real > a ) > set_real_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
member_quasi_borel_a: quasi_borel_a > set_quasi_borel_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_X____,type,
x: quasi_borel_a ).
thf(sy_v_Y____,type,
y: quasi_borel_a ).
thf(sy_v_Z____,type,
z: quasi_borel_a ).
% Relevant facts (1271)
thf(fact_0__C2_C,axiom,
ord_less_set_a @ ( qbs_space_a @ x ) @ ( qbs_space_a @ y ) ).
% "2"
thf(fact_1_h_I2_J,axiom,
ord_le1843388692487544644orel_a @ y @ z ).
% h(2)
thf(fact_2_h_I1_J,axiom,
ord_le1843388692487544644orel_a @ x @ z ).
% h(1)
thf(fact_3_le__sup__iff,axiom,
! [X: produc5346009699553737693_a_nat,Y: produc5346009699553737693_a_nat,Z: produc5346009699553737693_a_nat] :
( ( ord_le9077581563206736957_a_nat @ ( sup_su871486172591609097_a_nat @ X @ Y ) @ Z )
= ( ( ord_le9077581563206736957_a_nat @ X @ Z )
& ( ord_le9077581563206736957_a_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_4_le__sup__iff,axiom,
! [X: produc2875793189560775939_set_a,Y: produc2875793189560775939_set_a,Z: produc2875793189560775939_set_a] :
( ( ord_le6607365053213775203_set_a @ ( sup_su7624641699453423151_set_a @ X @ Y ) @ Z )
= ( ( ord_le6607365053213775203_set_a @ X @ Z )
& ( ord_le6607365053213775203_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_5_le__sup__iff,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
( ( ord_le8460144461188290721at_nat @ ( sup_su4120719815643632853at_nat @ X @ Y ) @ Z )
= ( ( ord_le8460144461188290721at_nat @ X @ Z )
& ( ord_le8460144461188290721at_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_6_le__sup__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ Z )
= ( ( ord_le746702958409616551od_a_a @ X @ Z )
& ( ord_le746702958409616551od_a_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_7_le__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z )
= ( ( ord_le3724670747650509150_set_a @ X @ Z )
& ( ord_le3724670747650509150_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_8_le__sup__iff,axiom,
! [X: set_real_a,Y: set_real_a,Z: set_real_a] :
( ( ord_le5743406823621094409real_a @ ( sup_sup_set_real_a @ X @ Y ) @ Z )
= ( ( ord_le5743406823621094409real_a @ X @ Z )
& ( ord_le5743406823621094409real_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_9_le__sup__iff,axiom,
! [X: $o > set_a,Y: $o > set_a,Z: $o > set_a] :
( ( ord_less_eq_o_set_a @ ( sup_sup_o_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_o_set_a @ X @ Z )
& ( ord_less_eq_o_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_10_le__sup__iff,axiom,
! [X: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_o_nat @ X @ Z )
& ( ord_less_eq_o_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_11_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_12_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_13_sup_Obounded__iff,axiom,
! [B: produc5346009699553737693_a_nat,C: produc5346009699553737693_a_nat,A: produc5346009699553737693_a_nat] :
( ( ord_le9077581563206736957_a_nat @ ( sup_su871486172591609097_a_nat @ B @ C ) @ A )
= ( ( ord_le9077581563206736957_a_nat @ B @ A )
& ( ord_le9077581563206736957_a_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_14_sup_Obounded__iff,axiom,
! [B: produc2875793189560775939_set_a,C: produc2875793189560775939_set_a,A: produc2875793189560775939_set_a] :
( ( ord_le6607365053213775203_set_a @ ( sup_su7624641699453423151_set_a @ B @ C ) @ A )
= ( ( ord_le6607365053213775203_set_a @ B @ A )
& ( ord_le6607365053213775203_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_15_sup_Obounded__iff,axiom,
! [B: product_prod_nat_nat,C: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( ord_le8460144461188290721at_nat @ ( sup_su4120719815643632853at_nat @ B @ C ) @ A )
= ( ( ord_le8460144461188290721at_nat @ B @ A )
& ( ord_le8460144461188290721at_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_16_sup_Obounded__iff,axiom,
! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ C ) @ A )
= ( ( ord_le746702958409616551od_a_a @ B @ A )
& ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_17_sup_Obounded__iff,axiom,
! [B: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B @ C ) @ A )
= ( ( ord_le3724670747650509150_set_a @ B @ A )
& ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_18_sup_Obounded__iff,axiom,
! [B: set_real_a,C: set_real_a,A: set_real_a] :
( ( ord_le5743406823621094409real_a @ ( sup_sup_set_real_a @ B @ C ) @ A )
= ( ( ord_le5743406823621094409real_a @ B @ A )
& ( ord_le5743406823621094409real_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_19_sup_Obounded__iff,axiom,
! [B: $o > set_a,C: $o > set_a,A: $o > set_a] :
( ( ord_less_eq_o_set_a @ ( sup_sup_o_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_o_set_a @ B @ A )
& ( ord_less_eq_o_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_20_sup_Obounded__iff,axiom,
! [B: $o > nat,C: $o > nat,A: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ B @ C ) @ A )
= ( ( ord_less_eq_o_nat @ B @ A )
& ( ord_less_eq_o_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_21_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_22_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_23_sup_Oidem,axiom,
! [A: produc5346009699553737693_a_nat] :
( ( sup_su871486172591609097_a_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_24_sup_Oidem,axiom,
! [A: produc2875793189560775939_set_a] :
( ( sup_su7624641699453423151_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_25_sup_Oidem,axiom,
! [A: product_prod_nat_nat] :
( ( sup_su4120719815643632853at_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_26_sup_Oidem,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_27_sup_Oidem,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_28_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_29_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_30_sup__idem,axiom,
! [X: produc5346009699553737693_a_nat] :
( ( sup_su871486172591609097_a_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_31_sup__idem,axiom,
! [X: produc2875793189560775939_set_a] :
( ( sup_su7624641699453423151_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_32_sup__idem,axiom,
! [X: product_prod_nat_nat] :
( ( sup_su4120719815643632853at_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_33_sup__idem,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_34_sup__idem,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_35_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_36_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_37_sup_Oleft__idem,axiom,
! [A: produc5346009699553737693_a_nat,B: produc5346009699553737693_a_nat] :
( ( sup_su871486172591609097_a_nat @ A @ ( sup_su871486172591609097_a_nat @ A @ B ) )
= ( sup_su871486172591609097_a_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_38_sup_Oleft__idem,axiom,
! [A: produc2875793189560775939_set_a,B: produc2875793189560775939_set_a] :
( ( sup_su7624641699453423151_set_a @ A @ ( sup_su7624641699453423151_set_a @ A @ B ) )
= ( sup_su7624641699453423151_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_39_sup_Oleft__idem,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( sup_su4120719815643632853at_nat @ A @ ( sup_su4120719815643632853at_nat @ A @ B ) )
= ( sup_su4120719815643632853at_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_40_sup_Oleft__idem,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_41_sup_Oleft__idem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_42_sup_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_43_sup_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_44_sup__left__idem,axiom,
! [X: produc5346009699553737693_a_nat,Y: produc5346009699553737693_a_nat] :
( ( sup_su871486172591609097_a_nat @ X @ ( sup_su871486172591609097_a_nat @ X @ Y ) )
= ( sup_su871486172591609097_a_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_45_sup__left__idem,axiom,
! [X: produc2875793189560775939_set_a,Y: produc2875793189560775939_set_a] :
( ( sup_su7624641699453423151_set_a @ X @ ( sup_su7624641699453423151_set_a @ X @ Y ) )
= ( sup_su7624641699453423151_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_46_sup__left__idem,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( sup_su4120719815643632853at_nat @ X @ ( sup_su4120719815643632853at_nat @ X @ Y ) )
= ( sup_su4120719815643632853at_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_47_sup__left__idem,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= ( sup_su3048258781599657691od_a_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_48_sup__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= ( sup_sup_set_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_49_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_50_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_51_sup_Oright__idem,axiom,
! [A: produc5346009699553737693_a_nat,B: produc5346009699553737693_a_nat] :
( ( sup_su871486172591609097_a_nat @ ( sup_su871486172591609097_a_nat @ A @ B ) @ B )
= ( sup_su871486172591609097_a_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_52_sup_Oright__idem,axiom,
! [A: produc2875793189560775939_set_a,B: produc2875793189560775939_set_a] :
( ( sup_su7624641699453423151_set_a @ ( sup_su7624641699453423151_set_a @ A @ B ) @ B )
= ( sup_su7624641699453423151_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_53_sup_Oright__idem,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( sup_su4120719815643632853at_nat @ ( sup_su4120719815643632853at_nat @ A @ B ) @ B )
= ( sup_su4120719815643632853at_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_54_sup_Oright__idem,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ B )
= ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_55_sup_Oright__idem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ B )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_56_sup_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_57_sup_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_58_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_59_order__refl,axiom,
! [X: set_real_a] : ( ord_le5743406823621094409real_a @ X @ X ) ).
% order_refl
thf(fact_60_order__refl,axiom,
! [X: $o > quasi_borel_a] : ( ord_le1636368021143733457orel_a @ X @ X ) ).
% order_refl
thf(fact_61_order__refl,axiom,
! [X: $o > set_a] : ( ord_less_eq_o_set_a @ X @ X ) ).
% order_refl
thf(fact_62_order__refl,axiom,
! [X: $o > nat] : ( ord_less_eq_o_nat @ X @ X ) ).
% order_refl
thf(fact_63_order__refl,axiom,
! [X: quasi_borel_a] : ( ord_le1843388692487544644orel_a @ X @ X ) ).
% order_refl
thf(fact_64_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_65_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_66_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_67_dual__order_Orefl,axiom,
! [A: set_real_a] : ( ord_le5743406823621094409real_a @ A @ A ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A: $o > quasi_borel_a] : ( ord_le1636368021143733457orel_a @ A @ A ) ).
% dual_order.refl
thf(fact_69_dual__order_Orefl,axiom,
! [A: $o > set_a] : ( ord_less_eq_o_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_70_dual__order_Orefl,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% dual_order.refl
thf(fact_71_dual__order_Orefl,axiom,
! [A: quasi_borel_a] : ( ord_le1843388692487544644orel_a @ A @ A ) ).
% dual_order.refl
thf(fact_72_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_73_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_74_inf__sup__ord_I4_J,axiom,
! [Y: produc5346009699553737693_a_nat,X: produc5346009699553737693_a_nat] : ( ord_le9077581563206736957_a_nat @ Y @ ( sup_su871486172591609097_a_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_75_inf__sup__ord_I4_J,axiom,
! [Y: produc2875793189560775939_set_a,X: produc2875793189560775939_set_a] : ( ord_le6607365053213775203_set_a @ Y @ ( sup_su7624641699453423151_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_76_inf__sup__ord_I4_J,axiom,
! [Y: product_prod_nat_nat,X: product_prod_nat_nat] : ( ord_le8460144461188290721at_nat @ Y @ ( sup_su4120719815643632853at_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_77_inf__sup__ord_I4_J,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_78_inf__sup__ord_I4_J,axiom,
! [Y: set_set_a,X: set_set_a] : ( ord_le3724670747650509150_set_a @ Y @ ( sup_sup_set_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_79_inf__sup__ord_I4_J,axiom,
! [Y: set_real_a,X: set_real_a] : ( ord_le5743406823621094409real_a @ Y @ ( sup_sup_set_real_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_80_inf__sup__ord_I4_J,axiom,
! [Y: $o > set_a,X: $o > set_a] : ( ord_less_eq_o_set_a @ Y @ ( sup_sup_o_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_81_inf__sup__ord_I4_J,axiom,
! [Y: $o > nat,X: $o > nat] : ( ord_less_eq_o_nat @ Y @ ( sup_sup_o_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_82_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_83_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_84_inf__sup__ord_I3_J,axiom,
! [X: produc5346009699553737693_a_nat,Y: produc5346009699553737693_a_nat] : ( ord_le9077581563206736957_a_nat @ X @ ( sup_su871486172591609097_a_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_85_inf__sup__ord_I3_J,axiom,
! [X: produc2875793189560775939_set_a,Y: produc2875793189560775939_set_a] : ( ord_le6607365053213775203_set_a @ X @ ( sup_su7624641699453423151_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_86_inf__sup__ord_I3_J,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : ( ord_le8460144461188290721at_nat @ X @ ( sup_su4120719815643632853at_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_87_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_88_inf__sup__ord_I3_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_89_inf__sup__ord_I3_J,axiom,
! [X: set_real_a,Y: set_real_a] : ( ord_le5743406823621094409real_a @ X @ ( sup_sup_set_real_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_90_inf__sup__ord_I3_J,axiom,
! [X: $o > set_a,Y: $o > set_a] : ( ord_less_eq_o_set_a @ X @ ( sup_sup_o_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_91_inf__sup__ord_I3_J,axiom,
! [X: $o > nat,Y: $o > nat] : ( ord_less_eq_o_nat @ X @ ( sup_sup_o_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_92_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_93_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_94__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062qbs__space_AX_A_061_Aqbs__space_AY_A_092_060Longrightarrow_062_Athesis_059_Aqbs__space_AX_A_092_060subset_062_Aqbs__space_AY_A_092_060Longrightarrow_062_Athesis_059_Aqbs__space_AY_A_092_060subset_062_Aqbs__space_AX_A_092_060Longrightarrow_062_Athesis_059_Aqbs__space_AX_A_092_060subset_062_Aqbs__space_AX_A_092_060union_062_Aqbs__space_AY_A_092_060and_062_Aqbs__space_AY_A_092_060subset_062_Aqbs__space_AX_A_092_060union_062_Aqbs__space_AY_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ( ( qbs_space_a @ x )
!= ( qbs_space_a @ y ) )
=> ( ~ ( ord_less_set_a @ ( qbs_space_a @ x ) @ ( qbs_space_a @ y ) )
=> ( ~ ( ord_less_set_a @ ( qbs_space_a @ y ) @ ( qbs_space_a @ x ) )
=> ( ( ord_less_set_a @ ( qbs_space_a @ x ) @ ( sup_sup_set_a @ ( qbs_space_a @ x ) @ ( qbs_space_a @ y ) ) )
& ( ord_less_set_a @ ( qbs_space_a @ y ) @ ( sup_sup_set_a @ ( qbs_space_a @ x ) @ ( qbs_space_a @ y ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. \<lbrakk>qbs_space X = qbs_space Y \<Longrightarrow> thesis; qbs_space X \<subset> qbs_space Y \<Longrightarrow> thesis; qbs_space Y \<subset> qbs_space X \<Longrightarrow> thesis; qbs_space X \<subset> qbs_space X \<union> qbs_space Y \<and> qbs_space Y \<subset> qbs_space X \<union> qbs_space Y \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_95_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_96_less__imp__neq,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_97_less__imp__neq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_98_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_99_order_Oasym,axiom,
! [A: quasi_borel_a,B: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ~ ( ord_le3788981553861827664orel_a @ B @ A ) ) ).
% order.asym
thf(fact_100_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_101_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_102_ord__eq__less__trans,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( A = B )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ord_le3788981553861827664orel_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_103_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_104_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_105_ord__less__eq__trans,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( B = C )
=> ( ord_le3788981553861827664orel_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_106_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_107_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_108_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X2 )
=> ( P @ Y2 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_109_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_110_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_111_dual__order_Oasym,axiom,
! [B: quasi_borel_a,A: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ B @ A )
=> ~ ( ord_le3788981553861827664orel_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_112_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_113_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_114_dual__order_Oirrefl,axiom,
! [A: quasi_borel_a] :
~ ( ord_le3788981553861827664orel_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_115_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_116_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_117_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X3: nat] : ( P2 @ X3 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_118_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_119_order_Ostrict__trans,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ord_le3788981553861827664orel_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_120_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_121_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_122_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_123_dual__order_Ostrict__trans,axiom,
! [B: quasi_borel_a,A: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ B @ A )
=> ( ( ord_le3788981553861827664orel_a @ C @ B )
=> ( ord_le3788981553861827664orel_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_124_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_125_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_126_order_Ostrict__implies__not__eq,axiom,
! [A: quasi_borel_a,B: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_127_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_128_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_129_dual__order_Ostrict__implies__not__eq,axiom,
! [B: quasi_borel_a,A: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_130_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_131_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_132_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_133_order__less__asym,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ~ ( ord_le3788981553861827664orel_a @ Y @ X ) ) ).
% order_less_asym
thf(fact_134_order__less__asym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_asym
thf(fact_135_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_136_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_137_order__less__asym_H,axiom,
! [A: quasi_borel_a,B: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ~ ( ord_le3788981553861827664orel_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_138_order__less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_139_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_140_order__less__trans,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a,Z: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ( ( ord_le3788981553861827664orel_a @ Y @ Z )
=> ( ord_le3788981553861827664orel_a @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_141_order__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_142_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_143_ord__eq__less__subst,axiom,
! [A: quasi_borel_a,F: set_a > quasi_borel_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_144_ord__eq__less__subst,axiom,
! [A: quasi_borel_a,F: nat > quasi_borel_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_145_ord__eq__less__subst,axiom,
! [A: set_a,F: quasi_borel_a > set_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_146_ord__eq__less__subst,axiom,
! [A: nat,F: quasi_borel_a > nat,B: quasi_borel_a,C: quasi_borel_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_147_ord__eq__less__subst,axiom,
! [A: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_148_ord__eq__less__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_149_ord__eq__less__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_150_ord__eq__less__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_151_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_152_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_153_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_154_ord__less__eq__subst,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > set_a,C: set_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_155_ord__less__eq__subst,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > nat,C: nat] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_156_ord__less__eq__subst,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_157_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_158_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_159_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_160_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_161_order__less__irrefl,axiom,
! [X: quasi_borel_a] :
~ ( ord_le3788981553861827664orel_a @ X @ X ) ).
% order_less_irrefl
thf(fact_162_order__less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_163_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_164_order__less__subst1,axiom,
! [A: set_a,F: quasi_borel_a > set_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_165_order__less__subst1,axiom,
! [A: nat,F: quasi_borel_a > nat,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_166_order__less__subst1,axiom,
! [A: quasi_borel_a,F: set_a > quasi_borel_a,B: set_a,C: set_a] :
( ( ord_le3788981553861827664orel_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_167_order__less__subst1,axiom,
! [A: quasi_borel_a,F: nat > quasi_borel_a,B: nat,C: nat] :
( ( ord_le3788981553861827664orel_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_168_order__less__subst1,axiom,
! [A: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_169_order__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_170_order__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_171_order__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_172_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_173_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_le3788981553861827664orel_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_174_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le3788981553861827664orel_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_175_order__less__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > set_a,C: set_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_176_order__less__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > nat,C: nat] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_177_order__less__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ord_le3788981553861827664orel_a @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_178_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_179_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_180_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_181_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_182_mem__Collect__eq,axiom,
! [A: set_set_a,P: set_set_a > $o] :
( ( member_set_set_a @ A @ ( collect_set_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_183_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_184_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_185_mem__Collect__eq,axiom,
! [A: real > a,P: ( real > a ) > $o] :
( ( member_real_a @ A @ ( collect_real_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_186_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_187_Collect__mem__eq,axiom,
! [A3: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X4: set_set_a] : ( member_set_set_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_188_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_189_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_190_Collect__mem__eq,axiom,
! [A3: set_real_a] :
( ( collect_real_a
@ ^ [X4: real > a] : ( member_real_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_191_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_192_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_193_order__less__not__sym,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ~ ( ord_le3788981553861827664orel_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_194_order__less__not__sym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_195_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_196_order__less__imp__triv,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a,P: $o] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ( ( ord_le3788981553861827664orel_a @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_197_order__less__imp__triv,axiom,
! [X: set_a,Y: set_a,P: $o] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_198_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_199_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_200_order__less__imp__not__eq,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_201_order__less__imp__not__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_202_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_203_order__less__imp__not__eq2,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_204_order__less__imp__not__eq2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_205_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_206_order__less__imp__not__less,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ~ ( ord_le3788981553861827664orel_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_207_order__less__imp__not__less,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_208_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_209_order__le__imp__less__or__eq,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_less_set_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_210_order__le__imp__less__or__eq,axiom,
! [X: set_real_a,Y: set_real_a] :
( ( ord_le5743406823621094409real_a @ X @ Y )
=> ( ( ord_less_set_real_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_211_order__le__imp__less__or__eq,axiom,
! [X: $o > quasi_borel_a,Y: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ X @ Y )
=> ( ( ord_le2255205531489447621orel_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_212_order__le__imp__less__or__eq,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X @ Y )
=> ( ( ord_less_o_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_213_order__le__imp__less__or__eq,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_o_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_214_order__le__imp__less__or__eq,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X @ Y )
=> ( ( ord_le3788981553861827664orel_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_215_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_216_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_217_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_218_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_219_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_220_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_221_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_222_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_223_order__less__le__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_224_order__less__le__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > nat,C: nat] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_225_order__less__le__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_226_order__less__le__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > set_a,C: set_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_227_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_228_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_229_order__less__le__subst1,axiom,
! [A: nat,F: quasi_borel_a > nat,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_230_order__less__le__subst1,axiom,
! [A: quasi_borel_a,F: nat > quasi_borel_a,B: nat,C: nat] :
( ( ord_le3788981553861827664orel_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_231_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_232_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_233_order__less__le__subst1,axiom,
! [A: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_234_order__less__le__subst1,axiom,
! [A: set_a,F: quasi_borel_a > set_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_235_order__less__le__subst1,axiom,
! [A: quasi_borel_a,F: set_a > quasi_borel_a,B: set_a,C: set_a] :
( ( ord_le3788981553861827664orel_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_236_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_237_order__less__le__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( ord_less_set_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_238_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_239_order__le__less__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > nat,C: nat] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_240_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3788981553861827664orel_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_241_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_242_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_243_order__le__less__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_le3788981553861827664orel_a @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_244_order__le__less__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > set_a,C: set_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_245_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le3788981553861827664orel_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_246_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_247_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_248_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_249_order__le__less__subst1,axiom,
! [A: quasi_borel_a,F: nat > quasi_borel_a,B: nat,C: nat] :
( ( ord_le1843388692487544644orel_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_250_order__le__less__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_251_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_252_order__le__less__subst1,axiom,
! [A: quasi_borel_a,F: set_a > quasi_borel_a,B: set_a,C: set_a] :
( ( ord_le1843388692487544644orel_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_253_order__le__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_254_order__le__less__subst1,axiom,
! [A: nat,F: quasi_borel_a > nat,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_255_order__le__less__subst1,axiom,
! [A: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_le3788981553861827664orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3788981553861827664orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_256_order__le__less__subst1,axiom,
! [A: set_a,F: quasi_borel_a > set_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X2 @ Y3 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_257_order__le__less__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_258_order__less__le__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_less_set_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_less_set_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_259_order__less__le__trans,axiom,
! [X: set_real_a,Y: set_real_a,Z: set_real_a] :
( ( ord_less_set_real_a @ X @ Y )
=> ( ( ord_le5743406823621094409real_a @ Y @ Z )
=> ( ord_less_set_real_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_260_order__less__le__trans,axiom,
! [X: $o > quasi_borel_a,Y: $o > quasi_borel_a,Z: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ X @ Y )
=> ( ( ord_le1636368021143733457orel_a @ Y @ Z )
=> ( ord_le2255205531489447621orel_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_261_order__less__le__trans,axiom,
! [X: $o > set_a,Y: $o > set_a,Z: $o > set_a] :
( ( ord_less_o_set_a @ X @ Y )
=> ( ( ord_less_eq_o_set_a @ Y @ Z )
=> ( ord_less_o_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_262_order__less__le__trans,axiom,
! [X: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z )
=> ( ord_less_o_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_263_order__less__le__trans,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a,Z: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ( ( ord_le1843388692487544644orel_a @ Y @ Z )
=> ( ord_le3788981553861827664orel_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_264_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_265_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_266_order__le__less__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_less_set_set_a @ Y @ Z )
=> ( ord_less_set_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_267_order__le__less__trans,axiom,
! [X: set_real_a,Y: set_real_a,Z: set_real_a] :
( ( ord_le5743406823621094409real_a @ X @ Y )
=> ( ( ord_less_set_real_a @ Y @ Z )
=> ( ord_less_set_real_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_268_order__le__less__trans,axiom,
! [X: $o > quasi_borel_a,Y: $o > quasi_borel_a,Z: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ X @ Y )
=> ( ( ord_le2255205531489447621orel_a @ Y @ Z )
=> ( ord_le2255205531489447621orel_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_269_order__le__less__trans,axiom,
! [X: $o > set_a,Y: $o > set_a,Z: $o > set_a] :
( ( ord_less_eq_o_set_a @ X @ Y )
=> ( ( ord_less_o_set_a @ Y @ Z )
=> ( ord_less_o_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_270_order__le__less__trans,axiom,
! [X: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_o_nat @ Y @ Z )
=> ( ord_less_o_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_271_order__le__less__trans,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a,Z: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X @ Y )
=> ( ( ord_le3788981553861827664orel_a @ Y @ Z )
=> ( ord_le3788981553861827664orel_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_272_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_273_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_274_order__neq__le__trans,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A != B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_275_order__neq__le__trans,axiom,
! [A: set_real_a,B: set_real_a] :
( ( A != B )
=> ( ( ord_le5743406823621094409real_a @ A @ B )
=> ( ord_less_set_real_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_276_order__neq__le__trans,axiom,
! [A: $o > quasi_borel_a,B: $o > quasi_borel_a] :
( ( A != B )
=> ( ( ord_le1636368021143733457orel_a @ A @ B )
=> ( ord_le2255205531489447621orel_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_277_order__neq__le__trans,axiom,
! [A: $o > set_a,B: $o > set_a] :
( ( A != B )
=> ( ( ord_less_eq_o_set_a @ A @ B )
=> ( ord_less_o_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_278_order__neq__le__trans,axiom,
! [A: $o > nat,B: $o > nat] :
( ( A != B )
=> ( ( ord_less_eq_o_nat @ A @ B )
=> ( ord_less_o_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_279_order__neq__le__trans,axiom,
! [A: quasi_borel_a,B: quasi_borel_a] :
( ( A != B )
=> ( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ord_le3788981553861827664orel_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_280_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_281_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_282_order__le__neq__trans,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_283_order__le__neq__trans,axiom,
! [A: set_real_a,B: set_real_a] :
( ( ord_le5743406823621094409real_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_real_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_284_order__le__neq__trans,axiom,
! [A: $o > quasi_borel_a,B: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ A @ B )
=> ( ( A != B )
=> ( ord_le2255205531489447621orel_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_285_order__le__neq__trans,axiom,
! [A: $o > set_a,B: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_o_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_286_order__le__neq__trans,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_o_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_287_order__le__neq__trans,axiom,
! [A: quasi_borel_a,B: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( A != B )
=> ( ord_le3788981553861827664orel_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_288_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_289_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_290_order__less__imp__le,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_less_set_set_a @ X @ Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_291_order__less__imp__le,axiom,
! [X: set_real_a,Y: set_real_a] :
( ( ord_less_set_real_a @ X @ Y )
=> ( ord_le5743406823621094409real_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_292_order__less__imp__le,axiom,
! [X: $o > quasi_borel_a,Y: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ X @ Y )
=> ( ord_le1636368021143733457orel_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_293_order__less__imp__le,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( ord_less_o_set_a @ X @ Y )
=> ( ord_less_eq_o_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_294_order__less__imp__le,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( ord_less_o_nat @ X @ Y )
=> ( ord_less_eq_o_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_295_order__less__imp__le,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X @ Y )
=> ( ord_le1843388692487544644orel_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_296_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_297_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_298_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_299_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_300_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_301_order__less__le,axiom,
( ord_less_set_real_a
= ( ^ [X4: set_real_a,Y4: set_real_a] :
( ( ord_le5743406823621094409real_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_302_order__less__le,axiom,
( ord_le2255205531489447621orel_a
= ( ^ [X4: $o > quasi_borel_a,Y4: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_303_order__less__le,axiom,
( ord_less_o_set_a
= ( ^ [X4: $o > set_a,Y4: $o > set_a] :
( ( ord_less_eq_o_set_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_304_order__less__le,axiom,
( ord_less_o_nat
= ( ^ [X4: $o > nat,Y4: $o > nat] :
( ( ord_less_eq_o_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_305_order__less__le,axiom,
( ord_le3788981553861827664orel_a
= ( ^ [X4: quasi_borel_a,Y4: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_306_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_307_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_308_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_309_order__le__less,axiom,
( ord_le5743406823621094409real_a
= ( ^ [X4: set_real_a,Y4: set_real_a] :
( ( ord_less_set_real_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_310_order__le__less,axiom,
( ord_le1636368021143733457orel_a
= ( ^ [X4: $o > quasi_borel_a,Y4: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_311_order__le__less,axiom,
( ord_less_eq_o_set_a
= ( ^ [X4: $o > set_a,Y4: $o > set_a] :
( ( ord_less_o_set_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_312_order__le__less,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y4: $o > nat] :
( ( ord_less_o_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_313_order__le__less,axiom,
( ord_le1843388692487544644orel_a
= ( ^ [X4: quasi_borel_a,Y4: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_314_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_315_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_316_dual__order_Ostrict__implies__order,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ B @ A )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_317_dual__order_Ostrict__implies__order,axiom,
! [B: set_real_a,A: set_real_a] :
( ( ord_less_set_real_a @ B @ A )
=> ( ord_le5743406823621094409real_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_318_dual__order_Ostrict__implies__order,axiom,
! [B: $o > quasi_borel_a,A: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ B @ A )
=> ( ord_le1636368021143733457orel_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_319_dual__order_Ostrict__implies__order,axiom,
! [B: $o > set_a,A: $o > set_a] :
( ( ord_less_o_set_a @ B @ A )
=> ( ord_less_eq_o_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_320_dual__order_Ostrict__implies__order,axiom,
! [B: $o > nat,A: $o > nat] :
( ( ord_less_o_nat @ B @ A )
=> ( ord_less_eq_o_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_321_dual__order_Ostrict__implies__order,axiom,
! [B: quasi_borel_a,A: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ B @ A )
=> ( ord_le1843388692487544644orel_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_322_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_323_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_324_order_Ostrict__implies__order,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_325_order_Ostrict__implies__order,axiom,
! [A: set_real_a,B: set_real_a] :
( ( ord_less_set_real_a @ A @ B )
=> ( ord_le5743406823621094409real_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_326_order_Ostrict__implies__order,axiom,
! [A: $o > quasi_borel_a,B: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ A @ B )
=> ( ord_le1636368021143733457orel_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_327_order_Ostrict__implies__order,axiom,
! [A: $o > set_a,B: $o > set_a] :
( ( ord_less_o_set_a @ A @ B )
=> ( ord_less_eq_o_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_328_order_Ostrict__implies__order,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ord_less_o_nat @ A @ B )
=> ( ord_less_eq_o_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_329_order_Ostrict__implies__order,axiom,
! [A: quasi_borel_a,B: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ord_le1843388692487544644orel_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_330_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_331_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_332_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A4 )
& ~ ( ord_le3724670747650509150_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_333_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_real_a
= ( ^ [B3: set_real_a,A4: set_real_a] :
( ( ord_le5743406823621094409real_a @ B3 @ A4 )
& ~ ( ord_le5743406823621094409real_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_334_dual__order_Ostrict__iff__not,axiom,
( ord_le2255205531489447621orel_a
= ( ^ [B3: $o > quasi_borel_a,A4: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ B3 @ A4 )
& ~ ( ord_le1636368021143733457orel_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_335_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_set_a
= ( ^ [B3: $o > set_a,A4: $o > set_a] :
( ( ord_less_eq_o_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_o_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_336_dual__order_Ostrict__iff__not,axiom,
( ord_less_o_nat
= ( ^ [B3: $o > nat,A4: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A4 )
& ~ ( ord_less_eq_o_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_337_dual__order_Ostrict__iff__not,axiom,
( ord_le3788981553861827664orel_a
= ( ^ [B3: quasi_borel_a,A4: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ B3 @ A4 )
& ~ ( ord_le1843388692487544644orel_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_338_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_339_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_340_dual__order_Ostrict__trans2,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_341_dual__order_Ostrict__trans2,axiom,
! [B: set_real_a,A: set_real_a,C: set_real_a] :
( ( ord_less_set_real_a @ B @ A )
=> ( ( ord_le5743406823621094409real_a @ C @ B )
=> ( ord_less_set_real_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_342_dual__order_Ostrict__trans2,axiom,
! [B: $o > quasi_borel_a,A: $o > quasi_borel_a,C: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ B @ A )
=> ( ( ord_le1636368021143733457orel_a @ C @ B )
=> ( ord_le2255205531489447621orel_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_343_dual__order_Ostrict__trans2,axiom,
! [B: $o > set_a,A: $o > set_a,C: $o > set_a] :
( ( ord_less_o_set_a @ B @ A )
=> ( ( ord_less_eq_o_set_a @ C @ B )
=> ( ord_less_o_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_344_dual__order_Ostrict__trans2,axiom,
! [B: $o > nat,A: $o > nat,C: $o > nat] :
( ( ord_less_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_o_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_345_dual__order_Ostrict__trans2,axiom,
! [B: quasi_borel_a,A: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ B @ A )
=> ( ( ord_le1843388692487544644orel_a @ C @ B )
=> ( ord_le3788981553861827664orel_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_346_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_347_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_348_dual__order_Ostrict__trans1,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_less_set_set_a @ C @ B )
=> ( ord_less_set_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_349_dual__order_Ostrict__trans1,axiom,
! [B: set_real_a,A: set_real_a,C: set_real_a] :
( ( ord_le5743406823621094409real_a @ B @ A )
=> ( ( ord_less_set_real_a @ C @ B )
=> ( ord_less_set_real_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_350_dual__order_Ostrict__trans1,axiom,
! [B: $o > quasi_borel_a,A: $o > quasi_borel_a,C: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ B @ A )
=> ( ( ord_le2255205531489447621orel_a @ C @ B )
=> ( ord_le2255205531489447621orel_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_351_dual__order_Ostrict__trans1,axiom,
! [B: $o > set_a,A: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ B @ A )
=> ( ( ord_less_o_set_a @ C @ B )
=> ( ord_less_o_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_352_dual__order_Ostrict__trans1,axiom,
! [B: $o > nat,A: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_o_nat @ C @ B )
=> ( ord_less_o_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_353_dual__order_Ostrict__trans1,axiom,
! [B: quasi_borel_a,A: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ B @ A )
=> ( ( ord_le3788981553861827664orel_a @ C @ B )
=> ( ord_le3788981553861827664orel_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_354_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_355_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_356_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_357_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_real_a
= ( ^ [B3: set_real_a,A4: set_real_a] :
( ( ord_le5743406823621094409real_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_358_dual__order_Ostrict__iff__order,axiom,
( ord_le2255205531489447621orel_a
= ( ^ [B3: $o > quasi_borel_a,A4: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_359_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_set_a
= ( ^ [B3: $o > set_a,A4: $o > set_a] :
( ( ord_less_eq_o_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_360_dual__order_Ostrict__iff__order,axiom,
( ord_less_o_nat
= ( ^ [B3: $o > nat,A4: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_361_dual__order_Ostrict__iff__order,axiom,
( ord_le3788981553861827664orel_a
= ( ^ [B3: quasi_borel_a,A4: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_362_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_363_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_364_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A4: set_set_a] :
( ( ord_less_set_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_365_dual__order_Oorder__iff__strict,axiom,
( ord_le5743406823621094409real_a
= ( ^ [B3: set_real_a,A4: set_real_a] :
( ( ord_less_set_real_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_366_dual__order_Oorder__iff__strict,axiom,
( ord_le1636368021143733457orel_a
= ( ^ [B3: $o > quasi_borel_a,A4: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_367_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_set_a
= ( ^ [B3: $o > set_a,A4: $o > set_a] :
( ( ord_less_o_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_368_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_o_nat
= ( ^ [B3: $o > nat,A4: $o > nat] :
( ( ord_less_o_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_369_dual__order_Oorder__iff__strict,axiom,
( ord_le1843388692487544644orel_a
= ( ^ [B3: quasi_borel_a,A4: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_370_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_371_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( ord_less_set_a @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_372_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B3 )
& ~ ( ord_le3724670747650509150_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_373_order_Ostrict__iff__not,axiom,
( ord_less_set_real_a
= ( ^ [A4: set_real_a,B3: set_real_a] :
( ( ord_le5743406823621094409real_a @ A4 @ B3 )
& ~ ( ord_le5743406823621094409real_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_374_order_Ostrict__iff__not,axiom,
( ord_le2255205531489447621orel_a
= ( ^ [A4: $o > quasi_borel_a,B3: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ A4 @ B3 )
& ~ ( ord_le1636368021143733457orel_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_375_order_Ostrict__iff__not,axiom,
( ord_less_o_set_a
= ( ^ [A4: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_o_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_376_order_Ostrict__iff__not,axiom,
( ord_less_o_nat
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A4 @ B3 )
& ~ ( ord_less_eq_o_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_377_order_Ostrict__iff__not,axiom,
( ord_le3788981553861827664orel_a
= ( ^ [A4: quasi_borel_a,B3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A4 @ B3 )
& ~ ( ord_le1843388692487544644orel_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_378_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_379_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_380_order_Ostrict__trans2,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_381_order_Ostrict__trans2,axiom,
! [A: set_real_a,B: set_real_a,C: set_real_a] :
( ( ord_less_set_real_a @ A @ B )
=> ( ( ord_le5743406823621094409real_a @ B @ C )
=> ( ord_less_set_real_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_382_order_Ostrict__trans2,axiom,
! [A: $o > quasi_borel_a,B: $o > quasi_borel_a,C: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ A @ B )
=> ( ( ord_le1636368021143733457orel_a @ B @ C )
=> ( ord_le2255205531489447621orel_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_383_order_Ostrict__trans2,axiom,
! [A: $o > set_a,B: $o > set_a,C: $o > set_a] :
( ( ord_less_o_set_a @ A @ B )
=> ( ( ord_less_eq_o_set_a @ B @ C )
=> ( ord_less_o_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_384_order_Ostrict__trans2,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_o_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_385_order_Ostrict__trans2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ord_le3788981553861827664orel_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_386_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_387_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_388_order_Ostrict__trans1,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C )
=> ( ord_less_set_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_389_order_Ostrict__trans1,axiom,
! [A: set_real_a,B: set_real_a,C: set_real_a] :
( ( ord_le5743406823621094409real_a @ A @ B )
=> ( ( ord_less_set_real_a @ B @ C )
=> ( ord_less_set_real_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_390_order_Ostrict__trans1,axiom,
! [A: $o > quasi_borel_a,B: $o > quasi_borel_a,C: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ A @ B )
=> ( ( ord_le2255205531489447621orel_a @ B @ C )
=> ( ord_le2255205531489447621orel_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_391_order_Ostrict__trans1,axiom,
! [A: $o > set_a,B: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ B )
=> ( ( ord_less_o_set_a @ B @ C )
=> ( ord_less_o_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_392_order_Ostrict__trans1,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_o_nat @ B @ C )
=> ( ord_less_o_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_393_order_Ostrict__trans1,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_le3788981553861827664orel_a @ B @ C )
=> ( ord_le3788981553861827664orel_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_394_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_395_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_396_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_397_order_Ostrict__iff__order,axiom,
( ord_less_set_real_a
= ( ^ [A4: set_real_a,B3: set_real_a] :
( ( ord_le5743406823621094409real_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_398_order_Ostrict__iff__order,axiom,
( ord_le2255205531489447621orel_a
= ( ^ [A4: $o > quasi_borel_a,B3: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_399_order_Ostrict__iff__order,axiom,
( ord_less_o_set_a
= ( ^ [A4: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_400_order_Ostrict__iff__order,axiom,
( ord_less_o_nat
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_401_order_Ostrict__iff__order,axiom,
( ord_le3788981553861827664orel_a
= ( ^ [A4: quasi_borel_a,B3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_402_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_403_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_404_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_405_order_Oorder__iff__strict,axiom,
( ord_le5743406823621094409real_a
= ( ^ [A4: set_real_a,B3: set_real_a] :
( ( ord_less_set_real_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_406_order_Oorder__iff__strict,axiom,
( ord_le1636368021143733457orel_a
= ( ^ [A4: $o > quasi_borel_a,B3: $o > quasi_borel_a] :
( ( ord_le2255205531489447621orel_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_407_order_Oorder__iff__strict,axiom,
( ord_less_eq_o_set_a
= ( ^ [A4: $o > set_a,B3: $o > set_a] :
( ( ord_less_o_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_408_order_Oorder__iff__strict,axiom,
( ord_less_eq_o_nat
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( ord_less_o_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_409_order_Oorder__iff__strict,axiom,
( ord_le1843388692487544644orel_a
= ( ^ [A4: quasi_borel_a,B3: quasi_borel_a] :
( ( ord_le3788981553861827664orel_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_410_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_411_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_412_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_413_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ~ ( ord_le3724670747650509150_set_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_414_less__le__not__le,axiom,
( ord_less_set_real_a
= ( ^ [X4: set_real_a,Y4: set_real_a] :
( ( ord_le5743406823621094409real_a @ X4 @ Y4 )
& ~ ( ord_le5743406823621094409real_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_415_less__le__not__le,axiom,
( ord_le2255205531489447621orel_a
= ( ^ [X4: $o > quasi_borel_a,Y4: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ X4 @ Y4 )
& ~ ( ord_le1636368021143733457orel_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_416_less__le__not__le,axiom,
( ord_less_o_set_a
= ( ^ [X4: $o > set_a,Y4: $o > set_a] :
( ( ord_less_eq_o_set_a @ X4 @ Y4 )
& ~ ( ord_less_eq_o_set_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_417_less__le__not__le,axiom,
( ord_less_o_nat
= ( ^ [X4: $o > nat,Y4: $o > nat] :
( ( ord_less_eq_o_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_o_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_418_less__le__not__le,axiom,
( ord_le3788981553861827664orel_a
= ( ^ [X4: quasi_borel_a,Y4: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X4 @ Y4 )
& ~ ( ord_le1843388692487544644orel_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_419_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_420_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_421_antisym__conv2,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_422_antisym__conv2,axiom,
! [X: set_real_a,Y: set_real_a] :
( ( ord_le5743406823621094409real_a @ X @ Y )
=> ( ( ~ ( ord_less_set_real_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_423_antisym__conv2,axiom,
! [X: $o > quasi_borel_a,Y: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ X @ Y )
=> ( ( ~ ( ord_le2255205531489447621orel_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_424_antisym__conv2,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X @ Y )
=> ( ( ~ ( ord_less_o_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_425_antisym__conv2,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ~ ( ord_less_o_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_426_antisym__conv2,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X @ Y )
=> ( ( ~ ( ord_le3788981553861827664orel_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_427_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_428_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_429_antisym__conv1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ~ ( ord_less_set_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_430_antisym__conv1,axiom,
! [X: set_real_a,Y: set_real_a] :
( ~ ( ord_less_set_real_a @ X @ Y )
=> ( ( ord_le5743406823621094409real_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_431_antisym__conv1,axiom,
! [X: $o > quasi_borel_a,Y: $o > quasi_borel_a] :
( ~ ( ord_le2255205531489447621orel_a @ X @ Y )
=> ( ( ord_le1636368021143733457orel_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_432_antisym__conv1,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ~ ( ord_less_o_set_a @ X @ Y )
=> ( ( ord_less_eq_o_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_433_antisym__conv1,axiom,
! [X: $o > nat,Y: $o > nat] :
( ~ ( ord_less_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_434_antisym__conv1,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ~ ( ord_le3788981553861827664orel_a @ X @ Y )
=> ( ( ord_le1843388692487544644orel_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_435_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_436_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_437_nless__le,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A @ B ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_438_nless__le,axiom,
! [A: set_real_a,B: set_real_a] :
( ( ~ ( ord_less_set_real_a @ A @ B ) )
= ( ~ ( ord_le5743406823621094409real_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_439_nless__le,axiom,
! [A: $o > quasi_borel_a,B: $o > quasi_borel_a] :
( ( ~ ( ord_le2255205531489447621orel_a @ A @ B ) )
= ( ~ ( ord_le1636368021143733457orel_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_440_nless__le,axiom,
! [A: $o > set_a,B: $o > set_a] :
( ( ~ ( ord_less_o_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_o_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_441_nless__le,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ~ ( ord_less_o_nat @ A @ B ) )
= ( ~ ( ord_less_eq_o_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_442_nless__le,axiom,
! [A: quasi_borel_a,B: quasi_borel_a] :
( ( ~ ( ord_le3788981553861827664orel_a @ A @ B ) )
= ( ~ ( ord_le1843388692487544644orel_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_443_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_444_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_445_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_446_leD,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ~ ( ord_less_set_set_a @ X @ Y ) ) ).
% leD
thf(fact_447_leD,axiom,
! [Y: set_real_a,X: set_real_a] :
( ( ord_le5743406823621094409real_a @ Y @ X )
=> ~ ( ord_less_set_real_a @ X @ Y ) ) ).
% leD
thf(fact_448_leD,axiom,
! [Y: $o > quasi_borel_a,X: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ Y @ X )
=> ~ ( ord_le2255205531489447621orel_a @ X @ Y ) ) ).
% leD
thf(fact_449_leD,axiom,
! [Y: $o > set_a,X: $o > set_a] :
( ( ord_less_eq_o_set_a @ Y @ X )
=> ~ ( ord_less_o_set_a @ X @ Y ) ) ).
% leD
thf(fact_450_leD,axiom,
! [Y: $o > nat,X: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X )
=> ~ ( ord_less_o_nat @ X @ Y ) ) ).
% leD
thf(fact_451_leD,axiom,
! [Y: quasi_borel_a,X: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ Y @ X )
=> ~ ( ord_le3788981553861827664orel_a @ X @ Y ) ) ).
% leD
thf(fact_452_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_453_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_454_sup_Ostrict__coboundedI2,axiom,
! [C: produc5346009699553737693_a_nat,B: produc5346009699553737693_a_nat,A: produc5346009699553737693_a_nat] :
( ( ord_le2894110756394707505_a_nat @ C @ B )
=> ( ord_le2894110756394707505_a_nat @ C @ ( sup_su871486172591609097_a_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_455_sup_Ostrict__coboundedI2,axiom,
! [C: produc2875793189560775939_set_a,B: produc2875793189560775939_set_a,A: produc2875793189560775939_set_a] :
( ( ord_le423894246401745751_set_a @ C @ B )
=> ( ord_le423894246401745751_set_a @ C @ ( sup_su7624641699453423151_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_456_sup_Ostrict__coboundedI2,axiom,
! [C: product_prod_nat_nat,B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ C @ B )
=> ( ord_le1203424502768444845at_nat @ C @ ( sup_su4120719815643632853at_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_457_sup_Ostrict__coboundedI2,axiom,
! [C: set_Product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ C @ B )
=> ( ord_le6819997720685908915od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_458_sup_Ostrict__coboundedI2,axiom,
! [C: set_set_a,B: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ C @ B )
=> ( ord_less_set_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_459_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_460_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_461_sup_Ostrict__coboundedI1,axiom,
! [C: produc5346009699553737693_a_nat,A: produc5346009699553737693_a_nat,B: produc5346009699553737693_a_nat] :
( ( ord_le2894110756394707505_a_nat @ C @ A )
=> ( ord_le2894110756394707505_a_nat @ C @ ( sup_su871486172591609097_a_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_462_sup_Ostrict__coboundedI1,axiom,
! [C: produc2875793189560775939_set_a,A: produc2875793189560775939_set_a,B: produc2875793189560775939_set_a] :
( ( ord_le423894246401745751_set_a @ C @ A )
=> ( ord_le423894246401745751_set_a @ C @ ( sup_su7624641699453423151_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_463_sup_Ostrict__coboundedI1,axiom,
! [C: product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ C @ A )
=> ( ord_le1203424502768444845at_nat @ C @ ( sup_su4120719815643632853at_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_464_sup_Ostrict__coboundedI1,axiom,
! [C: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ C @ A )
=> ( ord_le6819997720685908915od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_465_sup_Ostrict__coboundedI1,axiom,
! [C: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ C @ A )
=> ( ord_less_set_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_466_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ C @ A )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_467_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_468_sup_Ostrict__order__iff,axiom,
( ord_le2894110756394707505_a_nat
= ( ^ [B3: produc5346009699553737693_a_nat,A4: produc5346009699553737693_a_nat] :
( ( A4
= ( sup_su871486172591609097_a_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_469_sup_Ostrict__order__iff,axiom,
( ord_le423894246401745751_set_a
= ( ^ [B3: produc2875793189560775939_set_a,A4: produc2875793189560775939_set_a] :
( ( A4
= ( sup_su7624641699453423151_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_470_sup_Ostrict__order__iff,axiom,
( ord_le1203424502768444845at_nat
= ( ^ [B3: product_prod_nat_nat,A4: product_prod_nat_nat] :
( ( A4
= ( sup_su4120719815643632853at_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_471_sup_Ostrict__order__iff,axiom,
( ord_le6819997720685908915od_a_a
= ( ^ [B3: set_Product_prod_a_a,A4: set_Product_prod_a_a] :
( ( A4
= ( sup_su3048258781599657691od_a_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_472_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_a
= ( ^ [B3: set_set_a,A4: set_set_a] :
( ( A4
= ( sup_sup_set_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_473_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( A4
= ( sup_sup_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_474_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_475_sup_Ostrict__boundedE,axiom,
! [B: produc5346009699553737693_a_nat,C: produc5346009699553737693_a_nat,A: produc5346009699553737693_a_nat] :
( ( ord_le2894110756394707505_a_nat @ ( sup_su871486172591609097_a_nat @ B @ C ) @ A )
=> ~ ( ( ord_le2894110756394707505_a_nat @ B @ A )
=> ~ ( ord_le2894110756394707505_a_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_476_sup_Ostrict__boundedE,axiom,
! [B: produc2875793189560775939_set_a,C: produc2875793189560775939_set_a,A: produc2875793189560775939_set_a] :
( ( ord_le423894246401745751_set_a @ ( sup_su7624641699453423151_set_a @ B @ C ) @ A )
=> ~ ( ( ord_le423894246401745751_set_a @ B @ A )
=> ~ ( ord_le423894246401745751_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_477_sup_Ostrict__boundedE,axiom,
! [B: product_prod_nat_nat,C: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ ( sup_su4120719815643632853at_nat @ B @ C ) @ A )
=> ~ ( ( ord_le1203424502768444845at_nat @ B @ A )
=> ~ ( ord_le1203424502768444845at_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_478_sup_Ostrict__boundedE,axiom,
! [B: set_Product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ ( sup_su3048258781599657691od_a_a @ B @ C ) @ A )
=> ~ ( ( ord_le6819997720685908915od_a_a @ B @ A )
=> ~ ( ord_le6819997720685908915od_a_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_479_sup_Ostrict__boundedE,axiom,
! [B: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ ( sup_sup_set_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_set_set_a @ B @ A )
=> ~ ( ord_less_set_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_480_sup_Ostrict__boundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_481_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_482_sup_Oabsorb4,axiom,
! [A: produc5346009699553737693_a_nat,B: produc5346009699553737693_a_nat] :
( ( ord_le2894110756394707505_a_nat @ A @ B )
=> ( ( sup_su871486172591609097_a_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_483_sup_Oabsorb4,axiom,
! [A: produc2875793189560775939_set_a,B: produc2875793189560775939_set_a] :
( ( ord_le423894246401745751_set_a @ A @ B )
=> ( ( sup_su7624641699453423151_set_a @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_484_sup_Oabsorb4,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ A @ B )
=> ( ( sup_su4120719815643632853at_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_485_sup_Oabsorb4,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A @ B )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_486_sup_Oabsorb4,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( sup_sup_set_set_a @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_487_sup_Oabsorb4,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_488_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_489_sup_Oabsorb3,axiom,
! [B: produc5346009699553737693_a_nat,A: produc5346009699553737693_a_nat] :
( ( ord_le2894110756394707505_a_nat @ B @ A )
=> ( ( sup_su871486172591609097_a_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_490_sup_Oabsorb3,axiom,
! [B: produc2875793189560775939_set_a,A: produc2875793189560775939_set_a] :
( ( ord_le423894246401745751_set_a @ B @ A )
=> ( ( sup_su7624641699453423151_set_a @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_491_sup_Oabsorb3,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ B @ A )
=> ( ( sup_su4120719815643632853at_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_492_sup_Oabsorb3,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ B @ A )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_493_sup_Oabsorb3,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ B @ A )
=> ( ( sup_sup_set_set_a @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_494_sup_Oabsorb3,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_495_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_496_less__supI2,axiom,
! [X: produc5346009699553737693_a_nat,B: produc5346009699553737693_a_nat,A: produc5346009699553737693_a_nat] :
( ( ord_le2894110756394707505_a_nat @ X @ B )
=> ( ord_le2894110756394707505_a_nat @ X @ ( sup_su871486172591609097_a_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_497_less__supI2,axiom,
! [X: produc2875793189560775939_set_a,B: produc2875793189560775939_set_a,A: produc2875793189560775939_set_a] :
( ( ord_le423894246401745751_set_a @ X @ B )
=> ( ord_le423894246401745751_set_a @ X @ ( sup_su7624641699453423151_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_498_less__supI2,axiom,
! [X: product_prod_nat_nat,B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X @ B )
=> ( ord_le1203424502768444845at_nat @ X @ ( sup_su4120719815643632853at_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_499_less__supI2,axiom,
! [X: set_Product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X @ B )
=> ( ord_le6819997720685908915od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_500_less__supI2,axiom,
! [X: set_set_a,B: set_set_a,A: set_set_a] :
( ( ord_less_set_set_a @ X @ B )
=> ( ord_less_set_set_a @ X @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_501_less__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ X @ B )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_502_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_503_less__supI1,axiom,
! [X: produc5346009699553737693_a_nat,A: produc5346009699553737693_a_nat,B: produc5346009699553737693_a_nat] :
( ( ord_le2894110756394707505_a_nat @ X @ A )
=> ( ord_le2894110756394707505_a_nat @ X @ ( sup_su871486172591609097_a_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_504_less__supI1,axiom,
! [X: produc2875793189560775939_set_a,A: produc2875793189560775939_set_a,B: produc2875793189560775939_set_a] :
( ( ord_le423894246401745751_set_a @ X @ A )
=> ( ord_le423894246401745751_set_a @ X @ ( sup_su7624641699453423151_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_505_less__supI1,axiom,
! [X: product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X @ A )
=> ( ord_le1203424502768444845at_nat @ X @ ( sup_su4120719815643632853at_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_506_less__supI1,axiom,
! [X: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ X @ A )
=> ( ord_le6819997720685908915od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_507_less__supI1,axiom,
! [X: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ X @ A )
=> ( ord_less_set_set_a @ X @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_508_less__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ X @ A )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_509_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_510_order__antisym__conv,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_511_order__antisym__conv,axiom,
! [Y: set_real_a,X: set_real_a] :
( ( ord_le5743406823621094409real_a @ Y @ X )
=> ( ( ord_le5743406823621094409real_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_512_order__antisym__conv,axiom,
! [Y: $o > quasi_borel_a,X: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ Y @ X )
=> ( ( ord_le1636368021143733457orel_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_513_order__antisym__conv,axiom,
! [Y: $o > set_a,X: $o > set_a] :
( ( ord_less_eq_o_set_a @ Y @ X )
=> ( ( ord_less_eq_o_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_514_order__antisym__conv,axiom,
! [Y: $o > nat,X: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X )
=> ( ( ord_less_eq_o_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_515_order__antisym__conv,axiom,
! [Y: quasi_borel_a,X: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ Y @ X )
=> ( ( ord_le1843388692487544644orel_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_516_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_517_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_518_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_519_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_520_ord__le__eq__subst,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > nat,C: nat] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_521_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_522_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_523_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_524_ord__le__eq__subst,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_525_ord__le__eq__subst,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > set_a,C: set_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_526_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_527_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_528_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_529_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_530_ord__eq__le__subst,axiom,
! [A: nat,F: quasi_borel_a > nat,B: quasi_borel_a,C: quasi_borel_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_531_ord__eq__le__subst,axiom,
! [A: quasi_borel_a,F: nat > quasi_borel_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_532_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_533_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_534_ord__eq__le__subst,axiom,
! [A: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_535_ord__eq__le__subst,axiom,
! [A: set_a,F: quasi_borel_a > set_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_536_ord__eq__le__subst,axiom,
! [A: quasi_borel_a,F: set_a > quasi_borel_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_537_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_538_ord__eq__le__subst,axiom,
! [A: set_set_a,F: nat > set_set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_539_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_540_order__eq__refl,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( X = Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_541_order__eq__refl,axiom,
! [X: set_real_a,Y: set_real_a] :
( ( X = Y )
=> ( ord_le5743406823621094409real_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_542_order__eq__refl,axiom,
! [X: $o > quasi_borel_a,Y: $o > quasi_borel_a] :
( ( X = Y )
=> ( ord_le1636368021143733457orel_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_543_order__eq__refl,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( X = Y )
=> ( ord_less_eq_o_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_544_order__eq__refl,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( X = Y )
=> ( ord_less_eq_o_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_545_order__eq__refl,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( X = Y )
=> ( ord_le1843388692487544644orel_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_546_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_547_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_548_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_549_order__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > nat,C: nat] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_550_order__subst2,axiom,
! [A: nat,B: nat,F: nat > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_551_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_552_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_553_order__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_554_order__subst2,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,F: quasi_borel_a > set_a,C: set_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_555_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_556_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_557_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_558_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_559_order__subst1,axiom,
! [A: quasi_borel_a,F: nat > quasi_borel_a,B: nat,C: nat] :
( ( ord_le1843388692487544644orel_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_560_order__subst1,axiom,
! [A: nat,F: quasi_borel_a > nat,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_561_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_562_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_563_order__subst1,axiom,
! [A: quasi_borel_a,F: quasi_borel_a > quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_564_order__subst1,axiom,
! [A: quasi_borel_a,F: set_a > quasi_borel_a,B: set_a,C: set_a] :
( ( ord_le1843388692487544644orel_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1843388692487544644orel_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_565_order__subst1,axiom,
! [A: set_a,F: quasi_borel_a > set_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ! [X2: quasi_borel_a,Y3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_566_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_567_order__subst1,axiom,
! [A: nat,F: set_set_a > nat,B: set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_568_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_a,Z2: set_set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_569_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_real_a,Z2: set_real_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_real_a,B3: set_real_a] :
( ( ord_le5743406823621094409real_a @ A4 @ B3 )
& ( ord_le5743406823621094409real_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_570_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: $o > quasi_borel_a,Z2: $o > quasi_borel_a] : ( Y5 = Z2 ) )
= ( ^ [A4: $o > quasi_borel_a,B3: $o > quasi_borel_a] :
( ( ord_le1636368021143733457orel_a @ A4 @ B3 )
& ( ord_le1636368021143733457orel_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_571_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: $o > set_a,Z2: $o > set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ A4 @ B3 )
& ( ord_less_eq_o_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_572_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: $o > nat,Z2: $o > nat] : ( Y5 = Z2 ) )
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A4 @ B3 )
& ( ord_less_eq_o_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_573_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: quasi_borel_a,Z2: quasi_borel_a] : ( Y5 = Z2 ) )
= ( ^ [A4: quasi_borel_a,B3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A4 @ B3 )
& ( ord_le1843388692487544644orel_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_574_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_575_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_576_le__fun__def,axiom,
( ord_le1636368021143733457orel_a
= ( ^ [F2: $o > quasi_borel_a,G: $o > quasi_borel_a] :
! [X4: $o] : ( ord_le1843388692487544644orel_a @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_577_le__fun__def,axiom,
( ord_less_eq_o_set_a
= ( ^ [F2: $o > set_a,G: $o > set_a] :
! [X4: $o] : ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_578_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G: $o > nat] :
! [X4: $o] : ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_579_le__funI,axiom,
! [F: $o > quasi_borel_a,G2: $o > quasi_borel_a] :
( ! [X2: $o] : ( ord_le1843388692487544644orel_a @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_le1636368021143733457orel_a @ F @ G2 ) ) ).
% le_funI
thf(fact_580_le__funI,axiom,
! [F: $o > nat,G2: $o > nat] :
( ! [X2: $o] : ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_o_nat @ F @ G2 ) ) ).
% le_funI
thf(fact_581_le__funI,axiom,
! [F: $o > set_a,G2: $o > set_a] :
( ! [X2: $o] : ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_o_set_a @ F @ G2 ) ) ).
% le_funI
thf(fact_582_le__funE,axiom,
! [F: $o > quasi_borel_a,G2: $o > quasi_borel_a,X: $o] :
( ( ord_le1636368021143733457orel_a @ F @ G2 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_583_le__funE,axiom,
! [F: $o > set_a,G2: $o > set_a,X: $o] :
( ( ord_less_eq_o_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_584_le__funE,axiom,
! [F: $o > nat,G2: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_585_le__funD,axiom,
! [F: $o > nat,G2: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_586_antisym,axiom,
! [A: quasi_borel_a,B: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_587_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_588_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_589_dual__order_Otrans,axiom,
! [B: quasi_borel_a,A: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ B @ A )
=> ( ( ord_le1843388692487544644orel_a @ C @ B )
=> ( ord_le1843388692487544644orel_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_590_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_591_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_592_dual__order_Oantisym,axiom,
! [B: quasi_borel_a,A: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ B @ A )
=> ( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_593_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_594_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_595_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: quasi_borel_a,Z2: quasi_borel_a] : ( Y5 = Z2 ) )
= ( ^ [A4: quasi_borel_a,B3: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ B3 @ A4 )
& ( ord_le1843388692487544644orel_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_596_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_597_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_598_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_599_order__trans,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a,Z: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X @ Y )
=> ( ( ord_le1843388692487544644orel_a @ Y @ Z )
=> ( ord_le1843388692487544644orel_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_600_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_601_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_602_order_Otrans,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ord_le1843388692487544644orel_a @ A @ C ) ) ) ).
% order.trans
thf(fact_603_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_604_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_605_order__antisym,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X @ Y )
=> ( ( ord_le1843388692487544644orel_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_606_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_607_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_608_ord__le__eq__trans,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ B )
=> ( ( B = C )
=> ( ord_le1843388692487544644orel_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_609_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_610_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_611_ord__eq__le__trans,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a] :
( ( A = B )
=> ( ( ord_le1843388692487544644orel_a @ B @ C )
=> ( ord_le1843388692487544644orel_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_612_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_613_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_614_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: quasi_borel_a,Z2: quasi_borel_a] : ( Y5 = Z2 ) )
= ( ^ [X4: quasi_borel_a,Y4: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X4 @ Y4 )
& ( ord_le1843388692487544644orel_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_615_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_616_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_617_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_618_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_619_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_620_sup__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_621_sup_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_622_sup_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C ) )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_623_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_624_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X4: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_625_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_626_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A4: nat,B3: nat] : ( sup_sup_nat @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_627_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_628_sup__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_629_sup_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_630_sup_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_631_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_632_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X4: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_633_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_634_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_635_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_636_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_637_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_638_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_639_less__eq__quasi__borel_Ointros_I1_J,axiom,
! [X5: quasi_borel_a,Y6: quasi_borel_a] :
( ( ord_less_set_a @ ( qbs_space_a @ X5 ) @ ( qbs_space_a @ Y6 ) )
=> ( ord_le1843388692487544644orel_a @ X5 @ Y6 ) ) ).
% less_eq_quasi_borel.intros(1)
thf(fact_640_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_641_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_642_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_643_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_644_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_645_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_646_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_647_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_648_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_649_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_650_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_651_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_652_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_653_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_654_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_655_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_656_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_657_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_658_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_659_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_660_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_661_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_662_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_663_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_664_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_665_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_666_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_667_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_668_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_669_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_670_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_671_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_672_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( sup_sup_nat @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_673_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_674_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_675_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_676_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_677_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_678_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_679_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_680_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_681_le__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_682_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_683_le__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_684_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_685_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_686_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_687_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_688_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_689_le__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_690_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_691_le__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_692_sup__quasi__borel__def,axiom,
( sup_su6298519176299948920orel_a
= ( ^ [X6: quasi_borel_a,Y7: quasi_borel_a] :
( if_quasi_borel_a
@ ( ( qbs_space_a @ X6 )
= ( qbs_space_a @ Y7 ) )
@ ( inf_quasi_borel_a @ X6 @ Y7 )
@ ( if_quasi_borel_a @ ( ord_less_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) @ Y7 @ ( if_quasi_borel_a @ ( ord_less_set_a @ ( qbs_space_a @ Y7 ) @ ( qbs_space_a @ X6 ) ) @ X6 @ ( max_quasi_borel_a @ ( sup_sup_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) ) ) ) ) ) ) ).
% sup_quasi_borel_def
thf(fact_693_psubsetI,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( A3 != B4 )
=> ( ord_less_set_a @ A3 @ B4 ) ) ) ).
% psubsetI
thf(fact_694_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A @ X7 )
& ( ord_less_nat @ X7 @ C2 ) )
=> ( P @ X7 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_695_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_696_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_697_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_698_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_699_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_700_max__qbs__space,axiom,
! [X5: set_a] :
( ( qbs_space_a @ ( max_quasi_borel_a @ X5 ) )
= X5 ) ).
% max_qbs_space
thf(fact_701_QuasiBorel_Osup__quasi__borel__def,axiom,
( sup_su6298519176299948920orel_a
= ( ^ [X6: quasi_borel_a,Y7: quasi_borel_a] :
( if_quasi_borel_a
@ ( ( qbs_space_a @ X6 )
= ( qbs_space_a @ Y7 ) )
@ ( inf_quasi_borel_a @ X6 @ Y7 )
@ ( if_quasi_borel_a @ ( ord_less_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) @ Y7 @ ( if_quasi_borel_a @ ( ord_less_set_a @ ( qbs_space_a @ Y7 ) @ ( qbs_space_a @ X6 ) ) @ X6 @ ( max_quasi_borel_a @ ( sup_sup_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) ) ) ) ) ) ) ).
% QuasiBorel.sup_quasi_borel_def
thf(fact_702_le__quasi__borel__iff,axiom,
( ord_le1843388692487544644orel_a
= ( ^ [X6: quasi_borel_a,Y7: quasi_borel_a] :
( ( ( ( qbs_space_a @ X6 )
= ( qbs_space_a @ Y7 ) )
=> ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ Y7 ) @ ( qbs_Mx_a @ X6 ) ) )
& ( ( ( qbs_space_a @ X6 )
!= ( qbs_space_a @ Y7 ) )
=> ( ord_less_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) ) ) ) ) ).
% le_quasi_borel_iff
thf(fact_703_subsetI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B4 ) )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_704_subset__antisym,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_705_UnCI,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_706_Un__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_707_Un__subset__iff,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C3 )
= ( ( ord_less_eq_set_a @ A3 @ C3 )
& ( ord_less_eq_set_a @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_708_UnE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_709_UnI1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_710_UnI2,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_711_bex__Un,axiom,
! [A3: set_a,B4: set_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B4 ) )
& ( P @ X4 ) ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A3 )
& ( P @ X4 ) )
| ? [X4: a] :
( ( member_a @ X4 @ B4 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_712_Un__mono,axiom,
! [A3: set_a,C3: set_a,B4: set_a,D3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C3 )
=> ( ( ord_less_eq_set_a @ B4 @ D3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ ( sup_sup_set_a @ C3 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_713_ball__Un,axiom,
! [A3: set_a,B4: set_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ X4 ) )
& ! [X4: a] :
( ( member_a @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_714_in__mono,axiom,
! [A3: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_715_subsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_716_Un__assoc,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C3 )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C3 ) ) ) ).
% Un_assoc
thf(fact_717_Un__least,axiom,
! [A3: set_a,C3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ C3 )
=> ( ( ord_less_eq_set_a @ B4 @ C3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C3 ) ) ) ).
% Un_least
thf(fact_718_Un__absorb,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_719_Un__upper1,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_upper1
thf(fact_720_Un__upper2,axiom,
! [B4: set_a,A3: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_upper2
thf(fact_721_equalityE,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_722_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_723_Un__absorb1,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( sup_sup_set_a @ A3 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_724_Un__absorb2,axiom,
! [B4: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( sup_sup_set_a @ A3 @ B4 )
= A3 ) ) ).
% Un_absorb2
thf(fact_725_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_726_equalityD1,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_727_equalityD2,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_728_subset__UnE,axiom,
! [C3: set_a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A3 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B4 )
=> ( C3
!= ( sup_sup_set_a @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_729_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_730_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_731_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_732_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( sup_sup_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_733_subset__trans,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C3 )
=> ( ord_less_eq_set_a @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_734_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_735_Un__left__absorb,axiom,
! [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_736_Un__left__commute,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C3 ) )
= ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_737_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_738_qbs__space__eq__Mx,axiom,
! [X5: quasi_borel_a,Y6: quasi_borel_a] :
( ( ( qbs_Mx_a @ X5 )
= ( qbs_Mx_a @ Y6 ) )
=> ( ( qbs_space_a @ X5 )
= ( qbs_space_a @ Y6 ) ) ) ).
% qbs_space_eq_Mx
thf(fact_739_qbs__Mx__to__X_I2_J,axiom,
! [Alpha: real > a,X5: quasi_borel_a,R: real] :
( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X5 ) )
=> ( member_a @ ( Alpha @ R ) @ ( qbs_space_a @ X5 ) ) ) ).
% qbs_Mx_to_X(2)
thf(fact_740_less__quasi__borel__def,axiom,
( ord_le3788981553861827664orel_a
= ( ^ [X6: quasi_borel_a,Y7: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X6 @ Y7 )
& ~ ( ord_le1843388692487544644orel_a @ Y7 @ X6 ) ) ) ) ).
% less_quasi_borel_def
thf(fact_741_less__eq__quasi__borel_Ointros_I2_J,axiom,
! [X5: quasi_borel_a,Y6: quasi_borel_a] :
( ( ( qbs_space_a @ X5 )
= ( qbs_space_a @ Y6 ) )
=> ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ Y6 ) @ ( qbs_Mx_a @ X5 ) )
=> ( ord_le1843388692487544644orel_a @ X5 @ Y6 ) ) ) ).
% less_eq_quasi_borel.intros(2)
thf(fact_742_verit__comp__simplify1_I2_J,axiom,
! [A: quasi_borel_a] : ( ord_le1843388692487544644orel_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_743_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_744_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_745_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_746_verit__comp__simplify1_I1_J,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_747_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_748_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_749_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_750_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_751_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_752_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_753_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_754_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_755_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_756_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_757_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_758_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_759_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_760_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_761_subset__psubset__trans,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C3 )
=> ( ord_less_set_a @ A3 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_762_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_763_psubset__subset__trans,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C3 )
=> ( ord_less_set_a @ A3 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_764_psubset__imp__subset,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_765_psubset__trans,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C3 )
=> ( ord_less_set_a @ A3 @ C3 ) ) ) ).
% psubset_trans
thf(fact_766_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_767_psubsetE,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% psubsetE
thf(fact_768_psubsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_769_less__eq__quasi__borel_Ocases,axiom,
! [A1: quasi_borel_a,A22: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A1 @ A22 )
=> ( ~ ( ord_less_set_a @ ( qbs_space_a @ A1 ) @ ( qbs_space_a @ A22 ) )
=> ~ ( ( ( qbs_space_a @ A1 )
= ( qbs_space_a @ A22 ) )
=> ~ ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ A22 ) @ ( qbs_Mx_a @ A1 ) ) ) ) ) ).
% less_eq_quasi_borel.cases
thf(fact_770_less__eq__quasi__borel_Osimps,axiom,
( ord_le1843388692487544644orel_a
= ( ^ [A12: quasi_borel_a,A23: quasi_borel_a] :
( ? [X6: quasi_borel_a,Y7: quasi_borel_a] :
( ( A12 = X6 )
& ( A23 = Y7 )
& ( ord_less_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ Y7 ) ) )
| ? [X6: quasi_borel_a,Y7: quasi_borel_a] :
( ( A12 = X6 )
& ( A23 = Y7 )
& ( ( qbs_space_a @ X6 )
= ( qbs_space_a @ Y7 ) )
& ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ Y7 ) @ ( qbs_Mx_a @ X6 ) ) ) ) ) ) ).
% less_eq_quasi_borel.simps
thf(fact_771_inf__qbs__space,axiom,
! [X5: quasi_borel_a,X8: quasi_borel_a] :
( ( qbs_space_a @ ( inf_quasi_borel_a @ X5 @ X8 ) )
= ( inf_inf_set_a @ ( qbs_space_a @ X5 ) @ ( qbs_space_a @ X8 ) ) ) ).
% inf_qbs_space
thf(fact_772_GreatestI2__order,axiom,
! [P: quasi_borel_a > $o,X: quasi_borel_a,Q: quasi_borel_a > $o] :
( ( P @ X )
=> ( ! [Y3: quasi_borel_a] :
( ( P @ Y3 )
=> ( ord_le1843388692487544644orel_a @ Y3 @ X ) )
=> ( ! [X2: quasi_borel_a] :
( ( P @ X2 )
=> ( ! [Y2: quasi_borel_a] :
( ( P @ Y2 )
=> ( ord_le1843388692487544644orel_a @ Y2 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_2939245480137816317orel_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_773_GreatestI2__order,axiom,
! [P: set_a > $o,X: set_a,Q: set_a > $o] :
( ( P @ X )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ! [X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_774_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_775_Greatest__equality,axiom,
! [P: quasi_borel_a > $o,X: quasi_borel_a] :
( ( P @ X )
=> ( ! [Y3: quasi_borel_a] :
( ( P @ Y3 )
=> ( ord_le1843388692487544644orel_a @ Y3 @ X ) )
=> ( ( order_2939245480137816317orel_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_776_Greatest__equality,axiom,
! [P: set_a > $o,X: set_a] :
( ( P @ X )
=> ( ! [Y3: set_a] :
( ( P @ Y3 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ( order_Greatest_set_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_777_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_778_inf__qbs__Mx,axiom,
! [X5: quasi_borel_a,X8: quasi_borel_a] :
( ( qbs_Mx_a @ ( inf_quasi_borel_a @ X5 @ X8 ) )
= ( inf_inf_set_real_a @ ( qbs_Mx_a @ X5 ) @ ( qbs_Mx_a @ X8 ) ) ) ).
% inf_qbs_Mx
thf(fact_779_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_a,K: set_a,A: set_a,B: set_a] :
( ( A3
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A3 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_780_boolean__algebra__cancel_Osup1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( sup_sup_nat @ K @ A ) )
=> ( ( sup_sup_nat @ A3 @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_781_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_a,K: set_a,B: set_a,A: set_a] :
( ( B4
= ( sup_sup_set_a @ K @ B ) )
=> ( ( sup_sup_set_a @ A @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_782_boolean__algebra__cancel_Osup2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( sup_sup_nat @ K @ B ) )
=> ( ( sup_sup_nat @ A @ B4 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_783_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_784_inf_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_785_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_786_inf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_787_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_788_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_789_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_790_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_791_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_792_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_793_Int__subset__iff,axiom,
! [C3: set_a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A3 @ B4 ) )
= ( ( ord_less_eq_set_a @ C3 @ A3 )
& ( ord_less_eq_set_a @ C3 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_794_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_795_sup__inf__absorb,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_796_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_797_inf__sup__absorb,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_798_Un__Int__eq_I1_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_799_Un__Int__eq_I2_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_800_Un__Int__eq_I3_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_801_Un__Int__eq_I4_J,axiom,
! [T3: set_a,S: set_a] :
( ( inf_inf_set_a @ T3 @ ( sup_sup_set_a @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_802_Int__Un__eq_I1_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_803_Int__Un__eq_I2_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_804_Int__Un__eq_I3_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_805_Int__Un__eq_I4_J,axiom,
! [T3: set_a,S: set_a] :
( ( sup_sup_set_a @ T3 @ ( inf_inf_set_a @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_806_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_807_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_808_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_809_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_810_boolean__algebra__cancel_Oinf1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( inf_inf_nat @ K @ A ) )
=> ( ( inf_inf_nat @ A3 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_811_boolean__algebra__cancel_Oinf2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( inf_inf_nat @ K @ B ) )
=> ( ( inf_inf_nat @ A @ B4 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_812_inf__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_813_inf_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_814_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X4: nat,Y4: nat] : ( inf_inf_nat @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_815_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A4: nat,B3: nat] : ( inf_inf_nat @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_816_inf__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_817_inf_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_818_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X4: nat,Y4: nat] : ( inf_inf_nat @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_819_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_820_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_821_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_822_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_823_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_824_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_825_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_826_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_827_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_828_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_829_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( inf_inf_set_a @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_830_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_831_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_832_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_833_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_834_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_835_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_836_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_837_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_838_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_839_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_840_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_841_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_842_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_843_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_844_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_845_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_846_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_847_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_848_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_849_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_850_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( inf_inf_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_851_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_852_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_853_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_854_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_855_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_856_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_857_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_858_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_859_le__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_860_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_861_le__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_862_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_863_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_864_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_865_le__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_866_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_867_le__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_868_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_869_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_870_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_871_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_872_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_873_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_874_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_875_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_876_less__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_set_a @ A @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_877_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_878_less__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_set_a @ B @ X )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_879_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_880_inf_Oabsorb3,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_881_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_882_inf_Oabsorb4,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_883_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_884_inf_Ostrict__boundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_885_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_886_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_887_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_888_inf_Ostrict__coboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_set_a @ A @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_889_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_890_inf_Ostrict__coboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_891_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_892_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_893_sup__inf__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
= ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_894_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_895_sup__inf__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_896_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_897_inf__sup__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
= ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_898_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_899_inf__sup__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_900_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_901_distrib__imp2,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X2: nat,Y3: nat,Z3: nat] :
( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z3 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ ( sup_sup_nat @ X2 @ Z3 ) ) )
=> ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_902_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_903_distrib__imp1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X2: nat,Y3: nat,Z3: nat] :
( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z3 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ ( inf_inf_nat @ X2 @ Z3 ) ) )
=> ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_904_Int__Collect__mono,axiom,
! [A3: set_a,B4: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B4 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_905_Int__greatest,axiom,
! [C3: set_a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( ( ord_less_eq_set_a @ C3 @ B4 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A3 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_906_Int__absorb2,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( inf_inf_set_a @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_907_Int__absorb1,axiom,
! [B4: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_908_Int__lower2,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_909_Int__lower1,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_910_Int__mono,axiom,
! [A3: set_a,C3: set_a,B4: set_a,D3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C3 )
=> ( ( ord_less_eq_set_a @ B4 @ D3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ ( inf_inf_set_a @ C3 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_911_Un__Int__distrib2,axiom,
! [B4: set_a,C3: set_a,A3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B4 @ C3 ) @ A3 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B4 @ A3 ) @ ( sup_sup_set_a @ C3 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_912_Int__Un__distrib2,axiom,
! [B4: set_a,C3: set_a,A3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B4 @ C3 ) @ A3 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B4 @ A3 ) @ ( inf_inf_set_a @ C3 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_913_Un__Int__distrib,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( sup_sup_set_a @ A3 @ ( inf_inf_set_a @ B4 @ C3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ ( sup_sup_set_a @ A3 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_914_Int__Un__distrib,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( inf_inf_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ ( inf_inf_set_a @ A3 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_915_Un__Int__crazy,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ ( inf_inf_set_a @ B4 @ C3 ) ) @ ( inf_inf_set_a @ C3 @ A3 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ ( sup_sup_set_a @ B4 @ C3 ) ) @ ( sup_sup_set_a @ C3 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_916_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_917_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_918_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_919_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_920_Un__Int__assoc__eq,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ C3 )
= ( inf_inf_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C3 ) ) )
= ( ord_less_eq_set_a @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_921_sub__qbs__space,axiom,
! [X5: quasi_borel_a,U: set_a] :
( ( qbs_space_a @ ( sub_qbs_a @ X5 @ U ) )
= ( inf_inf_set_a @ ( qbs_space_a @ X5 ) @ U ) ) ).
% sub_qbs_space
thf(fact_922_inf__quasi__borel__def,axiom,
( inf_quasi_borel_a
= ( ^ [X6: quasi_borel_a,X9: quasi_borel_a] : ( quasi_2002468295286565184orel_a @ ( produc8198169892514940194real_a @ ( inf_inf_set_a @ ( qbs_space_a @ X6 ) @ ( qbs_space_a @ X9 ) ) @ ( inf_inf_set_real_a @ ( qbs_Mx_a @ X6 ) @ ( qbs_Mx_a @ X9 ) ) ) ) ) ) ).
% inf_quasi_borel_def
thf(fact_923_inf__quasi__borel__correct,axiom,
! [X5: quasi_borel_a,X8: quasi_borel_a] :
( ( quasi_4958298574314430517orel_a @ ( inf_quasi_borel_a @ X5 @ X8 ) )
= ( produc8198169892514940194real_a @ ( inf_inf_set_a @ ( qbs_space_a @ X5 ) @ ( qbs_space_a @ X8 ) ) @ ( inf_inf_set_real_a @ ( qbs_Mx_a @ X5 ) @ ( qbs_Mx_a @ X8 ) ) ) ) ).
% inf_quasi_borel_correct
thf(fact_924_inf_Osemilattice__order__axioms,axiom,
semila1248733672344298208er_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% inf.semilattice_order_axioms
thf(fact_925_inf_Osemilattice__order__axioms,axiom,
semila4706084620769370446_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% inf.semilattice_order_axioms
thf(fact_926_sup__Pair__Pair,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( sup_su2807591092881064699_set_a @ ( produc9088192753505129239_set_a @ A @ B ) @ ( produc9088192753505129239_set_a @ C @ D ) )
= ( produc9088192753505129239_set_a @ ( sup_sup_set_a @ A @ C ) @ ( sup_sup_set_a @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_927_sup__Pair__Pair,axiom,
! [A: set_a,B: nat,C: set_a,D: nat] :
( ( sup_su871486172591609097_a_nat @ ( produc8600956749907467479_a_nat @ A @ B ) @ ( produc8600956749907467479_a_nat @ C @ D ) )
= ( produc8600956749907467479_a_nat @ ( sup_sup_set_a @ A @ C ) @ ( sup_sup_nat @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_928_sup__Pair__Pair,axiom,
! [A: nat,B: set_a,C: nat,D: set_a] :
( ( sup_su7624641699453423151_set_a @ ( produc4534374738092020277_set_a @ A @ B ) @ ( produc4534374738092020277_set_a @ C @ D ) )
= ( produc4534374738092020277_set_a @ ( sup_sup_nat @ A @ C ) @ ( sup_sup_set_a @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_929_sup__Pair__Pair,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( sup_su4120719815643632853at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( sup_sup_nat @ A @ C ) @ ( sup_sup_nat @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_930_inf__Pair__Pair,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( inf_in4240866628268492783at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( inf_inf_nat @ A @ C ) @ ( inf_inf_nat @ B @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_931_Pair__le,axiom,
! [A: quasi_borel_a,B: quasi_borel_a,C: quasi_borel_a,D: quasi_borel_a] :
( ( ord_le5941759531670611655orel_a @ ( produc6592225216727209751orel_a @ A @ B ) @ ( produc6592225216727209751orel_a @ C @ D ) )
= ( ( ord_le1843388692487544644orel_a @ A @ C )
& ( ord_le1843388692487544644orel_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_932_Pair__le,axiom,
! [A: quasi_borel_a,B: nat,C: quasi_borel_a,D: nat] :
( ( ord_le1479402446426951683_a_nat @ ( produc831911609708260381_a_nat @ A @ B ) @ ( produc831911609708260381_a_nat @ C @ D ) )
= ( ( ord_le1843388692487544644orel_a @ A @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_933_Pair__le,axiom,
! [A: quasi_borel_a,B: set_a,C: quasi_borel_a,D: set_a] :
( ( ord_le5001701540438981249_set_a @ ( produc6811942899106390353_set_a @ A @ B ) @ ( produc6811942899106390353_set_a @ C @ D ) )
= ( ( ord_le1843388692487544644orel_a @ A @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_934_Pair__le,axiom,
! [A: nat,B: quasi_borel_a,C: nat,D: quasi_borel_a] :
( ( ord_le185366790440233129orel_a @ ( produc5438933203794904571orel_a @ A @ B ) @ ( produc5438933203794904571orel_a @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ C )
& ( ord_le1843388692487544644orel_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_935_Pair__le,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_936_Pair__le,axiom,
! [A: nat,B: set_a,C: nat,D: set_a] :
( ( ord_le6607365053213775203_set_a @ ( produc4534374738092020277_set_a @ A @ B ) @ ( produc4534374738092020277_set_a @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_937_Pair__le,axiom,
! [A: set_a,B: quasi_borel_a,C: set_a,D: quasi_borel_a] :
( ( ord_le7723843944399864333orel_a @ ( produc7808136018108345821orel_a @ A @ B ) @ ( produc7808136018108345821orel_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_le1843388692487544644orel_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_938_Pair__le,axiom,
! [A: set_a,B: nat,C: set_a,D: nat] :
( ( ord_le9077581563206736957_a_nat @ ( produc8600956749907467479_a_nat @ A @ B ) @ ( produc8600956749907467479_a_nat @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_nat @ B @ D ) ) ) ).
% Pair_le
thf(fact_939_Pair__le,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le7186201010124320967_set_a @ ( produc9088192753505129239_set_a @ A @ B ) @ ( produc9088192753505129239_set_a @ C @ D ) )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_set_a @ B @ D ) ) ) ).
% Pair_le
thf(fact_940_Pair__mono,axiom,
! [X: quasi_borel_a,X10: quasi_borel_a,Y: quasi_borel_a,Y8: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X @ X10 )
=> ( ( ord_le1843388692487544644orel_a @ Y @ Y8 )
=> ( ord_le5941759531670611655orel_a @ ( produc6592225216727209751orel_a @ X @ Y ) @ ( produc6592225216727209751orel_a @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_941_Pair__mono,axiom,
! [X: quasi_borel_a,X10: quasi_borel_a,Y: nat,Y8: nat] :
( ( ord_le1843388692487544644orel_a @ X @ X10 )
=> ( ( ord_less_eq_nat @ Y @ Y8 )
=> ( ord_le1479402446426951683_a_nat @ ( produc831911609708260381_a_nat @ X @ Y ) @ ( produc831911609708260381_a_nat @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_942_Pair__mono,axiom,
! [X: quasi_borel_a,X10: quasi_borel_a,Y: set_a,Y8: set_a] :
( ( ord_le1843388692487544644orel_a @ X @ X10 )
=> ( ( ord_less_eq_set_a @ Y @ Y8 )
=> ( ord_le5001701540438981249_set_a @ ( produc6811942899106390353_set_a @ X @ Y ) @ ( produc6811942899106390353_set_a @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_943_Pair__mono,axiom,
! [X: nat,X10: nat,Y: quasi_borel_a,Y8: quasi_borel_a] :
( ( ord_less_eq_nat @ X @ X10 )
=> ( ( ord_le1843388692487544644orel_a @ Y @ Y8 )
=> ( ord_le185366790440233129orel_a @ ( produc5438933203794904571orel_a @ X @ Y ) @ ( produc5438933203794904571orel_a @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_944_Pair__mono,axiom,
! [X: nat,X10: nat,Y: nat,Y8: nat] :
( ( ord_less_eq_nat @ X @ X10 )
=> ( ( ord_less_eq_nat @ Y @ Y8 )
=> ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_945_Pair__mono,axiom,
! [X: nat,X10: nat,Y: set_a,Y8: set_a] :
( ( ord_less_eq_nat @ X @ X10 )
=> ( ( ord_less_eq_set_a @ Y @ Y8 )
=> ( ord_le6607365053213775203_set_a @ ( produc4534374738092020277_set_a @ X @ Y ) @ ( produc4534374738092020277_set_a @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_946_Pair__mono,axiom,
! [X: set_a,X10: set_a,Y: quasi_borel_a,Y8: quasi_borel_a] :
( ( ord_less_eq_set_a @ X @ X10 )
=> ( ( ord_le1843388692487544644orel_a @ Y @ Y8 )
=> ( ord_le7723843944399864333orel_a @ ( produc7808136018108345821orel_a @ X @ Y ) @ ( produc7808136018108345821orel_a @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_947_Pair__mono,axiom,
! [X: set_a,X10: set_a,Y: nat,Y8: nat] :
( ( ord_less_eq_set_a @ X @ X10 )
=> ( ( ord_less_eq_nat @ Y @ Y8 )
=> ( ord_le9077581563206736957_a_nat @ ( produc8600956749907467479_a_nat @ X @ Y ) @ ( produc8600956749907467479_a_nat @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_948_Pair__mono,axiom,
! [X: set_a,X10: set_a,Y: set_a,Y8: set_a] :
( ( ord_less_eq_set_a @ X @ X10 )
=> ( ( ord_less_eq_set_a @ Y @ Y8 )
=> ( ord_le7186201010124320967_set_a @ ( produc9088192753505129239_set_a @ X @ Y ) @ ( produc9088192753505129239_set_a @ X10 @ Y8 ) ) ) ) ).
% Pair_mono
thf(fact_949_qbs__space__def,axiom,
( qbs_space_a
= ( ^ [X6: quasi_borel_a] : ( produc7517347076341637902real_a @ ( quasi_4958298574314430517orel_a @ X6 ) ) ) ) ).
% qbs_space_def
thf(fact_950_sup__prod__def,axiom,
( sup_su2807591092881064699_set_a
= ( ^ [X4: produc1703568184450464039_set_a,Y4: produc1703568184450464039_set_a] : ( produc9088192753505129239_set_a @ ( sup_sup_set_a @ ( produc9088895665703139587_set_a @ X4 ) @ ( produc9088895665703139587_set_a @ Y4 ) ) @ ( sup_sup_set_a @ ( produc1983107199584856133_set_a @ X4 ) @ ( produc1983107199584856133_set_a @ Y4 ) ) ) ) ) ).
% sup_prod_def
thf(fact_951_sup__prod__def,axiom,
( sup_su871486172591609097_a_nat
= ( ^ [X4: produc5346009699553737693_a_nat,Y4: produc5346009699553737693_a_nat] : ( produc8600956749907467479_a_nat @ ( sup_sup_set_a @ ( produc2726959211105985515_a_nat @ X4 ) @ ( produc2726959211105985515_a_nat @ Y4 ) ) @ ( sup_sup_nat @ ( produc1912305412385931305_a_nat @ X4 ) @ ( produc1912305412385931305_a_nat @ Y4 ) ) ) ) ) ).
% sup_prod_def
thf(fact_952_sup__prod__def,axiom,
( sup_su7624641699453423151_set_a
= ( ^ [X4: produc2875793189560775939_set_a,Y4: produc2875793189560775939_set_a] : ( produc4534374738092020277_set_a @ ( sup_sup_nat @ ( produc7883749236145314121_set_a @ X4 ) @ ( produc7883749236145314121_set_a @ Y4 ) ) @ ( sup_sup_set_a @ ( produc7069095437425259911_set_a @ X4 ) @ ( produc7069095437425259911_set_a @ Y4 ) ) ) ) ) ).
% sup_prod_def
thf(fact_953_sup__prod__def,axiom,
( sup_su4120719815643632853at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( sup_sup_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) ) @ ( sup_sup_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% sup_prod_def
thf(fact_954_inf__prod__def,axiom,
( inf_in4240866628268492783at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( inf_inf_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) ) @ ( inf_inf_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% inf_prod_def
thf(fact_955_less__eq__prod__def,axiom,
( ord_le5941759531670611655orel_a
= ( ^ [X4: produc149604174626918183orel_a,Y4: produc149604174626918183orel_a] :
( ( ord_le1843388692487544644orel_a @ ( produc2874185341511397635orel_a @ X4 ) @ ( produc2874185341511397635orel_a @ Y4 ) )
& ( ord_le1843388692487544644orel_a @ ( produc7830989722923372101orel_a @ X4 ) @ ( produc7830989722923372101orel_a @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_956_less__eq__prod__def,axiom,
( ord_le1479402446426951683_a_nat
= ( ^ [X4: produc1856989317385805731_a_nat,Y4: produc1856989317385805731_a_nat] :
( ( ord_le1843388692487544644orel_a @ ( produc4520212424918210865_a_nat @ X4 ) @ ( produc4520212424918210865_a_nat @ Y4 ) )
& ( ord_less_eq_nat @ ( produc5186514561800519279_a_nat @ X4 ) @ ( produc5186514561800519279_a_nat @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_957_less__eq__prod__def,axiom,
( ord_le5001701540438981249_set_a
= ( ^ [X4: produc2269578075601916641_set_a,Y4: produc2269578075601916641_set_a] :
( ( ord_le1843388692487544644orel_a @ ( produc5500841278744043837_set_a @ X4 ) @ ( produc5500841278744043837_set_a @ Y4 ) )
& ( ord_less_eq_set_a @ ( produc760345589714667391_set_a @ X4 ) @ ( produc760345589714667391_set_a @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_958_less__eq__prod__def,axiom,
( ord_le185366790440233129orel_a
= ( ^ [X4: produc562953661399087177orel_a,Y4: produc562953661399087177orel_a] :
( ( ord_less_eq_nat @ ( produc9127234019004855055orel_a @ X4 ) @ ( produc9127234019004855055orel_a @ Y4 ) )
& ( ord_le1843388692487544644orel_a @ ( produc570164119032387661orel_a @ X4 ) @ ( produc570164119032387661orel_a @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_959_less__eq__prod__def,axiom,
( ord_le8460144461188290721at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) )
& ( ord_less_eq_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_960_less__eq__prod__def,axiom,
( ord_le6607365053213775203_set_a
= ( ^ [X4: produc2875793189560775939_set_a,Y4: produc2875793189560775939_set_a] :
( ( ord_less_eq_nat @ ( produc7883749236145314121_set_a @ X4 ) @ ( produc7883749236145314121_set_a @ Y4 ) )
& ( ord_less_eq_set_a @ ( produc7069095437425259911_set_a @ X4 ) @ ( produc7069095437425259911_set_a @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_961_less__eq__prod__def,axiom,
( ord_le7723843944399864333orel_a
= ( ^ [X4: produc4991720479562799725orel_a,Y4: produc4991720479562799725orel_a] :
( ( ord_less_eq_set_a @ ( produc6497034397745999305orel_a @ X4 ) @ ( produc6497034397745999305orel_a @ Y4 ) )
& ( ord_le1843388692487544644orel_a @ ( produc1756538708716622859orel_a @ X4 ) @ ( produc1756538708716622859orel_a @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_962_less__eq__prod__def,axiom,
( ord_le9077581563206736957_a_nat
= ( ^ [X4: produc5346009699553737693_a_nat,Y4: produc5346009699553737693_a_nat] :
( ( ord_less_eq_set_a @ ( produc2726959211105985515_a_nat @ X4 ) @ ( produc2726959211105985515_a_nat @ Y4 ) )
& ( ord_less_eq_nat @ ( produc1912305412385931305_a_nat @ X4 ) @ ( produc1912305412385931305_a_nat @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_963_less__eq__prod__def,axiom,
( ord_le7186201010124320967_set_a
= ( ^ [X4: produc1703568184450464039_set_a,Y4: produc1703568184450464039_set_a] :
( ( ord_less_eq_set_a @ ( produc9088895665703139587_set_a @ X4 ) @ ( produc9088895665703139587_set_a @ Y4 ) )
& ( ord_less_eq_set_a @ ( produc1983107199584856133_set_a @ X4 ) @ ( produc1983107199584856133_set_a @ Y4 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_964_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C4: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ C4 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ C4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
| ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_965_inf_Osemilattice__axioms,axiom,
semilattice_nat @ inf_inf_nat ).
% inf.semilattice_axioms
thf(fact_966_sup_Osemilattice__axioms,axiom,
semilattice_set_a @ sup_sup_set_a ).
% sup.semilattice_axioms
thf(fact_967_sup_Osemilattice__axioms,axiom,
semilattice_nat @ sup_sup_nat ).
% sup.semilattice_axioms
thf(fact_968_increasingD,axiom,
! [M2: set_set_a,F: set_a > quasi_borel_a,X: set_a,Y: set_a] :
( ( measur6337775561432609113orel_a @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ( member_set_a @ X @ M2 )
=> ( ( member_set_a @ Y @ M2 )
=> ( ord_le1843388692487544644orel_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_969_increasingD,axiom,
! [M2: set_set_a,F: set_a > nat,X: set_a,Y: set_a] :
( ( measur8151441426001876059_a_nat @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ( member_set_a @ X @ M2 )
=> ( ( member_set_a @ Y @ M2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_970_increasingD,axiom,
! [M2: set_set_a,F: set_a > set_a,X: set_a,Y: set_a] :
( ( measur7842569353079325843_set_a @ M2 @ F )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ( member_set_a @ X @ M2 )
=> ( ( member_set_a @ Y @ M2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_971_increasing__def,axiom,
( measur6337775561432609113orel_a
= ( ^ [M3: set_set_a,Mu: set_a > quasi_borel_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ M3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_le1843388692487544644orel_a @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_972_increasing__def,axiom,
( measur8151441426001876059_a_nat
= ( ^ [M3: set_set_a,Mu: set_a > nat] :
! [X4: set_a] :
( ( member_set_a @ X4 @ M3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_973_increasing__def,axiom,
( measur7842569353079325843_set_a
= ( ^ [M3: set_set_a,Mu: set_a > set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ M3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ M3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( Mu @ X4 ) @ ( Mu @ Y4 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_974_bdd__above_Opreordering__bdd__axioms,axiom,
condit229665075952935227orel_a @ ord_le1843388692487544644orel_a @ ord_le3788981553861827664orel_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_975_bdd__above_Opreordering__bdd__axioms,axiom,
condit7935552474144124665dd_nat @ ord_less_eq_nat @ ord_less_nat ).
% bdd_above.preordering_bdd_axioms
thf(fact_976_bdd__above_Opreordering__bdd__axioms,axiom,
condit6315317455391067509_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_977_le__sup__lexord,axiom,
! [K: quasi_borel_a > quasi_borel_a,A3: quasi_borel_a,B4: quasi_borel_a,Ca: quasi_borel_a,C: quasi_borel_a,S2: quasi_borel_a] :
( ( ( ord_le3788981553861827664orel_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ B4 ) )
=> ( ( ( ord_le3788981553861827664orel_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ C ) )
=> ( ( ~ ( ord_le1843388692487544644orel_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_le1843388692487544644orel_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ S2 ) ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ ( measur7073608842307378588orel_a @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_978_le__sup__lexord,axiom,
! [K: quasi_borel_a > nat,A3: quasi_borel_a,B4: quasi_borel_a,Ca: quasi_borel_a,C: quasi_borel_a,S2: quasi_borel_a] :
( ( ( ord_less_nat @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ B4 ) )
=> ( ( ( ord_less_nat @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ S2 ) ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ ( measur1617175427248866264_a_nat @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_979_le__sup__lexord,axiom,
! [K: quasi_borel_a > set_a,A3: quasi_borel_a,B4: quasi_borel_a,Ca: quasi_borel_a,C: quasi_borel_a,S2: quasi_borel_a] :
( ( ( ord_less_set_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ B4 ) )
=> ( ( ( ord_less_set_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ S2 ) ) )
=> ( ord_le1843388692487544644orel_a @ Ca @ ( measur7302512200507534934_set_a @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_980_le__sup__lexord,axiom,
! [K: nat > quasi_borel_a,A3: nat,B4: nat,Ca: nat,C: nat,S2: nat] :
( ( ( ord_le3788981553861827664orel_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ B4 ) )
=> ( ( ( ord_le3788981553861827664orel_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_less_eq_nat @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_le1843388692487544644orel_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_le1843388692487544644orel_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ S2 ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur6224197021335510454orel_a @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_981_le__sup__lexord,axiom,
! [K: nat > nat,A3: nat,B4: nat,Ca: nat,C: nat,S2: nat] :
( ( ( ord_less_nat @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ B4 ) )
=> ( ( ( ord_less_nat @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_less_eq_nat @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ S2 ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur4601247141005857854at_nat @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_982_le__sup__lexord,axiom,
! [K: nat > set_a,A3: nat,B4: nat,Ca: nat,C: nat,S2: nat] :
( ( ( ord_less_set_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ B4 ) )
=> ( ( ( ord_less_set_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_less_eq_nat @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_nat @ Ca @ S2 ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur5940501938642862960_set_a @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_983_le__sup__lexord,axiom,
! [K: set_a > quasi_borel_a,A3: set_a,B4: set_a,Ca: set_a,C: set_a,S2: set_a] :
( ( ( ord_le3788981553861827664orel_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ B4 ) )
=> ( ( ( ord_le3788981553861827664orel_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_less_eq_set_a @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_le1843388692487544644orel_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_le1843388692487544644orel_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ S2 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur8298705319509490402orel_a @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_984_le__sup__lexord,axiom,
! [K: set_a > nat,A3: set_a,B4: set_a,Ca: set_a,C: set_a,S2: set_a] :
( ( ( ord_less_nat @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ B4 ) )
=> ( ( ( ord_less_nat @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_less_eq_set_a @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ S2 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur783711913603534354_a_nat @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_985_le__sup__lexord,axiom,
! [K: set_a > set_a,A3: set_a,B4: set_a,Ca: set_a,C: set_a,S2: set_a] :
( ( ( ord_less_set_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ B4 ) )
=> ( ( ( ord_less_set_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ord_less_eq_set_a @ Ca @ A3 ) )
=> ( ( ( ( K @ A3 )
= ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_set_a @ ( K @ B4 ) @ ( K @ A3 ) )
=> ( ~ ( ord_less_eq_set_a @ ( K @ A3 ) @ ( K @ B4 ) )
=> ( ord_less_eq_set_a @ Ca @ S2 ) ) )
=> ( ord_less_eq_set_a @ Ca @ ( measur758946168800011932_set_a @ A3 @ B4 @ K @ S2 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_986_preordering__bdd_Omono,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,B4: set_a,A3: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( condit6541519642617408243_bdd_a @ Less_eq @ A3 ) ) ) ) ).
% preordering_bdd.mono
thf(fact_987_lfp_Oleq__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% lfp.leq_refl
thf(fact_988_gfp_Oleq__trans,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ Y )
=> ( ord_less_eq_set_a @ Z @ X ) ) ) ).
% gfp.leq_trans
thf(fact_989_inf_Oabel__semigroup__axioms,axiom,
abel_semigroup_nat @ inf_inf_nat ).
% inf.abel_semigroup_axioms
thf(fact_990_sup_Oabel__semigroup__axioms,axiom,
abel_semigroup_set_a @ sup_sup_set_a ).
% sup.abel_semigroup_axioms
thf(fact_991_sup_Oabel__semigroup__axioms,axiom,
abel_semigroup_nat @ sup_sup_nat ).
% sup.abel_semigroup_axioms
thf(fact_992_lfp_Oleq__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% lfp.leq_antisym
thf(fact_993_gfp_Oleq__antisym,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( X = Y ) ) ) ).
% gfp.leq_antisym
thf(fact_994_lfp_Oleq__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% lfp.leq_trans
thf(fact_995_le__left__mono,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a,A: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X @ Y )
=> ( ( ord_le1843388692487544644orel_a @ Y @ A )
=> ( ord_le1843388692487544644orel_a @ X @ A ) ) ) ).
% le_left_mono
thf(fact_996_le__left__mono,axiom,
! [X: nat,Y: nat,A: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ A )
=> ( ord_less_eq_nat @ X @ A ) ) ) ).
% le_left_mono
thf(fact_997_le__left__mono,axiom,
! [X: set_a,Y: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ A )
=> ( ord_less_eq_set_a @ X @ A ) ) ) ).
% le_left_mono
thf(fact_998_le__rel__bool__arg__iff,axiom,
( ord_le1636368021143733457orel_a
= ( ^ [X6: $o > quasi_borel_a,Y7: $o > quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le1843388692487544644orel_a @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_999_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X6: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_1000_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X6: $o > set_a,Y7: $o > set_a] :
( ( ord_less_eq_set_a @ ( X6 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_a @ ( X6 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_1001_generating__Mx__Mx,axiom,
! [X5: quasi_borel_a] :
( ( generating_Mx_a @ ( qbs_space_a @ X5 ) @ ( qbs_Mx_a @ X5 ) )
= ( qbs_Mx_a @ X5 ) ) ).
% generating_Mx_Mx
thf(fact_1002_shunt1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ Y ) @ Z ) ) ) ).
% shunt1
thf(fact_1003_Compl__anti__mono,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B4 ) @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_1004_Compl__subset__Compl__iff,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B4 ) )
= ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1005_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1006_compl__less__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_set_a @ Y @ X ) ) ).
% compl_less_compl_iff
thf(fact_1007_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ X @ Y ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_1008_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ X @ Y ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_1009_compl__less__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_less_swap1
thf(fact_1010_compl__less__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_less_swap2
thf(fact_1011_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1012_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1013_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_1014_Compl__Int,axiom,
! [A3: set_a,B4: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B4 ) ) ) ).
% Compl_Int
thf(fact_1015_Compl__Un,axiom,
! [A3: set_a,B4: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ A3 ) @ ( uminus_uminus_set_a @ B4 ) ) ) ).
% Compl_Un
thf(fact_1016_sup__neg__inf,axiom,
! [P5: set_a,Q3: set_a,R: set_a] :
( ( ord_less_eq_set_a @ P5 @ ( sup_sup_set_a @ Q3 @ R ) )
= ( ord_less_eq_set_a @ ( inf_inf_set_a @ P5 @ ( uminus_uminus_set_a @ Q3 ) ) @ R ) ) ).
% sup_neg_inf
thf(fact_1017_shunt2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) @ Z )
= ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% shunt2
thf(fact_1018_disjoint__eq__subset__Compl,axiom,
! [A3: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1019_min_Osemilattice__order__axioms,axiom,
semila1248733672344298208er_nat @ ord_min_nat @ ord_less_eq_nat @ ord_less_nat ).
% min.semilattice_order_axioms
thf(fact_1020_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_1021_min_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
= ( ord_min_nat @ A @ B ) ) ).
% min.left_idem
thf(fact_1022_min_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
= ( ord_min_nat @ A @ B ) ) ).
% min.right_idem
thf(fact_1023_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_1024_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_1025_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1026_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1027_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1028_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1029_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1030_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1031_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_1032_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_1033_Un__empty,axiom,
! [A3: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_1034_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_1035_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_1036_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_1037_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_1038_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_1039_min__less__iff__conj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
& ( ord_less_nat @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_1040_min__bot,axiom,
! [X: nat] :
( ( ord_min_nat @ bot_bot_nat @ X )
= bot_bot_nat ) ).
% min_bot
thf(fact_1041_min__bot2,axiom,
! [X: nat] :
( ( ord_min_nat @ X @ bot_bot_nat )
= bot_bot_nat ) ).
% min_bot2
thf(fact_1042_min_Oabel__semigroup__axioms,axiom,
abel_semigroup_nat @ ord_min_nat ).
% min.abel_semigroup_axioms
thf(fact_1043_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_1044_Un__empty__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Un_empty_right
thf(fact_1045_Un__empty__left,axiom,
! [B4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1046_inf__min,axiom,
inf_inf_nat = ord_min_nat ).
% inf_min
thf(fact_1047_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1048_min_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.assoc
thf(fact_1049_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( ord_min_nat @ B3 @ A4 ) ) ) ).
% min.commute
thf(fact_1050_min_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.left_commute
thf(fact_1051_qbs__empty__equiv,axiom,
! [X5: quasi_borel_a] :
( ( ( qbs_space_a @ X5 )
= bot_bot_set_a )
= ( ( qbs_Mx_a @ X5 )
= bot_bot_set_real_a ) ) ).
% qbs_empty_equiv
thf(fact_1052_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_1053_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_1054_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( A4
= ( ord_min_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_1055_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_1056_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
| ( ord_less_nat @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_1057_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1058_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1059_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1060_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1061_min_Osemilattice__axioms,axiom,
semilattice_nat @ ord_min_nat ).
% min.semilattice_axioms
thf(fact_1062_min__def,axiom,
( ord_mi5312030764815082635orel_a
= ( ^ [A4: quasi_borel_a,B3: quasi_borel_a] : ( if_quasi_borel_a @ ( ord_le1843388692487544644orel_a @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_1063_min__def,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_1064_min__def,axiom,
( ord_min_set_a
= ( ^ [A4: set_a,B3: set_a] : ( if_set_a @ ( ord_less_eq_set_a @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_1065_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% min.mono
thf(fact_1066_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_1067_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_1068_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_1069_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_1070_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( ord_min_nat @ A4 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_1071_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_1072_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_1073_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_min_nat @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_1074_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_min_nat @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_1075_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_1076_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_1077_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
| ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_1078_min__absorb1,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ X @ Y )
=> ( ( ord_mi5312030764815082635orel_a @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_1079_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_1080_min__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_min_set_a @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_1081_min__absorb2,axiom,
! [Y: quasi_borel_a,X: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ Y @ X )
=> ( ( ord_mi5312030764815082635orel_a @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1082_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1083_min__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_min_set_a @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1084_bot_Oextremum,axiom,
! [A: quasi_borel_a] : ( ord_le1843388692487544644orel_a @ bot_bo450704735548830736orel_a @ A ) ).
% bot.extremum
thf(fact_1085_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1086_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_1087_bot_Oextremum__unique,axiom,
! [A: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ bot_bo450704735548830736orel_a )
= ( A = bot_bo450704735548830736orel_a ) ) ).
% bot.extremum_unique
thf(fact_1088_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1089_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_1090_bot_Oextremum__uniqueI,axiom,
! [A: quasi_borel_a] :
( ( ord_le1843388692487544644orel_a @ A @ bot_bo450704735548830736orel_a )
=> ( A = bot_bo450704735548830736orel_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1091_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1092_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1093_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1094_subset__Compl__self__eq,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1095_inf__shunt,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% inf_shunt
thf(fact_1096_eqb__space,axiom,
( ( qbs_space_a @ empty_quasi_borel_a )
= bot_bot_set_a ) ).
% eqb_space
thf(fact_1097_empty__quasi__borel__iff,axiom,
! [X5: quasi_borel_a] :
( ( ( qbs_space_a @ X5 )
= bot_bot_set_a )
= ( X5 = empty_quasi_borel_a ) ) ).
% empty_quasi_borel_iff
thf(fact_1098_subset__Compl__singleton,axiom,
! [A3: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_1099_insert__subset,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_1100_Un__insert__right,axiom,
! [A3: set_a,A: a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_1101_Un__insert__left,axiom,
! [A: a,B4: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B4 ) @ C3 )
= ( insert_a @ A @ ( sup_sup_set_a @ B4 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_1102_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1103_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1104_insert__subsetI,axiom,
! [X: a,A3: set_a,X5: set_a] :
( ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ X5 @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1105_subset__insertI2,axiom,
! [A3: set_a,B4: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1106_subset__insertI,axiom,
! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_1107_subset__insert,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_1108_Set_Oinsert__mono,axiom,
! [C3: set_a,D3: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D3 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D3 ) ) ) ).
% Set.insert_mono
thf(fact_1109_subset__singletonD,axiom,
! [A3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1110_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1111_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1112_Un__singleton__iff,axiom,
! [A3: set_a,B4: set_a,X: a] :
( ( ( sup_sup_set_a @ A3 @ B4 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B4
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B4 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B4
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1113_singleton__Un__iff,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A3 @ B4 ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B4
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B4 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B4
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1114_Field__insert,axiom,
! [A: a,B: a,R: set_Product_prod_a_a] :
( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) )
= ( sup_sup_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R ) ) ) ).
% Field_insert
thf(fact_1115_Field__Un,axiom,
! [R: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( field_a @ ( sup_su3048258781599657691od_a_a @ R @ S2 ) )
= ( sup_sup_set_a @ ( field_a @ R ) @ ( field_a @ S2 ) ) ) ).
% Field_Un
thf(fact_1116_image__Un,axiom,
! [F: a > a,A3: set_a,B4: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B4 ) ) ) ).
% image_Un
thf(fact_1117_refl__on__Un,axiom,
! [A3: set_a,R: set_Product_prod_a_a,B4: set_a,S2: set_Product_prod_a_a] :
( ( refl_on_a @ A3 @ R )
=> ( ( refl_on_a @ B4 @ S2 )
=> ( refl_on_a @ ( sup_sup_set_a @ A3 @ B4 ) @ ( sup_su3048258781599657691od_a_a @ R @ S2 ) ) ) ) ).
% refl_on_Un
thf(fact_1118_subset__image__iff,axiom,
! [B4: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B4
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1119_subset__imageE,axiom,
! [B4: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A3 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A3 )
=> ( B4
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1120_image__mono,axiom,
! [A3: set_a,B4: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_1121_mono__Field,axiom,
! [R: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S2 )
=> ( ord_less_eq_set_a @ ( field_a @ R ) @ ( field_a @ S2 ) ) ) ).
% mono_Field
thf(fact_1122_map__qbs__space,axiom,
! [F: a > a,X5: quasi_borel_a] :
( ( qbs_space_a @ ( map_qbs_a_a @ F @ X5 ) )
= ( image_a_a @ F @ ( qbs_space_a @ X5 ) ) ) ).
% map_qbs_space
thf(fact_1123_all__subset__image,axiom,
! [F: a > a,A3: set_a,P: set_a > $o] :
( ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A3 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A3 )
=> ( P @ ( image_a_a @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1124_Linear__order__Well__order__iff,axiom,
! [R: set_Product_prod_a_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( order_6972113574731384220r_on_a @ ( field_a @ R ) @ R )
= ( ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ ( field_a @ R ) )
=> ( ( A6 != bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ A6 )
& ! [Y4: a] :
( ( member_a @ Y4 @ A6 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1125_underS__incl__iff,axiom,
! [R: set_Product_prod_a_a,A: a,B: a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A @ ( field_a @ R ) )
=> ( ( member_a @ B @ ( field_a @ R ) )
=> ( ( ord_less_eq_set_a @ ( order_underS_a @ R @ A ) @ ( order_underS_a @ R @ B ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1126_Order__Relation_OunderS__Field,axiom,
! [R: set_Product_prod_a_a,A: a] : ( ord_less_eq_set_a @ ( order_underS_a @ R @ A ) @ ( field_a @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1127_underS__Field2,axiom,
! [A: a,R: set_Product_prod_a_a] :
( ( member_a @ A @ ( field_a @ R ) )
=> ( ord_less_set_a @ ( order_underS_a @ R @ A ) @ ( field_a @ R ) ) ) ).
% underS_Field2
thf(fact_1128_underS__Field3,axiom,
! [R: set_Product_prod_a_a,A: a] :
( ( ( field_a @ R )
!= bot_bot_set_a )
=> ( ord_less_set_a @ ( order_underS_a @ R @ A ) @ ( field_a @ R ) ) ) ).
% underS_Field3
thf(fact_1129_Field__def,axiom,
( field_a
= ( ^ [R2: set_Product_prod_a_a] : ( sup_sup_set_a @ ( domain_a_a @ R2 ) @ ( range_a_a @ R2 ) ) ) ) ).
% Field_def
thf(fact_1130_Refl__under__underS,axiom,
! [R: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ ( field_a @ R ) @ R )
=> ( ( member_a @ A @ ( field_a @ R ) )
=> ( ( order_under_a @ R @ A )
= ( sup_sup_set_a @ ( order_underS_a @ R @ A ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1131_psubset__insert__iff,axiom,
! [A3: set_a,X: a,B4: set_a] :
( ( ord_less_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ( ( member_a @ X @ B4 )
=> ( ord_less_set_a @ A3 @ B4 ) )
& ( ~ ( member_a @ X @ B4 )
=> ( ( ( member_a @ X @ A3 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1132_diff__Pair,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ D ) ) ) ).
% diff_Pair
thf(fact_1133_Un__Diff__cancel,axiom,
! [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B4 @ A3 ) )
= ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_1134_Un__Diff__cancel2,axiom,
! [B4: set_a,A3: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B4 @ A3 ) @ A3 )
= ( sup_sup_set_a @ B4 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_1135_Diff__eq__empty__iff,axiom,
! [A3: set_a,B4: set_a] :
( ( ( minus_minus_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1136_Compl__Diff__eq,axiom,
! [A3: set_a,B4: set_a] :
( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A3 ) @ B4 ) ) ).
% Compl_Diff_eq
thf(fact_1137_Un__Diff,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C3 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ C3 ) @ ( minus_minus_set_a @ B4 @ C3 ) ) ) ).
% Un_Diff
thf(fact_1138_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1139_subset__Diff__insert,axiom,
! [A3: set_a,B4: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B4 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B4 @ C3 ) )
& ~ ( member_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1140_Diff__subset__conv,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ C3 )
= ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1141_Diff__partition,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B4 @ A3 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1142_Diff__Un,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( minus_minus_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C3 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ ( minus_minus_set_a @ A3 @ C3 ) ) ) ).
% Diff_Un
thf(fact_1143_Diff__Int,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( minus_minus_set_a @ A3 @ ( inf_inf_set_a @ B4 @ C3 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ ( minus_minus_set_a @ A3 @ C3 ) ) ) ).
% Diff_Int
thf(fact_1144_Int__Diff__Un,axiom,
! [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A3 @ B4 ) @ ( minus_minus_set_a @ A3 @ B4 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1145_Un__Diff__Int,axiom,
! [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ ( inf_inf_set_a @ A3 @ B4 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1146_minus__prod__def,axiom,
( minus_4365393887724441320at_nat
= ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( minus_minus_nat @ ( product_fst_nat_nat @ X4 ) @ ( product_fst_nat_nat @ Y4 ) ) @ ( minus_minus_nat @ ( product_snd_nat_nat @ X4 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ).
% minus_prod_def
thf(fact_1147_psubset__imp__ex__mem,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1148_double__diff,axiom,
! [A3: set_a,B4: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C3 )
=> ( ( minus_minus_set_a @ B4 @ ( minus_minus_set_a @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1149_Diff__subset,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_1150_Diff__mono,axiom,
! [A3: set_a,C3: set_a,D3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ C3 )
=> ( ( ord_less_eq_set_a @ D3 @ B4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ ( minus_minus_set_a @ C3 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_1151_under__Field,axiom,
! [R: set_Product_prod_a_a,A: a] : ( ord_less_eq_set_a @ ( order_under_a @ R @ A ) @ ( field_a @ R ) ) ).
% under_Field
thf(fact_1152_underS__subset__under,axiom,
! [R: set_Product_prod_a_a,A: a] : ( ord_less_eq_set_a @ ( order_underS_a @ R @ A ) @ ( order_under_a @ R @ A ) ) ).
% underS_subset_under
thf(fact_1153_subset__insert__iff,axiom,
! [A3: set_a,X: a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ( ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1154_Diff__single__insert,axiom,
! [A3: set_a,X: a,B4: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B4 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_1155_Fpow__mono,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A3 ) @ ( finite_Fpow_a @ B4 ) ) ) ).
% Fpow_mono
thf(fact_1156_Linear__order__wf__diff__Id,axiom,
! [R: set_Product_prod_a_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R ) @ R )
=> ( ( wf_a @ ( minus_6817036919807184750od_a_a @ R @ id_a ) )
= ( ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ ( field_a @ R ) )
=> ( ( A6 != bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ A6 )
& ! [Y4: a] :
( ( member_a @ Y4 @ A6 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1157_total__on__subset,axiom,
! [A3: set_a,R: set_Product_prod_a_a,B4: set_a] :
( ( total_on_a @ A3 @ R )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( total_on_a @ B4 @ R ) ) ) ).
% total_on_subset
thf(fact_1158_wf__eq__minimal2,axiom,
( wf_a
= ( ^ [R2: set_Product_prod_a_a] :
! [A6: set_a] :
( ( ( ord_less_eq_set_a @ A6 @ ( field_a @ R2 ) )
& ( A6 != bot_bot_set_a ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A6 )
& ! [Y4: a] :
( ( member_a @ Y4 @ A6 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R2 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1159_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1160_diff__le__mono2,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1161_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1162_diff__le__self,axiom,
! [M4: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N2 ) @ M4 ) ).
% diff_le_self
thf(fact_1163_diff__le__mono,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1164_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1165_le__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1166_eq__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M4 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1167_less__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M4 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1168_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1169_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M4: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K2 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_1170_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1171_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_1172_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_1173_nat__le__linear,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
| ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% nat_le_linear
thf(fact_1174_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_1175_le__antisym,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M4 )
=> ( M4 = N2 ) ) ) ).
% le_antisym
thf(fact_1176_eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( M4 = N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% eq_imp_le
thf(fact_1177_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1178_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1179_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_1180_less__imp__le__nat,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1181_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1182_less__or__eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1183_le__neq__implies__less,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( M4 != N2 )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1184_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1185_diff__less__mono2,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ( ord_less_nat @ M4 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ) ).
% diff_less_mono2
thf(fact_1186_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1187_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1188_min__diff,axiom,
! [M4: nat,I: nat,N2: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M4 @ I ) @ ( minus_minus_nat @ N2 @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M4 @ N2 ) @ I ) ) ).
% min_diff
thf(fact_1189_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_1190_max__ext__additive,axiom,
! [A3: set_a,B4: set_a,R3: set_Product_prod_a_a,C3: set_a,D3: set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A3 @ B4 ) @ ( max_ext_a @ R3 ) )
=> ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ C3 @ D3 ) @ ( max_ext_a @ R3 ) )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ ( sup_sup_set_a @ A3 @ C3 ) @ ( sup_sup_set_a @ B4 @ D3 ) ) @ ( max_ext_a @ R3 ) ) ) ) ).
% max_ext_additive
thf(fact_1191_wo__rel_Ocases__Total3,axiom,
! [R: set_Product_prod_a_a,A: a,B: a,Phi: a > a > $o] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R ) )
=> ( ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( minus_6817036919807184750od_a_a @ R @ id_a ) )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ ( minus_6817036919807184750od_a_a @ R @ id_a ) ) )
=> ( Phi @ A @ B ) )
=> ( ( ( A = B )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1192_Union__Un__distrib,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% Union_Un_distrib
thf(fact_1193_complete__lattice__class_OSup__insert,axiom,
! [A: set_a,A3: set_set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ A3 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ A3 ) ) ) ).
% complete_lattice_class.Sup_insert
thf(fact_1194_Sup__union__distrib,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% Sup_union_distrib
thf(fact_1195_Union__insert,axiom,
! [A: set_a,B4: set_set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ A @ B4 ) )
= ( sup_sup_set_a @ A @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% Union_insert
thf(fact_1196_less__cSupD,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ X5 )
& ( ord_less_nat @ Z @ X2 ) ) ) ) ).
% less_cSupD
thf(fact_1197_less__cSupE,axiom,
! [Y: nat,X5: set_nat] :
( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) )
=> ( ( X5 != bot_bot_set_nat )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ) ) ).
% less_cSupE
thf(fact_1198_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1199_cSup__eq__maximum,axiom,
! [Z: set_a,X5: set_set_a] :
( ( member_set_a @ Z @ X5 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1200_Union__mono,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% Union_mono
thf(fact_1201_Union__least,axiom,
! [A3: set_set_a,C3: set_a] :
( ! [X11: set_a] :
( ( member_set_a @ X11 @ A3 )
=> ( ord_less_eq_set_a @ X11 @ C3 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ C3 ) ) ).
% Union_least
thf(fact_1202_Union__upper,axiom,
! [B4: set_a,A3: set_set_a] :
( ( member_set_a @ B4 @ A3 )
=> ( ord_less_eq_set_a @ B4 @ ( comple2307003609928055243_set_a @ A3 ) ) ) ).
% Union_upper
thf(fact_1203_Union__subsetI,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ? [Y2: set_a] :
( ( member_set_a @ Y2 @ B4 )
& ( ord_less_eq_set_a @ X2 @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% Union_subsetI
thf(fact_1204_Sup__eqI,axiom,
! [A3: set_set_a,X: set_a] :
( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ A3 )
=> ( ord_less_eq_set_a @ Y3 @ X ) )
=> ( ! [Y3: set_a] :
( ! [Z4: set_a] :
( ( member_set_a @ Z4 @ A3 )
=> ( ord_less_eq_set_a @ Z4 @ Y3 ) )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ A3 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1205_complete__lattice__class_OSup__mono,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ! [A2: set_a] :
( ( member_set_a @ A2 @ A3 )
=> ? [X7: set_a] :
( ( member_set_a @ X7 @ B4 )
& ( ord_less_eq_set_a @ A2 @ X7 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% complete_lattice_class.Sup_mono
thf(fact_1206_Sup__least,axiom,
! [A3: set_set_a,Z: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ Z ) ) ).
% Sup_least
thf(fact_1207_Sup__upper,axiom,
! [X: set_a,A3: set_set_a] :
( ( member_set_a @ X @ A3 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ A3 ) ) ) ).
% Sup_upper
thf(fact_1208_Sup__le__iff,axiom,
! [A3: set_set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ B )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( ord_less_eq_set_a @ X4 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1209_Sup__upper2,axiom,
! [U2: set_a,A3: set_set_a,V: set_a] :
( ( member_set_a @ U2 @ A3 )
=> ( ( ord_less_eq_set_a @ V @ U2 )
=> ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A3 ) ) ) ) ).
% Sup_upper2
thf(fact_1210_Sup__subset__mono,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% Sup_subset_mono
thf(fact_1211_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1212_cSup__least,axiom,
! [X5: set_set_a,Z: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1213_cSup__eq__non__empty,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ A ) )
=> ( ! [Y3: nat] :
( ! [X7: nat] :
( ( member_nat @ X7 @ X5 )
=> ( ord_less_eq_nat @ X7 @ Y3 ) )
=> ( ord_less_eq_nat @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1214_cSup__eq__non__empty,axiom,
! [X5: set_set_a,A: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ X2 @ A ) )
=> ( ! [Y3: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ X5 )
=> ( ord_less_eq_set_a @ X7 @ Y3 ) )
=> ( ord_less_eq_set_a @ A @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1215_less__eq__Sup,axiom,
! [A3: set_set_a,U2: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A3 )
=> ( ord_less_eq_set_a @ U2 @ V2 ) )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U2 @ ( comple2307003609928055243_set_a @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_1216_Union__Int__subset,axiom,
! [A3: set_set_a,B4: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% Union_Int_subset
thf(fact_1217_Sup__inter__less__eq,axiom,
! [A3: set_set_a,B4: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A3 @ B4 ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1218_Sup__image__sup,axiom,
! [Y6: set_set_a,X: set_a] :
( ( Y6 != bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( sup_sup_set_a @ X ) @ Y6 ) )
= ( sup_sup_set_a @ X @ ( comple2307003609928055243_set_a @ Y6 ) ) ) ) ).
% Sup_image_sup
thf(fact_1219_wo__rel_Ocases__Total,axiom,
! [R: set_Product_prod_a_a,A: a,B: a,Phi: a > a > $o] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R ) )
=> ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( Phi @ A @ B ) )
=> ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1220_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( B4 != bot_bot_set_a )
=> ? [X_1: a] : ( bNF_We6697304935525757620inim_a @ R @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1221_wo__rel_Ominim__least,axiom,
! [R: set_Product_prod_a_a,B4: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( member_a @ B @ B4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We5615626441682584778inim_a @ R @ B4 ) @ B ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1222_wo__rel_Ominim__isMinim,axiom,
! [R: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( B4 != bot_bot_set_a )
=> ( bNF_We6697304935525757620inim_a @ R @ B4 @ ( bNF_We5615626441682584778inim_a @ R @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1223_wo__rel_Ominim__in,axiom,
! [R: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( B4 != bot_bot_set_a )
=> ( member_a @ ( bNF_We5615626441682584778inim_a @ R @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1224_wo__rel_Ominim__inField,axiom,
! [R: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( B4 != bot_bot_set_a )
=> ( member_a @ ( bNF_We5615626441682584778inim_a @ R @ B4 ) @ ( field_a @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1225_wo__rel_Oequals__minim,axiom,
! [R: set_Product_prod_a_a,B4: set_a,A: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( member_a @ A @ B4 )
=> ( ! [B2: a] :
( ( member_a @ B2 @ B4 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ R ) )
=> ( A
= ( bNF_We5615626441682584778inim_a @ R @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1226_chains__extend,axiom,
! [C: set_set_a,S: set_set_a,Z: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ Z @ S )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ C )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1227_chainsD,axiom,
! [C: set_set_a,S: set_set_a,X: set_a,Y: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ X @ C )
=> ( ( member_set_a @ Y @ C )
=> ( ( ord_less_eq_set_a @ X @ Y )
| ( ord_less_eq_set_a @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_1228_Zorn__Lemma2,axiom,
! [A3: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A3 ) )
=> ? [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X2 )
=> ( ord_less_eq_set_a @ Xb @ Xa ) ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1229_Zorn__Lemma,axiom,
! [A3: set_set_a] :
( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ ( chains_a @ A3 ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ X2 ) @ A3 ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_1230_wo__rel_Oofilter__def,axiom,
! [R: set_Product_prod_a_a,A3: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( order_ofilter_a @ R @ A3 )
= ( ( ord_less_eq_set_a @ A3 @ ( field_a @ R ) )
& ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( ord_less_eq_set_a @ ( order_under_a @ R @ X4 ) @ A3 ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_1231_compact__bot,axiom,
! [X: set_a] :
( ( X
= ( comple2307003609928055243_set_a @ bot_bot_set_set_a ) )
=> ( comple704071332753839977_set_a @ comple2307003609928055243_set_a @ ord_less_eq_set_a @ X ) ) ).
% compact_bot
thf(fact_1232_wo__rel_Oofilter__linord,axiom,
! [R: set_Product_prod_a_a,A3: set_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( order_ofilter_a @ R @ A3 )
=> ( ( order_ofilter_a @ R @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
| ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_1233_ofilter__def,axiom,
( order_ofilter_a
= ( ^ [R2: set_Product_prod_a_a,A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ ( field_a @ R2 ) )
& ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( ord_less_eq_set_a @ ( order_under_a @ R2 @ X4 ) @ A6 ) ) ) ) ) ).
% ofilter_def
thf(fact_1234_wo__rel_Oofilter__AboveS__Field,axiom,
! [R: set_Product_prod_a_a,A3: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( order_ofilter_a @ R @ A3 )
=> ( ( sup_sup_set_a @ A3 @ ( order_AboveS_a @ R @ A3 ) )
= ( field_a @ R ) ) ) ) ).
% wo_rel.ofilter_AboveS_Field
thf(fact_1235_Min_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ ord_min_nat @ ord_less_eq_nat @ ord_less_nat ).
% Min.semilattice_order_set_axioms
thf(fact_1236_AboveS__Field,axiom,
! [R: set_Product_prod_a_a,A3: set_a] : ( ord_less_eq_set_a @ ( order_AboveS_a @ R @ A3 ) @ ( field_a @ R ) ) ).
% AboveS_Field
thf(fact_1237_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1238_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic8986249270076014136_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1239_wo__rel_Osuc__greater,axiom,
! [R: set_Product_prod_a_a,B4: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( ( order_AboveS_a @ R @ B4 )
!= bot_bot_set_a )
=> ( ( member_a @ B @ B4 )
=> ( ( ( bNF_We6154283375207884895_suc_a @ R @ B4 )
!= B )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We6154283375207884895_suc_a @ R @ B4 ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_1240_wo__rel_Osuc__inField,axiom,
! [R: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( ( order_AboveS_a @ R @ B4 )
!= bot_bot_set_a )
=> ( member_a @ ( bNF_We6154283375207884895_suc_a @ R @ B4 ) @ ( field_a @ R ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_1241_wo__rel_Osuc__AboveS,axiom,
! [R: set_Product_prod_a_a,B4: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( ( order_AboveS_a @ R @ B4 )
!= bot_bot_set_a )
=> ( member_a @ ( bNF_We6154283375207884895_suc_a @ R @ B4 ) @ ( order_AboveS_a @ R @ B4 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_1242_wo__rel_Oequals__suc__AboveS,axiom,
! [R: set_Product_prod_a_a,B4: set_a,A: a] :
( ( bNF_We1162827675446709994_rel_a @ R )
=> ( ( ord_less_eq_set_a @ B4 @ ( field_a @ R ) )
=> ( ( member_a @ A @ ( order_AboveS_a @ R @ B4 ) )
=> ( ! [A8: a] :
( ( member_a @ A8 @ ( order_AboveS_a @ R @ B4 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A8 ) @ R ) )
=> ( A
= ( bNF_We6154283375207884895_suc_a @ R @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1243_cSUP__union,axiom,
! [A3: set_a,F: a > set_a,B4: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ A3 ) )
=> ( ( B4 != bot_bot_set_a )
=> ( ( condit3373647341569784514_set_a @ ( image_a_set_a @ F @ B4 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( sup_sup_set_a @ A3 @ B4 ) ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A3 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1244_cSUP__union,axiom,
! [A3: set_a,F: a > nat,B4: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A3 ) )
=> ( ( B4 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ B4 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ ( sup_sup_set_a @ A3 @ B4 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1245_Sup__minus__bot,axiom,
! [A3: set_set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ A3 )
=> ( ( comple2307003609928055243_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ ( comple2307003609928055243_set_a @ bot_bot_set_set_a ) @ bot_bot_set_set_a ) ) )
= ( comple2307003609928055243_set_a @ A3 ) ) ) ).
% Sup_minus_bot
thf(fact_1246_linorder__chain,axiom,
! [Y6: set_nat] : ( comple7016393980872852640in_nat @ ord_less_eq_nat @ Y6 ) ).
% linorder_chain
thf(fact_1247_bdd__above_OI,axiom,
! [A3: set_quasi_borel_a,M2: quasi_borel_a] :
( ! [X2: quasi_borel_a] :
( ( member_quasi_borel_a @ X2 @ A3 )
=> ( ord_le1843388692487544644orel_a @ X2 @ M2 ) )
=> ( condit3516415644855960584orel_a @ A3 ) ) ).
% bdd_above.I
thf(fact_1248_bdd__above_OI,axiom,
! [A3: set_nat,M2: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ( condit2214826472909112428ve_nat @ A3 ) ) ).
% bdd_above.I
thf(fact_1249_bdd__above_OI,axiom,
! [A3: set_set_a,M2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ X2 @ M2 ) )
=> ( condit3373647341569784514_set_a @ A3 ) ) ).
% bdd_above.I
thf(fact_1250_bdd__above__primitive__def,axiom,
( condit3516415644855960584orel_a
= ( condit6163609524079627545orel_a @ ord_le1843388692487544644orel_a ) ) ).
% bdd_above_primitive_def
thf(fact_1251_bdd__above__primitive__def,axiom,
( condit2214826472909112428ve_nat
= ( condit4013746787832047771dd_nat @ ord_less_eq_nat ) ) ).
% bdd_above_primitive_def
thf(fact_1252_bdd__above__primitive__def,axiom,
( condit3373647341569784514_set_a
= ( condit4774827555938943059_set_a @ ord_less_eq_set_a ) ) ).
% bdd_above_primitive_def
thf(fact_1253_bdd__above_OE,axiom,
! [A3: set_quasi_borel_a] :
( ( condit3516415644855960584orel_a @ A3 )
=> ~ ! [M5: quasi_borel_a] :
~ ! [X7: quasi_borel_a] :
( ( member_quasi_borel_a @ X7 @ A3 )
=> ( ord_le1843388692487544644orel_a @ X7 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1254_bdd__above_OE,axiom,
! [A3: set_nat] :
( ( condit2214826472909112428ve_nat @ A3 )
=> ~ ! [M5: nat] :
~ ! [X7: nat] :
( ( member_nat @ X7 @ A3 )
=> ( ord_less_eq_nat @ X7 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1255_bdd__above_OE,axiom,
! [A3: set_set_a] :
( ( condit3373647341569784514_set_a @ A3 )
=> ~ ! [M5: set_a] :
~ ! [X7: set_a] :
( ( member_set_a @ X7 @ A3 )
=> ( ord_less_eq_set_a @ X7 @ M5 ) ) ) ).
% bdd_above.E
thf(fact_1256_bdd__above_Ounfold,axiom,
( condit3516415644855960584orel_a
= ( ^ [A6: set_quasi_borel_a] :
? [M3: quasi_borel_a] :
! [X4: quasi_borel_a] :
( ( member_quasi_borel_a @ X4 @ A6 )
=> ( ord_le1843388692487544644orel_a @ X4 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1257_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A6: set_nat] :
? [M3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1258_bdd__above_Ounfold,axiom,
( condit3373647341569784514_set_a
= ( ^ [A6: set_set_a] :
? [M3: set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A6 )
=> ( ord_less_eq_set_a @ X4 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1259_lfp_Olub__least,axiom,
! [A3: set_set_a,Z: set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ A3 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A3 ) @ Z ) ) ) ).
% lfp.lub_least
thf(fact_1260_lfp_Olub__upper,axiom,
! [A3: set_set_a,X: set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ A3 )
=> ( ( member_set_a @ X @ A3 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ A3 ) ) ) ) ).
% lfp.lub_upper
thf(fact_1261_ccpo__Sup__below__iff,axiom,
! [Y6: set_set_a,X: set_a] :
( ( comple4316259127148425102_set_a @ ord_less_eq_set_a @ Y6 )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ Y6 ) @ X )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ Y6 )
=> ( ord_less_eq_set_a @ X4 @ X ) ) ) ) ) ).
% ccpo_Sup_below_iff
thf(fact_1262_cSup__upper2,axiom,
! [X: nat,X5: set_nat,Y: nat] :
( ( member_nat @ X @ X5 )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1263_cSup__upper2,axiom,
! [X: set_a,X5: set_set_a,Y: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( ( condit3373647341569784514_set_a @ X5 )
=> ( ord_less_eq_set_a @ Y @ ( comple2307003609928055243_set_a @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1264_cSup__upper,axiom,
! [X: nat,X5: set_nat] :
( ( member_nat @ X @ X5 )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_1265_cSup__upper,axiom,
! [X: set_a,X5: set_set_a] :
( ( member_set_a @ X @ X5 )
=> ( ( condit3373647341569784514_set_a @ X5 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_1266_cSup__le__iff,axiom,
! [S: set_nat,A: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ( ord_less_eq_nat @ X4 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1267_cSup__le__iff,axiom,
! [S: set_set_a,A: set_a] :
( ( S != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ S )
=> ( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ S ) @ A )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ S )
=> ( ord_less_eq_set_a @ X4 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1268_cSup__mono,axiom,
! [B4: set_nat,A3: set_nat] :
( ( B4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A3 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ B4 )
=> ? [X7: nat] :
( ( member_nat @ X7 @ A3 )
& ( ord_less_eq_nat @ B2 @ X7 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B4 ) @ ( complete_Sup_Sup_nat @ A3 ) ) ) ) ) ).
% cSup_mono
thf(fact_1269_cSup__mono,axiom,
! [B4: set_set_a,A3: set_set_a] :
( ( B4 != bot_bot_set_set_a )
=> ( ( condit3373647341569784514_set_a @ A3 )
=> ( ! [B2: set_a] :
( ( member_set_a @ B2 @ B4 )
=> ? [X7: set_a] :
( ( member_set_a @ X7 @ A3 )
& ( ord_less_eq_set_a @ B2 @ X7 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ B4 ) @ ( comple2307003609928055243_set_a @ A3 ) ) ) ) ) ).
% cSup_mono
thf(fact_1270_less__cSup__iff,axiom,
! [X5: set_nat,Y: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( ord_less_nat @ Y @ X4 ) ) ) ) ) ) ).
% less_cSup_iff
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X: set_a,Y: set_a] :
( ( if_set_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__QuasiBorel__Oquasi____borel_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__QuasiBorel__Oquasi____borel_Itf__a_J_T,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( if_quasi_borel_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__QuasiBorel__Oquasi____borel_Itf__a_J_T,axiom,
! [X: quasi_borel_a,Y: quasi_borel_a] :
( ( if_quasi_borel_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_le1843388692487544644orel_a @ ( sup_su6298519176299948920orel_a @ x @ y ) @ z ).
%------------------------------------------------------------------------------