TPTP Problem File: SLH0497^1.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 : Separation_Logic_Unbounded/0003_FixedPoint/prob_00728_021650__6885160_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1435 ( 593 unt; 264 typ; 0 def)
% Number of atoms : 3290 (1350 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 10376 ( 380 ~; 36 |; 263 &;8591 @)
% ( 0 <=>;1106 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 6 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 1368 (1368 >; 0 *; 0 +; 0 <<)
% Number of symbols : 229 ( 226 usr; 17 con; 0-7 aty)
% Number of variables : 3440 ( 149 ^;3228 !; 63 ?;3440 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:05:34.507
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
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thf(ty_n_t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (226)
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001tf__a_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Finite__Set_OFpow_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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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_001_062_Itf__a_M_Eo_J,type,
uminus_uminus_a_o: ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
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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__Option__Ooption_Itf__a_J,type,
if_option_a: $o > option_a > option_a > option_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
inf_inf_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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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_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
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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_Map_Odom_001t__Option__Ooption_Itf__a_J_001tf__a,type,
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thf(sy_c_Map_Odom_001t__Set__Oset_Itf__a_J_001tf__a,type,
dom_set_a_a: ( set_a > option_a ) > set_set_a ).
thf(sy_c_Map_Odom_001tf__a_001tf__a,type,
dom_a_a: ( a > option_a ) > set_a ).
thf(sy_c_Map_Ograph_001tf__a_001tf__a,type,
graph_a_a: ( a > option_a ) > set_Product_prod_a_a ).
thf(sy_c_Map_Omap__add_001t__Option__Ooption_Itf__a_J_001tf__a,type,
map_add_option_a_a: ( option_a > option_a ) > ( option_a > option_a ) > option_a > option_a ).
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thf(sy_c_Map_Omap__add_001t__Set__Oset_Itf__a_J_001tf__a,type,
map_add_set_a_a: ( set_a > option_a ) > ( set_a > option_a ) > set_a > option_a ).
thf(sy_c_Map_Omap__add_001tf__a_001tf__a,type,
map_add_a_a: ( a > option_a ) > ( a > option_a ) > a > option_a ).
thf(sy_c_Map_Omap__le_001tf__a_001tf__a,type,
map_le_a_a: ( a > option_a ) > ( a > option_a ) > $o ).
thf(sy_c_Map_Oran_001t__Option__Ooption_Itf__a_J_001tf__a,type,
ran_option_a_a: ( option_a > option_a ) > set_a ).
thf(sy_c_Map_Oran_001tf__a_001tf__a,type,
ran_a_a: ( a > option_a ) > set_a ).
thf(sy_c_Map_Orestrict__map_001t__Option__Ooption_Itf__a_J_001tf__a,type,
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thf(sy_c_Map_Orestrict__map_001t__Set__Oset_Itf__a_J_001tf__a,type,
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thf(sy_c_Map_Orestrict__map_001tf__a_001tf__a,type,
restrict_map_a_a: ( a > option_a ) > set_a > a > option_a ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_Itf__a_J,type,
none_option_a: option_option_a ).
thf(sy_c_Option_Ooption_ONone_001tf__a,type,
none_a: option_a ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_Itf__a_J,type,
some_option_a: option_a > option_option_a ).
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_Itf__a_J,type,
some_set_a: set_a > option_set_a ).
thf(sy_c_Option_Ooption_OSome_001tf__a,type,
some_a: a > option_a ).
thf(sy_c_Option_Ooption_Oset__option_001t__Option__Ooption_Itf__a_J,type,
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thf(sy_c_Option_Ooption_Oset__option_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_Option_Ooption_Oset__option_001t__Set__Oset_Itf__a_J,type,
set_option_set_a2: option_set_a > set_set_a ).
thf(sy_c_Option_Ooption_Oset__option_001tf__a,type,
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the_elem_option_a: set_option_a > option_a ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_UnboundedLogic_Oassertion_OExists_001tf__a_001tf__a_001tf__b_001t__Option__Ooption_Itf__a_J,type,
exists7165000112504185261tion_a: a > assert1556940916145061938on_a_a > assert1556940916145061938on_a_a ).
thf(sy_c_UnboundedLogic_Oassertion_OForall_001tf__a_001tf__a_001tf__b_001t__Option__Ooption_Itf__a_J,type,
forall5484998627543102345tion_a: a > assert1556940916145061938on_a_a > assert1556940916145061938on_a_a ).
thf(sy_c_UnboundedLogic_Ologic_001tf__a_001tf__b,type,
logic_a_b: ( a > a > option_a ) > ( b > a > a ) > ( b > b > b ) > ( b > b > b ) > ( b > b ) > b > ( a > $o ) > $o ).
thf(sy_c_UnboundedLogic_Ologic_Osat_001tf__a_001tf__b_001tf__a_001t__Option__Ooption_Itf__a_J,type,
sat_a_b_a_option_a: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > a > ( a > option_a ) > ( ( a > option_a ) > set_a ) > assert1556940916145061938on_a_a > $o ).
thf(sy_c_UnboundedLogic_Opre__logic_Ocompatible_001tf__a,type,
pre_compatible_a: ( a > a > option_a ) > a > a > $o ).
thf(sy_c_UnboundedLogic_Opre__logic_Olarger_001tf__a,type,
pre_larger_a: ( a > a > option_a ) > a > a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
member5113800082084363315tion_a: option_option_a > set_option_option_a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member8183384484874023062od_a_a: option5210160422955383789od_a_a > set_op7160277562814721357od_a_a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_Itf__a_J_J,type,
member_option_set_a: option_set_a > set_option_set_a > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
member5498148017924304208tion_a: produc3509355604313844263tion_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member7490379427973688371od_a_a: produc8326286882927642762od_a_a > set_Pr5481487045354815082od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member2542914943202714858_set_a: produc589746269170340929_set_a > set_Pr5928443294945626017_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J,type,
member6056235002698166154on_a_a: produc3083010940779526881on_a_a > set_Pr6308966090954093121on_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
member6937434987665551382tion_a: produc3964210925746912109tion_a > set_Pr3411724424142761165tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member3071122053849602553od_a_a: produc4044097585999906000od_a_a > set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
member4771970882521526448_set_a: product_prod_a_set_a > set_Pr6393634178297680487_set_a > $o ).
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__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
member_set_option_a: set_option_a > set_set_option_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_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__092_060Delta_062,type,
delta: ( c > d ) > set_a ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b2: a ).
thf(sy_v_mult,type,
mult: b > a > a ).
thf(sy_v_one,type,
one: b ).
thf(sy_v_p,type,
p: b ).
thf(sy_v_plus,type,
plus: a > a > option_a ).
thf(sy_v_q,type,
q: b ).
thf(sy_v_s,type,
s: c > d ).
thf(sy_v_sadd,type,
sadd: b > b > b ).
thf(sy_v_sinv,type,
sinv: b > b ).
thf(sy_v_smult,type,
smult: b > b > b ).
thf(sy_v_valid,type,
valid: a > $o ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1167)
thf(fact_0_assms_I3_J,axiom,
member_a @ b2 @ ( delta @ s ) ).
% assms(3)
thf(fact_1_assms_I2_J,axiom,
member_a @ a2 @ ( delta @ s ) ).
% assms(2)
thf(fact_2_assms_I4_J,axiom,
( ( some_a @ x )
= ( plus @ ( mult @ p @ a2 ) @ ( mult @ q @ b2 ) ) ) ).
% assms(4)
thf(fact_3_logic_Osem__combinable_Ocong,axiom,
sem_co7516848414490435095_b_c_d = sem_co7516848414490435095_b_c_d ).
% logic.sem_combinable.cong
thf(fact_4_can__factorize,axiom,
! [Q: b,P: b] :
? [R: b] :
( Q
= ( smult @ R @ P ) ) ).
% can_factorize
thf(fact_5_smult__asso,axiom,
! [P: b,Q: b,R2: b] :
( ( smult @ ( smult @ P @ Q ) @ R2 )
= ( smult @ P @ ( smult @ Q @ R2 ) ) ) ).
% smult_asso
thf(fact_6_smult__comm,axiom,
! [P: b,Q: b] :
( ( smult @ P @ Q )
= ( smult @ Q @ P ) ) ).
% smult_comm
thf(fact_7_commutative,axiom,
! [A: a,B: a] :
( ( plus @ A @ B )
= ( plus @ B @ A ) ) ).
% commutative
thf(fact_8_can__divide,axiom,
! [P: b,A: a,B: a] :
( ( ( mult @ P @ A )
= ( mult @ P @ B ) )
=> ( A = B ) ) ).
% can_divide
thf(fact_9_asso1,axiom,
! [A: a,B: a,Ab: a,C: a,Bc: a] :
( ( ( ( plus @ A @ B )
= ( some_a @ Ab ) )
& ( ( plus @ B @ C )
= ( some_a @ Bc ) ) )
=> ( ( plus @ Ab @ C )
= ( plus @ A @ Bc ) ) ) ).
% asso1
thf(fact_10_move__sum,axiom,
! [A: a,A1: a,A2: a,B: a,B1: a,B2: a,X: a,X1: a,X2: a] :
( ( ( some_a @ A )
= ( plus @ A1 @ A2 ) )
=> ( ( ( some_a @ B )
= ( plus @ B1 @ B2 ) )
=> ( ( ( some_a @ X )
= ( plus @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( plus @ A1 @ B1 ) )
=> ( ( ( some_a @ X2 )
= ( plus @ A2 @ B2 ) )
=> ( ( some_a @ X )
= ( plus @ X1 @ X2 ) ) ) ) ) ) ) ).
% move_sum
thf(fact_11_double__mult,axiom,
! [P: b,Q: b,A: a] :
( ( mult @ P @ ( mult @ Q @ A ) )
= ( mult @ ( smult @ P @ Q ) @ A ) ) ).
% double_mult
thf(fact_12_plus__mult,axiom,
! [A: a,B: a,C: a,P: b] :
( ( ( some_a @ A )
= ( plus @ B @ C ) )
=> ( ( some_a @ ( mult @ P @ A ) )
= ( plus @ ( mult @ P @ B ) @ ( mult @ P @ C ) ) ) ) ).
% plus_mult
thf(fact_13_unique__inv,axiom,
! [A: a,P: b,B: a] :
( ( A
= ( mult @ P @ B ) )
= ( B
= ( mult @ ( sinv @ P ) @ A ) ) ) ).
% unique_inv
thf(fact_14_compatible__iff,axiom,
! [A: a,B: a,P: b] :
( ( pre_compatible_a @ plus @ A @ B )
= ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% compatible_iff
thf(fact_15_compatible__imp,axiom,
! [A: a,B: a,P: b] :
( ( pre_compatible_a @ plus @ A @ B )
=> ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% compatible_imp
thf(fact_16_compatible__multiples,axiom,
! [P: b,A: a,Q: b,B: a] :
( ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) )
=> ( pre_compatible_a @ plus @ A @ B ) ) ).
% compatible_multiples
thf(fact_17_larger__same,axiom,
! [A: a,B: a,P: b] :
( ( pre_larger_a @ plus @ A @ B )
= ( pre_larger_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% larger_same
thf(fact_18_asso2,axiom,
! [A: a,B: a,Ab: a,C: a] :
( ( ( ( plus @ A @ B )
= ( some_a @ Ab ) )
& ~ ( pre_compatible_a @ plus @ B @ C ) )
=> ~ ( pre_compatible_a @ plus @ Ab @ C ) ) ).
% asso2
thf(fact_19_asso3,axiom,
! [A: a,B: a,C: a,Bc: a] :
( ~ ( pre_compatible_a @ plus @ A @ B )
=> ( ( ( plus @ B @ C )
= ( some_a @ Bc ) )
=> ~ ( pre_compatible_a @ plus @ A @ Bc ) ) ) ).
% asso3
thf(fact_20_larger__def,axiom,
! [A: a,B: a] :
( ( pre_larger_a @ plus @ A @ B )
= ( ? [C2: a] :
( ( some_a @ A )
= ( plus @ B @ C2 ) ) ) ) ).
% larger_def
thf(fact_21_larger__first__sum,axiom,
! [Y: a,A: a,B: a,X: a] :
( ( ( some_a @ Y )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ X @ Y )
=> ? [A3: a] :
( ( ( some_a @ X )
= ( plus @ A3 @ B ) )
& ( pre_larger_a @ plus @ A3 @ A ) ) ) ) ).
% larger_first_sum
thf(fact_22_sum__both__larger,axiom,
! [X3: a,A4: a,B3: a,X: a,A: a,B: a] :
( ( ( some_a @ X3 )
= ( plus @ A4 @ B3 ) )
=> ( ( ( some_a @ X )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ A4 @ A )
=> ( ( pre_larger_a @ plus @ B3 @ B )
=> ( pre_larger_a @ plus @ X3 @ X ) ) ) ) ) ).
% sum_both_larger
thf(fact_23_option_Oinject,axiom,
! [X2: a,Y2: a] :
( ( ( some_a @ X2 )
= ( some_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_24_sone__neutral,axiom,
! [P: b] :
( ( smult @ one @ P )
= P ) ).
% sone_neutral
thf(fact_25_distrib__mult,axiom,
! [P: b,Q: b,X: a] :
( ( some_a @ ( mult @ ( sadd @ P @ Q ) @ X ) )
= ( plus @ ( mult @ P @ X ) @ ( mult @ Q @ X ) ) ) ).
% distrib_mult
thf(fact_26_sadd__comm,axiom,
! [P: b,Q: b] :
( ( sadd @ P @ Q )
= ( sadd @ Q @ P ) ) ).
% sadd_comm
thf(fact_27_one__neutral,axiom,
! [A: a] :
( ( mult @ one @ A )
= A ) ).
% one_neutral
thf(fact_28_smult__distrib,axiom,
! [P: b,Q: b,R2: b] :
( ( smult @ P @ ( sadd @ Q @ R2 ) )
= ( sadd @ ( smult @ P @ Q ) @ ( smult @ P @ R2 ) ) ) ).
% smult_distrib
thf(fact_29_larger__implies__compatible,axiom,
! [X: a,Y: a] :
( ( pre_larger_a @ plus @ X @ Y )
=> ( pre_compatible_a @ plus @ X @ Y ) ) ).
% larger_implies_compatible
thf(fact_30_compatible__smaller,axiom,
! [A: a,B: a,X: a] :
( ( pre_larger_a @ plus @ A @ B )
=> ( ( pre_compatible_a @ plus @ X @ A )
=> ( pre_compatible_a @ plus @ X @ B ) ) ) ).
% compatible_smaller
thf(fact_31_assms_I1_J,axiom,
sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ delta ).
% assms(1)
thf(fact_32_mem__Collect__eq,axiom,
! [A: option_a,P2: option_a > $o] :
( ( member_option_a @ A @ ( collect_option_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_33_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_34_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_35_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_36_Collect__mem__eq,axiom,
! [A5: set_option_a] :
( ( collect_option_a
@ ^ [X4: option_a] : ( member_option_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_37_Collect__mem__eq,axiom,
! [A5: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_38_Collect__mem__eq,axiom,
! [A5: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_39_Collect__mem__eq,axiom,
! [A5: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_40_Collect__cong,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X5: a] :
( ( P2 @ X5 )
= ( Q2 @ X5 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_41_assms_I5_J,axiom,
( ( sadd @ p @ q )
= one ) ).
% assms(5)
thf(fact_42_sinv__inverse,axiom,
! [P: b] :
( ( smult @ P @ ( sinv @ P ) )
= one ) ).
% sinv_inverse
thf(fact_43_sem__combinable__def,axiom,
! [Delta: ( c > d ) > set_a] :
( ( sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ Delta )
= ( ! [S: c > d,P3: b,Q3: b,A6: a,B4: a,X4: a] :
( ( ( ( sadd @ P3 @ Q3 )
= one )
& ( member_a @ A6 @ ( Delta @ S ) )
& ( member_a @ B4 @ ( Delta @ S ) )
& ( ( some_a @ X4 )
= ( plus @ ( mult @ P3 @ A6 ) @ ( mult @ Q3 @ B4 ) ) ) )
=> ( member_a @ X4 @ ( Delta @ S ) ) ) ) ) ).
% sem_combinable_def
thf(fact_44_sem__combinableI,axiom,
! [Delta: ( c > d ) > set_a] :
( ! [S2: c > d,P4: b,Q4: b,A7: a,B5: a,X5: a] :
( ( ( ( sadd @ P4 @ Q4 )
= one )
& ( member_a @ A7 @ ( Delta @ S2 ) )
& ( member_a @ B5 @ ( Delta @ S2 ) )
& ( ( some_a @ X5 )
= ( plus @ ( mult @ P4 @ A7 ) @ ( mult @ Q4 @ B5 ) ) ) )
=> ( member_a @ X5 @ ( Delta @ S2 ) ) )
=> ( sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ Delta ) ) ).
% sem_combinableI
thf(fact_45_compatible__def,axiom,
! [A: a,B: a] :
( ( pre_compatible_a @ plus @ A @ B )
= ( ( plus @ A @ B )
!= none_a ) ) ).
% compatible_def
thf(fact_46_pre__logic_Olarger__def,axiom,
( pre_larger_a
= ( ^ [Plus: a > a > option_a,A6: a,B4: a] :
? [C2: a] :
( ( some_a @ A6 )
= ( Plus @ B4 @ C2 ) ) ) ) ).
% pre_logic.larger_def
thf(fact_47_logic__axioms,axiom,
logic_a_b @ plus @ mult @ smult @ sadd @ sinv @ one @ valid ).
% logic_axioms
thf(fact_48_pre__logic_Ocompatible_Ocong,axiom,
pre_compatible_a = pre_compatible_a ).
% pre_logic.compatible.cong
thf(fact_49_pre__logic_Olarger_Ocong,axiom,
pre_larger_a = pre_larger_a ).
% pre_logic.larger.cong
thf(fact_50_valid__mono,axiom,
! [A: a,B: a] :
( ( ( valid @ A )
& ( pre_larger_a @ plus @ A @ B ) )
=> ( valid @ B ) ) ).
% valid_mono
thf(fact_51_not__None__eq,axiom,
! [X: option_a] :
( ( X != none_a )
= ( ? [Y3: a] :
( X
= ( some_a @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_52_not__Some__eq,axiom,
! [X: option_a] :
( ( ! [Y3: a] :
( X
!= ( some_a @ Y3 ) ) )
= ( X = none_a ) ) ).
% not_Some_eq
thf(fact_53_logic_Osadd__comm,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Sadd @ P @ Q )
= ( Sadd @ Q @ P ) ) ) ).
% logic.sadd_comm
thf(fact_54_logic_Ocan__divide,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( Mult @ P @ A )
= ( Mult @ P @ B ) )
=> ( A = B ) ) ) ).
% logic.can_divide
thf(fact_55_logic_Osmult__asso,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,R2: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ ( Smult @ P @ Q ) @ R2 )
= ( Smult @ P @ ( Smult @ Q @ R2 ) ) ) ) ).
% logic.smult_asso
thf(fact_56_logic_Osmult__comm,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ Q )
= ( Smult @ Q @ P ) ) ) ).
% logic.smult_comm
thf(fact_57_logic_Ounique__inv,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,P: b,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( A
= ( Mult @ P @ B ) )
= ( B
= ( Mult @ ( Sinv @ P ) @ A ) ) ) ) ).
% logic.unique_inv
thf(fact_58_logic_Ocommutative,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Plus2 @ A @ B )
= ( Plus2 @ B @ A ) ) ) ).
% logic.commutative
thf(fact_59_logic_Odouble__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,A: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Mult @ P @ ( Mult @ Q @ A ) )
= ( Mult @ ( Smult @ P @ Q ) @ A ) ) ) ).
% logic.double_mult
thf(fact_60_logic_Oone__neutral,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Mult @ One @ A )
= A ) ) ).
% logic.one_neutral
thf(fact_61_logic_Osinv__inverse,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ ( Sinv @ P ) )
= One ) ) ).
% logic.sinv_inverse
thf(fact_62_logic_Osone__neutral,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ One @ P )
= P ) ) ).
% logic.sone_neutral
thf(fact_63_logic_Osmult__distrib,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,R2: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ ( Sadd @ Q @ R2 ) )
= ( Sadd @ ( Smult @ P @ Q ) @ ( Smult @ P @ R2 ) ) ) ) ).
% logic.smult_distrib
thf(fact_64_logic_Oasso1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a,Bc: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( ( Plus2 @ A @ B )
= ( some_a @ Ab ) )
& ( ( Plus2 @ B @ C )
= ( some_a @ Bc ) ) )
=> ( ( Plus2 @ Ab @ C )
= ( Plus2 @ A @ Bc ) ) ) ) ).
% logic.asso1
thf(fact_65_logic_Omove__sum,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,A1: a,A2: a,B: a,B1: a,B2: a,X: a,X1: a,X2: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ A )
= ( Plus2 @ A1 @ A2 ) )
=> ( ( ( some_a @ B )
= ( Plus2 @ B1 @ B2 ) )
=> ( ( ( some_a @ X )
= ( Plus2 @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( Plus2 @ A1 @ B1 ) )
=> ( ( ( some_a @ X2 )
= ( Plus2 @ A2 @ B2 ) )
=> ( ( some_a @ X )
= ( Plus2 @ X1 @ X2 ) ) ) ) ) ) ) ) ).
% logic.move_sum
thf(fact_66_logic_Oplus__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ A )
= ( Plus2 @ B @ C ) )
=> ( ( some_a @ ( Mult @ P @ A ) )
= ( Plus2 @ ( Mult @ P @ B ) @ ( Mult @ P @ C ) ) ) ) ) ).
% logic.plus_mult
thf(fact_67_logic_Odistrib__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,X: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( some_a @ ( Mult @ ( Sadd @ P @ Q ) @ X ) )
= ( Plus2 @ ( Mult @ P @ X ) @ ( Mult @ Q @ X ) ) ) ) ).
% logic.distrib_mult
thf(fact_68_logic_Ovalid__mono,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( Valid @ A )
& ( pre_larger_a @ Plus2 @ A @ B ) )
=> ( Valid @ B ) ) ) ).
% logic.valid_mono
thf(fact_69_logic_Olarger__same,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ A @ B )
= ( pre_larger_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.larger_same
thf(fact_70_logic_Ocompatible__iff,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ A @ B )
= ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.compatible_iff
thf(fact_71_logic_Ocompatible__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ A @ B )
=> ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.compatible_imp
thf(fact_72_logic_Ocompatible__multiples,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,Q: b,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) )
=> ( pre_compatible_a @ Plus2 @ A @ B ) ) ) ).
% logic.compatible_multiples
thf(fact_73_pre__logic_Ocompatible__def,axiom,
( pre_compatible_a
= ( ^ [Plus: a > a > option_a,A6: a,B4: a] :
( ( Plus @ A6 @ B4 )
!= none_a ) ) ) ).
% pre_logic.compatible_def
thf(fact_74_option_Odistinct_I1_J,axiom,
! [X2: a] :
( none_a
!= ( some_a @ X2 ) ) ).
% option.distinct(1)
thf(fact_75_option_OdiscI,axiom,
! [Option: option_a,X2: a] :
( ( Option
= ( some_a @ X2 ) )
=> ( Option != none_a ) ) ).
% option.discI
thf(fact_76_option_Oexhaust,axiom,
! [Y: option_a] :
( ( Y != none_a )
=> ~ ! [X22: a] :
( Y
!= ( some_a @ X22 ) ) ) ).
% option.exhaust
thf(fact_77_split__option__ex,axiom,
( ( ^ [P5: option_a > $o] :
? [X6: option_a] : ( P5 @ X6 ) )
= ( ^ [P6: option_a > $o] :
( ( P6 @ none_a )
| ? [X4: a] : ( P6 @ ( some_a @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_78_split__option__all,axiom,
( ( ^ [P5: option_a > $o] :
! [X6: option_a] : ( P5 @ X6 ) )
= ( ^ [P6: option_a > $o] :
( ( P6 @ none_a )
& ! [X4: a] : ( P6 @ ( some_a @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_79_combine__options__cases,axiom,
! [X: option_a,P2: option_a > option_a > $o,Y: option_a] :
( ( ( X = none_a )
=> ( P2 @ X @ Y ) )
=> ( ( ( Y = none_a )
=> ( P2 @ X @ Y ) )
=> ( ! [A7: a,B5: a] :
( ( X
= ( some_a @ A7 ) )
=> ( ( Y
= ( some_a @ B5 ) )
=> ( P2 @ X @ Y ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_80_logic_Olarger__first__sum,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Y: a,A: a,B: a,X: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ Y )
= ( Plus2 @ A @ B ) )
=> ( ( pre_larger_a @ Plus2 @ X @ Y )
=> ? [A3: a] :
( ( ( some_a @ X )
= ( Plus2 @ A3 @ B ) )
& ( pre_larger_a @ Plus2 @ A3 @ A ) ) ) ) ) ).
% logic.larger_first_sum
thf(fact_81_logic_Osum__both__larger,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X3: a,A4: a,B3: a,X: a,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ X3 )
= ( Plus2 @ A4 @ B3 ) )
=> ( ( ( some_a @ X )
= ( Plus2 @ A @ B ) )
=> ( ( pre_larger_a @ Plus2 @ A4 @ A )
=> ( ( pre_larger_a @ Plus2 @ B3 @ B )
=> ( pre_larger_a @ Plus2 @ X3 @ X ) ) ) ) ) ) ).
% logic.sum_both_larger
thf(fact_82_logic_Oasso3,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,Bc: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ~ ( pre_compatible_a @ Plus2 @ A @ B )
=> ( ( ( Plus2 @ B @ C )
= ( some_a @ Bc ) )
=> ~ ( pre_compatible_a @ Plus2 @ A @ Bc ) ) ) ) ).
% logic.asso3
thf(fact_83_logic_Oasso2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( ( Plus2 @ A @ B )
= ( some_a @ Ab ) )
& ~ ( pre_compatible_a @ Plus2 @ B @ C ) )
=> ~ ( pre_compatible_a @ Plus2 @ Ab @ C ) ) ) ).
% logic.asso2
thf(fact_84_logic_Ocompatible__smaller,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,X: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ A @ B )
=> ( ( pre_compatible_a @ Plus2 @ X @ A )
=> ( pre_compatible_a @ Plus2 @ X @ B ) ) ) ) ).
% logic.compatible_smaller
thf(fact_85_logic_Olarger__implies__compatible,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: a,Y: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ X @ Y )
=> ( pre_compatible_a @ Plus2 @ X @ Y ) ) ) ).
% logic.larger_implies_compatible
thf(fact_86_logic__def,axiom,
( logic_a_b
= ( ^ [Plus: a > a > option_a,Mult2: b > a > a,Smult2: b > b > b,Sadd2: b > b > b,Sinv2: b > b,One2: b,Valid2: a > $o] :
( ! [A6: a,B4: a] :
( ( Plus @ A6 @ B4 )
= ( Plus @ B4 @ A6 ) )
& ! [A6: a,B4: a,Ab2: a,C2: a,Bc2: a] :
( ( ( ( Plus @ A6 @ B4 )
= ( some_a @ Ab2 ) )
& ( ( Plus @ B4 @ C2 )
= ( some_a @ Bc2 ) ) )
=> ( ( Plus @ Ab2 @ C2 )
= ( Plus @ A6 @ Bc2 ) ) )
& ! [A6: a,B4: a,Ab2: a,C2: a] :
( ( ( ( Plus @ A6 @ B4 )
= ( some_a @ Ab2 ) )
& ~ ( pre_compatible_a @ Plus @ B4 @ C2 ) )
=> ~ ( pre_compatible_a @ Plus @ Ab2 @ C2 ) )
& ! [P3: b] :
( ( Smult2 @ P3 @ ( Sinv2 @ P3 ) )
= One2 )
& ! [P3: b] :
( ( Smult2 @ One2 @ P3 )
= P3 )
& ! [P3: b,Q3: b] :
( ( Sadd2 @ P3 @ Q3 )
= ( Sadd2 @ Q3 @ P3 ) )
& ! [P3: b,Q3: b] :
( ( Smult2 @ P3 @ Q3 )
= ( Smult2 @ Q3 @ P3 ) )
& ! [P3: b,Q3: b,R3: b] :
( ( Smult2 @ P3 @ ( Sadd2 @ Q3 @ R3 ) )
= ( Sadd2 @ ( Smult2 @ P3 @ Q3 ) @ ( Smult2 @ P3 @ R3 ) ) )
& ! [P3: b,Q3: b,R3: b] :
( ( Smult2 @ ( Smult2 @ P3 @ Q3 ) @ R3 )
= ( Smult2 @ P3 @ ( Smult2 @ Q3 @ R3 ) ) )
& ! [P3: b,Q3: b,A6: a] :
( ( Mult2 @ P3 @ ( Mult2 @ Q3 @ A6 ) )
= ( Mult2 @ ( Smult2 @ P3 @ Q3 ) @ A6 ) )
& ! [A6: a,B4: a,C2: a,P3: b] :
( ( ( some_a @ A6 )
= ( Plus @ B4 @ C2 ) )
=> ( ( some_a @ ( Mult2 @ P3 @ A6 ) )
= ( Plus @ ( Mult2 @ P3 @ B4 ) @ ( Mult2 @ P3 @ C2 ) ) ) )
& ! [P3: b,Q3: b,X4: a] :
( ( some_a @ ( Mult2 @ ( Sadd2 @ P3 @ Q3 ) @ X4 ) )
= ( Plus @ ( Mult2 @ P3 @ X4 ) @ ( Mult2 @ Q3 @ X4 ) ) )
& ! [A6: a] :
( ( Mult2 @ One2 @ A6 )
= A6 )
& ! [A6: a,B4: a] :
( ( ( Valid2 @ A6 )
& ( pre_larger_a @ Plus @ A6 @ B4 ) )
=> ( Valid2 @ B4 ) ) ) ) ) ).
% logic_def
thf(fact_87_logic_Osem__combinableI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [S2: c > d,P4: b,Q4: b,A7: a,B5: a,X5: a] :
( ( ( ( Sadd @ P4 @ Q4 )
= One )
& ( member_a @ A7 @ ( Delta @ S2 ) )
& ( member_a @ B5 @ ( Delta @ S2 ) )
& ( ( some_a @ X5 )
= ( Plus2 @ ( Mult @ P4 @ A7 ) @ ( Mult @ Q4 @ B5 ) ) ) )
=> ( member_a @ X5 @ ( Delta @ S2 ) ) )
=> ( sem_co7516848414490435095_b_c_d @ Plus2 @ Mult @ Sadd @ One @ Delta ) ) ) ).
% logic.sem_combinableI
thf(fact_88_logic_Osem__combinable__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sem_co7516848414490435095_b_c_d @ Plus2 @ Mult @ Sadd @ One @ Delta )
= ( ! [S: c > d,P3: b,Q3: b,A6: a,B4: a,X4: a] :
( ( ( ( Sadd @ P3 @ Q3 )
= One )
& ( member_a @ A6 @ ( Delta @ S ) )
& ( member_a @ B4 @ ( Delta @ S ) )
& ( ( some_a @ X4 )
= ( Plus2 @ ( Mult @ P3 @ A6 ) @ ( Mult @ Q3 @ B4 ) ) ) )
=> ( member_a @ X4 @ ( Delta @ S ) ) ) ) ) ) ).
% logic.sem_combinable_def
thf(fact_89_subset__empty,axiom,
! [A5: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ bot_bot_set_option_a )
= ( A5 = bot_bot_set_option_a ) ) ).
% subset_empty
thf(fact_90_subset__empty,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_91_empty__subsetI,axiom,
! [A5: set_option_a] : ( ord_le1955136853071979460tion_a @ bot_bot_set_option_a @ A5 ) ).
% empty_subsetI
thf(fact_92_empty__subsetI,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A5 ) ).
% empty_subsetI
thf(fact_93_subsetI,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ! [X5: option_a] :
( ( member_option_a @ X5 @ A5 )
=> ( member_option_a @ X5 @ B6 ) )
=> ( ord_le1955136853071979460tion_a @ A5 @ B6 ) ) ).
% subsetI
thf(fact_94_subsetI,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A5 )
=> ( member1426531477525435216od_a_a @ X5 @ B6 ) )
=> ( ord_le746702958409616551od_a_a @ A5 @ B6 ) ) ).
% subsetI
thf(fact_95_subsetI,axiom,
! [A5: set_set_a,B6: set_set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( member_set_a @ X5 @ B6 ) )
=> ( ord_le3724670747650509150_set_a @ A5 @ B6 ) ) ).
% subsetI
thf(fact_96_subsetI,axiom,
! [A5: set_a,B6: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( member_a @ X5 @ B6 ) )
=> ( ord_less_eq_set_a @ A5 @ B6 ) ) ).
% subsetI
thf(fact_97_subset__antisym,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ A5 )
=> ( A5 = B6 ) ) ) ).
% subset_antisym
thf(fact_98_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_99_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_100_empty__iff,axiom,
! [C: option_a] :
~ ( member_option_a @ C @ bot_bot_set_option_a ) ).
% empty_iff
thf(fact_101_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_102_all__not__in__conv,axiom,
! [A5: set_Product_prod_a_a] :
( ( ! [X4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X4 @ A5 ) )
= ( A5 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_103_all__not__in__conv,axiom,
! [A5: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A5 ) )
= ( A5 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_104_all__not__in__conv,axiom,
! [A5: set_option_a] :
( ( ! [X4: option_a] :
~ ( member_option_a @ X4 @ A5 ) )
= ( A5 = bot_bot_set_option_a ) ) ).
% all_not_in_conv
thf(fact_105_all__not__in__conv,axiom,
! [A5: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_106_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_107_Collect__empty__eq,axiom,
! [P2: option_a > $o] :
( ( ( collect_option_a @ P2 )
= bot_bot_set_option_a )
= ( ! [X4: option_a] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_108_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X4: a] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_109_empty__Collect__eq,axiom,
! [P2: option_a > $o] :
( ( bot_bot_set_option_a
= ( collect_option_a @ P2 ) )
= ( ! [X4: option_a] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_110_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_111_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_112_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_113_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_114_bot__set__def,axiom,
( bot_bot_set_option_a
= ( collect_option_a @ bot_bot_option_a_o ) ) ).
% bot_set_def
thf(fact_115_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_116_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_117_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_119_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_121_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A6: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A6 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_122_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_123_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_124_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_125_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A6: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A6 )
& ( ord_less_eq_set_a @ A6 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_126_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_127_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_128_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_129_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_130_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_131_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_132_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_133_ex__in__conv,axiom,
! [A5: set_Product_prod_a_a] :
( ( ? [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A5 ) )
= ( A5 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_134_ex__in__conv,axiom,
! [A5: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A5 ) )
= ( A5 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_135_ex__in__conv,axiom,
! [A5: set_option_a] :
( ( ? [X4: option_a] : ( member_option_a @ X4 @ A5 ) )
= ( A5 != bot_bot_set_option_a ) ) ).
% ex_in_conv
thf(fact_136_ex__in__conv,axiom,
! [A5: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A5 ) )
= ( A5 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_137_equals0I,axiom,
! [A5: set_Product_prod_a_a] :
( ! [Y4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y4 @ A5 )
=> ( A5 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_138_equals0I,axiom,
! [A5: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A5 )
=> ( A5 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_139_equals0I,axiom,
! [A5: set_option_a] :
( ! [Y4: option_a] :
~ ( member_option_a @ Y4 @ A5 )
=> ( A5 = bot_bot_set_option_a ) ) ).
% equals0I
thf(fact_140_equals0I,axiom,
! [A5: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A5 )
=> ( A5 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_141_equals0D,axiom,
! [A5: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A5 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A5 ) ) ).
% equals0D
thf(fact_142_equals0D,axiom,
! [A5: set_set_a,A: set_a] :
( ( A5 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A5 ) ) ).
% equals0D
thf(fact_143_equals0D,axiom,
! [A5: set_option_a,A: option_a] :
( ( A5 = bot_bot_set_option_a )
=> ~ ( member_option_a @ A @ A5 ) ) ).
% equals0D
thf(fact_144_equals0D,axiom,
! [A5: set_a,A: a] :
( ( A5 = bot_bot_set_a )
=> ~ ( member_a @ A @ A5 ) ) ).
% equals0D
thf(fact_145_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_146_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_147_emptyE,axiom,
! [A: option_a] :
~ ( member_option_a @ A @ bot_bot_set_option_a ) ).
% emptyE
thf(fact_148_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_149_Collect__mono__iff,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) )
= ( ! [X4: a] :
( ( P2 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_150_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A8: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A8 @ B7 )
& ( ord_less_eq_set_a @ B7 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_151_subset__trans,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ord_less_eq_set_a @ A5 @ C3 ) ) ) ).
% subset_trans
thf(fact_152_Collect__mono,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X5: a] :
( ( P2 @ X5 )
=> ( Q2 @ X5 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_153_subset__refl,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ A5 @ A5 ) ).
% subset_refl
thf(fact_154_subset__iff,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A8: set_option_a,B7: set_option_a] :
! [T: option_a] :
( ( member_option_a @ T @ A8 )
=> ( member_option_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_155_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A8: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A8 )
=> ( member1426531477525435216od_a_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_156_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B7: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A8 )
=> ( member_set_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_157_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B7: set_a] :
! [T: a] :
( ( member_a @ T @ A8 )
=> ( member_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_158_equalityD2,axiom,
! [A5: set_a,B6: set_a] :
( ( A5 = B6 )
=> ( ord_less_eq_set_a @ B6 @ A5 ) ) ).
% equalityD2
thf(fact_159_equalityD1,axiom,
! [A5: set_a,B6: set_a] :
( ( A5 = B6 )
=> ( ord_less_eq_set_a @ A5 @ B6 ) ) ).
% equalityD1
thf(fact_160_subset__eq,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A8: set_option_a,B7: set_option_a] :
! [X4: option_a] :
( ( member_option_a @ X4 @ A8 )
=> ( member_option_a @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_161_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A8: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A8 )
=> ( member1426531477525435216od_a_a @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_162_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B7: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A8 )
=> ( member_set_a @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_163_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B7: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A8 )
=> ( member_a @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_164_equalityE,axiom,
! [A5: set_a,B6: set_a] :
( ( A5 = B6 )
=> ~ ( ( ord_less_eq_set_a @ A5 @ B6 )
=> ~ ( ord_less_eq_set_a @ B6 @ A5 ) ) ) ).
% equalityE
thf(fact_165_subsetD,axiom,
! [A5: set_option_a,B6: set_option_a,C: option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ B6 )
=> ( ( member_option_a @ C @ A5 )
=> ( member_option_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_166_subsetD,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B6 )
=> ( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_167_subsetD,axiom,
! [A5: set_set_a,B6: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B6 )
=> ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_168_subsetD,axiom,
! [A5: set_a,B6: set_a,C: a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_169_in__mono,axiom,
! [A5: set_option_a,B6: set_option_a,X: option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ B6 )
=> ( ( member_option_a @ X @ A5 )
=> ( member_option_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_170_in__mono,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B6 )
=> ( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( member1426531477525435216od_a_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_171_in__mono,axiom,
! [A5: set_set_a,B6: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B6 )
=> ( ( member_set_a @ X @ A5 )
=> ( member_set_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_172_in__mono,axiom,
! [A5: set_a,B6: set_a,X: a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( member_a @ X @ A5 )
=> ( member_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_173_bot_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
=> ( A = bot_bot_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_174_bot_Oextremum__uniqueI,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ bot_bot_set_option_a )
=> ( A = bot_bot_set_option_a ) ) ).
% bot.extremum_uniqueI
thf(fact_175_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_176_bot_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
= ( A = bot_bot_a_o ) ) ).
% bot.extremum_unique
thf(fact_177_bot_Oextremum__unique,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ bot_bot_set_option_a )
= ( A = bot_bot_set_option_a ) ) ).
% bot.extremum_unique
thf(fact_178_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_179_bot_Oextremum,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A ) ).
% bot.extremum
thf(fact_180_bot_Oextremum,axiom,
! [A: set_option_a] : ( ord_le1955136853071979460tion_a @ bot_bot_set_option_a @ A ) ).
% bot.extremum
thf(fact_181_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_182_sat__forall,axiom,
! [Sigma: a,S3: a > option_a,X: a,Delta: ( a > option_a ) > set_a,A5: assert1556940916145061938on_a_a] :
( ! [V: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ ( fun_upd_a_option_a @ S3 @ X @ V ) @ Delta @ A5 )
=> ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ S3 @ Delta @ ( forall5484998627543102345tion_a @ X @ A5 ) ) ) ).
% sat_forall
thf(fact_183_sat_Osimps_I9_J,axiom,
! [Sigma: a,S3: a > option_a,Delta: ( a > option_a ) > set_a,X: a,A5: assert1556940916145061938on_a_a] :
( ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ S3 @ Delta @ ( forall5484998627543102345tion_a @ X @ A5 ) )
= ( ! [V2: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ ( fun_upd_a_option_a @ S3 @ X @ V2 ) @ Delta @ A5 ) ) ) ).
% sat.simps(9)
thf(fact_184_sat_Osimps_I8_J,axiom,
! [Sigma: a,S3: a > option_a,Delta: ( a > option_a ) > set_a,X: a,A5: assert1556940916145061938on_a_a] :
( ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ S3 @ Delta @ ( exists7165000112504185261tion_a @ X @ A5 ) )
= ( ? [V2: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ ( fun_upd_a_option_a @ S3 @ X @ V2 ) @ Delta @ A5 ) ) ) ).
% sat.simps(8)
thf(fact_185_unambiguousI,axiom,
! [X: a,Delta: ( a > option_a ) > set_a,A5: assert1556940916145061938on_a_a] :
( ! [Sigma_1: a,Sigma_2: a,V1: option_a,V22: option_a,S2: a > option_a] :
( ( ( pre_compatible_a @ plus @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_1 @ ( fun_upd_a_option_a @ S2 @ X @ V1 ) @ Delta @ A5 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_2 @ ( fun_upd_a_option_a @ S2 @ X @ V22 ) @ Delta @ A5 ) )
=> ( V1 = V22 ) )
=> ( unambi704529886615442436tion_a @ plus @ mult @ valid @ Delta @ A5 @ X ) ) ).
% unambiguousI
thf(fact_186_unambiguous__def,axiom,
! [Delta: ( a > option_a ) > set_a,A5: assert1556940916145061938on_a_a,X: a] :
( ( unambi704529886615442436tion_a @ plus @ mult @ valid @ Delta @ A5 @ X )
= ( ! [Sigma_12: a,Sigma_22: a,V12: option_a,V23: option_a,S: a > option_a] :
( ( ( pre_compatible_a @ plus @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_12 @ ( fun_upd_a_option_a @ S @ X @ V12 ) @ Delta @ A5 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_22 @ ( fun_upd_a_option_a @ S @ X @ V23 ) @ Delta @ A5 ) )
=> ( V12 = V23 ) ) ) ) ).
% unambiguous_def
thf(fact_187_logic_Osat_Osimps_I8_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S3: a > option_a,Delta: ( a > option_a ) > set_a,X: a,A5: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ S3 @ Delta @ ( exists7165000112504185261tion_a @ X @ A5 ) )
= ( ? [V2: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_a_option_a @ S3 @ X @ V2 ) @ Delta @ A5 ) ) ) ) ).
% logic.sat.simps(8)
thf(fact_188_logic_Ounambiguous__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( a > option_a ) > set_a,A5: assert1556940916145061938on_a_a,X: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( unambi704529886615442436tion_a @ Plus2 @ Mult @ Valid @ Delta @ A5 @ X )
= ( ! [Sigma_12: a,Sigma_22: a,V12: option_a,V23: option_a,S: a > option_a] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_12 @ ( fun_upd_a_option_a @ S @ X @ V12 ) @ Delta @ A5 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_22 @ ( fun_upd_a_option_a @ S @ X @ V23 ) @ Delta @ A5 ) )
=> ( V12 = V23 ) ) ) ) ) ).
% logic.unambiguous_def
thf(fact_189_logic_OunambiguousI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: a,Delta: ( a > option_a ) > set_a,A5: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma_1: a,Sigma_2: a,V1: option_a,V22: option_a,S2: a > option_a] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_1 @ ( fun_upd_a_option_a @ S2 @ X @ V1 ) @ Delta @ A5 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_2 @ ( fun_upd_a_option_a @ S2 @ X @ V22 ) @ Delta @ A5 ) )
=> ( V1 = V22 ) )
=> ( unambi704529886615442436tion_a @ Plus2 @ Mult @ Valid @ Delta @ A5 @ X ) ) ) ).
% logic.unambiguousI
thf(fact_190_logic_Osat__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S3: a > option_a,X: a,Delta: ( a > option_a ) > set_a,A5: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [V: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_a_option_a @ S3 @ X @ V ) @ Delta @ A5 )
=> ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ S3 @ Delta @ ( forall5484998627543102345tion_a @ X @ A5 ) ) ) ) ).
% logic.sat_forall
thf(fact_191_logic_Osat_Osimps_I9_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S3: a > option_a,Delta: ( a > option_a ) > set_a,X: a,A5: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ S3 @ Delta @ ( forall5484998627543102345tion_a @ X @ A5 ) )
= ( ! [V2: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_a_option_a @ S3 @ X @ V2 ) @ Delta @ A5 ) ) ) ) ).
% logic.sat.simps(9)
thf(fact_192_logic_Ocan__factorize,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Q: b,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ? [R: b] :
( Q
= ( Smult @ R @ P ) ) ) ).
% logic.can_factorize
thf(fact_193_Collect__empty__eq__bot,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( P2 = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_194_Collect__empty__eq__bot,axiom,
! [P2: option_a > $o] :
( ( ( collect_option_a @ P2 )
= bot_bot_set_option_a )
= ( P2 = bot_bot_option_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_195_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_196_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_197_bot__empty__eq,axiom,
( bot_bot_option_a_o
= ( ^ [X4: option_a] : ( member_option_a @ X4 @ bot_bot_set_option_a ) ) ) ).
% bot_empty_eq
thf(fact_198_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_199_map__upd__nonempty,axiom,
! [T2: a > option_a,K: a,X: a] :
( ( fun_upd_a_option_a @ T2 @ K @ ( some_a @ X ) )
!= ( ^ [X4: a] : none_a ) ) ).
% map_upd_nonempty
thf(fact_200_subset__emptyI,axiom,
! [A5: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X5 @ A5 )
=> ( ord_le746702958409616551od_a_a @ A5 @ bot_bo3357376287454694259od_a_a ) ) ).
% subset_emptyI
thf(fact_201_subset__emptyI,axiom,
! [A5: set_set_a] :
( ! [X5: set_a] :
~ ( member_set_a @ X5 @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_202_subset__emptyI,axiom,
! [A5: set_option_a] :
( ! [X5: option_a] :
~ ( member_option_a @ X5 @ A5 )
=> ( ord_le1955136853071979460tion_a @ A5 @ bot_bot_set_option_a ) ) ).
% subset_emptyI
thf(fact_203_subset__emptyI,axiom,
! [A5: set_a] :
( ! [X5: a] :
~ ( member_a @ X5 @ A5 )
=> ( ord_less_eq_set_a @ A5 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_204_map__upd__Some__unfold,axiom,
! [M: a > option_a,A: a,B: a,X: a,Y: a] :
( ( ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) @ X )
= ( some_a @ Y ) )
= ( ( ( X = A )
& ( B = Y ) )
| ( ( X != A )
& ( ( M @ X )
= ( some_a @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_205_map__upd__triv,axiom,
! [T2: a > option_a,K: a,X: a] :
( ( ( T2 @ K )
= ( some_a @ X ) )
=> ( ( fun_upd_a_option_a @ T2 @ K @ ( some_a @ X ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_206_map__upd__eqD1,axiom,
! [M: a > option_a,A: a,X: a,N: a > option_a,Y: a] :
( ( ( fun_upd_a_option_a @ M @ A @ ( some_a @ X ) )
= ( fun_upd_a_option_a @ N @ A @ ( some_a @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_207_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A8: set_a] : ( A8 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_208_Set_Ois__empty__def,axiom,
( is_empty_option_a
= ( ^ [A8: set_option_a] : ( A8 = bot_bot_set_option_a ) ) ) ).
% Set.is_empty_def
thf(fact_209_map__le__imp__upd__le,axiom,
! [M1: a > option_a,M2: a > option_a,X: a,Y: a] :
( ( map_le_a_a @ M1 @ M2 )
=> ( map_le_a_a @ ( fun_upd_a_option_a @ M1 @ X @ none_a ) @ ( fun_upd_a_option_a @ M2 @ X @ ( some_a @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_210_Greatest__equality,axiom,
! [P2: set_a > $o,X: set_a] :
( ( P2 @ X )
=> ( ! [Y4: set_a] :
( ( P2 @ Y4 )
=> ( ord_less_eq_set_a @ Y4 @ X ) )
=> ( ( order_Greatest_set_a @ P2 )
= X ) ) ) ).
% Greatest_equality
thf(fact_211_subrelI,axiom,
! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ! [X5: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ S3 ) )
=> ( ord_le746702958409616551od_a_a @ R2 @ S3 ) ) ).
% subrelI
thf(fact_212_upd__None__map__le,axiom,
! [F: a > option_a,X: a] : ( map_le_a_a @ ( fun_upd_a_option_a @ F @ X @ none_a ) @ F ) ).
% upd_None_map_le
thf(fact_213_GreatestI2__order,axiom,
! [P2: set_a > $o,X: set_a,Q2: set_a > $o] :
( ( P2 @ X )
=> ( ! [Y4: set_a] :
( ( P2 @ Y4 )
=> ( ord_less_eq_set_a @ Y4 @ X ) )
=> ( ! [X5: set_a] :
( ( P2 @ X5 )
=> ( ! [Y6: set_a] :
( ( P2 @ Y6 )
=> ( ord_less_eq_set_a @ Y6 @ X5 ) )
=> ( Q2 @ X5 ) ) )
=> ( Q2 @ ( order_Greatest_set_a @ P2 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_214_relChain__def,axiom,
( bNF_Ca1905259564361943684_set_a
= ( ^ [R3: set_Product_prod_a_a,As: a > set_a] :
! [I: a,J: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_215_set__empty__eq,axiom,
! [Xo: option_option_a] :
( ( ( set_option_option_a2 @ Xo )
= bot_bot_set_option_a )
= ( Xo = none_option_a ) ) ).
% set_empty_eq
thf(fact_216_set__empty__eq,axiom,
! [Xo: option_a] :
( ( ( set_option_a2 @ Xo )
= bot_bot_set_a )
= ( Xo = none_a ) ) ).
% set_empty_eq
thf(fact_217_restrict__map__to__empty,axiom,
! [M: a > option_a] :
( ( restrict_map_a_a @ M @ bot_bot_set_a )
= ( ^ [X4: a] : none_a ) ) ).
% restrict_map_to_empty
thf(fact_218_restrict__map__to__empty,axiom,
! [M: option_a > option_a] :
( ( restri3984065703976872170on_a_a @ M @ bot_bot_set_option_a )
= ( ^ [X4: option_a] : none_a ) ) ).
% restrict_map_to_empty
thf(fact_219_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_220_elem__set,axiom,
! [X: option_a,Xo: option_option_a] :
( ( member_option_a @ X @ ( set_option_option_a2 @ Xo ) )
= ( Xo
= ( some_option_a @ X ) ) ) ).
% elem_set
thf(fact_221_elem__set,axiom,
! [X: product_prod_a_a,Xo: option5210160422955383789od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_op3367984300228426933od_a_a @ Xo ) )
= ( Xo
= ( some_P3592067295195376908od_a_a @ X ) ) ) ).
% elem_set
thf(fact_222_elem__set,axiom,
! [X: set_a,Xo: option_set_a] :
( ( member_set_a @ X @ ( set_option_set_a2 @ Xo ) )
= ( Xo
= ( some_set_a @ X ) ) ) ).
% elem_set
thf(fact_223_elem__set,axiom,
! [X: a,Xo: option_a] :
( ( member_a @ X @ ( set_option_a2 @ Xo ) )
= ( Xo
= ( some_a @ X ) ) ) ).
% elem_set
thf(fact_224_restrict__out,axiom,
! [X: option_a,A5: set_option_a,M: option_a > option_a] :
( ~ ( member_option_a @ X @ A5 )
=> ( ( restri3984065703976872170on_a_a @ M @ A5 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_225_restrict__out,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,M: product_prod_a_a > option_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( restri3846233575700461639_a_a_a @ M @ A5 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_226_restrict__out,axiom,
! [X: set_a,A5: set_set_a,M: set_a > option_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ( restrict_map_set_a_a @ M @ A5 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_227_restrict__out,axiom,
! [X: a,A5: set_a,M: a > option_a] :
( ~ ( member_a @ X @ A5 )
=> ( ( restrict_map_a_a @ M @ A5 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_228_option_Oset__cases,axiom,
! [E: option_a,A: option_option_a] :
( ( member_option_a @ E @ ( set_option_option_a2 @ A ) )
=> ( A
= ( some_option_a @ E ) ) ) ).
% option.set_cases
thf(fact_229_option_Oset__cases,axiom,
! [E: product_prod_a_a,A: option5210160422955383789od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( set_op3367984300228426933od_a_a @ A ) )
=> ( A
= ( some_P3592067295195376908od_a_a @ E ) ) ) ).
% option.set_cases
thf(fact_230_option_Oset__cases,axiom,
! [E: set_a,A: option_set_a] :
( ( member_set_a @ E @ ( set_option_set_a2 @ A ) )
=> ( A
= ( some_set_a @ E ) ) ) ).
% option.set_cases
thf(fact_231_option_Oset__cases,axiom,
! [E: a,A: option_a] :
( ( member_a @ E @ ( set_option_a2 @ A ) )
=> ( A
= ( some_a @ E ) ) ) ).
% option.set_cases
thf(fact_232_option_Oset__intros,axiom,
! [X2: option_a] : ( member_option_a @ X2 @ ( set_option_option_a2 @ ( some_option_a @ X2 ) ) ) ).
% option.set_intros
thf(fact_233_option_Oset__intros,axiom,
! [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( set_op3367984300228426933od_a_a @ ( some_P3592067295195376908od_a_a @ X2 ) ) ) ).
% option.set_intros
thf(fact_234_option_Oset__intros,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ ( set_option_set_a2 @ ( some_set_a @ X2 ) ) ) ).
% option.set_intros
thf(fact_235_option_Oset__intros,axiom,
! [X2: a] : ( member_a @ X2 @ ( set_option_a2 @ ( some_a @ X2 ) ) ) ).
% option.set_intros
thf(fact_236_ospec,axiom,
! [A5: option_a,P2: a > $o,X: a] :
( ! [X5: a] :
( ( member_a @ X5 @ ( set_option_a2 @ A5 ) )
=> ( P2 @ X5 ) )
=> ( ( A5
= ( some_a @ X ) )
=> ( P2 @ X ) ) ) ).
% ospec
thf(fact_237_restrict__map__def,axiom,
( restri3984065703976872170on_a_a
= ( ^ [M3: option_a > option_a,A8: set_option_a,X4: option_a] : ( if_option_a @ ( member_option_a @ X4 @ A8 ) @ ( M3 @ X4 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_238_restrict__map__def,axiom,
( restri3846233575700461639_a_a_a
= ( ^ [M3: product_prod_a_a > option_a,A8: set_Product_prod_a_a,X4: product_prod_a_a] : ( if_option_a @ ( member1426531477525435216od_a_a @ X4 @ A8 ) @ ( M3 @ X4 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_239_restrict__map__def,axiom,
( restrict_map_set_a_a
= ( ^ [M3: set_a > option_a,A8: set_set_a,X4: set_a] : ( if_option_a @ ( member_set_a @ X4 @ A8 ) @ ( M3 @ X4 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_240_restrict__map__def,axiom,
( restrict_map_a_a
= ( ^ [M3: a > option_a,A8: set_a,X4: a] : ( if_option_a @ ( member_a @ X4 @ A8 ) @ ( M3 @ X4 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_241_option_Osimps_I14_J,axiom,
( ( set_option_option_a2 @ none_option_a )
= bot_bot_set_option_a ) ).
% option.simps(14)
thf(fact_242_option_Osimps_I14_J,axiom,
( ( set_option_a2 @ none_a )
= bot_bot_set_a ) ).
% option.simps(14)
thf(fact_243_graph__restrictD_I2_J,axiom,
! [K: a,V3: a,M: a > option_a,A5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K @ V3 ) @ ( graph_a_a @ ( restrict_map_a_a @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some_a @ V3 ) ) ) ).
% graph_restrictD(2)
thf(fact_244_option_Osimps_I15_J,axiom,
! [X2: option_a] :
( ( set_option_option_a2 @ ( some_option_a @ X2 ) )
= ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ).
% option.simps(15)
thf(fact_245_option_Osimps_I15_J,axiom,
! [X2: a] :
( ( set_option_a2 @ ( some_a @ X2 ) )
= ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% option.simps(15)
thf(fact_246_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X7: $o > set_a,Y7: $o > set_a] :
( ( ord_less_eq_set_a @ ( X7 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_a @ ( X7 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_247_insert__absorb2,axiom,
! [X: a,A5: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A5 ) )
= ( insert_a @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_248_insert__absorb2,axiom,
! [X: option_a,A5: set_option_a] :
( ( insert_option_a @ X @ ( insert_option_a @ X @ A5 ) )
= ( insert_option_a @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_249_insert__iff,axiom,
! [A: option_a,B: option_a,A5: set_option_a] :
( ( member_option_a @ A @ ( insert_option_a @ B @ A5 ) )
= ( ( A = B )
| ( member_option_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_250_insert__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A5 ) )
= ( ( A = B )
| ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_251_insert__iff,axiom,
! [A: set_a,B: set_a,A5: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A5 ) )
= ( ( A = B )
| ( member_set_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_252_insert__iff,axiom,
! [A: a,B: a,A5: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A5 ) )
= ( ( A = B )
| ( member_a @ A @ A5 ) ) ) ).
% insert_iff
thf(fact_253_insertCI,axiom,
! [A: option_a,B6: set_option_a,B: option_a] :
( ( ~ ( member_option_a @ A @ B6 )
=> ( A = B ) )
=> ( member_option_a @ A @ ( insert_option_a @ B @ B6 ) ) ) ).
% insertCI
thf(fact_254_insertCI,axiom,
! [A: product_prod_a_a,B6: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A @ B6 )
=> ( A = B ) )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B6 ) ) ) ).
% insertCI
thf(fact_255_insertCI,axiom,
! [A: set_a,B6: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B6 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B6 ) ) ) ).
% insertCI
thf(fact_256_insertCI,axiom,
! [A: a,B6: set_a,B: a] :
( ( ~ ( member_a @ A @ B6 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B6 ) ) ) ).
% insertCI
thf(fact_257_singletonI,axiom,
! [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_258_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_259_singletonI,axiom,
! [A: option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).
% singletonI
thf(fact_260_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_261_insert__subset,axiom,
! [X: option_a,A5: set_option_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A5 ) @ B6 )
= ( ( member_option_a @ X @ B6 )
& ( ord_le1955136853071979460tion_a @ A5 @ B6 ) ) ) ).
% insert_subset
thf(fact_262_insert__subset,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ A5 ) @ B6 )
= ( ( member1426531477525435216od_a_a @ X @ B6 )
& ( ord_le746702958409616551od_a_a @ A5 @ B6 ) ) ) ).
% insert_subset
thf(fact_263_insert__subset,axiom,
! [X: set_a,A5: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A5 ) @ B6 )
= ( ( member_set_a @ X @ B6 )
& ( ord_le3724670747650509150_set_a @ A5 @ B6 ) ) ) ).
% insert_subset
thf(fact_264_insert__subset,axiom,
! [X: a,A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A5 ) @ B6 )
= ( ( member_a @ X @ B6 )
& ( ord_less_eq_set_a @ A5 @ B6 ) ) ) ).
% insert_subset
thf(fact_265_singleton__insert__inj__eq,axiom,
! [B: option_a,A: option_a,A5: set_option_a] :
( ( ( insert_option_a @ B @ bot_bot_set_option_a )
= ( insert_option_a @ A @ A5 ) )
= ( ( A = B )
& ( ord_le1955136853071979460tion_a @ A5 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_266_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A5: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A5 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A5 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_267_singleton__insert__inj__eq_H,axiom,
! [A: option_a,A5: set_option_a,B: option_a] :
( ( ( insert_option_a @ A @ A5 )
= ( insert_option_a @ B @ bot_bot_set_option_a ) )
= ( ( A = B )
& ( ord_le1955136853071979460tion_a @ A5 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_268_singleton__insert__inj__eq_H,axiom,
! [A: a,A5: set_a,B: a] :
( ( ( insert_a @ A @ A5 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A5 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_269_graph__map__upd,axiom,
! [M: a > option_a,K: a,V3: a] :
( ( graph_a_a @ ( fun_upd_a_option_a @ M @ K @ ( some_a @ V3 ) ) )
= ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ K @ V3 ) @ ( graph_a_a @ ( fun_upd_a_option_a @ M @ K @ none_a ) ) ) ) ).
% graph_map_upd
thf(fact_270_insert__subsetI,axiom,
! [X: option_a,A5: set_option_a,X8: set_option_a] :
( ( member_option_a @ X @ A5 )
=> ( ( ord_le1955136853071979460tion_a @ X8 @ A5 )
=> ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ X8 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_271_insert__subsetI,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,X8: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( ord_le746702958409616551od_a_a @ X8 @ A5 )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ X8 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_272_insert__subsetI,axiom,
! [X: set_a,A5: set_set_a,X8: set_set_a] :
( ( member_set_a @ X @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ X8 @ A5 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ X8 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_273_insert__subsetI,axiom,
! [X: a,A5: set_a,X8: set_a] :
( ( member_a @ X @ A5 )
=> ( ( ord_less_eq_set_a @ X8 @ A5 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X8 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_274_insert__mono,axiom,
! [C3: set_option_a,D: set_option_a,A: option_a] :
( ( ord_le1955136853071979460tion_a @ C3 @ D )
=> ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ C3 ) @ ( insert_option_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_275_insert__mono,axiom,
! [C3: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_276_subset__insert,axiom,
! [X: option_a,A5: set_option_a,B6: set_option_a] :
( ~ ( member_option_a @ X @ A5 )
=> ( ( ord_le1955136853071979460tion_a @ A5 @ ( insert_option_a @ X @ B6 ) )
= ( ord_le1955136853071979460tion_a @ A5 @ B6 ) ) ) ).
% subset_insert
thf(fact_277_subset__insert,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( ord_le746702958409616551od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( ord_le746702958409616551od_a_a @ A5 @ B6 ) ) ) ).
% subset_insert
thf(fact_278_subset__insert,axiom,
! [X: set_a,A5: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ X @ B6 ) )
= ( ord_le3724670747650509150_set_a @ A5 @ B6 ) ) ) ).
% subset_insert
thf(fact_279_subset__insert,axiom,
! [X: a,A5: set_a,B6: set_a] :
( ~ ( member_a @ X @ A5 )
=> ( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X @ B6 ) )
= ( ord_less_eq_set_a @ A5 @ B6 ) ) ) ).
% subset_insert
thf(fact_280_subset__insertI,axiom,
! [B6: set_option_a,A: option_a] : ( ord_le1955136853071979460tion_a @ B6 @ ( insert_option_a @ A @ B6 ) ) ).
% subset_insertI
thf(fact_281_subset__insertI,axiom,
! [B6: set_a,A: a] : ( ord_less_eq_set_a @ B6 @ ( insert_a @ A @ B6 ) ) ).
% subset_insertI
thf(fact_282_subset__insertI2,axiom,
! [A5: set_option_a,B6: set_option_a,B: option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ B6 )
=> ( ord_le1955136853071979460tion_a @ A5 @ ( insert_option_a @ B @ B6 ) ) ) ).
% subset_insertI2
thf(fact_283_subset__insertI2,axiom,
! [A5: set_a,B6: set_a,B: a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ord_less_eq_set_a @ A5 @ ( insert_a @ B @ B6 ) ) ) ).
% subset_insertI2
thf(fact_284_mk__disjoint__insert,axiom,
! [A: option_a,A5: set_option_a] :
( ( member_option_a @ A @ A5 )
=> ? [B8: set_option_a] :
( ( A5
= ( insert_option_a @ A @ B8 ) )
& ~ ( member_option_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_285_mk__disjoint__insert,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A5 )
=> ? [B8: set_Product_prod_a_a] :
( ( A5
= ( insert4534936382041156343od_a_a @ A @ B8 ) )
& ~ ( member1426531477525435216od_a_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_286_mk__disjoint__insert,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ? [B8: set_set_a] :
( ( A5
= ( insert_set_a @ A @ B8 ) )
& ~ ( member_set_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_287_mk__disjoint__insert,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ? [B8: set_a] :
( ( A5
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_288_insert__commute,axiom,
! [X: a,Y: a,A5: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A5 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_289_insert__commute,axiom,
! [X: option_a,Y: option_a,A5: set_option_a] :
( ( insert_option_a @ X @ ( insert_option_a @ Y @ A5 ) )
= ( insert_option_a @ Y @ ( insert_option_a @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_290_insert__eq__iff,axiom,
! [A: option_a,A5: set_option_a,B: option_a,B6: set_option_a] :
( ~ ( member_option_a @ A @ A5 )
=> ( ~ ( member_option_a @ B @ B6 )
=> ( ( ( insert_option_a @ A @ A5 )
= ( insert_option_a @ B @ B6 ) )
= ( ( ( A = B )
=> ( A5 = B6 ) )
& ( ( A != B )
=> ? [C4: set_option_a] :
( ( A5
= ( insert_option_a @ B @ C4 ) )
& ~ ( member_option_a @ B @ C4 )
& ( B6
= ( insert_option_a @ A @ C4 ) )
& ~ ( member_option_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_291_insert__eq__iff,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a,B: product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ~ ( member1426531477525435216od_a_a @ B @ B6 )
=> ( ( ( insert4534936382041156343od_a_a @ A @ A5 )
= ( insert4534936382041156343od_a_a @ B @ B6 ) )
= ( ( ( A = B )
=> ( A5 = B6 ) )
& ( ( A != B )
=> ? [C4: set_Product_prod_a_a] :
( ( A5
= ( insert4534936382041156343od_a_a @ B @ C4 ) )
& ~ ( member1426531477525435216od_a_a @ B @ C4 )
& ( B6
= ( insert4534936382041156343od_a_a @ A @ C4 ) )
& ~ ( member1426531477525435216od_a_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_292_insert__eq__iff,axiom,
! [A: set_a,A5: set_set_a,B: set_a,B6: set_set_a] :
( ~ ( member_set_a @ A @ A5 )
=> ( ~ ( member_set_a @ B @ B6 )
=> ( ( ( insert_set_a @ A @ A5 )
= ( insert_set_a @ B @ B6 ) )
= ( ( ( A = B )
=> ( A5 = B6 ) )
& ( ( A != B )
=> ? [C4: set_set_a] :
( ( A5
= ( insert_set_a @ B @ C4 ) )
& ~ ( member_set_a @ B @ C4 )
& ( B6
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_293_insert__eq__iff,axiom,
! [A: a,A5: set_a,B: a,B6: set_a] :
( ~ ( member_a @ A @ A5 )
=> ( ~ ( member_a @ B @ B6 )
=> ( ( ( insert_a @ A @ A5 )
= ( insert_a @ B @ B6 ) )
= ( ( ( A = B )
=> ( A5 = B6 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A5
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B6
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_294_insert__absorb,axiom,
! [A: option_a,A5: set_option_a] :
( ( member_option_a @ A @ A5 )
=> ( ( insert_option_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_295_insert__absorb,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ( insert4534936382041156343od_a_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_296_insert__absorb,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( insert_set_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_297_insert__absorb,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ( ( insert_a @ A @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_298_insert__ident,axiom,
! [X: option_a,A5: set_option_a,B6: set_option_a] :
( ~ ( member_option_a @ X @ A5 )
=> ( ~ ( member_option_a @ X @ B6 )
=> ( ( ( insert_option_a @ X @ A5 )
= ( insert_option_a @ X @ B6 ) )
= ( A5 = B6 ) ) ) ) ).
% insert_ident
thf(fact_299_insert__ident,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( ( insert4534936382041156343od_a_a @ X @ A5 )
= ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( A5 = B6 ) ) ) ) ).
% insert_ident
thf(fact_300_insert__ident,axiom,
! [X: set_a,A5: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ~ ( member_set_a @ X @ B6 )
=> ( ( ( insert_set_a @ X @ A5 )
= ( insert_set_a @ X @ B6 ) )
= ( A5 = B6 ) ) ) ) ).
% insert_ident
thf(fact_301_insert__ident,axiom,
! [X: a,A5: set_a,B6: set_a] :
( ~ ( member_a @ X @ A5 )
=> ( ~ ( member_a @ X @ B6 )
=> ( ( ( insert_a @ X @ A5 )
= ( insert_a @ X @ B6 ) )
= ( A5 = B6 ) ) ) ) ).
% insert_ident
thf(fact_302_Set_Oset__insert,axiom,
! [X: option_a,A5: set_option_a] :
( ( member_option_a @ X @ A5 )
=> ~ ! [B8: set_option_a] :
( ( A5
= ( insert_option_a @ X @ B8 ) )
=> ( member_option_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_303_Set_Oset__insert,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ~ ! [B8: set_Product_prod_a_a] :
( ( A5
= ( insert4534936382041156343od_a_a @ X @ B8 ) )
=> ( member1426531477525435216od_a_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_304_Set_Oset__insert,axiom,
! [X: set_a,A5: set_set_a] :
( ( member_set_a @ X @ A5 )
=> ~ ! [B8: set_set_a] :
( ( A5
= ( insert_set_a @ X @ B8 ) )
=> ( member_set_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_305_Set_Oset__insert,axiom,
! [X: a,A5: set_a] :
( ( member_a @ X @ A5 )
=> ~ ! [B8: set_a] :
( ( A5
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_306_insertI2,axiom,
! [A: option_a,B6: set_option_a,B: option_a] :
( ( member_option_a @ A @ B6 )
=> ( member_option_a @ A @ ( insert_option_a @ B @ B6 ) ) ) ).
% insertI2
thf(fact_307_insertI2,axiom,
! [A: product_prod_a_a,B6: set_Product_prod_a_a,B: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ B6 )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B6 ) ) ) ).
% insertI2
thf(fact_308_insertI2,axiom,
! [A: set_a,B6: set_set_a,B: set_a] :
( ( member_set_a @ A @ B6 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B6 ) ) ) ).
% insertI2
thf(fact_309_insertI2,axiom,
! [A: a,B6: set_a,B: a] :
( ( member_a @ A @ B6 )
=> ( member_a @ A @ ( insert_a @ B @ B6 ) ) ) ).
% insertI2
thf(fact_310_insertI1,axiom,
! [A: option_a,B6: set_option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ B6 ) ) ).
% insertI1
thf(fact_311_insertI1,axiom,
! [A: product_prod_a_a,B6: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ B6 ) ) ).
% insertI1
thf(fact_312_insertI1,axiom,
! [A: set_a,B6: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B6 ) ) ).
% insertI1
thf(fact_313_insertI1,axiom,
! [A: a,B6: set_a] : ( member_a @ A @ ( insert_a @ A @ B6 ) ) ).
% insertI1
thf(fact_314_insertE,axiom,
! [A: option_a,B: option_a,A5: set_option_a] :
( ( member_option_a @ A @ ( insert_option_a @ B @ A5 ) )
=> ( ( A != B )
=> ( member_option_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_315_insertE,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A5 ) )
=> ( ( A != B )
=> ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_316_insertE,axiom,
! [A: set_a,B: set_a,A5: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A5 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_317_insertE,axiom,
! [A: a,B: a,A5: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A5 ) )
=> ( ( A != B )
=> ( member_a @ A @ A5 ) ) ) ).
% insertE
thf(fact_318_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_319_singleton__inject,axiom,
! [A: option_a,B: option_a] :
( ( ( insert_option_a @ A @ bot_bot_set_option_a )
= ( insert_option_a @ B @ bot_bot_set_option_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_320_insert__not__empty,axiom,
! [A: a,A5: set_a] :
( ( insert_a @ A @ A5 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_321_insert__not__empty,axiom,
! [A: option_a,A5: set_option_a] :
( ( insert_option_a @ A @ A5 )
!= bot_bot_set_option_a ) ).
% insert_not_empty
thf(fact_322_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_323_doubleton__eq__iff,axiom,
! [A: option_a,B: option_a,C: option_a,D2: option_a] :
( ( ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) )
= ( insert_option_a @ C @ ( insert_option_a @ D2 @ bot_bot_set_option_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_324_singleton__iff,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_325_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_326_singleton__iff,axiom,
! [B: option_a,A: option_a] :
( ( member_option_a @ B @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_327_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_328_singletonD,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_329_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_330_singletonD,axiom,
! [B: option_a,A: option_a] :
( ( member_option_a @ B @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_331_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_332_subset__singletonD,axiom,
! [A5: set_option_a,X: option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
=> ( ( A5 = bot_bot_set_option_a )
| ( A5
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ).
% subset_singletonD
thf(fact_333_subset__singletonD,axiom,
! [A5: set_a,X: a] :
( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A5 = bot_bot_set_a )
| ( A5
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_334_subset__singleton__iff,axiom,
! [X8: set_option_a,A: option_a] :
( ( ord_le1955136853071979460tion_a @ X8 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( ( X8 = bot_bot_set_option_a )
| ( X8
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_335_subset__singleton__iff,axiom,
! [X8: set_a,A: a] :
( ( ord_less_eq_set_a @ X8 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X8 = bot_bot_set_a )
| ( X8
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_336_in__graphI,axiom,
! [M: a > option_a,K: a,V3: a] :
( ( ( M @ K )
= ( some_a @ V3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K @ V3 ) @ ( graph_a_a @ M ) ) ) ).
% in_graphI
thf(fact_337_in__graphD,axiom,
! [K: a,V3: a,M: a > option_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K @ V3 ) @ ( graph_a_a @ M ) )
=> ( ( M @ K )
= ( some_a @ V3 ) ) ) ).
% in_graphD
thf(fact_338_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_339_the__elem__eq,axiom,
! [X: option_a] :
( ( the_elem_option_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
= X ) ).
% the_elem_eq
thf(fact_340_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_341_is__singletonI,axiom,
! [X: option_a] : ( is_sin3348965821858909752tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ).
% is_singletonI
thf(fact_342_ran__map__upd,axiom,
! [M: a > option_a,A: a,B: a] :
( ( ( M @ A )
= none_a )
=> ( ( ran_a_a @ ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) ) )
= ( insert_a @ B @ ( ran_a_a @ M ) ) ) ) ).
% ran_map_upd
thf(fact_343_restrict__upd__same,axiom,
! [M: option_a > option_a,X: option_a,Y: a] :
( ( restri3984065703976872170on_a_a @ ( fun_up1079276522633388797tion_a @ M @ X @ ( some_a @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
= ( restri3984065703976872170on_a_a @ M @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ).
% restrict_upd_same
thf(fact_344_restrict__upd__same,axiom,
! [M: a > option_a,X: a,Y: a] :
( ( restrict_map_a_a @ ( fun_upd_a_option_a @ M @ X @ ( some_a @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( restrict_map_a_a @ M @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% restrict_upd_same
thf(fact_345_ComplI,axiom,
! [C: option_a,A5: set_option_a] :
( ~ ( member_option_a @ C @ A5 )
=> ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A5 ) ) ) ).
% ComplI
thf(fact_346_ComplI,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A5 ) ) ) ).
% ComplI
thf(fact_347_ComplI,axiom,
! [C: set_a,A5: set_set_a] :
( ~ ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A5 ) ) ) ).
% ComplI
thf(fact_348_ComplI,axiom,
! [C: a,A5: set_a] :
( ~ ( member_a @ C @ A5 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A5 ) ) ) ).
% ComplI
thf(fact_349_Compl__iff,axiom,
! [C: option_a,A5: set_option_a] :
( ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A5 ) )
= ( ~ ( member_option_a @ C @ A5 ) ) ) ).
% Compl_iff
thf(fact_350_Compl__iff,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A5 ) )
= ( ~ ( member1426531477525435216od_a_a @ C @ A5 ) ) ) ).
% Compl_iff
thf(fact_351_Compl__iff,axiom,
! [C: set_a,A5: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A5 ) )
= ( ~ ( member_set_a @ C @ A5 ) ) ) ).
% Compl_iff
thf(fact_352_Compl__iff,axiom,
! [C: a,A5: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A5 ) )
= ( ~ ( member_a @ C @ A5 ) ) ) ).
% Compl_iff
thf(fact_353_Compl__eq__Compl__iff,axiom,
! [A5: set_a,B6: set_a] :
( ( ( uminus_uminus_set_a @ A5 )
= ( uminus_uminus_set_a @ B6 ) )
= ( A5 = B6 ) ) ).
% Compl_eq_Compl_iff
thf(fact_354_Compl__anti__mono,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B6 ) @ ( uminus_uminus_set_a @ A5 ) ) ) ).
% Compl_anti_mono
thf(fact_355_Compl__subset__Compl__iff,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A5 ) @ ( uminus_uminus_set_a @ B6 ) )
= ( ord_less_eq_set_a @ B6 @ A5 ) ) ).
% Compl_subset_Compl_iff
thf(fact_356_subset__Compl__singleton,axiom,
! [A5: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ ( uminus5530930396987473918od_a_a @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_357_subset__Compl__singleton,axiom,
! [A5: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) )
= ( ~ ( member_set_a @ B @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_358_subset__Compl__singleton,axiom,
! [A5: set_option_a,B: option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
= ( ~ ( member_option_a @ B @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_359_subset__Compl__singleton,axiom,
! [A5: set_a,B: a] :
( ( ord_less_eq_set_a @ A5 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_360_ComplD,axiom,
! [C: option_a,A5: set_option_a] :
( ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A5 ) )
=> ~ ( member_option_a @ C @ A5 ) ) ).
% ComplD
thf(fact_361_ComplD,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A5 ) )
=> ~ ( member1426531477525435216od_a_a @ C @ A5 ) ) ).
% ComplD
thf(fact_362_ComplD,axiom,
! [C: set_a,A5: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A5 ) )
=> ~ ( member_set_a @ C @ A5 ) ) ).
% ComplD
thf(fact_363_ComplD,axiom,
! [C: a,A5: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A5 ) )
=> ~ ( member_a @ C @ A5 ) ) ).
% ComplD
thf(fact_364_double__complement,axiom,
! [A5: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A5 ) )
= A5 ) ).
% double_complement
thf(fact_365_subset__Compl__self__eq,axiom,
! [A5: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ ( uminus6205308855922866075tion_a @ A5 ) )
= ( A5 = bot_bot_set_option_a ) ) ).
% subset_Compl_self_eq
thf(fact_366_subset__Compl__self__eq,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( uminus_uminus_set_a @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_367_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A8: set_a] :
( A8
= ( insert_a @ ( the_elem_a @ A8 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_368_is__singleton__the__elem,axiom,
( is_sin3348965821858909752tion_a
= ( ^ [A8: set_option_a] :
( A8
= ( insert_option_a @ ( the_elem_option_a @ A8 ) @ bot_bot_set_option_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_369_is__singletonI_H,axiom,
! [A5: set_Product_prod_a_a] :
( ( A5 != bot_bo3357376287454694259od_a_a )
=> ( ! [X5: product_prod_a_a,Y4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A5 )
=> ( ( member1426531477525435216od_a_a @ Y4 @ A5 )
=> ( X5 = Y4 ) ) )
=> ( is_sin3171834905898671131od_a_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_370_is__singletonI_H,axiom,
! [A5: set_set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ! [X5: set_a,Y4: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( ( member_set_a @ Y4 @ A5 )
=> ( X5 = Y4 ) ) )
=> ( is_singleton_set_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_371_is__singletonI_H,axiom,
! [A5: set_option_a] :
( ( A5 != bot_bot_set_option_a )
=> ( ! [X5: option_a,Y4: option_a] :
( ( member_option_a @ X5 @ A5 )
=> ( ( member_option_a @ Y4 @ A5 )
=> ( X5 = Y4 ) ) )
=> ( is_sin3348965821858909752tion_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_372_is__singletonI_H,axiom,
! [A5: set_a] :
( ( A5 != bot_bot_set_a )
=> ( ! [X5: a,Y4: a] :
( ( member_a @ X5 @ A5 )
=> ( ( member_a @ Y4 @ A5 )
=> ( X5 = Y4 ) ) )
=> ( is_singleton_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_373_ran__restrictD,axiom,
! [Y: a,M: a > option_a,A5: set_a] :
( ( member_a @ Y @ ( ran_a_a @ ( restrict_map_a_a @ M @ A5 ) ) )
=> ? [X5: a] :
( ( member_a @ X5 @ A5 )
& ( ( M @ X5 )
= ( some_a @ Y ) ) ) ) ).
% ran_restrictD
thf(fact_374_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A8: set_a] :
? [X4: a] :
( A8
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_375_is__singleton__def,axiom,
( is_sin3348965821858909752tion_a
= ( ^ [A8: set_option_a] :
? [X4: option_a] :
( A8
= ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) ) ).
% is_singleton_def
thf(fact_376_is__singletonE,axiom,
! [A5: set_a] :
( ( is_singleton_a @ A5 )
=> ~ ! [X5: a] :
( A5
!= ( insert_a @ X5 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_377_is__singletonE,axiom,
! [A5: set_option_a] :
( ( is_sin3348965821858909752tion_a @ A5 )
=> ~ ! [X5: option_a] :
( A5
!= ( insert_option_a @ X5 @ bot_bot_set_option_a ) ) ) ).
% is_singletonE
thf(fact_378_restrict__complement__singleton__eq,axiom,
! [F: option_a > option_a,X: option_a] :
( ( restri3984065703976872170on_a_a @ F @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
= ( fun_up1079276522633388797tion_a @ F @ X @ none_a ) ) ).
% restrict_complement_singleton_eq
thf(fact_379_restrict__complement__singleton__eq,axiom,
! [F: a > option_a,X: a] :
( ( restrict_map_a_a @ F @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( fun_upd_a_option_a @ F @ X @ none_a ) ) ).
% restrict_complement_singleton_eq
thf(fact_380_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_381_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_382_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_383_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_384_fun__upd__None__restrict,axiom,
! [X: product_prod_a_a,D: set_Product_prod_a_a,M: product_prod_a_a > option_a] :
( ( ( member1426531477525435216od_a_a @ X @ D )
=> ( ( fun_up8298456451713467738tion_a @ ( restri3846233575700461639_a_a_a @ M @ D ) @ X @ none_a )
= ( restri3846233575700461639_a_a_a @ M @ ( minus_6817036919807184750od_a_a @ D @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ D )
=> ( ( fun_up8298456451713467738tion_a @ ( restri3846233575700461639_a_a_a @ M @ D ) @ X @ none_a )
= ( restri3846233575700461639_a_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_385_fun__upd__None__restrict,axiom,
! [X: set_a,D: set_set_a,M: set_a > option_a] :
( ( ( member_set_a @ X @ D )
=> ( ( fun_up3663993102702442083tion_a @ ( restrict_map_set_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_set_a_a @ M @ ( minus_5736297505244876581_set_a @ D @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) )
& ( ~ ( member_set_a @ X @ D )
=> ( ( fun_up3663993102702442083tion_a @ ( restrict_map_set_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_set_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_386_fun__upd__None__restrict,axiom,
! [X: option_a,D: set_option_a,M: option_a > option_a] :
( ( ( member_option_a @ X @ D )
=> ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D ) @ X @ none_a )
= ( restri3984065703976872170on_a_a @ M @ ( minus_1574173051537231627tion_a @ D @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) )
& ( ~ ( member_option_a @ X @ D )
=> ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D ) @ X @ none_a )
= ( restri3984065703976872170on_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_387_fun__upd__None__restrict,axiom,
! [X: a,D: set_a,M: a > option_a] :
( ( ( member_a @ X @ D )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) )
& ( ~ ( member_a @ X @ D )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_388_refl__on__singleton,axiom,
! [X: a] : ( refl_on_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X @ X ) @ bot_bo3357376287454694259od_a_a ) ) ).
% refl_on_singleton
thf(fact_389_refl__on__singleton,axiom,
! [X: option_a] : ( refl_on_option_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X @ X ) @ bot_bo235252021745139059tion_a ) ) ).
% refl_on_singleton
thf(fact_390_fun__upd__restrict__conv,axiom,
! [X: a,D: set_a,M: a > option_a,Y: option_a] :
( ( member_a @ X @ D )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D ) @ X @ Y )
= ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_391_DiffI,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ A5 )
=> ( ~ ( member_option_a @ C @ B6 )
=> ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A5 @ B6 ) ) ) ) ).
% DiffI
thf(fact_392_DiffI,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( ~ ( member1426531477525435216od_a_a @ C @ B6 )
=> ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A5 @ B6 ) ) ) ) ).
% DiffI
thf(fact_393_DiffI,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( ~ ( member_set_a @ C @ B6 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) ) ) ) ).
% DiffI
thf(fact_394_DiffI,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ A5 )
=> ( ~ ( member_a @ C @ B6 )
=> ( member_a @ C @ ( minus_minus_set_a @ A5 @ B6 ) ) ) ) ).
% DiffI
thf(fact_395_Diff__iff,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A5 @ B6 ) )
= ( ( member_option_a @ C @ A5 )
& ~ ( member_option_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_396_Diff__iff,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A5 @ B6 ) )
= ( ( member1426531477525435216od_a_a @ C @ A5 )
& ~ ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_397_Diff__iff,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) )
= ( ( member_set_a @ C @ A5 )
& ~ ( member_set_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_398_Diff__iff,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B6 ) )
= ( ( member_a @ C @ A5 )
& ~ ( member_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_399_Diff__idemp,axiom,
! [A5: set_a,B6: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ B6 )
= ( minus_minus_set_a @ A5 @ B6 ) ) ).
% Diff_idemp
thf(fact_400_Diff__cancel,axiom,
! [A5: set_option_a] :
( ( minus_1574173051537231627tion_a @ A5 @ A5 )
= bot_bot_set_option_a ) ).
% Diff_cancel
thf(fact_401_Diff__cancel,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ A5 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_402_empty__Diff,axiom,
! [A5: set_option_a] :
( ( minus_1574173051537231627tion_a @ bot_bot_set_option_a @ A5 )
= bot_bot_set_option_a ) ).
% empty_Diff
thf(fact_403_empty__Diff,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A5 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_404_Diff__empty,axiom,
! [A5: set_option_a] :
( ( minus_1574173051537231627tion_a @ A5 @ bot_bot_set_option_a )
= A5 ) ).
% Diff_empty
thf(fact_405_Diff__empty,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ bot_bot_set_a )
= A5 ) ).
% Diff_empty
thf(fact_406_Diff__insert0,axiom,
! [X: option_a,A5: set_option_a,B6: set_option_a] :
( ~ ( member_option_a @ X @ A5 )
=> ( ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X @ B6 ) )
= ( minus_1574173051537231627tion_a @ A5 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_407_Diff__insert0,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( minus_6817036919807184750od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( minus_6817036919807184750od_a_a @ A5 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_408_Diff__insert0,axiom,
! [X: set_a,A5: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ B6 ) )
= ( minus_5736297505244876581_set_a @ A5 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_409_Diff__insert0,axiom,
! [X: a,A5: set_a,B6: set_a] :
( ~ ( member_a @ X @ A5 )
=> ( ( minus_minus_set_a @ A5 @ ( insert_a @ X @ B6 ) )
= ( minus_minus_set_a @ A5 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_410_insert__Diff1,axiom,
! [X: option_a,B6: set_option_a,A5: set_option_a] :
( ( member_option_a @ X @ B6 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A5 ) @ B6 )
= ( minus_1574173051537231627tion_a @ A5 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_411_insert__Diff1,axiom,
! [X: product_prod_a_a,B6: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X @ A5 ) @ B6 )
= ( minus_6817036919807184750od_a_a @ A5 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_412_insert__Diff1,axiom,
! [X: set_a,B6: set_set_a,A5: set_set_a] :
( ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A5 ) @ B6 )
= ( minus_5736297505244876581_set_a @ A5 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_413_insert__Diff1,axiom,
! [X: a,B6: set_a,A5: set_a] :
( ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A5 ) @ B6 )
= ( minus_minus_set_a @ A5 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_414_Diff__eq__empty__iff,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( ( minus_1574173051537231627tion_a @ A5 @ B6 )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ A5 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_415_Diff__eq__empty__iff,axiom,
! [A5: set_a,B6: set_a] :
( ( ( minus_minus_set_a @ A5 @ B6 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A5 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_416_insert__Diff__single,axiom,
! [A: option_a,A5: set_option_a] :
( ( insert_option_a @ A @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( insert_option_a @ A @ A5 ) ) ).
% insert_Diff_single
thf(fact_417_insert__Diff__single,axiom,
! [A: a,A5: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A5 ) ) ).
% insert_Diff_single
thf(fact_418_restrict__fun__upd,axiom,
! [X: a,D: set_a,M: a > option_a,Y: option_a] :
( ( ( member_a @ X @ D )
=> ( ( restrict_map_a_a @ ( fun_upd_a_option_a @ M @ X @ Y ) @ D )
= ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ X @ Y ) ) )
& ( ~ ( member_a @ X @ D )
=> ( ( restrict_map_a_a @ ( fun_upd_a_option_a @ M @ X @ Y ) @ D )
= ( restrict_map_a_a @ M @ D ) ) ) ) ).
% restrict_fun_upd
thf(fact_419_Diff__mono,axiom,
! [A5: set_a,C3: set_a,D: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ D @ B6 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ ( minus_minus_set_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_420_Diff__subset,axiom,
! [A5: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ A5 ) ).
% Diff_subset
thf(fact_421_double__diff,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( minus_minus_set_a @ B6 @ ( minus_minus_set_a @ C3 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_422_DiffE,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A5 @ B6 ) )
=> ~ ( ( member_option_a @ C @ A5 )
=> ( member_option_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_423_DiffE,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A5 @ B6 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_424_DiffE,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) )
=> ~ ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_425_DiffE,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B6 ) )
=> ~ ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_426_DiffD1,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A5 @ B6 ) )
=> ( member_option_a @ C @ A5 ) ) ).
% DiffD1
thf(fact_427_DiffD1,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A5 @ B6 ) )
=> ( member1426531477525435216od_a_a @ C @ A5 ) ) ).
% DiffD1
thf(fact_428_DiffD1,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) )
=> ( member_set_a @ C @ A5 ) ) ).
% DiffD1
thf(fact_429_DiffD1,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B6 ) )
=> ( member_a @ C @ A5 ) ) ).
% DiffD1
thf(fact_430_DiffD2,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A5 @ B6 ) )
=> ~ ( member_option_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_431_DiffD2,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A5 @ B6 ) )
=> ~ ( member1426531477525435216od_a_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_432_DiffD2,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) )
=> ~ ( member_set_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_433_DiffD2,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B6 ) )
=> ~ ( member_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_434_insert__Diff__if,axiom,
! [X: option_a,B6: set_option_a,A5: set_option_a] :
( ( ( member_option_a @ X @ B6 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A5 ) @ B6 )
= ( minus_1574173051537231627tion_a @ A5 @ B6 ) ) )
& ( ~ ( member_option_a @ X @ B6 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A5 ) @ B6 )
= ( insert_option_a @ X @ ( minus_1574173051537231627tion_a @ A5 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_435_insert__Diff__if,axiom,
! [X: product_prod_a_a,B6: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X @ A5 ) @ B6 )
= ( minus_6817036919807184750od_a_a @ A5 @ B6 ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X @ A5 ) @ B6 )
= ( insert4534936382041156343od_a_a @ X @ ( minus_6817036919807184750od_a_a @ A5 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_436_insert__Diff__if,axiom,
! [X: set_a,B6: set_set_a,A5: set_set_a] :
( ( ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A5 ) @ B6 )
= ( minus_5736297505244876581_set_a @ A5 @ B6 ) ) )
& ( ~ ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A5 ) @ B6 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_437_insert__Diff__if,axiom,
! [X: a,B6: set_a,A5: set_a] :
( ( ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A5 ) @ B6 )
= ( minus_minus_set_a @ A5 @ B6 ) ) )
& ( ~ ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A5 ) @ B6 )
= ( insert_a @ X @ ( minus_minus_set_a @ A5 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_438_diff__shunt__var,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( minus_minus_a_o @ X @ Y )
= bot_bot_a_o )
= ( ord_less_eq_a_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_439_diff__shunt__var,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( minus_1574173051537231627tion_a @ X @ Y )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_440_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_441_Diff__insert,axiom,
! [A5: set_option_a,A: option_a,B6: set_option_a] :
( ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ B6 ) )
= ( minus_1574173051537231627tion_a @ ( minus_1574173051537231627tion_a @ A5 @ B6 ) @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ).
% Diff_insert
thf(fact_442_Diff__insert,axiom,
! [A5: set_a,A: a,B6: set_a] :
( ( minus_minus_set_a @ A5 @ ( insert_a @ A @ B6 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_443_insert__Diff,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ( insert4534936382041156343od_a_a @ A @ ( minus_6817036919807184750od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_444_insert__Diff,axiom,
! [A: set_a,A5: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_445_insert__Diff,axiom,
! [A: option_a,A5: set_option_a] :
( ( member_option_a @ A @ A5 )
=> ( ( insert_option_a @ A @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_446_insert__Diff,axiom,
! [A: a,A5: set_a] :
( ( member_a @ A @ A5 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_447_Diff__insert2,axiom,
! [A5: set_option_a,A: option_a,B6: set_option_a] :
( ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ B6 ) )
= ( minus_1574173051537231627tion_a @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_448_Diff__insert2,axiom,
! [A5: set_a,A: a,B6: set_a] :
( ( minus_minus_set_a @ A5 @ ( insert_a @ A @ B6 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_449_Diff__insert__absorb,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X @ A5 ) @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_450_Diff__insert__absorb,axiom,
! [X: set_a,A5: set_set_a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A5 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_451_Diff__insert__absorb,axiom,
! [X: option_a,A5: set_option_a] :
( ~ ( member_option_a @ X @ A5 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A5 ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_452_Diff__insert__absorb,axiom,
! [X: a,A5: set_a] :
( ~ ( member_a @ X @ A5 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A5 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_453_subset__Diff__insert,axiom,
! [A5: set_option_a,B6: set_option_a,X: option_a,C3: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ ( minus_1574173051537231627tion_a @ B6 @ ( insert_option_a @ X @ C3 ) ) )
= ( ( ord_le1955136853071979460tion_a @ A5 @ ( minus_1574173051537231627tion_a @ B6 @ C3 ) )
& ~ ( member_option_a @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_454_subset__Diff__insert,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a,X: product_prod_a_a,C3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ ( minus_6817036919807184750od_a_a @ B6 @ ( insert4534936382041156343od_a_a @ X @ C3 ) ) )
= ( ( ord_le746702958409616551od_a_a @ A5 @ ( minus_6817036919807184750od_a_a @ B6 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_455_subset__Diff__insert,axiom,
! [A5: set_set_a,B6: set_set_a,X: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( minus_5736297505244876581_set_a @ B6 @ ( insert_set_a @ X @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A5 @ ( minus_5736297505244876581_set_a @ B6 @ C3 ) )
& ~ ( member_set_a @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_456_subset__Diff__insert,axiom,
! [A5: set_a,B6: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( minus_minus_set_a @ B6 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A5 @ ( minus_minus_set_a @ B6 @ C3 ) )
& ~ ( member_a @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_457_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% refl_on_empty
thf(fact_458_refl__on__empty,axiom,
refl_on_option_a @ bot_bot_set_option_a @ bot_bo235252021745139059tion_a ).
% refl_on_empty
thf(fact_459_Diff__single__insert,axiom,
! [A5: set_option_a,X: option_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B6 )
=> ( ord_le1955136853071979460tion_a @ A5 @ ( insert_option_a @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_460_Diff__single__insert,axiom,
! [A5: set_a,X: a,B6: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 )
=> ( ord_less_eq_set_a @ A5 @ ( insert_a @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_461_subset__insert__iff,axiom,
! [A5: set_Product_prod_a_a,X: product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( ( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) @ B6 ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ord_le746702958409616551od_a_a @ A5 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_462_subset__insert__iff,axiom,
! [A5: set_set_a,X: set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( insert_set_a @ X @ B6 ) )
= ( ( ( member_set_a @ X @ A5 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B6 ) )
& ( ~ ( member_set_a @ X @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_463_subset__insert__iff,axiom,
! [A5: set_option_a,X: option_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A5 @ ( insert_option_a @ X @ B6 ) )
= ( ( ( member_option_a @ X @ A5 )
=> ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B6 ) )
& ( ~ ( member_option_a @ X @ A5 )
=> ( ord_le1955136853071979460tion_a @ A5 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_464_subset__insert__iff,axiom,
! [A5: set_a,X: a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X @ B6 ) )
= ( ( ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 ) )
& ( ~ ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ A5 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_465_Compl__insert,axiom,
! [X: option_a,A5: set_option_a] :
( ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ A5 ) )
= ( minus_1574173051537231627tion_a @ ( uminus6205308855922866075tion_a @ A5 ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ).
% Compl_insert
thf(fact_466_Compl__insert,axiom,
! [X: a,A5: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X @ A5 ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A5 ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_467_fun__upd__restrict,axiom,
! [M: a > option_a,D: set_a,X: a,Y: option_a] :
( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D ) @ X @ Y )
= ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_468_remove__def,axiom,
( remove_option_a
= ( ^ [X4: option_a,A8: set_option_a] : ( minus_1574173051537231627tion_a @ A8 @ ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) ) ).
% remove_def
thf(fact_469_remove__def,axiom,
( remove_a
= ( ^ [X4: a,A8: set_a] : ( minus_minus_set_a @ A8 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_470_dom__fun__upd,axiom,
! [Y: option_a,F: option_a > option_a,X: option_a] :
( ( ( Y = none_a )
=> ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) )
= ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) )
& ( ( Y != none_a )
=> ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) )
= ( insert_option_a @ X @ ( dom_option_a_a @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_471_dom__fun__upd,axiom,
! [Y: option_a,F: a > option_a,X: a] :
( ( ( Y = none_a )
=> ( ( dom_a_a @ ( fun_upd_a_option_a @ F @ X @ Y ) )
= ( minus_minus_set_a @ ( dom_a_a @ F ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
& ( ( Y != none_a )
=> ( ( dom_a_a @ ( fun_upd_a_option_a @ F @ X @ Y ) )
= ( insert_a @ X @ ( dom_a_a @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_472_these__insert__Some,axiom,
! [X: option_a,A5: set_option_option_a] :
( ( these_option_a @ ( insert605063979879581146tion_a @ ( some_option_a @ X ) @ A5 ) )
= ( insert_option_a @ X @ ( these_option_a @ A5 ) ) ) ).
% these_insert_Some
thf(fact_473_these__insert__Some,axiom,
! [X: a,A5: set_option_a] :
( ( these_a @ ( insert_option_a @ ( some_a @ X ) @ A5 ) )
= ( insert_a @ X @ ( these_a @ A5 ) ) ) ).
% these_insert_Some
thf(fact_474_member__remove,axiom,
! [X: option_a,Y: option_a,A5: set_option_a] :
( ( member_option_a @ X @ ( remove_option_a @ Y @ A5 ) )
= ( ( member_option_a @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_475_member__remove,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( remove8198300757409973004od_a_a @ Y @ A5 ) )
= ( ( member1426531477525435216od_a_a @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_476_member__remove,axiom,
! [X: set_a,Y: set_a,A5: set_set_a] :
( ( member_set_a @ X @ ( remove_set_a @ Y @ A5 ) )
= ( ( member_set_a @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_477_member__remove,axiom,
! [X: a,Y: a,A5: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A5 ) )
= ( ( member_a @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_478_these__empty,axiom,
( ( these_option_a @ bot_bo4163488203964334806tion_a )
= bot_bot_set_option_a ) ).
% these_empty
thf(fact_479_these__empty,axiom,
( ( these_a @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% these_empty
thf(fact_480_dom__eq__empty__conv,axiom,
! [F: a > option_a] :
( ( ( dom_a_a @ F )
= bot_bot_set_a )
= ( F
= ( ^ [X4: a] : none_a ) ) ) ).
% dom_eq_empty_conv
thf(fact_481_dom__eq__empty__conv,axiom,
! [F: option_a > option_a] :
( ( ( dom_option_a_a @ F )
= bot_bot_set_option_a )
= ( F
= ( ^ [X4: option_a] : none_a ) ) ) ).
% dom_eq_empty_conv
thf(fact_482_fun__upd__None__if__notin__dom,axiom,
! [K: option_a,M: option_a > option_a] :
( ~ ( member_option_a @ K @ ( dom_option_a_a @ M ) )
=> ( ( fun_up1079276522633388797tion_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_483_fun__upd__None__if__notin__dom,axiom,
! [K: product_prod_a_a,M: product_prod_a_a > option_a] :
( ~ ( member1426531477525435216od_a_a @ K @ ( dom_Pr6378187988305063785_a_a_a @ M ) )
=> ( ( fun_up8298456451713467738tion_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_484_fun__upd__None__if__notin__dom,axiom,
! [K: set_a,M: set_a > option_a] :
( ~ ( member_set_a @ K @ ( dom_set_a_a @ M ) )
=> ( ( fun_up3663993102702442083tion_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_485_fun__upd__None__if__notin__dom,axiom,
! [K: a,M: a > option_a] :
( ~ ( member_a @ K @ ( dom_a_a @ M ) )
=> ( ( fun_upd_a_option_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_486_these__insert__None,axiom,
! [A5: set_option_a] :
( ( these_a @ ( insert_option_a @ none_a @ A5 ) )
= ( these_a @ A5 ) ) ).
% these_insert_None
thf(fact_487_domD,axiom,
! [A: option_a,M: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
=> ? [B5: a] :
( ( M @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_488_domD,axiom,
! [A: product_prod_a_a,M: product_prod_a_a > option_a] :
( ( member1426531477525435216od_a_a @ A @ ( dom_Pr6378187988305063785_a_a_a @ M ) )
=> ? [B5: a] :
( ( M @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_489_domD,axiom,
! [A: set_a,M: set_a > option_a] :
( ( member_set_a @ A @ ( dom_set_a_a @ M ) )
=> ? [B5: a] :
( ( M @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_490_domD,axiom,
! [A: a,M: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M ) )
=> ? [B5: a] :
( ( M @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_491_domI,axiom,
! [M: option_a > option_a,A: option_a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_option_a @ A @ ( dom_option_a_a @ M ) ) ) ).
% domI
thf(fact_492_domI,axiom,
! [M: product_prod_a_a > option_a,A: product_prod_a_a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member1426531477525435216od_a_a @ A @ ( dom_Pr6378187988305063785_a_a_a @ M ) ) ) ).
% domI
thf(fact_493_domI,axiom,
! [M: set_a > option_a,A: set_a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_set_a @ A @ ( dom_set_a_a @ M ) ) ) ).
% domI
thf(fact_494_domI,axiom,
! [M: a > option_a,A: a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_a @ A @ ( dom_a_a @ M ) ) ) ).
% domI
thf(fact_495_domIff,axiom,
! [A: option_a,M: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_496_domIff,axiom,
! [A: product_prod_a_a,M: product_prod_a_a > option_a] :
( ( member1426531477525435216od_a_a @ A @ ( dom_Pr6378187988305063785_a_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_497_domIff,axiom,
! [A: set_a,M: set_a > option_a] :
( ( member_set_a @ A @ ( dom_set_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_498_domIff,axiom,
! [A: a,M: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_499_insert__dom,axiom,
! [F: a > option_a,X: a,Y: a] :
( ( ( F @ X )
= ( some_a @ Y ) )
=> ( ( insert_a @ X @ ( dom_a_a @ F ) )
= ( dom_a_a @ F ) ) ) ).
% insert_dom
thf(fact_500_insert__dom,axiom,
! [F: option_a > option_a,X: option_a,Y: a] :
( ( ( F @ X )
= ( some_a @ Y ) )
=> ( ( insert_option_a @ X @ ( dom_option_a_a @ F ) )
= ( dom_option_a_a @ F ) ) ) ).
% insert_dom
thf(fact_501_in__these__eq,axiom,
! [X: option_a,A5: set_option_option_a] :
( ( member_option_a @ X @ ( these_option_a @ A5 ) )
= ( member5113800082084363315tion_a @ ( some_option_a @ X ) @ A5 ) ) ).
% in_these_eq
thf(fact_502_in__these__eq,axiom,
! [X: product_prod_a_a,A5: set_op7160277562814721357od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( these_5100388957577570148od_a_a @ A5 ) )
= ( member8183384484874023062od_a_a @ ( some_P3592067295195376908od_a_a @ X ) @ A5 ) ) ).
% in_these_eq
thf(fact_503_in__these__eq,axiom,
! [X: set_a,A5: set_option_set_a] :
( ( member_set_a @ X @ ( these_set_a @ A5 ) )
= ( member_option_set_a @ ( some_set_a @ X ) @ A5 ) ) ).
% in_these_eq
thf(fact_504_in__these__eq,axiom,
! [X: a,A5: set_option_a] :
( ( member_a @ X @ ( these_a @ A5 ) )
= ( member_option_a @ ( some_a @ X ) @ A5 ) ) ).
% in_these_eq
thf(fact_505_map__le__implies__dom__le,axiom,
! [F: a > option_a,G: a > option_a] :
( ( map_le_a_a @ F @ G )
=> ( ord_less_eq_set_a @ ( dom_a_a @ F ) @ ( dom_a_a @ G ) ) ) ).
% map_le_implies_dom_le
thf(fact_506_dom__minus,axiom,
! [F: option_a > option_a,X: option_a,A5: set_option_a] :
( ( ( F @ X )
= none_a )
=> ( ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X @ A5 ) )
= ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_507_dom__minus,axiom,
! [F: a > option_a,X: a,A5: set_a] :
( ( ( F @ X )
= none_a )
=> ( ( minus_minus_set_a @ ( dom_a_a @ F ) @ ( insert_a @ X @ A5 ) )
= ( minus_minus_set_a @ ( dom_a_a @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_508_these__not__empty__eq,axiom,
! [B6: set_option_option_a] :
( ( ( these_option_a @ B6 )
!= bot_bot_set_option_a )
= ( ( B6 != bot_bo4163488203964334806tion_a )
& ( B6
!= ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_509_these__not__empty__eq,axiom,
! [B6: set_option_a] :
( ( ( these_a @ B6 )
!= bot_bot_set_a )
= ( ( B6 != bot_bot_set_option_a )
& ( B6
!= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_510_these__empty__eq,axiom,
! [B6: set_option_option_a] :
( ( ( these_option_a @ B6 )
= bot_bot_set_option_a )
= ( ( B6 = bot_bo4163488203964334806tion_a )
| ( B6
= ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).
% these_empty_eq
thf(fact_511_these__empty__eq,axiom,
! [B6: set_option_a] :
( ( ( these_a @ B6 )
= bot_bot_set_a )
= ( ( B6 = bot_bot_set_option_a )
| ( B6
= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_empty_eq
thf(fact_512_linear__order__on__singleton,axiom,
! [X: a] : ( order_8768733634509060147r_on_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X @ X ) @ bot_bo3357376287454694259od_a_a ) ) ).
% linear_order_on_singleton
thf(fact_513_linear__order__on__singleton,axiom,
! [X: option_a] : ( order_7850372301378808569tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X @ X ) @ bot_bo235252021745139059tion_a ) ) ).
% linear_order_on_singleton
thf(fact_514_pairwise__alt,axiom,
( pairwise_option_a
= ( ^ [R4: option_a > option_a > $o,S4: set_option_a] :
! [X4: option_a] :
( ( member_option_a @ X4 @ S4 )
=> ! [Y3: option_a] :
( ( member_option_a @ Y3 @ ( minus_1574173051537231627tion_a @ S4 @ ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) )
=> ( R4 @ X4 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_515_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R4: a > a > $o,S4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ S4 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( minus_minus_set_a @ S4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
=> ( R4 @ X4 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_516_psubset__insert__iff,axiom,
! [A5: set_Product_prod_a_a,X: product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( ( ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ord_le6819997720685908915od_a_a @ A5 @ B6 ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ord_le6819997720685908915od_a_a @ ( minus_6817036919807184750od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) @ B6 ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ord_le746702958409616551od_a_a @ A5 @ B6 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_517_psubset__insert__iff,axiom,
! [A5: set_set_a,X: set_a,B6: set_set_a] :
( ( ord_less_set_set_a @ A5 @ ( insert_set_a @ X @ B6 ) )
= ( ( ( member_set_a @ X @ B6 )
=> ( ord_less_set_set_a @ A5 @ B6 ) )
& ( ~ ( member_set_a @ X @ B6 )
=> ( ( ( member_set_a @ X @ A5 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B6 ) )
& ( ~ ( member_set_a @ X @ A5 )
=> ( ord_le3724670747650509150_set_a @ A5 @ B6 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_518_psubset__insert__iff,axiom,
! [A5: set_option_a,X: option_a,B6: set_option_a] :
( ( ord_le5631237216984945872tion_a @ A5 @ ( insert_option_a @ X @ B6 ) )
= ( ( ( member_option_a @ X @ B6 )
=> ( ord_le5631237216984945872tion_a @ A5 @ B6 ) )
& ( ~ ( member_option_a @ X @ B6 )
=> ( ( ( member_option_a @ X @ A5 )
=> ( ord_le5631237216984945872tion_a @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B6 ) )
& ( ~ ( member_option_a @ X @ A5 )
=> ( ord_le1955136853071979460tion_a @ A5 @ B6 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_519_psubset__insert__iff,axiom,
! [A5: set_a,X: a,B6: set_a] :
( ( ord_less_set_a @ A5 @ ( insert_a @ X @ B6 ) )
= ( ( ( member_a @ X @ B6 )
=> ( ord_less_set_a @ A5 @ B6 ) )
& ( ~ ( member_a @ X @ B6 )
=> ( ( ( member_a @ X @ A5 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 ) )
& ( ~ ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ A5 @ B6 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_520_psubsetI,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( A5 != B6 )
=> ( ord_less_set_a @ A5 @ B6 ) ) ) ).
% psubsetI
thf(fact_521_psubset__imp__ex__mem,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( ord_le5631237216984945872tion_a @ A5 @ B6 )
=> ? [B5: option_a] : ( member_option_a @ B5 @ ( minus_1574173051537231627tion_a @ B6 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_522_psubset__imp__ex__mem,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A5 @ B6 )
=> ? [B5: product_prod_a_a] : ( member1426531477525435216od_a_a @ B5 @ ( minus_6817036919807184750od_a_a @ B6 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_523_psubset__imp__ex__mem,axiom,
! [A5: set_set_a,B6: set_set_a] :
( ( ord_less_set_set_a @ A5 @ B6 )
=> ? [B5: set_a] : ( member_set_a @ B5 @ ( minus_5736297505244876581_set_a @ B6 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_524_psubset__imp__ex__mem,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_set_a @ A5 @ B6 )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B6 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_525_pairwise__insert,axiom,
! [R2: option_a > option_a > $o,X: option_a,S3: set_option_a] :
( ( pairwise_option_a @ R2 @ ( insert_option_a @ X @ S3 ) )
= ( ! [Y3: option_a] :
( ( ( member_option_a @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise_option_a @ R2 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_526_pairwise__insert,axiom,
! [R2: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a,S3: set_Product_prod_a_a] :
( ( pairwi8942280066358138514od_a_a @ R2 @ ( insert4534936382041156343od_a_a @ X @ S3 ) )
= ( ! [Y3: product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwi8942280066358138514od_a_a @ R2 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_527_pairwise__insert,axiom,
! [R2: set_a > set_a > $o,X: set_a,S3: set_set_a] :
( ( pairwise_set_a @ R2 @ ( insert_set_a @ X @ S3 ) )
= ( ! [Y3: set_a] :
( ( ( member_set_a @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise_set_a @ R2 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_528_pairwise__insert,axiom,
! [R2: a > a > $o,X: a,S3: set_a] :
( ( pairwise_a @ R2 @ ( insert_a @ X @ S3 ) )
= ( ! [Y3: a] :
( ( ( member_a @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise_a @ R2 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_529_lnear__order__on__empty,axiom,
order_8768733634509060147r_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% lnear_order_on_empty
thf(fact_530_lnear__order__on__empty,axiom,
order_7850372301378808569tion_a @ bot_bot_set_option_a @ bot_bo235252021745139059tion_a ).
% lnear_order_on_empty
thf(fact_531_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_532_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_533_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_534_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_535_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
& ~ ( ord_less_eq_set_a @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_536_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
( ( ord_less_set_a @ A6 @ B4 )
| ( A6 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_537_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A6 @ B4 )
& ( A6 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_538_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_539_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_540_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A6 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_541_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A6: set_a] :
( ( ord_less_set_a @ B4 @ A6 )
| ( A6 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_542_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A6: set_a] :
( ( ord_less_eq_set_a @ B4 @ A6 )
& ( A6 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_543_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_544_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_545_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A6: set_a] :
( ( ord_less_eq_set_a @ B4 @ A6 )
& ~ ( ord_less_eq_set_a @ A6 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_546_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_547_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_548_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y3: set_a] :
( ( ord_less_set_a @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_549_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_550_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_551_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_552_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_553_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_554_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_555_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_set_a @ X5 @ Y4 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_556_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_557_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_558_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_set_a @ X5 @ Y4 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_559_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_560_psubsetE,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_set_a @ A5 @ B6 )
=> ~ ( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ord_less_eq_set_a @ B6 @ A5 ) ) ) ).
% psubsetE
thf(fact_561_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A8: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A8 @ B7 )
& ( A8 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_562_psubset__imp__subset,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_set_a @ A5 @ B6 )
=> ( ord_less_eq_set_a @ A5 @ B6 ) ) ).
% psubset_imp_subset
thf(fact_563_psubset__subset__trans,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( ord_less_set_a @ A5 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ord_less_set_a @ A5 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_564_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A8: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A8 @ B7 )
& ~ ( ord_less_eq_set_a @ B7 @ A8 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_565_subset__psubset__trans,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( ord_less_set_a @ B6 @ C3 )
=> ( ord_less_set_a @ A5 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_566_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B7: set_a] :
( ( ord_less_set_a @ A8 @ B7 )
| ( A8 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_567_not__psubset__empty,axiom,
! [A5: set_option_a] :
~ ( ord_le5631237216984945872tion_a @ A5 @ bot_bot_set_option_a ) ).
% not_psubset_empty
thf(fact_568_not__psubset__empty,axiom,
! [A5: set_a] :
~ ( ord_less_set_a @ A5 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_569_bot_Onot__eq__extremum,axiom,
! [A: a > $o] :
( ( A != bot_bot_a_o )
= ( ord_less_a_o @ bot_bot_a_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_570_bot_Onot__eq__extremum,axiom,
! [A: set_option_a] :
( ( A != bot_bot_set_option_a )
= ( ord_le5631237216984945872tion_a @ bot_bot_set_option_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_571_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_572_bot_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_573_bot_Oextremum__strict,axiom,
! [A: set_option_a] :
~ ( ord_le5631237216984945872tion_a @ A @ bot_bot_set_option_a ) ).
% bot.extremum_strict
thf(fact_574_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_575_less__imp__neq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_576_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_577_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_578_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_579_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_580_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_581_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_582_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_583_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_584_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_585_psubsetD,axiom,
! [A5: set_option_a,B6: set_option_a,C: option_a] :
( ( ord_le5631237216984945872tion_a @ A5 @ B6 )
=> ( ( member_option_a @ C @ A5 )
=> ( member_option_a @ C @ B6 ) ) ) ).
% psubsetD
thf(fact_586_psubsetD,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A5 @ B6 )
=> ( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% psubsetD
thf(fact_587_psubsetD,axiom,
! [A5: set_set_a,B6: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A5 @ B6 )
=> ( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B6 ) ) ) ).
% psubsetD
thf(fact_588_psubsetD,axiom,
! [A5: set_a,B6: set_a,C: a] :
( ( ord_less_set_a @ A5 @ B6 )
=> ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B6 ) ) ) ).
% psubsetD
thf(fact_589_pairwiseD,axiom,
! [R5: option_a > option_a > $o,S5: set_option_a,X: option_a,Y: option_a] :
( ( pairwise_option_a @ R5 @ S5 )
=> ( ( member_option_a @ X @ S5 )
=> ( ( member_option_a @ Y @ S5 )
=> ( ( X != Y )
=> ( R5 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_590_pairwiseD,axiom,
! [R5: product_prod_a_a > product_prod_a_a > $o,S5: set_Product_prod_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( pairwi8942280066358138514od_a_a @ R5 @ S5 )
=> ( ( member1426531477525435216od_a_a @ X @ S5 )
=> ( ( member1426531477525435216od_a_a @ Y @ S5 )
=> ( ( X != Y )
=> ( R5 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_591_pairwiseD,axiom,
! [R5: set_a > set_a > $o,S5: set_set_a,X: set_a,Y: set_a] :
( ( pairwise_set_a @ R5 @ S5 )
=> ( ( member_set_a @ X @ S5 )
=> ( ( member_set_a @ Y @ S5 )
=> ( ( X != Y )
=> ( R5 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_592_pairwiseD,axiom,
! [R5: a > a > $o,S5: set_a,X: a,Y: a] :
( ( pairwise_a @ R5 @ S5 )
=> ( ( member_a @ X @ S5 )
=> ( ( member_a @ Y @ S5 )
=> ( ( X != Y )
=> ( R5 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_593_pairwiseI,axiom,
! [S5: set_option_a,R5: option_a > option_a > $o] :
( ! [X5: option_a,Y4: option_a] :
( ( member_option_a @ X5 @ S5 )
=> ( ( member_option_a @ Y4 @ S5 )
=> ( ( X5 != Y4 )
=> ( R5 @ X5 @ Y4 ) ) ) )
=> ( pairwise_option_a @ R5 @ S5 ) ) ).
% pairwiseI
thf(fact_594_pairwiseI,axiom,
! [S5: set_Product_prod_a_a,R5: product_prod_a_a > product_prod_a_a > $o] :
( ! [X5: product_prod_a_a,Y4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ S5 )
=> ( ( member1426531477525435216od_a_a @ Y4 @ S5 )
=> ( ( X5 != Y4 )
=> ( R5 @ X5 @ Y4 ) ) ) )
=> ( pairwi8942280066358138514od_a_a @ R5 @ S5 ) ) ).
% pairwiseI
thf(fact_595_pairwiseI,axiom,
! [S5: set_set_a,R5: set_a > set_a > $o] :
( ! [X5: set_a,Y4: set_a] :
( ( member_set_a @ X5 @ S5 )
=> ( ( member_set_a @ Y4 @ S5 )
=> ( ( X5 != Y4 )
=> ( R5 @ X5 @ Y4 ) ) ) )
=> ( pairwise_set_a @ R5 @ S5 ) ) ).
% pairwiseI
thf(fact_596_pairwiseI,axiom,
! [S5: set_a,R5: a > a > $o] :
( ! [X5: a,Y4: a] :
( ( member_a @ X5 @ S5 )
=> ( ( member_a @ Y4 @ S5 )
=> ( ( X5 != Y4 )
=> ( R5 @ X5 @ Y4 ) ) ) )
=> ( pairwise_a @ R5 @ S5 ) ) ).
% pairwiseI
thf(fact_597_pairwise__def,axiom,
( pairwise_a
= ( ^ [R4: a > a > $o,S4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ S4 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ S4 )
=> ( ( X4 != Y3 )
=> ( R4 @ X4 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_598_psubset__trans,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( ord_less_set_a @ A5 @ B6 )
=> ( ( ord_less_set_a @ B6 @ C3 )
=> ( ord_less_set_a @ A5 @ C3 ) ) ) ).
% psubset_trans
thf(fact_599_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_600_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_601_order__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_602_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_set_a @ X5 @ Y4 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_603_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_set_a @ X5 @ Y4 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_604_order__less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% order_less_irrefl
thf(fact_605_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_set_a @ X5 @ Y4 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_606_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 )
=> ( ! [X5: set_a,Y4: set_a] :
( ( ord_less_set_a @ X5 @ Y4 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_607_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_608_order__less__imp__triv,axiom,
! [X: set_a,Y: set_a,P2: $o] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_609_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_610_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_611_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_612_pairwise__mono,axiom,
! [P2: a > a > $o,A5: set_a,Q2: a > a > $o,B6: set_a] :
( ( pairwise_a @ P2 @ A5 )
=> ( ! [X5: a,Y4: a] :
( ( P2 @ X5 @ Y4 )
=> ( Q2 @ X5 @ Y4 ) )
=> ( ( ord_less_eq_set_a @ B6 @ A5 )
=> ( pairwise_a @ Q2 @ B6 ) ) ) ) ).
% pairwise_mono
thf(fact_613_pairwise__subset,axiom,
! [P2: a > a > $o,S5: set_a,T3: set_a] :
( ( pairwise_a @ P2 @ S5 )
=> ( ( ord_less_eq_set_a @ T3 @ S5 )
=> ( pairwise_a @ P2 @ T3 ) ) ) ).
% pairwise_subset
thf(fact_614_pairwise__empty,axiom,
! [P2: a > a > $o] : ( pairwise_a @ P2 @ bot_bot_set_a ) ).
% pairwise_empty
thf(fact_615_pairwise__empty,axiom,
! [P2: option_a > option_a > $o] : ( pairwise_option_a @ P2 @ bot_bot_set_option_a ) ).
% pairwise_empty
thf(fact_616_pairwise__singleton,axiom,
! [P2: a > a > $o,A5: a] : ( pairwise_a @ P2 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ).
% pairwise_singleton
thf(fact_617_pairwise__singleton,axiom,
! [P2: option_a > option_a > $o,A5: option_a] : ( pairwise_option_a @ P2 @ ( insert_option_a @ A5 @ bot_bot_set_option_a ) ) ).
% pairwise_singleton
thf(fact_618_Id__on__empty,axiom,
( ( id_on_a @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% Id_on_empty
thf(fact_619_Id__on__empty,axiom,
( ( id_on_option_a @ bot_bot_set_option_a )
= bot_bo235252021745139059tion_a ) ).
% Id_on_empty
thf(fact_620_preorder__on__empty,axiom,
order_4134995541221112539tion_a @ bot_bot_set_option_a @ bot_bo235252021745139059tion_a ).
% preorder_on_empty
thf(fact_621_preorder__on__empty,axiom,
order_preorder_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% preorder_on_empty
thf(fact_622_map__add__upd__left,axiom,
! [M: option_a,E2: option_a > option_a,E1: option_a > option_a,U1: a] :
( ~ ( member_option_a @ M @ ( dom_option_a_a @ E2 ) )
=> ( ( map_add_option_a_a @ ( fun_up1079276522633388797tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
= ( fun_up1079276522633388797tion_a @ ( map_add_option_a_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_623_map__add__upd__left,axiom,
! [M: product_prod_a_a,E2: product_prod_a_a > option_a,E1: product_prod_a_a > option_a,U1: a] :
( ~ ( member1426531477525435216od_a_a @ M @ ( dom_Pr6378187988305063785_a_a_a @ E2 ) )
=> ( ( map_ad5780899907112351436_a_a_a @ ( fun_up8298456451713467738tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
= ( fun_up8298456451713467738tion_a @ ( map_ad5780899907112351436_a_a_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_624_map__add__upd__left,axiom,
! [M: set_a,E2: set_a > option_a,E1: set_a > option_a,U1: a] :
( ~ ( member_set_a @ M @ ( dom_set_a_a @ E2 ) )
=> ( ( map_add_set_a_a @ ( fun_up3663993102702442083tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
= ( fun_up3663993102702442083tion_a @ ( map_add_set_a_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_625_map__add__upd__left,axiom,
! [M: a,E2: a > option_a,E1: a > option_a,U1: a] :
( ~ ( member_a @ M @ ( dom_a_a @ E2 ) )
=> ( ( map_add_a_a @ ( fun_upd_a_option_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
= ( fun_upd_a_option_a @ ( map_add_a_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_626_ran__map__upd__Some,axiom,
! [M: a > option_a,X: a,Y: a,Z2: a] :
( ( ( M @ X )
= ( some_a @ Y ) )
=> ( ( inj_on_a_option_a @ M @ ( dom_a_a @ M ) )
=> ( ~ ( member_a @ Z2 @ ( ran_a_a @ M ) )
=> ( ( ran_a_a @ ( fun_upd_a_option_a @ M @ X @ ( some_a @ Z2 ) ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ ( ran_a_a @ M ) @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( insert_a @ Z2 @ bot_bot_set_a ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_627_Un__iff,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( sup_sup_set_option_a @ A5 @ B6 ) )
= ( ( member_option_a @ C @ A5 )
| ( member_option_a @ C @ B6 ) ) ) ).
% Un_iff
thf(fact_628_Un__iff,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B6 ) )
= ( ( member1426531477525435216od_a_a @ C @ A5 )
| ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% Un_iff
thf(fact_629_Un__iff,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B6 ) )
= ( ( member_set_a @ C @ A5 )
| ( member_set_a @ C @ B6 ) ) ) ).
% Un_iff
thf(fact_630_Un__iff,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A5 @ B6 ) )
= ( ( member_a @ C @ A5 )
| ( member_a @ C @ B6 ) ) ) ).
% Un_iff
thf(fact_631_UnCI,axiom,
! [C: option_a,B6: set_option_a,A5: set_option_a] :
( ( ~ ( member_option_a @ C @ B6 )
=> ( member_option_a @ C @ A5 ) )
=> ( member_option_a @ C @ ( sup_sup_set_option_a @ A5 @ B6 ) ) ) ).
% UnCI
thf(fact_632_UnCI,axiom,
! [C: product_prod_a_a,B6: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B6 )
=> ( member1426531477525435216od_a_a @ C @ A5 ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B6 ) ) ) ).
% UnCI
thf(fact_633_UnCI,axiom,
! [C: set_a,B6: set_set_a,A5: set_set_a] :
( ( ~ ( member_set_a @ C @ B6 )
=> ( member_set_a @ C @ A5 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B6 ) ) ) ).
% UnCI
thf(fact_634_UnCI,axiom,
! [C: a,B6: set_a,A5: set_a] :
( ( ~ ( member_a @ C @ B6 )
=> ( member_a @ C @ A5 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B6 ) ) ) ).
% UnCI
thf(fact_635_Un__empty,axiom,
! [A5: set_a,B6: set_a] :
( ( ( sup_sup_set_a @ A5 @ B6 )
= bot_bot_set_a )
= ( ( A5 = bot_bot_set_a )
& ( B6 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_636_Un__empty,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( ( sup_sup_set_option_a @ A5 @ B6 )
= bot_bot_set_option_a )
= ( ( A5 = bot_bot_set_option_a )
& ( B6 = bot_bot_set_option_a ) ) ) ).
% Un_empty
thf(fact_637_Un__subset__iff,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B6 ) @ C3 )
= ( ( ord_less_eq_set_a @ A5 @ C3 )
& ( ord_less_eq_set_a @ B6 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_638_Un__insert__left,axiom,
! [A: option_a,B6: set_option_a,C3: set_option_a] :
( ( sup_sup_set_option_a @ ( insert_option_a @ A @ B6 ) @ C3 )
= ( insert_option_a @ A @ ( sup_sup_set_option_a @ B6 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_639_Un__insert__left,axiom,
! [A: a,B6: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B6 ) @ C3 )
= ( insert_a @ A @ ( sup_sup_set_a @ B6 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_640_Un__insert__right,axiom,
! [A5: set_option_a,A: option_a,B6: set_option_a] :
( ( sup_sup_set_option_a @ A5 @ ( insert_option_a @ A @ B6 ) )
= ( insert_option_a @ A @ ( sup_sup_set_option_a @ A5 @ B6 ) ) ) ).
% Un_insert_right
thf(fact_641_Un__insert__right,axiom,
! [A5: set_a,A: a,B6: set_a] :
( ( sup_sup_set_a @ A5 @ ( insert_a @ A @ B6 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A5 @ B6 ) ) ) ).
% Un_insert_right
thf(fact_642_Un__Diff__cancel2,axiom,
! [B6: set_a,A5: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B6 @ A5 ) @ A5 )
= ( sup_sup_set_a @ B6 @ A5 ) ) ).
% Un_Diff_cancel2
thf(fact_643_Un__Diff__cancel,axiom,
! [A5: set_a,B6: set_a] :
( ( sup_sup_set_a @ A5 @ ( minus_minus_set_a @ B6 @ A5 ) )
= ( sup_sup_set_a @ A5 @ B6 ) ) ).
% Un_Diff_cancel
thf(fact_644_map__add__find__right,axiom,
! [N: a > option_a,K: a,Xx: a,M: a > option_a] :
( ( ( N @ K )
= ( some_a @ Xx ) )
=> ( ( map_add_a_a @ M @ N @ K )
= ( some_a @ Xx ) ) ) ).
% map_add_find_right
thf(fact_645_map__add__None,axiom,
! [M: a > option_a,N: a > option_a,K: a] :
( ( ( map_add_a_a @ M @ N @ K )
= none_a )
= ( ( ( N @ K )
= none_a )
& ( ( M @ K )
= none_a ) ) ) ).
% map_add_None
thf(fact_646_map__add__eq__empty__iff,axiom,
! [F: a > option_a,G: a > option_a] :
( ( ( map_add_a_a @ F @ G )
= ( ^ [X4: a] : none_a ) )
= ( ( F
= ( ^ [X4: a] : none_a ) )
& ( G
= ( ^ [X4: a] : none_a ) ) ) ) ).
% map_add_eq_empty_iff
thf(fact_647_Compl__Diff__eq,axiom,
! [A5: set_a,B6: set_a] :
( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A5 @ B6 ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A5 ) @ B6 ) ) ).
% Compl_Diff_eq
thf(fact_648_map__add__upd,axiom,
! [F: a > option_a,G: a > option_a,X: a,Y: a] :
( ( map_add_a_a @ F @ ( fun_upd_a_option_a @ G @ X @ ( some_a @ Y ) ) )
= ( fun_upd_a_option_a @ ( map_add_a_a @ F @ G ) @ X @ ( some_a @ Y ) ) ) ).
% map_add_upd
thf(fact_649_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_650_boolean__algebra_Odisj__zero__right,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ bot_bot_a_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_651_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ X @ bot_bot_set_option_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_652_Un__empty__right,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ bot_bot_set_a )
= A5 ) ).
% Un_empty_right
thf(fact_653_Un__empty__right,axiom,
! [A5: set_option_a] :
( ( sup_sup_set_option_a @ A5 @ bot_bot_set_option_a )
= A5 ) ).
% Un_empty_right
thf(fact_654_Un__empty__left,axiom,
! [B6: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B6 )
= B6 ) ).
% Un_empty_left
thf(fact_655_Un__empty__left,axiom,
! [B6: set_option_a] :
( ( sup_sup_set_option_a @ bot_bot_set_option_a @ B6 )
= B6 ) ).
% Un_empty_left
thf(fact_656_inj__Some,axiom,
! [A5: set_a] : ( inj_on_a_option_a @ some_a @ A5 ) ).
% inj_Some
thf(fact_657_Un__left__commute,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ B6 @ C3 ) )
= ( sup_sup_set_a @ B6 @ ( sup_sup_set_a @ A5 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_658_Un__left__absorb,axiom,
! [A5: set_a,B6: set_a] :
( ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ A5 @ B6 ) )
= ( sup_sup_set_a @ A5 @ B6 ) ) ).
% Un_left_absorb
thf(fact_659_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A8: set_a,B7: set_a] : ( sup_sup_set_a @ B7 @ A8 ) ) ) ).
% Un_commute
thf(fact_660_Un__absorb,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_661_Un__assoc,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A5 @ B6 ) @ C3 )
= ( sup_sup_set_a @ A5 @ ( sup_sup_set_a @ B6 @ C3 ) ) ) ).
% Un_assoc
thf(fact_662_ball__Un,axiom,
! [A5: set_a,B6: set_a,P2: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A5 @ B6 ) )
=> ( P2 @ X4 ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P2 @ X4 ) )
& ! [X4: a] :
( ( member_a @ X4 @ B6 )
=> ( P2 @ X4 ) ) ) ) ).
% ball_Un
thf(fact_663_bex__Un,axiom,
! [A5: set_a,B6: set_a,P2: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A5 @ B6 ) )
& ( P2 @ X4 ) ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( P2 @ X4 ) )
| ? [X4: a] :
( ( member_a @ X4 @ B6 )
& ( P2 @ X4 ) ) ) ) ).
% bex_Un
thf(fact_664_UnI2,axiom,
! [C: option_a,B6: set_option_a,A5: set_option_a] :
( ( member_option_a @ C @ B6 )
=> ( member_option_a @ C @ ( sup_sup_set_option_a @ A5 @ B6 ) ) ) ).
% UnI2
thf(fact_665_UnI2,axiom,
! [C: product_prod_a_a,B6: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B6 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B6 ) ) ) ).
% UnI2
thf(fact_666_UnI2,axiom,
! [C: set_a,B6: set_set_a,A5: set_set_a] :
( ( member_set_a @ C @ B6 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B6 ) ) ) ).
% UnI2
thf(fact_667_UnI2,axiom,
! [C: a,B6: set_a,A5: set_a] :
( ( member_a @ C @ B6 )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B6 ) ) ) ).
% UnI2
thf(fact_668_UnI1,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ A5 )
=> ( member_option_a @ C @ ( sup_sup_set_option_a @ A5 @ B6 ) ) ) ).
% UnI1
thf(fact_669_UnI1,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B6 ) ) ) ).
% UnI1
thf(fact_670_UnI1,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B6 ) ) ) ).
% UnI1
thf(fact_671_UnI1,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ A5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B6 ) ) ) ).
% UnI1
thf(fact_672_UnE,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( sup_sup_set_option_a @ A5 @ B6 ) )
=> ( ~ ( member_option_a @ C @ A5 )
=> ( member_option_a @ C @ B6 ) ) ) ).
% UnE
thf(fact_673_UnE,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B6 ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% UnE
thf(fact_674_UnE,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A5 @ B6 ) )
=> ( ~ ( member_set_a @ C @ A5 )
=> ( member_set_a @ C @ B6 ) ) ) ).
% UnE
thf(fact_675_UnE,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A5 @ B6 ) )
=> ( ~ ( member_a @ C @ A5 )
=> ( member_a @ C @ B6 ) ) ) ).
% UnE
thf(fact_676_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B7: set_a] :
( ( sup_sup_set_a @ A8 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_677_subset__UnE,axiom,
! [C3: set_a,A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A5 @ B6 ) )
=> ~ ! [A9: set_a] :
( ( ord_less_eq_set_a @ A9 @ A5 )
=> ! [B9: set_a] :
( ( ord_less_eq_set_a @ B9 @ B6 )
=> ( C3
!= ( sup_sup_set_a @ A9 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_678_Un__absorb2,axiom,
! [B6: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B6 @ A5 )
=> ( ( sup_sup_set_a @ A5 @ B6 )
= A5 ) ) ).
% Un_absorb2
thf(fact_679_Un__absorb1,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( sup_sup_set_a @ A5 @ B6 )
= B6 ) ) ).
% Un_absorb1
thf(fact_680_Un__upper2,axiom,
! [B6: set_a,A5: set_a] : ( ord_less_eq_set_a @ B6 @ ( sup_sup_set_a @ A5 @ B6 ) ) ).
% Un_upper2
thf(fact_681_Un__upper1,axiom,
! [A5: set_a,B6: set_a] : ( ord_less_eq_set_a @ A5 @ ( sup_sup_set_a @ A5 @ B6 ) ) ).
% Un_upper1
thf(fact_682_Un__least,axiom,
! [A5: set_a,C3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B6 ) @ C3 ) ) ) ).
% Un_least
thf(fact_683_Un__mono,axiom,
! [A5: set_a,C3: set_a,B6: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A5 @ B6 ) @ ( sup_sup_set_a @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_684_Un__Diff,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A5 @ B6 ) @ C3 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ C3 ) @ ( minus_minus_set_a @ B6 @ C3 ) ) ) ).
% Un_Diff
thf(fact_685_Diff__partition,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( sup_sup_set_a @ A5 @ ( minus_minus_set_a @ B6 @ A5 ) )
= B6 ) ) ).
% Diff_partition
thf(fact_686_Diff__subset__conv,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ C3 )
= ( ord_less_eq_set_a @ A5 @ ( sup_sup_set_a @ B6 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_687_insert__is__Un,axiom,
( insert_a
= ( ^ [A6: a] : ( sup_sup_set_a @ ( insert_a @ A6 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_688_insert__is__Un,axiom,
( insert_option_a
= ( ^ [A6: option_a] : ( sup_sup_set_option_a @ ( insert_option_a @ A6 @ bot_bot_set_option_a ) ) ) ) ).
% insert_is_Un
thf(fact_689_Un__singleton__iff,axiom,
! [A5: set_a,B6: set_a,X: a] :
( ( ( sup_sup_set_a @ A5 @ B6 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A5 = bot_bot_set_a )
& ( B6
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A5
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B6 = bot_bot_set_a ) )
| ( ( A5
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B6
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_690_Un__singleton__iff,axiom,
! [A5: set_option_a,B6: set_option_a,X: option_a] :
( ( ( sup_sup_set_option_a @ A5 @ B6 )
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
= ( ( ( A5 = bot_bot_set_option_a )
& ( B6
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
| ( ( A5
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
& ( B6 = bot_bot_set_option_a ) )
| ( ( A5
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
& ( B6
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_691_singleton__Un__iff,axiom,
! [X: a,A5: set_a,B6: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A5 @ B6 ) )
= ( ( ( A5 = bot_bot_set_a )
& ( B6
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A5
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B6 = bot_bot_set_a ) )
| ( ( A5
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B6
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_692_singleton__Un__iff,axiom,
! [X: option_a,A5: set_option_a,B6: set_option_a] :
( ( ( insert_option_a @ X @ bot_bot_set_option_a )
= ( sup_sup_set_option_a @ A5 @ B6 ) )
= ( ( ( A5 = bot_bot_set_option_a )
& ( B6
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
| ( ( A5
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
& ( B6 = bot_bot_set_option_a ) )
| ( ( A5
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
& ( B6
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_693_map__add__SomeD,axiom,
! [M: a > option_a,N: a > option_a,K: a,X: a] :
( ( ( map_add_a_a @ M @ N @ K )
= ( some_a @ X ) )
=> ( ( ( N @ K )
= ( some_a @ X ) )
| ( ( ( N @ K )
= none_a )
& ( ( M @ K )
= ( some_a @ X ) ) ) ) ) ).
% map_add_SomeD
thf(fact_694_map__add__Some__iff,axiom,
! [M: a > option_a,N: a > option_a,K: a,X: a] :
( ( ( map_add_a_a @ M @ N @ K )
= ( some_a @ X ) )
= ( ( ( N @ K )
= ( some_a @ X ) )
| ( ( ( N @ K )
= none_a )
& ( ( M @ K )
= ( some_a @ X ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_695_inj__on__empty,axiom,
! [F: a > option_a] : ( inj_on_a_option_a @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_696_inj__on__empty,axiom,
! [F: a > a] : ( inj_on_a_a @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_697_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_698_sup__bot_Oright__neutral,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ A @ bot_bot_a_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_699_sup__bot_Oright__neutral,axiom,
! [A: set_option_a] :
( ( sup_sup_set_option_a @ A @ bot_bot_set_option_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_700_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_701_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_702_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_703_sup__bot__left,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X )
= X ) ).
% sup_bot_left
thf(fact_704_sup__bot__left,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ bot_bot_set_option_a @ X )
= X ) ).
% sup_bot_left
thf(fact_705_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_706_sup__bot__right,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ bot_bot_a_o )
= X ) ).
% sup_bot_right
thf(fact_707_sup__bot__right,axiom,
! [X: set_option_a] :
( ( sup_sup_set_option_a @ X @ bot_bot_set_option_a )
= X ) ).
% sup_bot_right
thf(fact_708_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_709_bot__eq__sup__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X @ Y ) )
= ( ( X = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_710_bot__eq__sup__iff,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( bot_bot_set_option_a
= ( sup_sup_set_option_a @ X @ Y ) )
= ( ( X = bot_bot_set_option_a )
& ( Y = bot_bot_set_option_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_711_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_712_sup__eq__bot__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( sup_sup_a_o @ X @ Y )
= bot_bot_a_o )
= ( ( X = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_713_sup__eq__bot__iff,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( sup_sup_set_option_a @ X @ Y )
= bot_bot_set_option_a )
= ( ( X = bot_bot_set_option_a )
& ( Y = bot_bot_set_option_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_714_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_715_sup__bot_Oeq__neutr__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( ( sup_sup_a_o @ A @ B )
= bot_bot_a_o )
= ( ( A = bot_bot_a_o )
& ( B = bot_bot_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_716_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_option_a,B: set_option_a] :
( ( ( sup_sup_set_option_a @ A @ B )
= bot_bot_set_option_a )
= ( ( A = bot_bot_set_option_a )
& ( B = bot_bot_set_option_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_717_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_718_sup__bot_Oleft__neutral,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_719_sup__bot_Oleft__neutral,axiom,
! [A: set_option_a] :
( ( sup_sup_set_option_a @ bot_bot_set_option_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_720_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_721_sup__bot_Oneutr__eq__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ A @ B ) )
= ( ( A = bot_bot_a_o )
& ( B = bot_bot_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_722_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_option_a,B: set_option_a] :
( ( bot_bot_set_option_a
= ( sup_sup_set_option_a @ A @ B ) )
= ( ( A = bot_bot_set_option_a )
& ( B = bot_bot_set_option_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_723_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_724_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_725_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
( ( sup_sup_set_a @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_726_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A6: set_a] :
( ( sup_sup_set_a @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_727_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_728_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_729_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A6: set_a] :
( A6
= ( sup_sup_set_a @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_730_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_731_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_732_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_733_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_734_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_735_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_736_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X5 @ ( F @ X5 @ Y4 ) )
=> ( ! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F @ X5 @ Y4 ) )
=> ( ! [X5: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X5 )
=> ( ( ord_less_eq_set_a @ Z3 @ X5 )
=> ( ord_less_eq_set_a @ ( F @ Y4 @ Z3 ) @ X5 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_737_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_738_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_739_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_740_sup__least,axiom,
! [Y: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_741_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_742_sup_Omono,axiom,
! [C: set_a,A: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_subset__inj__on,axiom,
! [F: a > option_a,B6: set_a,A5: set_a] :
( ( inj_on_a_option_a @ F @ B6 )
=> ( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( inj_on_a_option_a @ F @ A5 ) ) ) ).
% subset_inj_on
thf(fact_752_subset__inj__on,axiom,
! [F: a > a,B6: set_a,A5: set_a] :
( ( inj_on_a_a @ F @ B6 )
=> ( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( inj_on_a_a @ F @ A5 ) ) ) ).
% subset_inj_on
thf(fact_753_inj__on__subset,axiom,
! [F: a > option_a,A5: set_a,B6: set_a] :
( ( inj_on_a_option_a @ F @ A5 )
=> ( ( ord_less_eq_set_a @ B6 @ A5 )
=> ( inj_on_a_option_a @ F @ B6 ) ) ) ).
% inj_on_subset
thf(fact_754_inj__on__subset,axiom,
! [F: a > a,A5: set_a,B6: set_a] :
( ( inj_on_a_a @ F @ A5 )
=> ( ( ord_less_eq_set_a @ B6 @ A5 )
=> ( inj_on_a_a @ F @ B6 ) ) ) ).
% inj_on_subset
thf(fact_755_Field__insert,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a] :
( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) )
= ( sup_sup_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) ) ) ).
% Field_insert
thf(fact_756_Field__insert,axiom,
! [A: option_a,B: option_a,R2: set_Pr7585778909603769095tion_a] :
( ( field_option_a @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) )
= ( sup_sup_set_option_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R2 ) ) ) ).
% Field_insert
thf(fact_757_ran__map__add,axiom,
! [M1: a > option_a,M2: a > option_a] :
( ( ( inf_inf_set_a @ ( dom_a_a @ M1 ) @ ( dom_a_a @ M2 ) )
= bot_bot_set_a )
=> ( ( ran_a_a @ ( map_add_a_a @ M1 @ M2 ) )
= ( sup_sup_set_a @ ( ran_a_a @ M1 ) @ ( ran_a_a @ M2 ) ) ) ) ).
% ran_map_add
thf(fact_758_ran__map__add,axiom,
! [M1: option_a > option_a,M2: option_a > option_a] :
( ( ( inf_inf_set_option_a @ ( dom_option_a_a @ M1 ) @ ( dom_option_a_a @ M2 ) )
= bot_bot_set_option_a )
=> ( ( ran_option_a_a @ ( map_add_option_a_a @ M1 @ M2 ) )
= ( sup_sup_set_a @ ( ran_option_a_a @ M1 ) @ ( ran_option_a_a @ M2 ) ) ) ) ).
% ran_map_add
thf(fact_759_Int__iff,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( inf_inf_set_option_a @ A5 @ B6 ) )
= ( ( member_option_a @ C @ A5 )
& ( member_option_a @ C @ B6 ) ) ) ).
% Int_iff
thf(fact_760_Int__iff,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) )
= ( ( member1426531477525435216od_a_a @ C @ A5 )
& ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% Int_iff
thf(fact_761_Int__iff,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B6 ) )
= ( ( member_set_a @ C @ A5 )
& ( member_set_a @ C @ B6 ) ) ) ).
% Int_iff
thf(fact_762_Int__iff,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B6 ) )
= ( ( member_a @ C @ A5 )
& ( member_a @ C @ B6 ) ) ) ).
% Int_iff
thf(fact_763_IntI,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ A5 )
=> ( ( member_option_a @ C @ B6 )
=> ( member_option_a @ C @ ( inf_inf_set_option_a @ A5 @ B6 ) ) ) ) ).
% IntI
thf(fact_764_IntI,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( ( member1426531477525435216od_a_a @ C @ B6 )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) ) ) ) ).
% IntI
thf(fact_765_IntI,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( ( member_set_a @ C @ B6 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B6 ) ) ) ) ).
% IntI
thf(fact_766_IntI,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ A5 )
=> ( ( member_a @ C @ B6 )
=> ( member_a @ C @ ( inf_inf_set_a @ A5 @ B6 ) ) ) ) ).
% IntI
thf(fact_767_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_768_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_769_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_770_inf__bot__left,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X )
= bot_bot_a_o ) ).
% inf_bot_left
thf(fact_771_inf__bot__left,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ bot_bot_set_option_a @ X )
= bot_bot_set_option_a ) ).
% inf_bot_left
thf(fact_772_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_773_inf__bot__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ bot_bot_a_o )
= bot_bot_a_o ) ).
% inf_bot_right
thf(fact_774_inf__bot__right,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ X @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% inf_bot_right
thf(fact_775_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_776_boolean__algebra_Oconj__zero__left,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_777_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ bot_bot_set_option_a @ X )
= bot_bot_set_option_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_778_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_779_boolean__algebra_Oconj__zero__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ bot_bot_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_780_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ X @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_781_Int__subset__iff,axiom,
! [C3: set_a,A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A5 @ B6 ) )
= ( ( ord_less_eq_set_a @ C3 @ A5 )
& ( ord_less_eq_set_a @ C3 @ B6 ) ) ) ).
% Int_subset_iff
thf(fact_782_Int__insert__left__if0,axiom,
! [A: option_a,C3: set_option_a,B6: set_option_a] :
( ~ ( member_option_a @ A @ C3 )
=> ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ B6 ) @ C3 )
= ( inf_inf_set_option_a @ B6 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_783_Int__insert__left__if0,axiom,
! [A: product_prod_a_a,C3: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ C3 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B6 ) @ C3 )
= ( inf_in8905007599844390133od_a_a @ B6 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_784_Int__insert__left__if0,axiom,
! [A: set_a,C3: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B6 ) @ C3 )
= ( inf_inf_set_set_a @ B6 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_785_Int__insert__left__if0,axiom,
! [A: a,C3: set_a,B6: set_a] :
( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B6 ) @ C3 )
= ( inf_inf_set_a @ B6 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_786_Int__insert__left__if1,axiom,
! [A: option_a,C3: set_option_a,B6: set_option_a] :
( ( member_option_a @ A @ C3 )
=> ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ B6 ) @ C3 )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ B6 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_787_Int__insert__left__if1,axiom,
! [A: product_prod_a_a,C3: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ C3 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B6 ) @ C3 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B6 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_788_Int__insert__left__if1,axiom,
! [A: set_a,C3: set_set_a,B6: set_set_a] :
( ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B6 ) @ C3 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B6 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_789_Int__insert__left__if1,axiom,
! [A: a,C3: set_a,B6: set_a] :
( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B6 ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B6 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_790_insert__inter__insert,axiom,
! [A: option_a,A5: set_option_a,B6: set_option_a] :
( ( inf_inf_set_option_a @ ( insert_option_a @ A @ A5 ) @ ( insert_option_a @ A @ B6 ) )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ A5 @ B6 ) ) ) ).
% insert_inter_insert
thf(fact_791_insert__inter__insert,axiom,
! [A: a,A5: set_a,B6: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ ( insert_a @ A @ B6 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B6 ) ) ) ).
% insert_inter_insert
thf(fact_792_Int__insert__right__if0,axiom,
! [A: option_a,A5: set_option_a,B6: set_option_a] :
( ~ ( member_option_a @ A @ A5 )
=> ( ( inf_inf_set_option_a @ A5 @ ( insert_option_a @ A @ B6 ) )
= ( inf_inf_set_option_a @ A5 @ B6 ) ) ) ).
% Int_insert_right_if0
thf(fact_793_Int__insert__right__if0,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A @ B6 ) )
= ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) ) ) ).
% Int_insert_right_if0
thf(fact_794_Int__insert__right__if0,axiom,
! [A: set_a,A5: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B6 ) )
= ( inf_inf_set_set_a @ A5 @ B6 ) ) ) ).
% Int_insert_right_if0
thf(fact_795_Int__insert__right__if0,axiom,
! [A: a,A5: set_a,B6: set_a] :
( ~ ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B6 ) )
= ( inf_inf_set_a @ A5 @ B6 ) ) ) ).
% Int_insert_right_if0
thf(fact_796_Int__insert__right__if1,axiom,
! [A: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ A @ A5 )
=> ( ( inf_inf_set_option_a @ A5 @ ( insert_option_a @ A @ B6 ) )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ A5 @ B6 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_797_Int__insert__right__if1,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A @ B6 ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_798_Int__insert__right__if1,axiom,
! [A: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B6 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A5 @ B6 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_799_Int__insert__right__if1,axiom,
! [A: a,A5: set_a,B6: set_a] :
( ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B6 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B6 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_800_Un__Int__eq_I1_J,axiom,
! [S5: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S5 @ T3 ) @ S5 )
= S5 ) ).
% Un_Int_eq(1)
thf(fact_801_Un__Int__eq_I2_J,axiom,
! [S5: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S5 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_802_Un__Int__eq_I3_J,axiom,
! [S5: set_a,T3: set_a] :
( ( inf_inf_set_a @ S5 @ ( sup_sup_set_a @ S5 @ T3 ) )
= S5 ) ).
% Un_Int_eq(3)
thf(fact_803_Un__Int__eq_I4_J,axiom,
! [T3: set_a,S5: set_a] :
( ( inf_inf_set_a @ T3 @ ( sup_sup_set_a @ S5 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_804_Int__Un__eq_I1_J,axiom,
! [S5: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S5 @ T3 ) @ S5 )
= S5 ) ).
% Int_Un_eq(1)
thf(fact_805_Int__Un__eq_I2_J,axiom,
! [S5: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S5 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_806_Int__Un__eq_I3_J,axiom,
! [S5: set_a,T3: set_a] :
( ( sup_sup_set_a @ S5 @ ( inf_inf_set_a @ S5 @ T3 ) )
= S5 ) ).
% Int_Un_eq(3)
thf(fact_807_Int__Un__eq_I4_J,axiom,
! [T3: set_a,S5: set_a] :
( ( sup_sup_set_a @ T3 @ ( inf_inf_set_a @ S5 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_808_inf__compl__bot__left1,axiom,
! [X: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ ( uminus_uminus_a_o @ X ) @ ( inf_inf_a_o @ X @ Y ) )
= bot_bot_a_o ) ).
% inf_compl_bot_left1
thf(fact_809_inf__compl__bot__left1,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ ( inf_inf_set_option_a @ X @ Y ) )
= bot_bot_set_option_a ) ).
% inf_compl_bot_left1
thf(fact_810_inf__compl__bot__left1,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( inf_inf_set_a @ X @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_811_inf__compl__bot__left2,axiom,
! [X: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ X @ ( inf_inf_a_o @ ( uminus_uminus_a_o @ X ) @ Y ) )
= bot_bot_a_o ) ).
% inf_compl_bot_left2
thf(fact_812_inf__compl__bot__left2,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( inf_inf_set_option_a @ X @ ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ Y ) )
= bot_bot_set_option_a ) ).
% inf_compl_bot_left2
thf(fact_813_inf__compl__bot__left2,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_814_inf__compl__bot__right,axiom,
! [X: a > $o,Y: a > $o] :
( ( inf_inf_a_o @ X @ ( inf_inf_a_o @ Y @ ( uminus_uminus_a_o @ X ) ) )
= bot_bot_a_o ) ).
% inf_compl_bot_right
thf(fact_815_inf__compl__bot__right,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( inf_inf_set_option_a @ X @ ( inf_inf_set_option_a @ Y @ ( uminus6205308855922866075tion_a @ X ) ) )
= bot_bot_set_option_a ) ).
% inf_compl_bot_right
thf(fact_816_inf__compl__bot__right,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ ( uminus_uminus_set_a @ X ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_817_boolean__algebra_Oconj__cancel__left,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ ( uminus_uminus_a_o @ X ) @ X )
= bot_bot_a_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_818_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ X )
= bot_bot_set_option_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_819_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_820_boolean__algebra_Oconj__cancel__right,axiom,
! [X: a > $o] :
( ( inf_inf_a_o @ X @ ( uminus_uminus_a_o @ X ) )
= bot_bot_a_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_821_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_option_a] :
( ( inf_inf_set_option_a @ X @ ( uminus6205308855922866075tion_a @ X ) )
= bot_bot_set_option_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_822_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ X ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_823_insert__disjoint_I1_J,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A5 ) @ B6 )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B6 )
& ( ( inf_in8905007599844390133od_a_a @ A5 @ B6 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_824_insert__disjoint_I1_J,axiom,
! [A: set_a,A5: set_set_a,B6: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A5 ) @ B6 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B6 )
& ( ( inf_inf_set_set_a @ A5 @ B6 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_825_insert__disjoint_I1_J,axiom,
! [A: option_a,A5: set_option_a,B6: set_option_a] :
( ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ A5 ) @ B6 )
= bot_bot_set_option_a )
= ( ~ ( member_option_a @ A @ B6 )
& ( ( inf_inf_set_option_a @ A5 @ B6 )
= bot_bot_set_option_a ) ) ) ).
% insert_disjoint(1)
thf(fact_826_insert__disjoint_I1_J,axiom,
! [A: a,A5: set_a,B6: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ B6 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B6 )
& ( ( inf_inf_set_a @ A5 @ B6 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_827_insert__disjoint_I2_J,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A5 ) @ B6 ) )
= ( ~ ( member1426531477525435216od_a_a @ A @ B6 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_828_insert__disjoint_I2_J,axiom,
! [A: set_a,A5: set_set_a,B6: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A5 ) @ B6 ) )
= ( ~ ( member_set_a @ A @ B6 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ B6 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_829_insert__disjoint_I2_J,axiom,
! [A: option_a,A5: set_option_a,B6: set_option_a] :
( ( bot_bot_set_option_a
= ( inf_inf_set_option_a @ ( insert_option_a @ A @ A5 ) @ B6 ) )
= ( ~ ( member_option_a @ A @ B6 )
& ( bot_bot_set_option_a
= ( inf_inf_set_option_a @ A5 @ B6 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_830_insert__disjoint_I2_J,axiom,
! [A: a,A5: set_a,B6: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A5 ) @ B6 ) )
= ( ~ ( member_a @ A @ B6 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ B6 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_831_disjoint__insert_I1_J,axiom,
! [B6: set_Product_prod_a_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B6 @ ( insert4534936382041156343od_a_a @ A @ A5 ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B6 )
& ( ( inf_in8905007599844390133od_a_a @ B6 @ A5 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_832_disjoint__insert_I1_J,axiom,
! [B6: set_set_a,A: set_a,A5: set_set_a] :
( ( ( inf_inf_set_set_a @ B6 @ ( insert_set_a @ A @ A5 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B6 )
& ( ( inf_inf_set_set_a @ B6 @ A5 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_833_disjoint__insert_I1_J,axiom,
! [B6: set_option_a,A: option_a,A5: set_option_a] :
( ( ( inf_inf_set_option_a @ B6 @ ( insert_option_a @ A @ A5 ) )
= bot_bot_set_option_a )
= ( ~ ( member_option_a @ A @ B6 )
& ( ( inf_inf_set_option_a @ B6 @ A5 )
= bot_bot_set_option_a ) ) ) ).
% disjoint_insert(1)
thf(fact_834_disjoint__insert_I1_J,axiom,
! [B6: set_a,A: a,A5: set_a] :
( ( ( inf_inf_set_a @ B6 @ ( insert_a @ A @ A5 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B6 )
& ( ( inf_inf_set_a @ B6 @ A5 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_835_disjoint__insert_I2_J,axiom,
! [A5: set_Product_prod_a_a,B: product_prod_a_a,B6: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ B @ B6 ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B @ A5 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_836_disjoint__insert_I2_J,axiom,
! [A5: set_set_a,B: set_a,B6: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ B @ B6 ) ) )
= ( ~ ( member_set_a @ B @ A5 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A5 @ B6 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_837_disjoint__insert_I2_J,axiom,
! [A5: set_option_a,B: option_a,B6: set_option_a] :
( ( bot_bot_set_option_a
= ( inf_inf_set_option_a @ A5 @ ( insert_option_a @ B @ B6 ) ) )
= ( ~ ( member_option_a @ B @ A5 )
& ( bot_bot_set_option_a
= ( inf_inf_set_option_a @ A5 @ B6 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_838_disjoint__insert_I2_J,axiom,
! [A5: set_a,B: a,B6: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ ( insert_a @ B @ B6 ) ) )
= ( ~ ( member_a @ B @ A5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ B6 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_839_Diff__disjoint,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( inf_inf_set_option_a @ A5 @ ( minus_1574173051537231627tion_a @ B6 @ A5 ) )
= bot_bot_set_option_a ) ).
% Diff_disjoint
thf(fact_840_Diff__disjoint,axiom,
! [A5: set_a,B6: set_a] :
( ( inf_inf_set_a @ A5 @ ( minus_minus_set_a @ B6 @ A5 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_841_Field__empty,axiom,
( ( field_a @ bot_bo3357376287454694259od_a_a )
= bot_bot_set_a ) ).
% Field_empty
thf(fact_842_Field__empty,axiom,
( ( field_option_a @ bot_bo235252021745139059tion_a )
= bot_bot_set_option_a ) ).
% Field_empty
thf(fact_843_Compl__disjoint,axiom,
! [A5: set_option_a] :
( ( inf_inf_set_option_a @ A5 @ ( uminus6205308855922866075tion_a @ A5 ) )
= bot_bot_set_option_a ) ).
% Compl_disjoint
thf(fact_844_Compl__disjoint,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ ( uminus_uminus_set_a @ A5 ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_845_Compl__disjoint2,axiom,
! [A5: set_option_a] :
( ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ A5 ) @ A5 )
= bot_bot_set_option_a ) ).
% Compl_disjoint2
thf(fact_846_Compl__disjoint2,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A5 ) @ A5 )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_847_Diff__Compl,axiom,
! [A5: set_a,B6: set_a] :
( ( minus_minus_set_a @ A5 @ ( uminus_uminus_set_a @ B6 ) )
= ( inf_inf_set_a @ A5 @ B6 ) ) ).
% Diff_Compl
thf(fact_848_Un__Int__distrib2,axiom,
! [B6: set_a,C3: set_a,A5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B6 @ C3 ) @ A5 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B6 @ A5 ) @ ( sup_sup_set_a @ C3 @ A5 ) ) ) ).
% Un_Int_distrib2
thf(fact_849_Int__Un__distrib2,axiom,
! [B6: set_a,C3: set_a,A5: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B6 @ C3 ) @ A5 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B6 @ A5 ) @ ( inf_inf_set_a @ C3 @ A5 ) ) ) ).
% Int_Un_distrib2
thf(fact_850_Un__Int__distrib,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( sup_sup_set_a @ A5 @ ( inf_inf_set_a @ B6 @ C3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A5 @ B6 ) @ ( sup_sup_set_a @ A5 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_851_Int__Un__distrib,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( inf_inf_set_a @ A5 @ ( sup_sup_set_a @ B6 @ C3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ ( inf_inf_set_a @ A5 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_852_Un__Int__crazy,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ ( inf_inf_set_a @ B6 @ C3 ) ) @ ( inf_inf_set_a @ C3 @ A5 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A5 @ B6 ) @ ( sup_sup_set_a @ B6 @ C3 ) ) @ ( sup_sup_set_a @ C3 @ A5 ) ) ) ).
% Un_Int_crazy
thf(fact_853_Un__Int__assoc__eq,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ C3 )
= ( inf_inf_set_a @ A5 @ ( sup_sup_set_a @ B6 @ C3 ) ) )
= ( ord_less_eq_set_a @ C3 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_854_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_855_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_856_Compl__Int,axiom,
! [A5: set_a,B6: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ A5 @ B6 ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A5 ) @ ( uminus_uminus_set_a @ B6 ) ) ) ).
% Compl_Int
thf(fact_857_Compl__Un,axiom,
! [A5: set_a,B6: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ A5 @ B6 ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ A5 ) @ ( uminus_uminus_set_a @ B6 ) ) ) ).
% Compl_Un
thf(fact_858_Un__Diff__Int,axiom,
! [A5: set_a,B6: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ ( inf_inf_set_a @ A5 @ B6 ) )
= A5 ) ).
% Un_Diff_Int
thf(fact_859_Int__Diff__Un,axiom,
! [A5: set_a,B6: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ ( minus_minus_set_a @ A5 @ B6 ) )
= A5 ) ).
% Int_Diff_Un
thf(fact_860_Diff__Int,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( minus_minus_set_a @ A5 @ ( inf_inf_set_a @ B6 @ C3 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ ( minus_minus_set_a @ A5 @ C3 ) ) ) ).
% Diff_Int
thf(fact_861_Diff__Un,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( minus_minus_set_a @ A5 @ ( sup_sup_set_a @ B6 @ C3 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ ( minus_minus_set_a @ A5 @ C3 ) ) ) ).
% Diff_Un
thf(fact_862_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_863_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_864_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A6: set_a] :
( ( inf_inf_set_a @ A6 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_865_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
( ( inf_inf_set_a @ A6 @ B4 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_866_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_867_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_868_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
( A6
= ( inf_inf_set_a @ A6 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_869_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_870_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_871_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_872_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_873_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_874_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_875_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_876_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_877_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( ord_less_eq_set_a @ X5 @ Z3 )
=> ( ord_less_eq_set_a @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_878_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_879_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_880_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_881_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_882_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_883_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_884_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_885_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_886_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_887_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_888_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_889_disjoint__iff__not__equal,axiom,
! [A5: set_a,B6: set_a] :
( ( ( inf_inf_set_a @ A5 @ B6 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B6 )
=> ( X4 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_890_disjoint__iff__not__equal,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( ( inf_inf_set_option_a @ A5 @ B6 )
= bot_bot_set_option_a )
= ( ! [X4: option_a] :
( ( member_option_a @ X4 @ A5 )
=> ! [Y3: option_a] :
( ( member_option_a @ Y3 @ B6 )
=> ( X4 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_891_Int__empty__right,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_892_Int__empty__right,axiom,
! [A5: set_option_a] :
( ( inf_inf_set_option_a @ A5 @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% Int_empty_right
thf(fact_893_Int__empty__left,axiom,
! [B6: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B6 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_894_Int__empty__left,axiom,
! [B6: set_option_a] :
( ( inf_inf_set_option_a @ bot_bot_set_option_a @ B6 )
= bot_bot_set_option_a ) ).
% Int_empty_left
thf(fact_895_disjoint__iff,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A5 @ B6 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ~ ( member1426531477525435216od_a_a @ X4 @ B6 ) ) ) ) ).
% disjoint_iff
thf(fact_896_disjoint__iff,axiom,
! [A5: set_set_a,B6: set_set_a] :
( ( ( inf_inf_set_set_a @ A5 @ B6 )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ~ ( member_set_a @ X4 @ B6 ) ) ) ) ).
% disjoint_iff
thf(fact_897_disjoint__iff,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( ( inf_inf_set_option_a @ A5 @ B6 )
= bot_bot_set_option_a )
= ( ! [X4: option_a] :
( ( member_option_a @ X4 @ A5 )
=> ~ ( member_option_a @ X4 @ B6 ) ) ) ) ).
% disjoint_iff
thf(fact_898_disjoint__iff,axiom,
! [A5: set_a,B6: set_a] :
( ( ( inf_inf_set_a @ A5 @ B6 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ~ ( member_a @ X4 @ B6 ) ) ) ) ).
% disjoint_iff
thf(fact_899_Int__emptyI,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A5 )
=> ~ ( member1426531477525435216od_a_a @ X5 @ B6 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ B6 )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_900_Int__emptyI,axiom,
! [A5: set_set_a,B6: set_set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ~ ( member_set_a @ X5 @ B6 ) )
=> ( ( inf_inf_set_set_a @ A5 @ B6 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_901_Int__emptyI,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ! [X5: option_a] :
( ( member_option_a @ X5 @ A5 )
=> ~ ( member_option_a @ X5 @ B6 ) )
=> ( ( inf_inf_set_option_a @ A5 @ B6 )
= bot_bot_set_option_a ) ) ).
% Int_emptyI
thf(fact_902_Int__emptyI,axiom,
! [A5: set_a,B6: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ~ ( member_a @ X5 @ B6 ) )
=> ( ( inf_inf_set_a @ A5 @ B6 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_903_Int__left__commute,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B6 @ C3 ) )
= ( inf_inf_set_a @ B6 @ ( inf_inf_set_a @ A5 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_904_Int__left__absorb,axiom,
! [A5: set_a,B6: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ A5 @ B6 ) )
= ( inf_inf_set_a @ A5 @ B6 ) ) ).
% Int_left_absorb
thf(fact_905_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A8: set_a,B7: set_a] : ( inf_inf_set_a @ B7 @ A8 ) ) ) ).
% Int_commute
thf(fact_906_Int__absorb,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_907_Int__assoc,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ C3 )
= ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B6 @ C3 ) ) ) ).
% Int_assoc
thf(fact_908_IntD2,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( inf_inf_set_option_a @ A5 @ B6 ) )
=> ( member_option_a @ C @ B6 ) ) ).
% IntD2
thf(fact_909_IntD2,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) )
=> ( member1426531477525435216od_a_a @ C @ B6 ) ) ).
% IntD2
thf(fact_910_IntD2,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B6 ) )
=> ( member_set_a @ C @ B6 ) ) ).
% IntD2
thf(fact_911_IntD2,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B6 ) )
=> ( member_a @ C @ B6 ) ) ).
% IntD2
thf(fact_912_IntD1,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( inf_inf_set_option_a @ A5 @ B6 ) )
=> ( member_option_a @ C @ A5 ) ) ).
% IntD1
thf(fact_913_IntD1,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) )
=> ( member1426531477525435216od_a_a @ C @ A5 ) ) ).
% IntD1
thf(fact_914_IntD1,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B6 ) )
=> ( member_set_a @ C @ A5 ) ) ).
% IntD1
thf(fact_915_IntD1,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B6 ) )
=> ( member_a @ C @ A5 ) ) ).
% IntD1
thf(fact_916_IntE,axiom,
! [C: option_a,A5: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( inf_inf_set_option_a @ A5 @ B6 ) )
=> ~ ( ( member_option_a @ C @ A5 )
=> ~ ( member_option_a @ C @ B6 ) ) ) ).
% IntE
thf(fact_917_IntE,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A5 )
=> ~ ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% IntE
thf(fact_918_IntE,axiom,
! [C: set_a,A5: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B6 ) )
=> ~ ( ( member_set_a @ C @ A5 )
=> ~ ( member_set_a @ C @ B6 ) ) ) ).
% IntE
thf(fact_919_IntE,axiom,
! [C: a,A5: set_a,B6: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B6 ) )
=> ~ ( ( member_a @ C @ A5 )
=> ~ ( member_a @ C @ B6 ) ) ) ).
% IntE
thf(fact_920_Int__Collect__mono,axiom,
! [A5: set_option_a,B6: set_option_a,P2: option_a > $o,Q2: option_a > $o] :
( ( ord_le1955136853071979460tion_a @ A5 @ B6 )
=> ( ! [X5: option_a] :
( ( member_option_a @ X5 @ A5 )
=> ( ( P2 @ X5 )
=> ( Q2 @ X5 ) ) )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ A5 @ ( collect_option_a @ P2 ) ) @ ( inf_inf_set_option_a @ B6 @ ( collect_option_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_921_Int__Collect__mono,axiom,
! [A5: set_Product_prod_a_a,B6: set_Product_prod_a_a,P2: product_prod_a_a > $o,Q2: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A5 @ B6 )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A5 )
=> ( ( P2 @ X5 )
=> ( Q2 @ X5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A5 @ ( collec3336397797384452498od_a_a @ P2 ) ) @ ( inf_in8905007599844390133od_a_a @ B6 @ ( collec3336397797384452498od_a_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_922_Int__Collect__mono,axiom,
! [A5: set_set_a,B6: set_set_a,P2: set_a > $o,Q2: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A5 @ B6 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( ( P2 @ X5 )
=> ( Q2 @ X5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A5 @ ( collect_set_a @ P2 ) ) @ ( inf_inf_set_set_a @ B6 @ ( collect_set_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_923_Int__Collect__mono,axiom,
! [A5: set_a,B6: set_a,P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( ( P2 @ X5 )
=> ( Q2 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ ( collect_a @ P2 ) ) @ ( inf_inf_set_a @ B6 @ ( collect_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_924_Int__greatest,axiom,
! [C3: set_a,A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ C3 @ A5 )
=> ( ( ord_less_eq_set_a @ C3 @ B6 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A5 @ B6 ) ) ) ) ).
% Int_greatest
thf(fact_925_Int__absorb2,axiom,
! [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( inf_inf_set_a @ A5 @ B6 )
= A5 ) ) ).
% Int_absorb2
thf(fact_926_Int__absorb1,axiom,
! [B6: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B6 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ B6 )
= B6 ) ) ).
% Int_absorb1
thf(fact_927_Int__lower2,axiom,
! [A5: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ B6 ) ).
% Int_lower2
thf(fact_928_Int__lower1,axiom,
! [A5: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ A5 ) ).
% Int_lower1
thf(fact_929_Int__mono,axiom,
! [A5: set_a,C3: set_a,B6: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ ( inf_inf_set_a @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_930_Int__insert__left,axiom,
! [A: option_a,C3: set_option_a,B6: set_option_a] :
( ( ( member_option_a @ A @ C3 )
=> ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ B6 ) @ C3 )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ B6 @ C3 ) ) ) )
& ( ~ ( member_option_a @ A @ C3 )
=> ( ( inf_inf_set_option_a @ ( insert_option_a @ A @ B6 ) @ C3 )
= ( inf_inf_set_option_a @ B6 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_931_Int__insert__left,axiom,
! [A: product_prod_a_a,C3: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ C3 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B6 ) @ C3 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B6 @ C3 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ C3 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B6 ) @ C3 )
= ( inf_in8905007599844390133od_a_a @ B6 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_932_Int__insert__left,axiom,
! [A: set_a,C3: set_set_a,B6: set_set_a] :
( ( ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B6 ) @ C3 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B6 @ C3 ) ) ) )
& ( ~ ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B6 ) @ C3 )
= ( inf_inf_set_set_a @ B6 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_933_Int__insert__left,axiom,
! [A: a,C3: set_a,B6: set_a] :
( ( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B6 ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B6 @ C3 ) ) ) )
& ( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B6 ) @ C3 )
= ( inf_inf_set_a @ B6 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_934_Int__insert__right,axiom,
! [A: option_a,A5: set_option_a,B6: set_option_a] :
( ( ( member_option_a @ A @ A5 )
=> ( ( inf_inf_set_option_a @ A5 @ ( insert_option_a @ A @ B6 ) )
= ( insert_option_a @ A @ ( inf_inf_set_option_a @ A5 @ B6 ) ) ) )
& ( ~ ( member_option_a @ A @ A5 )
=> ( ( inf_inf_set_option_a @ A5 @ ( insert_option_a @ A @ B6 ) )
= ( inf_inf_set_option_a @ A5 @ B6 ) ) ) ) ).
% Int_insert_right
thf(fact_935_Int__insert__right,axiom,
! [A: product_prod_a_a,A5: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A @ B6 ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A @ B6 ) )
= ( inf_in8905007599844390133od_a_a @ A5 @ B6 ) ) ) ) ).
% Int_insert_right
thf(fact_936_Int__insert__right,axiom,
! [A: set_a,A5: set_set_a,B6: set_set_a] :
( ( ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B6 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A5 @ B6 ) ) ) )
& ( ~ ( member_set_a @ A @ A5 )
=> ( ( inf_inf_set_set_a @ A5 @ ( insert_set_a @ A @ B6 ) )
= ( inf_inf_set_set_a @ A5 @ B6 ) ) ) ) ).
% Int_insert_right
thf(fact_937_Int__insert__right,axiom,
! [A: a,A5: set_a,B6: set_a] :
( ( ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B6 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A5 @ B6 ) ) ) )
& ( ~ ( member_a @ A @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A @ B6 ) )
= ( inf_inf_set_a @ A5 @ B6 ) ) ) ) ).
% Int_insert_right
thf(fact_938_Diff__Int__distrib2,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ C3 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C3 ) @ ( inf_inf_set_a @ B6 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_939_Diff__Int__distrib,axiom,
! [C3: set_a,A5: set_a,B6: set_a] :
( ( inf_inf_set_a @ C3 @ ( minus_minus_set_a @ A5 @ B6 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C3 @ A5 ) @ ( inf_inf_set_a @ C3 @ B6 ) ) ) ).
% Diff_Int_distrib
thf(fact_940_Diff__Diff__Int,axiom,
! [A5: set_a,B6: set_a] :
( ( minus_minus_set_a @ A5 @ ( minus_minus_set_a @ A5 @ B6 ) )
= ( inf_inf_set_a @ A5 @ B6 ) ) ).
% Diff_Diff_Int
thf(fact_941_Diff__Int2,axiom,
! [A5: set_a,C3: set_a,B6: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C3 ) @ ( inf_inf_set_a @ B6 @ C3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ C3 ) @ B6 ) ) ).
% Diff_Int2
thf(fact_942_Int__Diff,axiom,
! [A5: set_a,B6: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ C3 )
= ( inf_inf_set_a @ A5 @ ( minus_minus_set_a @ B6 @ C3 ) ) ) ).
% Int_Diff
thf(fact_943_Diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A8: set_a,B7: set_a] : ( inf_inf_set_a @ A8 @ ( uminus_uminus_set_a @ B7 ) ) ) ) ).
% Diff_eq
thf(fact_944_mono__Field,axiom,
! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R2 @ S3 )
=> ( ord_less_eq_set_a @ ( field_a @ R2 ) @ ( field_a @ S3 ) ) ) ).
% mono_Field
thf(fact_945_Int__Diff__disjoint,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( inf_inf_set_option_a @ ( inf_inf_set_option_a @ A5 @ B6 ) @ ( minus_1574173051537231627tion_a @ A5 @ B6 ) )
= bot_bot_set_option_a ) ).
% Int_Diff_disjoint
thf(fact_946_Int__Diff__disjoint,axiom,
! [A5: set_a,B6: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B6 ) @ ( minus_minus_set_a @ A5 @ B6 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_947_Diff__triv,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( ( inf_inf_set_option_a @ A5 @ B6 )
= bot_bot_set_option_a )
=> ( ( minus_1574173051537231627tion_a @ A5 @ B6 )
= A5 ) ) ).
% Diff_triv
thf(fact_948_Diff__triv,axiom,
! [A5: set_a,B6: set_a] :
( ( ( inf_inf_set_a @ A5 @ B6 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A5 @ B6 )
= A5 ) ) ).
% Diff_triv
thf(fact_949_inf__cancel__left1,axiom,
! [X: a > $o,A: a > $o,B: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X @ A ) @ ( inf_inf_a_o @ ( uminus_uminus_a_o @ X ) @ B ) )
= bot_bot_a_o ) ).
% inf_cancel_left1
thf(fact_950_inf__cancel__left1,axiom,
! [X: set_option_a,A: set_option_a,B: set_option_a] :
( ( inf_inf_set_option_a @ ( inf_inf_set_option_a @ X @ A ) @ ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ B ) )
= bot_bot_set_option_a ) ).
% inf_cancel_left1
thf(fact_951_inf__cancel__left1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ A ) @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left1
thf(fact_952_inf__cancel__left2,axiom,
! [X: a > $o,A: a > $o,B: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ ( uminus_uminus_a_o @ X ) @ A ) @ ( inf_inf_a_o @ X @ B ) )
= bot_bot_a_o ) ).
% inf_cancel_left2
thf(fact_953_inf__cancel__left2,axiom,
! [X: set_option_a,A: set_option_a,B: set_option_a] :
( ( inf_inf_set_option_a @ ( inf_inf_set_option_a @ ( uminus6205308855922866075tion_a @ X ) @ A ) @ ( inf_inf_set_option_a @ X @ B ) )
= bot_bot_set_option_a ) ).
% inf_cancel_left2
thf(fact_954_inf__cancel__left2,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ A ) @ ( inf_inf_set_a @ X @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left2
thf(fact_955_inf__shunt,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( inf_inf_a_o @ X @ Y )
= bot_bot_a_o )
= ( ord_less_eq_a_o @ X @ ( uminus_uminus_a_o @ Y ) ) ) ).
% inf_shunt
thf(fact_956_inf__shunt,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( inf_inf_set_option_a @ X @ Y )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ X @ ( uminus6205308855922866075tion_a @ Y ) ) ) ).
% inf_shunt
thf(fact_957_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_958_sup__neg__inf,axiom,
! [P: set_a,Q: set_a,R2: set_a] :
( ( ord_less_eq_set_a @ P @ ( sup_sup_set_a @ Q @ R2 ) )
= ( ord_less_eq_set_a @ ( inf_inf_set_a @ P @ ( uminus_uminus_set_a @ Q ) ) @ R2 ) ) ).
% sup_neg_inf
thf(fact_959_shunt2,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) @ Z2 )
= ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% shunt2
thf(fact_960_shunt1,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z2 )
= ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ Y ) @ Z2 ) ) ) ).
% shunt1
thf(fact_961_disjoint__eq__subset__Compl,axiom,
! [A5: set_option_a,B6: set_option_a] :
( ( ( inf_inf_set_option_a @ A5 @ B6 )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ A5 @ ( uminus6205308855922866075tion_a @ B6 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_962_disjoint__eq__subset__Compl,axiom,
! [A5: set_a,B6: set_a] :
( ( ( inf_inf_set_a @ A5 @ B6 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A5 @ ( uminus_uminus_set_a @ B6 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_963_map__add__comm,axiom,
! [M1: a > option_a,M2: a > option_a] :
( ( ( inf_inf_set_a @ ( dom_a_a @ M1 ) @ ( dom_a_a @ M2 ) )
= bot_bot_set_a )
=> ( ( map_add_a_a @ M1 @ M2 )
= ( map_add_a_a @ M2 @ M1 ) ) ) ).
% map_add_comm
thf(fact_964_graph__map__add,axiom,
! [M1: a > option_a,M2: a > option_a] :
( ( ( inf_inf_set_a @ ( dom_a_a @ M1 ) @ ( dom_a_a @ M2 ) )
= bot_bot_set_a )
=> ( ( graph_a_a @ ( map_add_a_a @ M1 @ M2 ) )
= ( sup_su3048258781599657691od_a_a @ ( graph_a_a @ M1 ) @ ( graph_a_a @ M2 ) ) ) ) ).
% graph_map_add
thf(fact_965_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr8600417178894128327od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
( ( order_3202267349275844158od_a_a @ ( field_1126092520709947252od_a_a @ R2 ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ A @ ( field_1126092520709947252od_a_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ B @ ( field_1126092520709947252od_a_a @ R2 ) )
=> ( ( ord_le746702958409616551od_a_a @ ( image_9076584400576816019od_a_a @ R2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) @ ( image_9076584400576816019od_a_a @ R2 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_966_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr5845495582615845127_set_a,A: set_a,B: set_a] :
( ( order_4854350610611212405_set_a @ ( field_set_a @ R2 ) @ R2 )
=> ( ( member_set_a @ A @ ( field_set_a @ R2 ) )
=> ( ( member_set_a @ B @ ( field_set_a @ R2 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ R2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) @ ( image_set_a_set_a @ R2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) )
= ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_967_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( order_4134995541221112539tion_a @ ( field_option_a @ R2 ) @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_968_subset__Image1__Image1__iff,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( order_preorder_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_969_inj__on__Un,axiom,
! [F: set_a > set_a,A5: set_set_a,B6: set_set_a] :
( ( inj_on_set_a_set_a @ F @ ( sup_sup_set_set_a @ A5 @ B6 ) )
= ( ( inj_on_set_a_set_a @ F @ A5 )
& ( inj_on_set_a_set_a @ F @ B6 )
& ( ( inf_inf_set_set_a @ ( image_set_a_set_a2 @ F @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) ) @ ( image_set_a_set_a2 @ F @ ( minus_5736297505244876581_set_a @ B6 @ A5 ) ) )
= bot_bot_set_set_a ) ) ) ).
% inj_on_Un
thf(fact_970_inj__on__Un,axiom,
! [F: a > a,A5: set_a,B6: set_a] :
( ( inj_on_a_a @ F @ ( sup_sup_set_a @ A5 @ B6 ) )
= ( ( inj_on_a_a @ F @ A5 )
& ( inj_on_a_a @ F @ B6 )
& ( ( inf_inf_set_a @ ( image_a_a2 @ F @ ( minus_minus_set_a @ A5 @ B6 ) ) @ ( image_a_a2 @ F @ ( minus_minus_set_a @ B6 @ A5 ) ) )
= bot_bot_set_a ) ) ) ).
% inj_on_Un
thf(fact_971_inj__on__Un,axiom,
! [F: a > option_a,A5: set_a,B6: set_a] :
( ( inj_on_a_option_a @ F @ ( sup_sup_set_a @ A5 @ B6 ) )
= ( ( inj_on_a_option_a @ F @ A5 )
& ( inj_on_a_option_a @ F @ B6 )
& ( ( inf_inf_set_option_a @ ( image_a_option_a2 @ F @ ( minus_minus_set_a @ A5 @ B6 ) ) @ ( image_a_option_a2 @ F @ ( minus_minus_set_a @ B6 @ A5 ) ) )
= bot_bot_set_option_a ) ) ) ).
% inj_on_Un
thf(fact_972_image__eqI,axiom,
! [B: a,F: a > a,X: a,A5: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A5 )
=> ( member_a @ B @ ( image_a_a2 @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_973_image__eqI,axiom,
! [B: option_a,F: a > option_a,X: a,A5: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A5 )
=> ( member_option_a @ B @ ( image_a_option_a2 @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_974_image__eqI,axiom,
! [B: set_a,F: a > set_a,X: a,A5: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A5 )
=> ( member_set_a @ B @ ( image_a_set_a2 @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_975_image__eqI,axiom,
! [B: a,F: option_a > a,X: option_a,A5: set_option_a] :
( ( B
= ( F @ X ) )
=> ( ( member_option_a @ X @ A5 )
=> ( member_a @ B @ ( image_option_a_a2 @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_976_image__eqI,axiom,
! [B: a,F: set_a > a,X: set_a,A5: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A5 )
=> ( member_a @ B @ ( image_set_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_977_image__eqI,axiom,
! [B: product_prod_a_a,F: a > product_prod_a_a,X: a,A5: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A5 )
=> ( member1426531477525435216od_a_a @ B @ ( image_7400625782589995694od_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_978_image__eqI,axiom,
! [B: option_a,F: option_a > option_a,X: option_a,A5: set_option_a] :
( ( B
= ( F @ X ) )
=> ( ( member_option_a @ X @ A5 )
=> ( member_option_a @ B @ ( image_7439109396645324421tion_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_979_image__eqI,axiom,
! [B: set_a,F: option_a > set_a,X: option_a,A5: set_option_a] :
( ( B
= ( F @ X ) )
=> ( ( member_option_a @ X @ A5 )
=> ( member_set_a @ B @ ( image_option_a_set_a2 @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_980_image__eqI,axiom,
! [B: a,F: product_prod_a_a > a,X: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( B
= ( F @ X ) )
=> ( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( member_a @ B @ ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_981_image__eqI,axiom,
! [B: option_a,F: set_a > option_a,X: set_a,A5: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A5 )
=> ( member_option_a @ B @ ( image_set_a_option_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_982_image__empty,axiom,
! [F: set_a > set_a] :
( ( image_set_a_set_a2 @ F @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_983_image__empty,axiom,
! [F: a > a] :
( ( image_a_a2 @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_984_image__empty,axiom,
! [F: a > option_a] :
( ( image_a_option_a2 @ F @ bot_bot_set_a )
= bot_bot_set_option_a ) ).
% image_empty
thf(fact_985_image__empty,axiom,
! [F: option_a > a] :
( ( image_option_a_a2 @ F @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_986_image__empty,axiom,
! [F: option_a > option_a] :
( ( image_7439109396645324421tion_a @ F @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% image_empty
thf(fact_987_empty__is__image,axiom,
! [F: set_a > set_a,A5: set_set_a] :
( ( bot_bot_set_set_a
= ( image_set_a_set_a2 @ F @ A5 ) )
= ( A5 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_988_empty__is__image,axiom,
! [F: a > a,A5: set_a] :
( ( bot_bot_set_a
= ( image_a_a2 @ F @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_989_empty__is__image,axiom,
! [F: option_a > a,A5: set_option_a] :
( ( bot_bot_set_a
= ( image_option_a_a2 @ F @ A5 ) )
= ( A5 = bot_bot_set_option_a ) ) ).
% empty_is_image
thf(fact_990_empty__is__image,axiom,
! [F: a > option_a,A5: set_a] :
( ( bot_bot_set_option_a
= ( image_a_option_a2 @ F @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_991_empty__is__image,axiom,
! [F: option_a > option_a,A5: set_option_a] :
( ( bot_bot_set_option_a
= ( image_7439109396645324421tion_a @ F @ A5 ) )
= ( A5 = bot_bot_set_option_a ) ) ).
% empty_is_image
thf(fact_992_image__is__empty,axiom,
! [F: set_a > set_a,A5: set_set_a] :
( ( ( image_set_a_set_a2 @ F @ A5 )
= bot_bot_set_set_a )
= ( A5 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_993_image__is__empty,axiom,
! [F: a > a,A5: set_a] :
( ( ( image_a_a2 @ F @ A5 )
= bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_994_image__is__empty,axiom,
! [F: option_a > a,A5: set_option_a] :
( ( ( image_option_a_a2 @ F @ A5 )
= bot_bot_set_a )
= ( A5 = bot_bot_set_option_a ) ) ).
% image_is_empty
thf(fact_995_image__is__empty,axiom,
! [F: a > option_a,A5: set_a] :
( ( ( image_a_option_a2 @ F @ A5 )
= bot_bot_set_option_a )
= ( A5 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_996_image__is__empty,axiom,
! [F: option_a > option_a,A5: set_option_a] :
( ( ( image_7439109396645324421tion_a @ F @ A5 )
= bot_bot_set_option_a )
= ( A5 = bot_bot_set_option_a ) ) ).
% image_is_empty
thf(fact_997_insert__image,axiom,
! [X: a,A5: set_a,F: a > a] :
( ( member_a @ X @ A5 )
=> ( ( insert_a @ ( F @ X ) @ ( image_a_a2 @ F @ A5 ) )
= ( image_a_a2 @ F @ A5 ) ) ) ).
% insert_image
thf(fact_998_insert__image,axiom,
! [X: a,A5: set_a,F: a > option_a] :
( ( member_a @ X @ A5 )
=> ( ( insert_option_a @ ( F @ X ) @ ( image_a_option_a2 @ F @ A5 ) )
= ( image_a_option_a2 @ F @ A5 ) ) ) ).
% insert_image
thf(fact_999_insert__image,axiom,
! [X: option_a,A5: set_option_a,F: option_a > a] :
( ( member_option_a @ X @ A5 )
=> ( ( insert_a @ ( F @ X ) @ ( image_option_a_a2 @ F @ A5 ) )
= ( image_option_a_a2 @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1000_insert__image,axiom,
! [X: option_a,A5: set_option_a,F: option_a > option_a] :
( ( member_option_a @ X @ A5 )
=> ( ( insert_option_a @ ( F @ X ) @ ( image_7439109396645324421tion_a @ F @ A5 ) )
= ( image_7439109396645324421tion_a @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1001_insert__image,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,F: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( insert_a @ ( F @ X ) @ ( image_3437945252899457948_a_a_a @ F @ A5 ) )
= ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1002_insert__image,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,F: product_prod_a_a > option_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( insert_option_a @ ( F @ X ) @ ( image_4859188117451336930tion_a @ F @ A5 ) )
= ( image_4859188117451336930tion_a @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1003_insert__image,axiom,
! [X: set_a,A5: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A5 )
=> ( ( insert_set_a @ ( F @ X ) @ ( image_set_a_set_a2 @ F @ A5 ) )
= ( image_set_a_set_a2 @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1004_insert__image,axiom,
! [X: set_a,A5: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A5 )
=> ( ( insert_a @ ( F @ X ) @ ( image_set_a_a @ F @ A5 ) )
= ( image_set_a_a @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1005_insert__image,axiom,
! [X: set_a,A5: set_set_a,F: set_a > option_a] :
( ( member_set_a @ X @ A5 )
=> ( ( insert_option_a @ ( F @ X ) @ ( image_set_a_option_a @ F @ A5 ) )
= ( image_set_a_option_a @ F @ A5 ) ) ) ).
% insert_image
thf(fact_1006_image__insert,axiom,
! [F: set_a > set_a,A: set_a,B6: set_set_a] :
( ( image_set_a_set_a2 @ F @ ( insert_set_a @ A @ B6 ) )
= ( insert_set_a @ ( F @ A ) @ ( image_set_a_set_a2 @ F @ B6 ) ) ) ).
% image_insert
thf(fact_1007_image__insert,axiom,
! [F: a > a,A: a,B6: set_a] :
( ( image_a_a2 @ F @ ( insert_a @ A @ B6 ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a2 @ F @ B6 ) ) ) ).
% image_insert
thf(fact_1008_image__insert,axiom,
! [F: a > option_a,A: a,B6: set_a] :
( ( image_a_option_a2 @ F @ ( insert_a @ A @ B6 ) )
= ( insert_option_a @ ( F @ A ) @ ( image_a_option_a2 @ F @ B6 ) ) ) ).
% image_insert
thf(fact_1009_image__insert,axiom,
! [F: option_a > a,A: option_a,B6: set_option_a] :
( ( image_option_a_a2 @ F @ ( insert_option_a @ A @ B6 ) )
= ( insert_a @ ( F @ A ) @ ( image_option_a_a2 @ F @ B6 ) ) ) ).
% image_insert
thf(fact_1010_image__insert,axiom,
! [F: option_a > option_a,A: option_a,B6: set_option_a] :
( ( image_7439109396645324421tion_a @ F @ ( insert_option_a @ A @ B6 ) )
= ( insert_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ B6 ) ) ) ).
% image_insert
thf(fact_1011_Image__empty2,axiom,
! [R5: set_Product_prod_a_a] :
( ( image_a_a @ R5 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Image_empty2
thf(fact_1012_Image__empty2,axiom,
! [R5: set_Pr3411724424142761165tion_a] :
( ( image_a_option_a @ R5 @ bot_bot_set_a )
= bot_bot_set_option_a ) ).
% Image_empty2
thf(fact_1013_Image__empty2,axiom,
! [R5: set_Pr6308966090954093121on_a_a] :
( ( image_option_a_a @ R5 @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% Image_empty2
thf(fact_1014_Image__empty2,axiom,
! [R5: set_Pr7585778909603769095tion_a] :
( ( image_4442594622209975379tion_a @ R5 @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% Image_empty2
thf(fact_1015_image__map__upd,axiom,
! [X: option_a,A5: set_option_a,M: option_a > option_a,Y: a] :
( ~ ( member_option_a @ X @ A5 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ M @ X @ ( some_a @ Y ) ) @ A5 )
= ( image_7439109396645324421tion_a @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_1016_image__map__upd,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,M: product_prod_a_a > option_a,Y: a] :
( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( image_4859188117451336930tion_a @ ( fun_up8298456451713467738tion_a @ M @ X @ ( some_a @ Y ) ) @ A5 )
= ( image_4859188117451336930tion_a @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_1017_image__map__upd,axiom,
! [X: set_a,A5: set_set_a,M: set_a > option_a,Y: a] :
( ~ ( member_set_a @ X @ A5 )
=> ( ( image_set_a_option_a @ ( fun_up3663993102702442083tion_a @ M @ X @ ( some_a @ Y ) ) @ A5 )
= ( image_set_a_option_a @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_1018_image__map__upd,axiom,
! [X: a,A5: set_a,M: a > option_a,Y: a] :
( ~ ( member_a @ X @ A5 )
=> ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ M @ X @ ( some_a @ Y ) ) @ A5 )
= ( image_a_option_a2 @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_1019_these__image__Some__eq,axiom,
! [A5: set_a] :
( ( these_a @ ( image_a_option_a2 @ some_a @ A5 ) )
= A5 ) ).
% these_image_Some_eq
thf(fact_1020_Image__empty1,axiom,
! [X8: set_a] :
( ( image_a_a @ bot_bo3357376287454694259od_a_a @ X8 )
= bot_bot_set_a ) ).
% Image_empty1
thf(fact_1021_Image__singleton__iff,axiom,
! [B: option_a,R2: set_Pr3411724424142761165tion_a,A: a] :
( ( member_option_a @ B @ ( image_a_option_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1022_Image__singleton__iff,axiom,
! [B: product_prod_a_a,R2: set_Pr5530083903271594800od_a_a,A: a] :
( ( member1426531477525435216od_a_a @ B @ ( image_2799180466780705916od_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1023_Image__singleton__iff,axiom,
! [B: set_a,R2: set_Pr6393634178297680487_set_a,A: a] :
( ( member_set_a @ B @ ( image_a_set_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( member4771970882521526448_set_a @ ( product_Pair_a_set_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1024_Image__singleton__iff,axiom,
! [B: a,R2: set_Product_prod_a_a,A: a] :
( ( member_a @ B @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1025_Image__singleton__iff,axiom,
! [B: a,R2: set_Pr6308966090954093121on_a_a,A: option_a] :
( ( member_a @ B @ ( image_option_a_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1026_Image__singleton__iff,axiom,
! [B: option_a,R2: set_Pr7585778909603769095tion_a,A: option_a] :
( ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1027_Image__singleton__iff,axiom,
! [B: product_prod_a_a,R2: set_Pr5481487045354815082od_a_a,A: option_a] :
( ( member1426531477525435216od_a_a @ B @ ( image_6066027731942400950od_a_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( member7490379427973688371od_a_a @ ( produc798408195413729658od_a_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1028_Image__singleton__iff,axiom,
! [B: set_a,R2: set_Pr5928443294945626017_set_a,A: option_a] :
( ( member_set_a @ B @ ( image_option_a_set_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( member2542914943202714858_set_a @ ( produc1750493463454711729_set_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_1029_inj__on__insert,axiom,
! [F: set_a > set_a,A: set_a,A5: set_set_a] :
( ( inj_on_set_a_set_a @ F @ ( insert_set_a @ A @ A5 ) )
= ( ( inj_on_set_a_set_a @ F @ A5 )
& ~ ( member_set_a @ ( F @ A ) @ ( image_set_a_set_a2 @ F @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1030_inj__on__insert,axiom,
! [F: option_a > a,A: option_a,A5: set_option_a] :
( ( inj_on_option_a_a @ F @ ( insert_option_a @ A @ A5 ) )
= ( ( inj_on_option_a_a @ F @ A5 )
& ~ ( member_a @ ( F @ A ) @ ( image_option_a_a2 @ F @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1031_inj__on__insert,axiom,
! [F: option_a > option_a,A: option_a,A5: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ ( insert_option_a @ A @ A5 ) )
= ( ( inj_on8559383841115902449tion_a @ F @ A5 )
& ~ ( member_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1032_inj__on__insert,axiom,
! [F: option_a > product_prod_a_a,A: option_a,A5: set_option_a] :
( ( inj_on6713162466721990740od_a_a @ F @ ( insert_option_a @ A @ A5 ) )
= ( ( inj_on6713162466721990740od_a_a @ F @ A5 )
& ~ ( member1426531477525435216od_a_a @ ( F @ A ) @ ( image_7456799861883459304od_a_a @ F @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1033_inj__on__insert,axiom,
! [F: option_a > set_a,A: option_a,A5: set_option_a] :
( ( inj_on2187968386393286795_set_a @ F @ ( insert_option_a @ A @ A5 ) )
= ( ( inj_on2187968386393286795_set_a @ F @ A5 )
& ~ ( member_set_a @ ( F @ A ) @ ( image_option_a_set_a2 @ F @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1034_inj__on__insert,axiom,
! [F: a > a,A: a,A5: set_a] :
( ( inj_on_a_a @ F @ ( insert_a @ A @ A5 ) )
= ( ( inj_on_a_a @ F @ A5 )
& ~ ( member_a @ ( F @ A ) @ ( image_a_a2 @ F @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1035_inj__on__insert,axiom,
! [F: a > option_a,A: a,A5: set_a] :
( ( inj_on_a_option_a @ F @ ( insert_a @ A @ A5 ) )
= ( ( inj_on_a_option_a @ F @ A5 )
& ~ ( member_option_a @ ( F @ A ) @ ( image_a_option_a2 @ F @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1036_inj__on__insert,axiom,
! [F: a > product_prod_a_a,A: a,A5: set_a] :
( ( inj_on8941660083241582106od_a_a @ F @ ( insert_a @ A @ A5 ) )
= ( ( inj_on8941660083241582106od_a_a @ F @ A5 )
& ~ ( member1426531477525435216od_a_a @ ( F @ A ) @ ( image_7400625782589995694od_a_a @ F @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1037_inj__on__insert,axiom,
! [F: a > set_a,A: a,A5: set_a] :
( ( inj_on_a_set_a @ F @ ( insert_a @ A @ A5 ) )
= ( ( inj_on_a_set_a @ F @ A5 )
& ~ ( member_set_a @ ( F @ A ) @ ( image_a_set_a2 @ F @ ( minus_minus_set_a @ A5 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_1038_image__Int__subset,axiom,
! [F: set_a > set_a,A5: set_set_a,B6: set_set_a] : ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a2 @ F @ ( inf_inf_set_set_a @ A5 @ B6 ) ) @ ( inf_inf_set_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ ( image_set_a_set_a2 @ F @ B6 ) ) ) ).
% image_Int_subset
thf(fact_1039_image__Int__subset,axiom,
! [F: a > option_a,A5: set_a,B6: set_a] : ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F @ ( inf_inf_set_a @ A5 @ B6 ) ) @ ( inf_inf_set_option_a @ ( image_a_option_a2 @ F @ A5 ) @ ( image_a_option_a2 @ F @ B6 ) ) ) ).
% image_Int_subset
thf(fact_1040_image__Int__subset,axiom,
! [F: a > a,A5: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( image_a_a2 @ F @ ( inf_inf_set_a @ A5 @ B6 ) ) @ ( inf_inf_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B6 ) ) ) ).
% image_Int_subset
thf(fact_1041_Image__Int__subset,axiom,
! [R5: set_Product_prod_a_a,A5: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ R5 @ ( inf_inf_set_a @ A5 @ B6 ) ) @ ( inf_inf_set_a @ ( image_a_a @ R5 @ A5 ) @ ( image_a_a @ R5 @ B6 ) ) ) ).
% Image_Int_subset
thf(fact_1042_the__elem__image__unique,axiom,
! [A5: set_set_a,F: set_a > set_a,X: set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ! [Y4: set_a] :
( ( member_set_a @ Y4 @ A5 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_set_a @ ( image_set_a_set_a2 @ F @ A5 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1043_the__elem__image__unique,axiom,
! [A5: set_a,F: a > a,X: a] :
( ( A5 != bot_bot_set_a )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ A5 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_a @ ( image_a_a2 @ F @ A5 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1044_the__elem__image__unique,axiom,
! [A5: set_a,F: a > option_a,X: a] :
( ( A5 != bot_bot_set_a )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ A5 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_option_a @ ( image_a_option_a2 @ F @ A5 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1045_Image__mono,axiom,
! [R6: set_Product_prod_a_a,R2: set_Product_prod_a_a,A10: set_a,A5: set_a] :
( ( ord_le746702958409616551od_a_a @ R6 @ R2 )
=> ( ( ord_less_eq_set_a @ A10 @ A5 )
=> ( ord_less_eq_set_a @ ( image_a_a @ R6 @ A10 ) @ ( image_a_a @ R2 @ A5 ) ) ) ) ).
% Image_mono
thf(fact_1046_image__diff__subset,axiom,
! [F: set_a > set_a,A5: set_set_a,B6: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ ( image_set_a_set_a2 @ F @ B6 ) ) @ ( image_set_a_set_a2 @ F @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) ) ) ).
% image_diff_subset
thf(fact_1047_image__diff__subset,axiom,
! [F: a > option_a,A5: set_a,B6: set_a] : ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ ( image_a_option_a2 @ F @ A5 ) @ ( image_a_option_a2 @ F @ B6 ) ) @ ( image_a_option_a2 @ F @ ( minus_minus_set_a @ A5 @ B6 ) ) ) ).
% image_diff_subset
thf(fact_1048_image__diff__subset,axiom,
! [F: a > a,A5: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B6 ) ) @ ( image_a_a2 @ F @ ( minus_minus_set_a @ A5 @ B6 ) ) ) ).
% image_diff_subset
thf(fact_1049_None__notin__image__Some,axiom,
! [A5: set_a] :
~ ( member_option_a @ none_a @ ( image_a_option_a2 @ some_a @ A5 ) ) ).
% None_notin_image_Some
thf(fact_1050_image__mono,axiom,
! [A5: set_set_a,B6: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B6 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ ( image_set_a_set_a2 @ F @ B6 ) ) ) ).
% image_mono
thf(fact_1051_image__mono,axiom,
! [A5: set_a,B6: set_a,F: a > option_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F @ A5 ) @ ( image_a_option_a2 @ F @ B6 ) ) ) ).
% image_mono
thf(fact_1052_image__mono,axiom,
! [A5: set_a,B6: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ord_less_eq_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B6 ) ) ) ).
% image_mono
thf(fact_1053_image__subsetI,axiom,
! [A5: set_a,F: a > a,B6: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( member_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_less_eq_set_a @ ( image_a_a2 @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1054_image__subsetI,axiom,
! [A5: set_a,F: a > option_a,B6: set_option_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( member_option_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1055_image__subsetI,axiom,
! [A5: set_a,F: a > set_a,B6: set_set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( member_set_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a2 @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1056_image__subsetI,axiom,
! [A5: set_option_a,F: option_a > a,B6: set_a] :
( ! [X5: option_a] :
( ( member_option_a @ X5 @ A5 )
=> ( member_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_less_eq_set_a @ ( image_option_a_a2 @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1057_image__subsetI,axiom,
! [A5: set_set_a,F: set_a > a,B6: set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( member_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1058_image__subsetI,axiom,
! [A5: set_a,F: a > product_prod_a_a,B6: set_Product_prod_a_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( member1426531477525435216od_a_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_le746702958409616551od_a_a @ ( image_7400625782589995694od_a_a @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1059_image__subsetI,axiom,
! [A5: set_option_a,F: option_a > option_a,B6: set_option_a] :
( ! [X5: option_a] :
( ( member_option_a @ X5 @ A5 )
=> ( member_option_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_le1955136853071979460tion_a @ ( image_7439109396645324421tion_a @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1060_image__subsetI,axiom,
! [A5: set_option_a,F: option_a > set_a,B6: set_set_a] :
( ! [X5: option_a] :
( ( member_option_a @ X5 @ A5 )
=> ( member_set_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_option_a_set_a2 @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1061_image__subsetI,axiom,
! [A5: set_set_a,F: set_a > option_a,B6: set_option_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( member_option_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_le1955136853071979460tion_a @ ( image_set_a_option_a @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1062_image__subsetI,axiom,
! [A5: set_set_a,F: set_a > set_a,B6: set_set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( member_set_a @ ( F @ X5 ) @ B6 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1063_subset__imageE,axiom,
! [B6: set_set_a,F: set_a > set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ ( image_set_a_set_a2 @ F @ A5 ) )
=> ~ ! [C5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C5 @ A5 )
=> ( B6
!= ( image_set_a_set_a2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1064_subset__imageE,axiom,
! [B6: set_option_a,F: a > option_a,A5: set_a] :
( ( ord_le1955136853071979460tion_a @ B6 @ ( image_a_option_a2 @ F @ A5 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A5 )
=> ( B6
!= ( image_a_option_a2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1065_subset__imageE,axiom,
! [B6: set_a,F: a > a,A5: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a2 @ F @ A5 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A5 )
=> ( B6
!= ( image_a_a2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1066_image__subset__iff,axiom,
! [F: a > option_a,A5: set_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F @ A5 ) @ B6 )
= ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( member_option_a @ ( F @ X4 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_1067_image__subset__iff,axiom,
! [F: set_a > set_a,A5: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ B6 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( member_set_a @ ( F @ X4 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_1068_image__subset__iff,axiom,
! [F: a > a,A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a2 @ F @ A5 ) @ B6 )
= ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( member_a @ ( F @ X4 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_1069_subset__image__iff,axiom,
! [B6: set_set_a,F: set_a > set_a,A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ ( image_set_a_set_a2 @ F @ A5 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A5 )
& ( B6
= ( image_set_a_set_a2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1070_subset__image__iff,axiom,
! [B6: set_option_a,F: a > option_a,A5: set_a] :
( ( ord_le1955136853071979460tion_a @ B6 @ ( image_a_option_a2 @ F @ A5 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A5 )
& ( B6
= ( image_a_option_a2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1071_subset__image__iff,axiom,
! [B6: set_a,F: a > a,A5: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a2 @ F @ A5 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A5 )
& ( B6
= ( image_a_a2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1072_imageI,axiom,
! [X: a,A5: set_a,F: a > a] :
( ( member_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ ( image_a_a2 @ F @ A5 ) ) ) ).
% imageI
thf(fact_1073_imageI,axiom,
! [X: a,A5: set_a,F: a > option_a] :
( ( member_a @ X @ A5 )
=> ( member_option_a @ ( F @ X ) @ ( image_a_option_a2 @ F @ A5 ) ) ) ).
% imageI
thf(fact_1074_imageI,axiom,
! [X: a,A5: set_a,F: a > set_a] :
( ( member_a @ X @ A5 )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a2 @ F @ A5 ) ) ) ).
% imageI
thf(fact_1075_imageI,axiom,
! [X: option_a,A5: set_option_a,F: option_a > a] :
( ( member_option_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ ( image_option_a_a2 @ F @ A5 ) ) ) ).
% imageI
thf(fact_1076_imageI,axiom,
! [X: set_a,A5: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ ( image_set_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1077_imageI,axiom,
! [X: a,A5: set_a,F: a > product_prod_a_a] :
( ( member_a @ X @ A5 )
=> ( member1426531477525435216od_a_a @ ( F @ X ) @ ( image_7400625782589995694od_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1078_imageI,axiom,
! [X: option_a,A5: set_option_a,F: option_a > option_a] :
( ( member_option_a @ X @ A5 )
=> ( member_option_a @ ( F @ X ) @ ( image_7439109396645324421tion_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1079_imageI,axiom,
! [X: option_a,A5: set_option_a,F: option_a > set_a] :
( ( member_option_a @ X @ A5 )
=> ( member_set_a @ ( F @ X ) @ ( image_option_a_set_a2 @ F @ A5 ) ) ) ).
% imageI
thf(fact_1080_imageI,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,F: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1081_imageI,axiom,
! [X: set_a,A5: set_set_a,F: set_a > option_a] :
( ( member_set_a @ X @ A5 )
=> ( member_option_a @ ( F @ X ) @ ( image_set_a_option_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_1082_image__iff,axiom,
! [Z2: a,F: a > a,A5: set_a] :
( ( member_a @ Z2 @ ( image_a_a2 @ F @ A5 ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_1083_image__iff,axiom,
! [Z2: option_a,F: a > option_a,A5: set_a] :
( ( member_option_a @ Z2 @ ( image_a_option_a2 @ F @ A5 ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_1084_image__iff,axiom,
! [Z2: set_a,F: set_a > set_a,A5: set_set_a] :
( ( member_set_a @ Z2 @ ( image_set_a_set_a2 @ F @ A5 ) )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_1085_bex__imageD,axiom,
! [F: a > a,A5: set_a,P2: a > $o] :
( ? [X9: a] :
( ( member_a @ X9 @ ( image_a_a2 @ F @ A5 ) )
& ( P2 @ X9 ) )
=> ? [X5: a] :
( ( member_a @ X5 @ A5 )
& ( P2 @ ( F @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_1086_bex__imageD,axiom,
! [F: a > option_a,A5: set_a,P2: option_a > $o] :
( ? [X9: option_a] :
( ( member_option_a @ X9 @ ( image_a_option_a2 @ F @ A5 ) )
& ( P2 @ X9 ) )
=> ? [X5: a] :
( ( member_a @ X5 @ A5 )
& ( P2 @ ( F @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_1087_bex__imageD,axiom,
! [F: set_a > set_a,A5: set_set_a,P2: set_a > $o] :
( ? [X9: set_a] :
( ( member_set_a @ X9 @ ( image_set_a_set_a2 @ F @ A5 ) )
& ( P2 @ X9 ) )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
& ( P2 @ ( F @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_1088_image__cong,axiom,
! [M4: set_a,N2: set_a,F: a > a,G: a > a] :
( ( M4 = N2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ N2 )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( image_a_a2 @ F @ M4 )
= ( image_a_a2 @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_1089_image__cong,axiom,
! [M4: set_a,N2: set_a,F: a > option_a,G: a > option_a] :
( ( M4 = N2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ N2 )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( image_a_option_a2 @ F @ M4 )
= ( image_a_option_a2 @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_1090_image__cong,axiom,
! [M4: set_set_a,N2: set_set_a,F: set_a > set_a,G: set_a > set_a] :
( ( M4 = N2 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ N2 )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( image_set_a_set_a2 @ F @ M4 )
= ( image_set_a_set_a2 @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_1091_ball__imageD,axiom,
! [F: a > a,A5: set_a,P2: a > $o] :
( ! [X5: a] :
( ( member_a @ X5 @ ( image_a_a2 @ F @ A5 ) )
=> ( P2 @ X5 ) )
=> ! [X9: a] :
( ( member_a @ X9 @ A5 )
=> ( P2 @ ( F @ X9 ) ) ) ) ).
% ball_imageD
thf(fact_1092_ball__imageD,axiom,
! [F: a > option_a,A5: set_a,P2: option_a > $o] :
( ! [X5: option_a] :
( ( member_option_a @ X5 @ ( image_a_option_a2 @ F @ A5 ) )
=> ( P2 @ X5 ) )
=> ! [X9: a] :
( ( member_a @ X9 @ A5 )
=> ( P2 @ ( F @ X9 ) ) ) ) ).
% ball_imageD
thf(fact_1093_ball__imageD,axiom,
! [F: set_a > set_a,A5: set_set_a,P2: set_a > $o] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( image_set_a_set_a2 @ F @ A5 ) )
=> ( P2 @ X5 ) )
=> ! [X9: set_a] :
( ( member_set_a @ X9 @ A5 )
=> ( P2 @ ( F @ X9 ) ) ) ) ).
% ball_imageD
thf(fact_1094_rev__image__eqI,axiom,
! [X: a,A5: set_a,B: a,F: a > a] :
( ( member_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a2 @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1095_rev__image__eqI,axiom,
! [X: a,A5: set_a,B: option_a,F: a > option_a] :
( ( member_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_option_a @ B @ ( image_a_option_a2 @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1096_rev__image__eqI,axiom,
! [X: a,A5: set_a,B: set_a,F: a > set_a] :
( ( member_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_a_set_a2 @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1097_rev__image__eqI,axiom,
! [X: option_a,A5: set_option_a,B: a,F: option_a > a] :
( ( member_option_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_option_a_a2 @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1098_rev__image__eqI,axiom,
! [X: set_a,A5: set_set_a,B: a,F: set_a > a] :
( ( member_set_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1099_rev__image__eqI,axiom,
! [X: a,A5: set_a,B: product_prod_a_a,F: a > product_prod_a_a] :
( ( member_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member1426531477525435216od_a_a @ B @ ( image_7400625782589995694od_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1100_rev__image__eqI,axiom,
! [X: option_a,A5: set_option_a,B: option_a,F: option_a > option_a] :
( ( member_option_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_option_a @ B @ ( image_7439109396645324421tion_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1101_rev__image__eqI,axiom,
! [X: option_a,A5: set_option_a,B: set_a,F: option_a > set_a] :
( ( member_option_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_option_a_set_a2 @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1102_rev__image__eqI,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,B: a,F: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1103_rev__image__eqI,axiom,
! [X: set_a,A5: set_set_a,B: option_a,F: set_a > option_a] :
( ( member_set_a @ X @ A5 )
=> ( ( B
= ( F @ X ) )
=> ( member_option_a @ B @ ( image_set_a_option_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_1104_pairwise__imageI,axiom,
! [A5: set_a,F: a > option_a,P2: option_a > option_a > $o] :
( ! [X5: a,Y4: a] :
( ( member_a @ X5 @ A5 )
=> ( ( member_a @ Y4 @ A5 )
=> ( ( X5 != Y4 )
=> ( ( ( F @ X5 )
!= ( F @ Y4 ) )
=> ( P2 @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) )
=> ( pairwise_option_a @ P2 @ ( image_a_option_a2 @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1105_pairwise__imageI,axiom,
! [A5: set_set_a,F: set_a > set_a,P2: set_a > set_a > $o] :
( ! [X5: set_a,Y4: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( ( member_set_a @ Y4 @ A5 )
=> ( ( X5 != Y4 )
=> ( ( ( F @ X5 )
!= ( F @ Y4 ) )
=> ( P2 @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) )
=> ( pairwise_set_a @ P2 @ ( image_set_a_set_a2 @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1106_pairwise__imageI,axiom,
! [A5: set_a,F: a > a,P2: a > a > $o] :
( ! [X5: a,Y4: a] :
( ( member_a @ X5 @ A5 )
=> ( ( member_a @ Y4 @ A5 )
=> ( ( X5 != Y4 )
=> ( ( ( F @ X5 )
!= ( F @ Y4 ) )
=> ( P2 @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) )
=> ( pairwise_a @ P2 @ ( image_a_a2 @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1107_pairwise__imageI,axiom,
! [A5: set_option_a,F: option_a > a,P2: a > a > $o] :
( ! [X5: option_a,Y4: option_a] :
( ( member_option_a @ X5 @ A5 )
=> ( ( member_option_a @ Y4 @ A5 )
=> ( ( X5 != Y4 )
=> ( ( ( F @ X5 )
!= ( F @ Y4 ) )
=> ( P2 @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) )
=> ( pairwise_a @ P2 @ ( image_option_a_a2 @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1108_pairwise__imageI,axiom,
! [A5: set_Product_prod_a_a,F: product_prod_a_a > a,P2: a > a > $o] :
( ! [X5: product_prod_a_a,Y4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A5 )
=> ( ( member1426531477525435216od_a_a @ Y4 @ A5 )
=> ( ( X5 != Y4 )
=> ( ( ( F @ X5 )
!= ( F @ Y4 ) )
=> ( P2 @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) )
=> ( pairwise_a @ P2 @ ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1109_pairwise__imageI,axiom,
! [A5: set_set_a,F: set_a > a,P2: a > a > $o] :
( ! [X5: set_a,Y4: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( ( member_set_a @ Y4 @ A5 )
=> ( ( X5 != Y4 )
=> ( ( ( F @ X5 )
!= ( F @ Y4 ) )
=> ( P2 @ ( F @ X5 ) @ ( F @ Y4 ) ) ) ) ) )
=> ( pairwise_a @ P2 @ ( image_set_a_a @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_1110_inj__on__image__eq__iff,axiom,
! [F: set_a > set_a,C3: set_set_a,A5: set_set_a,B6: set_set_a] :
( ( inj_on_set_a_set_a @ F @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ C3 )
=> ( ( ( image_set_a_set_a2 @ F @ A5 )
= ( image_set_a_set_a2 @ F @ B6 ) )
= ( A5 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1111_inj__on__image__eq__iff,axiom,
! [F: a > option_a,C3: set_a,A5: set_a,B6: set_a] :
( ( inj_on_a_option_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( ( image_a_option_a2 @ F @ A5 )
= ( image_a_option_a2 @ F @ B6 ) )
= ( A5 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1112_inj__on__image__eq__iff,axiom,
! [F: a > a,C3: set_a,A5: set_a,B6: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( ( image_a_a2 @ F @ A5 )
= ( image_a_a2 @ F @ B6 ) )
= ( A5 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1113_inj__on__image__mem__iff,axiom,
! [F: a > a,B6: set_a,A: a,A5: set_a] :
( ( inj_on_a_a @ F @ B6 )
=> ( ( member_a @ A @ B6 )
=> ( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a2 @ F @ A5 ) )
= ( member_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1114_inj__on__image__mem__iff,axiom,
! [F: option_a > a,B6: set_option_a,A: option_a,A5: set_option_a] :
( ( inj_on_option_a_a @ F @ B6 )
=> ( ( member_option_a @ A @ B6 )
=> ( ( ord_le1955136853071979460tion_a @ A5 @ B6 )
=> ( ( member_a @ ( F @ A ) @ ( image_option_a_a2 @ F @ A5 ) )
= ( member_option_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1115_inj__on__image__mem__iff,axiom,
! [F: set_a > a,B6: set_set_a,A: set_a,A5: set_set_a] :
( ( inj_on_set_a_a @ F @ B6 )
=> ( ( member_set_a @ A @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B6 )
=> ( ( member_a @ ( F @ A ) @ ( image_set_a_a @ F @ A5 ) )
= ( member_set_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1116_inj__on__image__mem__iff,axiom,
! [F: a > option_a,B6: set_a,A: a,A5: set_a] :
( ( inj_on_a_option_a @ F @ B6 )
=> ( ( member_a @ A @ B6 )
=> ( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( member_option_a @ ( F @ A ) @ ( image_a_option_a2 @ F @ A5 ) )
= ( member_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1117_inj__on__image__mem__iff,axiom,
! [F: a > set_a,B6: set_a,A: a,A5: set_a] :
( ( inj_on_a_set_a @ F @ B6 )
=> ( ( member_a @ A @ B6 )
=> ( ( ord_less_eq_set_a @ A5 @ B6 )
=> ( ( member_set_a @ ( F @ A ) @ ( image_a_set_a2 @ F @ A5 ) )
= ( member_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1118_inj__on__image__mem__iff,axiom,
! [F: option_a > option_a,B6: set_option_a,A: option_a,A5: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ B6 )
=> ( ( member_option_a @ A @ B6 )
=> ( ( ord_le1955136853071979460tion_a @ A5 @ B6 )
=> ( ( member_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ A5 ) )
= ( member_option_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1119_inj__on__image__mem__iff,axiom,
! [F: option_a > set_a,B6: set_option_a,A: option_a,A5: set_option_a] :
( ( inj_on2187968386393286795_set_a @ F @ B6 )
=> ( ( member_option_a @ A @ B6 )
=> ( ( ord_le1955136853071979460tion_a @ A5 @ B6 )
=> ( ( member_set_a @ ( F @ A ) @ ( image_option_a_set_a2 @ F @ A5 ) )
= ( member_option_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1120_inj__on__image__mem__iff,axiom,
! [F: product_prod_a_a > a,B6: set_Product_prod_a_a,A: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( inj_on4978979553551044360_a_a_a @ F @ B6 )
=> ( ( member1426531477525435216od_a_a @ A @ B6 )
=> ( ( ord_le746702958409616551od_a_a @ A5 @ B6 )
=> ( ( member_a @ ( F @ A ) @ ( image_3437945252899457948_a_a_a @ F @ A5 ) )
= ( member1426531477525435216od_a_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1121_inj__on__image__mem__iff,axiom,
! [F: set_a > option_a,B6: set_set_a,A: set_a,A5: set_set_a] :
( ( inj_on7003276264704398167tion_a @ F @ B6 )
=> ( ( member_set_a @ A @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B6 )
=> ( ( member_option_a @ ( F @ A ) @ ( image_set_a_option_a @ F @ A5 ) )
= ( member_set_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1122_inj__on__image__mem__iff,axiom,
! [F: set_a > set_a,B6: set_set_a,A: set_a,A5: set_set_a] :
( ( inj_on_set_a_set_a @ F @ B6 )
=> ( ( member_set_a @ A @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B6 )
=> ( ( member_set_a @ ( F @ A ) @ ( image_set_a_set_a2 @ F @ A5 ) )
= ( member_set_a @ A @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1123_image__Un,axiom,
! [F: set_a > set_a,A5: set_set_a,B6: set_set_a] :
( ( image_set_a_set_a2 @ F @ ( sup_sup_set_set_a @ A5 @ B6 ) )
= ( sup_sup_set_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ ( image_set_a_set_a2 @ F @ B6 ) ) ) ).
% image_Un
thf(fact_1124_image__Un,axiom,
! [F: a > option_a,A5: set_a,B6: set_a] :
( ( image_a_option_a2 @ F @ ( sup_sup_set_a @ A5 @ B6 ) )
= ( sup_sup_set_option_a @ ( image_a_option_a2 @ F @ A5 ) @ ( image_a_option_a2 @ F @ B6 ) ) ) ).
% image_Un
thf(fact_1125_image__Un,axiom,
! [F: a > a,A5: set_a,B6: set_a] :
( ( image_a_a2 @ F @ ( sup_sup_set_a @ A5 @ B6 ) )
= ( sup_sup_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B6 ) ) ) ).
% image_Un
thf(fact_1126_in__image__insert__iff,axiom,
! [B6: set_se5735800977113168103od_a_a,X: product_prod_a_a,A5: set_Product_prod_a_a] :
( ! [C5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ C5 @ B6 )
=> ~ ( member1426531477525435216od_a_a @ X @ C5 ) )
=> ( ( member1816616512716248880od_a_a @ A5 @ ( image_4506799131697958853od_a_a @ ( insert4534936382041156343od_a_a @ X ) @ B6 ) )
= ( ( member1426531477525435216od_a_a @ X @ A5 )
& ( member1816616512716248880od_a_a @ ( minus_6817036919807184750od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1127_in__image__insert__iff,axiom,
! [B6: set_set_set_a,X: set_a,A5: set_set_a] :
( ! [C5: set_set_a] :
( ( member_set_set_a @ C5 @ B6 )
=> ~ ( member_set_a @ X @ C5 ) )
=> ( ( member_set_set_a @ A5 @ ( image_1042221919965026181_set_a @ ( insert_set_a @ X ) @ B6 ) )
= ( ( member_set_a @ X @ A5 )
& ( member_set_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1128_in__image__insert__iff,axiom,
! [B6: set_set_option_a,X: option_a,A5: set_option_a] :
( ! [C5: set_option_a] :
( ( member_set_option_a @ C5 @ B6 )
=> ~ ( member_option_a @ X @ C5 ) )
=> ( ( member_set_option_a @ A5 @ ( image_4886974585486332549tion_a @ ( insert_option_a @ X ) @ B6 ) )
= ( ( member_option_a @ X @ A5 )
& ( member_set_option_a @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1129_in__image__insert__iff,axiom,
! [B6: set_set_a,X: a,A5: set_a] :
( ! [C5: set_a] :
( ( member_set_a @ C5 @ B6 )
=> ~ ( member_a @ X @ C5 ) )
=> ( ( member_set_a @ A5 @ ( image_set_a_set_a2 @ ( insert_a @ X ) @ B6 ) )
= ( ( member_a @ X @ A5 )
& ( member_set_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1130_inj__on__image__Int,axiom,
! [F: set_a > set_a,C3: set_set_a,A5: set_set_a,B6: set_set_a] :
( ( inj_on_set_a_set_a @ F @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ C3 )
=> ( ( image_set_a_set_a2 @ F @ ( inf_inf_set_set_a @ A5 @ B6 ) )
= ( inf_inf_set_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ ( image_set_a_set_a2 @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1131_inj__on__image__Int,axiom,
! [F: a > option_a,C3: set_a,A5: set_a,B6: set_a] :
( ( inj_on_a_option_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( image_a_option_a2 @ F @ ( inf_inf_set_a @ A5 @ B6 ) )
= ( inf_inf_set_option_a @ ( image_a_option_a2 @ F @ A5 ) @ ( image_a_option_a2 @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1132_inj__on__image__Int,axiom,
! [F: a > a,C3: set_a,A5: set_a,B6: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( image_a_a2 @ F @ ( inf_inf_set_a @ A5 @ B6 ) )
= ( inf_inf_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1133_inj__on__image__set__diff,axiom,
! [F: set_a > set_a,C3: set_set_a,A5: set_set_a,B6: set_set_a] :
( ( inj_on_set_a_set_a @ F @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ C3 )
=> ( ( image_set_a_set_a2 @ F @ ( minus_5736297505244876581_set_a @ A5 @ B6 ) )
= ( minus_5736297505244876581_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ ( image_set_a_set_a2 @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1134_inj__on__image__set__diff,axiom,
! [F: a > option_a,C3: set_a,A5: set_a,B6: set_a] :
( ( inj_on_a_option_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( image_a_option_a2 @ F @ ( minus_minus_set_a @ A5 @ B6 ) )
= ( minus_1574173051537231627tion_a @ ( image_a_option_a2 @ F @ A5 ) @ ( image_a_option_a2 @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1135_inj__on__image__set__diff,axiom,
! [F: a > a,C3: set_a,A5: set_a,B6: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A5 @ B6 ) @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( image_a_a2 @ F @ ( minus_minus_set_a @ A5 @ B6 ) )
= ( minus_minus_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1136_fun__upd__image,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,F: product_prod_a_a > a,Y: a] :
( ( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( image_3437945252899457948_a_a_a @ ( fun_up5162090726179321876_a_a_a @ F @ X @ Y ) @ A5 )
= ( insert_a @ Y @ ( image_3437945252899457948_a_a_a @ F @ ( minus_6817036919807184750od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( image_3437945252899457948_a_a_a @ ( fun_up5162090726179321876_a_a_a @ F @ X @ Y ) @ A5 )
= ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1137_fun__upd__image,axiom,
! [X: product_prod_a_a,A5: set_Product_prod_a_a,F: product_prod_a_a > option_a,Y: option_a] :
( ( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( image_4859188117451336930tion_a @ ( fun_up8298456451713467738tion_a @ F @ X @ Y ) @ A5 )
= ( insert_option_a @ Y @ ( image_4859188117451336930tion_a @ F @ ( minus_6817036919807184750od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( image_4859188117451336930tion_a @ ( fun_up8298456451713467738tion_a @ F @ X @ Y ) @ A5 )
= ( image_4859188117451336930tion_a @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1138_fun__upd__image,axiom,
! [X: set_a,A5: set_set_a,F: set_a > set_a,Y: set_a] :
( ( ( member_set_a @ X @ A5 )
=> ( ( image_set_a_set_a2 @ ( fun_upd_set_a_set_a @ F @ X @ Y ) @ A5 )
= ( insert_set_a @ Y @ ( image_set_a_set_a2 @ F @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) )
& ( ~ ( member_set_a @ X @ A5 )
=> ( ( image_set_a_set_a2 @ ( fun_upd_set_a_set_a @ F @ X @ Y ) @ A5 )
= ( image_set_a_set_a2 @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1139_fun__upd__image,axiom,
! [X: set_a,A5: set_set_a,F: set_a > a,Y: a] :
( ( ( member_set_a @ X @ A5 )
=> ( ( image_set_a_a @ ( fun_upd_set_a_a @ F @ X @ Y ) @ A5 )
= ( insert_a @ Y @ ( image_set_a_a @ F @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) )
& ( ~ ( member_set_a @ X @ A5 )
=> ( ( image_set_a_a @ ( fun_upd_set_a_a @ F @ X @ Y ) @ A5 )
= ( image_set_a_a @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1140_fun__upd__image,axiom,
! [X: set_a,A5: set_set_a,F: set_a > option_a,Y: option_a] :
( ( ( member_set_a @ X @ A5 )
=> ( ( image_set_a_option_a @ ( fun_up3663993102702442083tion_a @ F @ X @ Y ) @ A5 )
= ( insert_option_a @ Y @ ( image_set_a_option_a @ F @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) )
& ( ~ ( member_set_a @ X @ A5 )
=> ( ( image_set_a_option_a @ ( fun_up3663993102702442083tion_a @ F @ X @ Y ) @ A5 )
= ( image_set_a_option_a @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1141_fun__upd__image,axiom,
! [X: option_a,A5: set_option_a,F: option_a > a,Y: a] :
( ( ( member_option_a @ X @ A5 )
=> ( ( image_option_a_a2 @ ( fun_upd_option_a_a @ F @ X @ Y ) @ A5 )
= ( insert_a @ Y @ ( image_option_a_a2 @ F @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) )
& ( ~ ( member_option_a @ X @ A5 )
=> ( ( image_option_a_a2 @ ( fun_upd_option_a_a @ F @ X @ Y ) @ A5 )
= ( image_option_a_a2 @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1142_fun__upd__image,axiom,
! [X: option_a,A5: set_option_a,F: option_a > option_a,Y: option_a] :
( ( ( member_option_a @ X @ A5 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) @ A5 )
= ( insert_option_a @ Y @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A5 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) )
& ( ~ ( member_option_a @ X @ A5 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) @ A5 )
= ( image_7439109396645324421tion_a @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1143_fun__upd__image,axiom,
! [X: a,A5: set_a,F: a > a,Y: a] :
( ( ( member_a @ X @ A5 )
=> ( ( image_a_a2 @ ( fun_upd_a_a @ F @ X @ Y ) @ A5 )
= ( insert_a @ Y @ ( image_a_a2 @ F @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A5 )
=> ( ( image_a_a2 @ ( fun_upd_a_a @ F @ X @ Y ) @ A5 )
= ( image_a_a2 @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1144_fun__upd__image,axiom,
! [X: a,A5: set_a,F: a > option_a,Y: option_a] :
( ( ( member_a @ X @ A5 )
=> ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ F @ X @ Y ) @ A5 )
= ( insert_option_a @ Y @ ( image_a_option_a2 @ F @ ( minus_minus_set_a @ A5 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A5 )
=> ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ F @ X @ Y ) @ A5 )
= ( image_a_option_a2 @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_1145_subset__Image__Image__iff,axiom,
! [R2: set_Product_prod_a_a,A5: set_a,B6: set_a] :
( ( order_preorder_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( ord_less_eq_set_a @ A5 @ ( field_a @ R2 ) )
=> ( ( ord_less_eq_set_a @ B6 @ ( field_a @ R2 ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R2 @ A5 ) @ ( image_a_a @ R2 @ B6 ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ? [Y3: a] :
( ( member_a @ Y3 @ B6 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1146_inj__on__iff__surj,axiom,
! [A5: set_set_a,A10: set_set_a] :
( ( A5 != bot_bot_set_set_a )
=> ( ( ? [F2: set_a > set_a] :
( ( inj_on_set_a_set_a @ F2 @ A5 )
& ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a2 @ F2 @ A5 ) @ A10 ) ) )
= ( ? [G2: set_a > set_a] :
( ( image_set_a_set_a2 @ G2 @ A10 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1147_inj__on__iff__surj,axiom,
! [A5: set_a,A10: set_option_a] :
( ( A5 != bot_bot_set_a )
=> ( ( ? [F2: a > option_a] :
( ( inj_on_a_option_a @ F2 @ A5 )
& ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F2 @ A5 ) @ A10 ) ) )
= ( ? [G2: option_a > a] :
( ( image_option_a_a2 @ G2 @ A10 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1148_inj__on__iff__surj,axiom,
! [A5: set_a,A10: set_a] :
( ( A5 != bot_bot_set_a )
=> ( ( ? [F2: a > a] :
( ( inj_on_a_a @ F2 @ A5 )
& ( ord_less_eq_set_a @ ( image_a_a2 @ F2 @ A5 ) @ A10 ) ) )
= ( ? [G2: a > a] :
( ( image_a_a2 @ G2 @ A10 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1149_inj__on__iff__surj,axiom,
! [A5: set_option_a,A10: set_a] :
( ( A5 != bot_bot_set_option_a )
=> ( ( ? [F2: option_a > a] :
( ( inj_on_option_a_a @ F2 @ A5 )
& ( ord_less_eq_set_a @ ( image_option_a_a2 @ F2 @ A5 ) @ A10 ) ) )
= ( ? [G2: a > option_a] :
( ( image_a_option_a2 @ G2 @ A10 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1150_subset__image__inj,axiom,
! [S5: set_set_a,F: set_a > set_a,T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S5 @ ( image_set_a_set_a2 @ F @ T3 ) )
= ( ? [U: set_set_a] :
( ( ord_le3724670747650509150_set_a @ U @ T3 )
& ( inj_on_set_a_set_a @ F @ U )
& ( S5
= ( image_set_a_set_a2 @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_1151_subset__image__inj,axiom,
! [S5: set_option_a,F: a > option_a,T3: set_a] :
( ( ord_le1955136853071979460tion_a @ S5 @ ( image_a_option_a2 @ F @ T3 ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T3 )
& ( inj_on_a_option_a @ F @ U )
& ( S5
= ( image_a_option_a2 @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_1152_subset__image__inj,axiom,
! [S5: set_a,F: a > a,T3: set_a] :
( ( ord_less_eq_set_a @ S5 @ ( image_a_a2 @ F @ T3 ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T3 )
& ( inj_on_a_a @ F @ U )
& ( S5
= ( image_a_a2 @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_1153_all__subset__image,axiom,
! [F: set_a > set_a,A5: set_set_a,P2: set_set_a > $o] :
( ( ! [B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ ( image_set_a_set_a2 @ F @ A5 ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ A5 )
=> ( P2 @ ( image_set_a_set_a2 @ F @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_1154_all__subset__image,axiom,
! [F: a > option_a,A5: set_a,P2: set_option_a > $o] :
( ( ! [B7: set_option_a] :
( ( ord_le1955136853071979460tion_a @ B7 @ ( image_a_option_a2 @ F @ A5 ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ A5 )
=> ( P2 @ ( image_a_option_a2 @ F @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_1155_all__subset__image,axiom,
! [F: a > a,A5: set_a,P2: set_a > $o] :
( ( ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ ( image_a_a2 @ F @ A5 ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ A5 )
=> ( P2 @ ( image_a_a2 @ F @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_1156_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr8600417178894128327od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
( ( order_7408868903334687516od_a_a @ ( field_1126092520709947252od_a_a @ R2 ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ A @ ( field_1126092520709947252od_a_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ B @ ( field_1126092520709947252od_a_a @ R2 ) )
=> ( ( ( image_9076584400576816019od_a_a @ R2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R2 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1157_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr5845495582615845127_set_a,A: set_a,B: set_a] :
( ( order_8908102307174497107_set_a @ ( field_set_a @ R2 ) @ R2 )
=> ( ( member_set_a @ A @ ( field_set_a @ R2 ) )
=> ( ( member_set_a @ B @ ( field_set_a @ R2 ) )
=> ( ( ( image_set_a_set_a @ R2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( image_set_a_set_a @ R2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1158_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( order_1124164593023023289tion_a @ ( field_option_a @ R2 ) @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1159_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( order_5272072345360262643r_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1160_partial__order__on__empty,axiom,
order_1124164593023023289tion_a @ bot_bot_set_option_a @ bot_bo235252021745139059tion_a ).
% partial_order_on_empty
thf(fact_1161_partial__order__on__empty,axiom,
order_5272072345360262643r_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% partial_order_on_empty
thf(fact_1162_image__Fpow__mono,axiom,
! [F: a > option_a,A5: set_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F @ A5 ) @ B6 )
=> ( ord_le7761976607055303332tion_a @ ( image_2200789291716222155tion_a @ ( image_a_option_a2 @ F ) @ ( finite_Fpow_a @ A5 ) ) @ ( finite_Fpow_option_a @ B6 ) ) ) ).
% image_Fpow_mono
thf(fact_1163_image__Fpow__mono,axiom,
! [F: set_a > set_a,A5: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a2 @ F @ A5 ) @ B6 )
=> ( ord_le5722252365846178494_set_a @ ( image_1042221919965026181_set_a @ ( image_set_a_set_a2 @ F ) @ ( finite_Fpow_set_a @ A5 ) ) @ ( finite_Fpow_set_a @ B6 ) ) ) ).
% image_Fpow_mono
thf(fact_1164_image__Fpow__mono,axiom,
! [F: a > a,A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a2 @ F @ A5 ) @ B6 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a2 @ ( image_a_a2 @ F ) @ ( finite_Fpow_a @ A5 ) ) @ ( finite_Fpow_a @ B6 ) ) ) ).
% image_Fpow_mono
thf(fact_1165_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
=> ( ( order_4821795997958563554tion_a @ ( field_option_a @ R2 ) @ R2 )
= ( ! [A8: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A8 @ ( field_option_a @ R2 ) )
=> ( ( A8 != bot_bot_set_option_a )
=> ? [X4: option_a] :
( ( member_option_a @ X4 @ A8 )
& ! [Y3: option_a] :
( ( member_option_a @ Y3 @ A8 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X4 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1166_Linear__order__Well__order__iff,axiom,
! [R2: set_Product_prod_a_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( order_6972113574731384220r_on_a @ ( field_a @ R2 ) @ R2 )
= ( ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ ( field_a @ R2 ) )
=> ( ( A8 != bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ A8 )
& ! [Y3: a] :
( ( member_a @ Y3 @ A8 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X: option_a,Y: option_a] :
( ( if_option_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X: option_a,Y: option_a] :
( ( if_option_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_a @ x @ ( delta @ s ) ).
%------------------------------------------------------------------------------