TPTP Problem File: ITP042^1.p
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%------------------------------------------------------------------------------
% File : ITP042^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_99__7210728_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Coincidence/prob_99__7210728_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 482 ( 196 unt; 123 typ; 0 def)
% Number of atoms : 940 ( 301 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2316 ( 34 ~; 10 |; 49 &;1961 @)
% ( 0 <=>; 262 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 29 ( 28 usr)
% Number of type conns : 289 ( 289 >; 0 *; 0 +; 0 <<)
% Number of symbols : 98 ( 95 usr; 13 con; 0-3 aty)
% Number of variables : 876 ( 138 ^; 720 !; 18 ?; 876 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:36:49.138
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
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% Explicit typings (95)
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thf(sy_c_Sum__Type_OInl_001tf__a_001t__Sum____Type__Osum_Itf__c_Mtf__b_J,type,
sum_In811702836um_c_b: a > sum_su1965225555um_c_b ).
thf(sy_c_Syntax_Odfree_001tf__a_001tf__b,type,
dfree_a_b: trm_a_b > $o ).
thf(sy_c_Syntax_Odfree_001tf__sf_001tf__sz,type,
dfree_sf_sz: trm_sf_sz > $o ).
thf(sy_c_Syntax_Otrm_OTimes_001tf__a_001tf__b,type,
times_a_b: trm_a_b > trm_a_b > trm_a_b ).
thf(sy_c_Syntax_Otrm_OTimes_001tf__sf_001tf__sz,type,
times_sf_sz: trm_sf_sz > trm_sf_sz > trm_sf_sz ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__b_J_001t__Real__Oreal,type,
topolo238266006b_real: set_Fi268318752real_b > ( finite1390225514real_b > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_001t__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J,type,
topolo885751137z_real: set_Fi291318197eal_sz > ( finite824932053eal_sz > bounde472938360z_real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_001t__Real__Oreal,type,
topolo1348430467z_real: set_Fi291318197eal_sz > ( finite824932053eal_sz > real ) > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Frechet____Correctness__Oids__Ogood____interp_Itf__sf_Mtf__sc_Mtf__sz_J_001t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J,type,
type_d1092523951t_unit: ( freche2095075217_sc_sz > denota610675952t_unit ) > ( denota610675952t_unit > freche2095075217_sc_sz ) > set_De208295462t_unit > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Frechet____Correctness__Oids__Ostrm_Itf__a_Mtf__b_J_001t__Syntax__Otrm_Itf__a_Mtf__b_J,type,
type_d2045870413rm_a_b: ( frechet_strm_a_b > trm_a_b ) > ( trm_a_b > frechet_strm_a_b ) > set_trm_a_b > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Frechet____Correctness__Oids__Ostrm_Itf__sf_Mtf__sz_J_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J,type,
type_d46389773_sf_sz: ( frechet_strm_sf_sz > trm_sf_sz ) > ( trm_sf_sz > frechet_strm_sf_sz ) > set_trm_sf_sz > $o ).
thf(sy_c_member_001t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__a_Mtf__c_Mtf__b_Mt__Product____Type__Ounit_J,type,
member122424677t_unit: denota723907260t_unit > set_De153000604t_unit > $o ).
thf(sy_c_member_001t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J,type,
member754147591t_unit: denota610675952t_unit > set_De208295462t_unit > $o ).
thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J,type,
member1693951742eal_sz: finite824932053eal_sz > set_Fi291318197eal_sz > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Sum____Type__Osum_Itf__c_Mtf__b_J_J,type,
member973578748um_c_b: sum_su1965225555um_c_b > set_Su111624115um_c_b > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__b_Mtf__b_J,type,
member_Sum_sum_b_b: sum_sum_b_b > set_Sum_sum_b_b > $o ).
thf(sy_c_member_001t__Syntax__Otrm_Itf__a_Mtf__b_J,type,
member_trm_a_b: trm_a_b > set_trm_a_b > $o ).
thf(sy_c_member_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J,type,
member_trm_sf_sz: trm_sf_sz > set_trm_sf_sz > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: denota723907260t_unit ).
thf(sy_v_J,type,
j: denota723907260t_unit ).
thf(sy_v__092_060nu_062,type,
nu: produc133005365real_b ).
thf(sy_v__092_060nu_062_H,type,
nu2: produc133005365real_b ).
thf(sy_v__092_060theta_062_092_060_094sub_0621____,type,
theta_1: trm_a_b ).
thf(sy_v__092_060theta_062_092_060_094sub_0622____,type,
theta_2: trm_a_b ).
% Relevant facts (355)
thf(fact_0__092_060open_062_092_060And_062J_AI_O_AIagree_AI_AJ_A_123Inl_Ax_A_124x_O_Ax_A_092_060in_062_ASIGT_A_ITimes_A_092_060theta_062_092_060_094sub_0621_A_092_060theta_062_092_060_094sub_0622_J_125_A_092_060Longrightarrow_062_AIagree_AI_AJ_A_123Inl_Ax_A_124x_O_Ax_A_092_060in_062_ASIGT_A_092_060theta_062_092_060_094sub_0621_125_092_060close_062,axiom,
! [I: denota723907260t_unit,J: denota723907260t_unit] :
( ( denota896875430_a_c_b @ I @ J
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ ( times_a_b @ theta_1 @ theta_2 ) ) ) ) ) )
=> ( denota896875430_a_c_b @ I @ J
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ theta_1 ) ) ) ) ) ) ).
% \<open>\<And>J I. Iagree I J {Inl x |x. x \<in> SIGT (Times \<theta>\<^sub>1 \<theta>\<^sub>2)} \<Longrightarrow> Iagree I J {Inl x |x. x \<in> SIGT \<theta>\<^sub>1}\<close>
thf(fact_1__092_060open_062_092_060And_062J_AI_O_AIagree_AI_AJ_A_123Inl_Ax_A_124x_O_Ax_A_092_060in_062_ASIGT_A_ITimes_A_092_060theta_062_092_060_094sub_0621_A_092_060theta_062_092_060_094sub_0622_J_125_A_092_060Longrightarrow_062_AIagree_AI_AJ_A_123Inl_Ax_A_124x_O_Ax_A_092_060in_062_ASIGT_A_092_060theta_062_092_060_094sub_0622_125_092_060close_062,axiom,
! [I: denota723907260t_unit,J: denota723907260t_unit] :
( ( denota896875430_a_c_b @ I @ J
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ ( times_a_b @ theta_1 @ theta_2 ) ) ) ) ) )
=> ( denota896875430_a_c_b @ I @ J
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ theta_2 ) ) ) ) ) ) ).
% \<open>\<And>J I. Iagree I J {Inl x |x. x \<in> SIGT (Times \<theta>\<^sub>1 \<theta>\<^sub>2)} \<Longrightarrow> Iagree I J {Inl x |x. x \<in> SIGT \<theta>\<^sub>2}\<close>
thf(fact_2_IA,axiom,
( denota896875430_a_c_b @ i @ j
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( ( member_a @ X @ ( static_SIGT_a_b @ theta_1 ) )
| ( member_a @ X @ ( static_SIGT_a_b @ theta_2 ) ) ) ) ) ) ).
% IA
thf(fact_3_dfree__Times_Oprems_I2_J,axiom,
( denota896875430_a_c_b @ i @ j
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ ( times_a_b @ theta_1 @ theta_2 ) ) ) ) ) ) ).
% dfree_Times.prems(2)
thf(fact_4_dfree__Times_Ohyps_I1_J,axiom,
dfree_a_b @ theta_1 ).
% dfree_Times.hyps(1)
thf(fact_5_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_In811702836um_c_b @ X1 )
= ( sum_In811702836um_c_b @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_6_old_Osum_Oinject_I1_J,axiom,
! [A: a,A2: a] :
( ( ( sum_In811702836um_c_b @ A )
= ( sum_In811702836um_c_b @ A2 ) )
= ( A = A2 ) ) ).
% old.sum.inject(1)
thf(fact_7_Isubs_I2_J,axiom,
( ord_le1404954451um_c_b
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ theta_2 ) ) ) )
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ ( times_a_b @ theta_1 @ theta_2 ) ) ) ) ) ) ).
% Isubs(2)
thf(fact_8_Isubs_I1_J,axiom,
( ord_le1404954451um_c_b
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ theta_1 ) ) ) )
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ ( times_a_b @ theta_1 @ theta_2 ) ) ) ) ) ) ).
% Isubs(1)
thf(fact_9_Iagree__comm,axiom,
! [A3: denota723907260t_unit,B: denota723907260t_unit,V: set_Su111624115um_c_b] :
( ( denota896875430_a_c_b @ A3 @ B @ V )
=> ( denota896875430_a_c_b @ B @ A3 @ V ) ) ).
% Iagree_comm
thf(fact_10_Iagree__refl,axiom,
! [I: denota723907260t_unit,A3: set_Su111624115um_c_b] : ( denota896875430_a_c_b @ I @ I @ A3 ) ).
% Iagree_refl
thf(fact_11_Inl__inject,axiom,
! [X2: a,Y: a] :
( ( ( sum_In811702836um_c_b @ X2 )
= ( sum_In811702836um_c_b @ Y ) )
=> ( X2 = Y ) ) ).
% Inl_inject
thf(fact_12_IH1,axiom,
( ( denota1997846517gree_b @ nu @ nu2 @ ( static_FVT_a_b @ theta_1 ) )
=> ( ( denota896875430_a_c_b @ i @ j
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ theta_1 ) ) ) ) )
=> ( ( denota722380397_a_c_b @ i @ theta_1 @ ( produc936751193real_b @ nu ) )
= ( denota722380397_a_c_b @ j @ theta_1 @ ( produc936751193real_b @ nu2 ) ) ) ) ) ).
% IH1
thf(fact_13_IH2,axiom,
( ( denota1997846517gree_b @ nu @ nu2 @ ( static_FVT_a_b @ theta_2 ) )
=> ( ( denota896875430_a_c_b @ i @ j
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ theta_2 ) ) ) ) )
=> ( ( denota722380397_a_c_b @ i @ theta_2 @ ( produc936751193real_b @ nu ) )
= ( denota722380397_a_c_b @ j @ theta_2 @ ( produc936751193real_b @ nu2 ) ) ) ) ) ).
% IH2
thf(fact_14_Iagree__Func,axiom,
! [I: denota723907260t_unit,J: denota723907260t_unit,V: set_Su111624115um_c_b,F: a] :
( ( denota896875430_a_c_b @ I @ J @ V )
=> ( ( member973578748um_c_b @ ( sum_In811702836um_c_b @ F ) @ V )
=> ( ( denota102976244t_unit @ I @ F )
= ( denota102976244t_unit @ J @ F ) ) ) ) ).
% Iagree_Func
thf(fact_15_raw__interp__inject,axiom,
! [X2: freche2095075217_sc_sz,Y: freche2095075217_sc_sz] :
( ( ( freche1421597129_sc_sz @ X2 )
= ( freche1421597129_sc_sz @ Y ) )
= ( X2 = Y ) ) ).
% raw_interp_inject
thf(fact_16_dfree__Times_Ohyps_I2_J,axiom,
dfree_a_b @ theta_2 ).
% dfree_Times.hyps(2)
thf(fact_17_coincidence__sterm,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,Theta: trm_sf_sz,I: denota610675952t_unit] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVT_sf_sz @ Theta ) )
=> ( ( denota1179238309_sc_sz @ I @ Theta @ ( produc1111759555eal_sz @ Nu ) )
= ( denota1179238309_sc_sz @ I @ Theta @ ( produc1111759555eal_sz @ Nu2 ) ) ) ) ).
% coincidence_sterm
thf(fact_18_coincidence__sterm,axiom,
! [Nu: produc133005365real_b,Nu2: produc133005365real_b,Theta: trm_a_b,I: denota723907260t_unit] :
( ( denota1997846517gree_b @ Nu @ Nu2 @ ( static_FVT_a_b @ Theta ) )
=> ( ( denota722380397_a_c_b @ I @ Theta @ ( produc936751193real_b @ Nu ) )
= ( denota722380397_a_c_b @ I @ Theta @ ( produc936751193real_b @ Nu2 ) ) ) ) ).
% coincidence_sterm
thf(fact_19_dfree__Times_Oprems_I1_J,axiom,
denota1997846517gree_b @ nu @ nu2 @ ( static_FVT_a_b @ ( times_a_b @ theta_1 @ theta_2 ) ) ).
% dfree_Times.prems(1)
thf(fact_20_VAs_I2_J,axiom,
denota1997846517gree_b @ nu @ nu2 @ ( static_FVT_a_b @ theta_2 ) ).
% VAs(2)
thf(fact_21_VAs_I1_J,axiom,
denota1997846517gree_b @ nu @ nu2 @ ( static_FVT_a_b @ theta_1 ) ).
% VAs(1)
thf(fact_22_VA,axiom,
denota1997846517gree_b @ nu @ nu2 @ ( sup_su2137030215um_b_b @ ( static_FVT_a_b @ theta_1 ) @ ( static_FVT_a_b @ theta_2 ) ) ).
% VA
thf(fact_23_agree__sub,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b,Nu: produc133005365real_b,Omega: produc133005365real_b] :
( ( ord_le412705147um_b_b @ A3 @ B )
=> ( ( denota1997846517gree_b @ Nu @ Omega @ B )
=> ( denota1997846517gree_b @ Nu @ Omega @ A3 ) ) ) ).
% agree_sub
thf(fact_24_agree__comm,axiom,
! [A3: produc133005365real_b,B: produc133005365real_b,V: set_Sum_sum_b_b] :
( ( denota1997846517gree_b @ A3 @ B @ V )
=> ( denota1997846517gree_b @ B @ A3 @ V ) ) ).
% agree_comm
thf(fact_25_agree__refl,axiom,
! [Nu: produc133005365real_b,A3: set_Sum_sum_b_b] : ( denota1997846517gree_b @ Nu @ Nu @ A3 ) ).
% agree_refl
thf(fact_26_agree__supset,axiom,
! [B: set_Sum_sum_b_b,A3: set_Sum_sum_b_b,Nu: produc133005365real_b,Nu2: produc133005365real_b] :
( ( ord_le412705147um_b_b @ B @ A3 )
=> ( ( denota1997846517gree_b @ Nu @ Nu2 @ A3 )
=> ( denota1997846517gree_b @ Nu @ Nu2 @ B ) ) ) ).
% agree_supset
thf(fact_27_cr__good__interp__def,axiom,
( freche58918398_sc_sz
= ( ^ [X: denota610675952t_unit,Y2: freche2095075217_sc_sz] :
( X
= ( freche1421597129_sc_sz @ Y2 ) ) ) ) ).
% cr_good_interp_def
thf(fact_28_dfree__Times__simps,axiom,
! [A: trm_sf_sz,B2: trm_sf_sz] :
( ( dfree_sf_sz @ ( times_sf_sz @ A @ B2 ) )
= ( ( dfree_sf_sz @ A )
& ( dfree_sf_sz @ B2 ) ) ) ).
% dfree_Times_simps
thf(fact_29_dfree__Times__simps,axiom,
! [A: trm_a_b,B2: trm_a_b] :
( ( dfree_a_b @ ( times_a_b @ A @ B2 ) )
= ( ( dfree_a_b @ A )
& ( dfree_a_b @ B2 ) ) ) ).
% dfree_Times_simps
thf(fact_30_simple__term__inject,axiom,
! [X2: trm_a_b,Y: trm_a_b] :
( ( member_trm_a_b @ X2 @ ( collect_trm_a_b @ dfree_a_b ) )
=> ( ( member_trm_a_b @ Y @ ( collect_trm_a_b @ dfree_a_b ) )
=> ( ( ( freche665377491rm_a_b @ X2 )
= ( freche665377491rm_a_b @ Y ) )
= ( X2 = Y ) ) ) ) ).
% simple_term_inject
thf(fact_31_simple__term__inject,axiom,
! [X2: trm_sf_sz,Y: trm_sf_sz] :
( ( member_trm_sf_sz @ X2 @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( ( member_trm_sf_sz @ Y @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( ( ( freche1046279700_sf_sz @ X2 )
= ( freche1046279700_sf_sz @ Y ) )
= ( X2 = Y ) ) ) ) ).
% simple_term_inject
thf(fact_32_simple__term__induct,axiom,
! [P: frechet_strm_a_b > $o,X2: frechet_strm_a_b] :
( ! [Y3: trm_a_b] :
( ( member_trm_a_b @ Y3 @ ( collect_trm_a_b @ dfree_a_b ) )
=> ( P @ ( freche665377491rm_a_b @ Y3 ) ) )
=> ( P @ X2 ) ) ).
% simple_term_induct
thf(fact_33_simple__term__induct,axiom,
! [P: frechet_strm_sf_sz > $o,X2: frechet_strm_sf_sz] :
( ! [Y3: trm_sf_sz] :
( ( member_trm_sf_sz @ Y3 @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( P @ ( freche1046279700_sf_sz @ Y3 ) ) )
=> ( P @ X2 ) ) ).
% simple_term_induct
thf(fact_34_simple__term__cases,axiom,
! [X2: frechet_strm_a_b] :
~ ! [Y3: trm_a_b] :
( ( X2
= ( freche665377491rm_a_b @ Y3 ) )
=> ~ ( member_trm_a_b @ Y3 @ ( collect_trm_a_b @ dfree_a_b ) ) ) ).
% simple_term_cases
thf(fact_35_simple__term__cases,axiom,
! [X2: frechet_strm_sf_sz] :
~ ! [Y3: trm_sf_sz] :
( ( X2
= ( freche1046279700_sf_sz @ Y3 ) )
=> ~ ( member_trm_sf_sz @ Y3 @ ( collect_trm_sf_sz @ dfree_sf_sz ) ) ) ).
% simple_term_cases
thf(fact_36_raw__interp__inverse,axiom,
! [X2: freche2095075217_sc_sz] :
( ( freche1784963216_sc_sz @ ( freche1421597129_sc_sz @ X2 ) )
= X2 ) ).
% raw_interp_inverse
thf(fact_37_raw__term__induct,axiom,
! [Y: trm_a_b,P: trm_a_b > $o] :
( ( member_trm_a_b @ Y @ ( collect_trm_a_b @ dfree_a_b ) )
=> ( ! [X3: frechet_strm_a_b] : ( P @ ( frechet_raw_term_a_b @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_term_induct
thf(fact_38_raw__term__induct,axiom,
! [Y: trm_sf_sz,P: trm_sf_sz > $o] :
( ( member_trm_sf_sz @ Y @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( ! [X3: frechet_strm_sf_sz] : ( P @ ( freche782854530_sf_sz @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_term_induct
thf(fact_39_raw__term__cases,axiom,
! [Y: trm_a_b] :
( ( member_trm_a_b @ Y @ ( collect_trm_a_b @ dfree_a_b ) )
=> ~ ! [X3: frechet_strm_a_b] :
( Y
!= ( frechet_raw_term_a_b @ X3 ) ) ) ).
% raw_term_cases
thf(fact_40_raw__term__cases,axiom,
! [Y: trm_sf_sz] :
( ( member_trm_sf_sz @ Y @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ~ ! [X3: frechet_strm_sf_sz] :
( Y
!= ( freche782854530_sf_sz @ X3 ) ) ) ).
% raw_term_cases
thf(fact_41_raw__term,axiom,
! [X2: frechet_strm_a_b] : ( member_trm_a_b @ ( frechet_raw_term_a_b @ X2 ) @ ( collect_trm_a_b @ dfree_a_b ) ) ).
% raw_term
thf(fact_42_raw__term,axiom,
! [X2: frechet_strm_sf_sz] : ( member_trm_sf_sz @ ( freche782854530_sf_sz @ X2 ) @ ( collect_trm_sf_sz @ dfree_sf_sz ) ) ).
% raw_term
thf(fact_43_trm_Oinject_I5_J,axiom,
! [X51: trm_sf_sz,X52: trm_sf_sz,Y51: trm_sf_sz,Y52: trm_sf_sz] :
( ( ( times_sf_sz @ X51 @ X52 )
= ( times_sf_sz @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_44_trm_Oinject_I5_J,axiom,
! [X51: trm_a_b,X52: trm_a_b,Y51: trm_a_b,Y52: trm_a_b] :
( ( ( times_a_b @ X51 @ X52 )
= ( times_a_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_45_raw__term__inject,axiom,
! [X2: frechet_strm_sf_sz,Y: frechet_strm_sf_sz] :
( ( ( freche782854530_sf_sz @ X2 )
= ( freche782854530_sf_sz @ Y ) )
= ( X2 = Y ) ) ).
% raw_term_inject
thf(fact_46_raw__term__inject,axiom,
! [X2: frechet_strm_a_b,Y: frechet_strm_a_b] :
( ( ( frechet_raw_term_a_b @ X2 )
= ( frechet_raw_term_a_b @ Y ) )
= ( X2 = Y ) ) ).
% raw_term_inject
thf(fact_47_raw__term__inverse,axiom,
! [X2: frechet_strm_a_b] :
( ( freche665377491rm_a_b @ ( frechet_raw_term_a_b @ X2 ) )
= X2 ) ).
% raw_term_inverse
thf(fact_48_raw__term__inverse,axiom,
! [X2: frechet_strm_sf_sz] :
( ( freche1046279700_sf_sz @ ( freche782854530_sf_sz @ X2 ) )
= X2 ) ).
% raw_term_inverse
thf(fact_49_simple__term__inverse,axiom,
! [Y: trm_a_b] :
( ( member_trm_a_b @ Y @ ( collect_trm_a_b @ dfree_a_b ) )
=> ( ( frechet_raw_term_a_b @ ( freche665377491rm_a_b @ Y ) )
= Y ) ) ).
% simple_term_inverse
thf(fact_50_simple__term__inverse,axiom,
! [Y: trm_sf_sz] :
( ( member_trm_sf_sz @ Y @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( ( freche782854530_sf_sz @ ( freche1046279700_sf_sz @ Y ) )
= Y ) ) ).
% simple_term_inverse
thf(fact_51_agree__union,axiom,
! [Nu: produc133005365real_b,Omega: produc133005365real_b,A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( denota1997846517gree_b @ Nu @ Omega @ A3 )
=> ( ( denota1997846517gree_b @ Nu @ Omega @ B )
=> ( denota1997846517gree_b @ Nu @ Omega @ ( sup_su2137030215um_b_b @ A3 @ B ) ) ) ) ).
% agree_union
thf(fact_52_Iagree__sub,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b,I: denota723907260t_unit,J: denota723907260t_unit] :
( ( ord_le1404954451um_c_b @ A3 @ B )
=> ( ( denota896875430_a_c_b @ I @ J @ B )
=> ( denota896875430_a_c_b @ I @ J @ A3 ) ) ) ).
% Iagree_sub
thf(fact_53_dfree_Odfree__Times,axiom,
! [Theta_1: trm_sf_sz,Theta_2: trm_sf_sz] :
( ( dfree_sf_sz @ Theta_1 )
=> ( ( dfree_sf_sz @ Theta_2 )
=> ( dfree_sf_sz @ ( times_sf_sz @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree.dfree_Times
thf(fact_54_dfree_Odfree__Times,axiom,
! [Theta_1: trm_a_b,Theta_2: trm_a_b] :
( ( dfree_a_b @ Theta_1 )
=> ( ( dfree_a_b @ Theta_2 )
=> ( dfree_a_b @ ( times_a_b @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree.dfree_Times
thf(fact_55_cr__strm__def,axiom,
( freche1244000341_sf_sz
= ( ^ [X: trm_sf_sz,Y2: frechet_strm_sf_sz] :
( X
= ( freche782854530_sf_sz @ Y2 ) ) ) ) ).
% cr_strm_def
thf(fact_56_cr__strm__def,axiom,
( frechet_cr_strm_a_b
= ( ^ [X: trm_a_b,Y2: frechet_strm_a_b] :
( X
= ( frechet_raw_term_a_b @ Y2 ) ) ) ) ).
% cr_strm_def
thf(fact_57_fst__sup,axiom,
! [X2: produc133005365real_b,Y: produc133005365real_b] :
( ( produc936751193real_b @ ( sup_su1581595657real_b @ X2 @ Y ) )
= ( sup_su2133361046real_b @ ( produc936751193real_b @ X2 ) @ ( produc936751193real_b @ Y ) ) ) ).
% fst_sup
thf(fact_58_mem__Collect__eq,axiom,
! [A: finite824932053eal_sz,P: finite824932053eal_sz > $o] :
( ( member1693951742eal_sz @ A @ ( collec1002602816eal_sz @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
! [A: sum_sum_b_b,P: sum_sum_b_b > $o] :
( ( member_Sum_sum_b_b @ A @ ( collect_Sum_sum_b_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: trm_sf_sz,P: trm_sf_sz > $o] :
( ( member_trm_sf_sz @ A @ ( collect_trm_sf_sz @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: trm_a_b,P: trm_a_b > $o] :
( ( member_trm_a_b @ A @ ( collect_trm_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: sum_su1965225555um_c_b,P: sum_su1965225555um_c_b > $o] :
( ( member973578748um_c_b @ A @ ( collec1166375742um_c_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A3: set_Fi291318197eal_sz] :
( ( collec1002602816eal_sz
@ ^ [X: finite824932053eal_sz] : ( member1693951742eal_sz @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A3: set_Sum_sum_b_b] :
( ( collect_Sum_sum_b_b
@ ^ [X: sum_sum_b_b] : ( member_Sum_sum_b_b @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A3: set_trm_sf_sz] :
( ( collect_trm_sf_sz
@ ^ [X: trm_sf_sz] : ( member_trm_sf_sz @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A3: set_trm_a_b] :
( ( collect_trm_a_b
@ ^ [X: trm_a_b] : ( member_trm_a_b @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A3: set_Su111624115um_c_b] :
( ( collec1166375742um_c_b
@ ^ [X: sum_su1965225555um_c_b] : ( member973578748um_c_b @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_70_Collect__cong,axiom,
! [P: finite824932053eal_sz > $o,Q: finite824932053eal_sz > $o] :
( ! [X3: finite824932053eal_sz] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1002602816eal_sz @ P )
= ( collec1002602816eal_sz @ Q ) ) ) ).
% Collect_cong
thf(fact_71_Collect__cong,axiom,
! [P: sum_sum_b_b > $o,Q: sum_sum_b_b > $o] :
( ! [X3: sum_sum_b_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_Sum_sum_b_b @ P )
= ( collect_Sum_sum_b_b @ Q ) ) ) ).
% Collect_cong
thf(fact_72_Collect__cong,axiom,
! [P: trm_sf_sz > $o,Q: trm_sf_sz > $o] :
( ! [X3: trm_sf_sz] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_trm_sf_sz @ P )
= ( collect_trm_sf_sz @ Q ) ) ) ).
% Collect_cong
thf(fact_73_Collect__cong,axiom,
! [P: trm_a_b > $o,Q: trm_a_b > $o] :
( ! [X3: trm_a_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_trm_a_b @ P )
= ( collect_trm_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_74_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_75_Collect__cong,axiom,
! [P: sum_su1965225555um_c_b > $o,Q: sum_su1965225555um_c_b > $o] :
( ! [X3: sum_su1965225555um_c_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1166375742um_c_b @ P )
= ( collec1166375742um_c_b @ Q ) ) ) ).
% Collect_cong
thf(fact_76_Un__subset__iff,axiom,
! [A3: set_Fi291318197eal_sz,B: set_Fi291318197eal_sz,C: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ ( sup_su1321388937eal_sz @ A3 @ B ) @ C )
= ( ( ord_le41263445eal_sz @ A3 @ C )
& ( ord_le41263445eal_sz @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_77_Un__subset__iff,axiom,
! [A3: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A3 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_78_Un__subset__iff,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b,C: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ A3 @ B ) @ C )
= ( ( ord_le412705147um_b_b @ A3 @ C )
& ( ord_le412705147um_b_b @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_79_Un__subset__iff,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b,C: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ A3 @ B ) @ C )
= ( ( ord_le1404954451um_c_b @ A3 @ C )
& ( ord_le1404954451um_c_b @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_80_sup_Obounded__iff,axiom,
! [B2: sum_sum_b_b > $o,C2: sum_sum_b_b > $o,A: sum_sum_b_b > $o] :
( ( ord_le1306331338_b_b_o @ ( sup_su2024788734_b_b_o @ B2 @ C2 ) @ A )
= ( ( ord_le1306331338_b_b_o @ B2 @ A )
& ( ord_le1306331338_b_b_o @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_81_sup_Obounded__iff,axiom,
! [B2: set_Fi291318197eal_sz,C2: set_Fi291318197eal_sz,A: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ ( sup_su1321388937eal_sz @ B2 @ C2 ) @ A )
= ( ( ord_le41263445eal_sz @ B2 @ A )
& ( ord_le41263445eal_sz @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_82_sup_Obounded__iff,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_83_sup_Obounded__iff,axiom,
! [B2: sum_su1965225555um_c_b > $o,C2: sum_su1965225555um_c_b > $o,A: sum_su1965225555um_c_b > $o] :
( ( ord_le1534094666_c_b_o @ ( sup_su1211201046_c_b_o @ B2 @ C2 ) @ A )
= ( ( ord_le1534094666_c_b_o @ B2 @ A )
& ( ord_le1534094666_c_b_o @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_84_sup_Obounded__iff,axiom,
! [B2: a > $o,C2: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_a_o @ B2 @ A )
& ( ord_less_eq_a_o @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_85_sup_Obounded__iff,axiom,
! [B2: set_Sum_sum_b_b,C2: set_Sum_sum_b_b,A: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ B2 @ C2 ) @ A )
= ( ( ord_le412705147um_b_b @ B2 @ A )
& ( ord_le412705147um_b_b @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_86_sup_Obounded__iff,axiom,
! [B2: set_Su111624115um_c_b,C2: set_Su111624115um_c_b,A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ B2 @ C2 ) @ A )
= ( ( ord_le1404954451um_c_b @ B2 @ A )
& ( ord_le1404954451um_c_b @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_87_le__sup__iff,axiom,
! [X2: sum_sum_b_b > $o,Y: sum_sum_b_b > $o,Z: sum_sum_b_b > $o] :
( ( ord_le1306331338_b_b_o @ ( sup_su2024788734_b_b_o @ X2 @ Y ) @ Z )
= ( ( ord_le1306331338_b_b_o @ X2 @ Z )
& ( ord_le1306331338_b_b_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_88_le__sup__iff,axiom,
! [X2: set_Fi291318197eal_sz,Y: set_Fi291318197eal_sz,Z: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ ( sup_su1321388937eal_sz @ X2 @ Y ) @ Z )
= ( ( ord_le41263445eal_sz @ X2 @ Z )
& ( ord_le41263445eal_sz @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_89_le__sup__iff,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X2 @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_90_le__sup__iff,axiom,
! [X2: sum_su1965225555um_c_b > $o,Y: sum_su1965225555um_c_b > $o,Z: sum_su1965225555um_c_b > $o] :
( ( ord_le1534094666_c_b_o @ ( sup_su1211201046_c_b_o @ X2 @ Y ) @ Z )
= ( ( ord_le1534094666_c_b_o @ X2 @ Z )
& ( ord_le1534094666_c_b_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_91_le__sup__iff,axiom,
! [X2: a > $o,Y: a > $o,Z: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_a_o @ X2 @ Z )
& ( ord_less_eq_a_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_92_le__sup__iff,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b,Z: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ X2 @ Y ) @ Z )
= ( ( ord_le412705147um_b_b @ X2 @ Z )
& ( ord_le412705147um_b_b @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_93_le__sup__iff,axiom,
! [X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b,Z: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ X2 @ Y ) @ Z )
= ( ( ord_le1404954451um_c_b @ X2 @ Z )
& ( ord_le1404954451um_c_b @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_94_FVT_Osimps_I5_J,axiom,
! [F: trm_sf_sz,G: trm_sf_sz] :
( ( static_FVT_sf_sz @ ( times_sf_sz @ F @ G ) )
= ( sup_su1670953621_sz_sz @ ( static_FVT_sf_sz @ F ) @ ( static_FVT_sf_sz @ G ) ) ) ).
% FVT.simps(5)
thf(fact_95_FVT_Osimps_I5_J,axiom,
! [F: trm_a_b,G: trm_a_b] :
( ( static_FVT_a_b @ ( times_a_b @ F @ G ) )
= ( sup_su2137030215um_b_b @ ( static_FVT_a_b @ F ) @ ( static_FVT_a_b @ G ) ) ) ).
% FVT.simps(5)
thf(fact_96_SIGT_Osimps_I5_J,axiom,
! [T1: trm_sf_sz,T2: trm_sf_sz] :
( ( static_SIGT_sf_sz @ ( times_sf_sz @ T1 @ T2 ) )
= ( sup_sup_set_sf @ ( static_SIGT_sf_sz @ T1 ) @ ( static_SIGT_sf_sz @ T2 ) ) ) ).
% SIGT.simps(5)
thf(fact_97_SIGT_Osimps_I5_J,axiom,
! [T1: trm_a_b,T2: trm_a_b] :
( ( static_SIGT_a_b @ ( times_a_b @ T1 @ T2 ) )
= ( sup_sup_set_a @ ( static_SIGT_a_b @ T1 ) @ ( static_SIGT_a_b @ T2 ) ) ) ).
% SIGT.simps(5)
thf(fact_98_type__definition__strm,axiom,
type_d2045870413rm_a_b @ frechet_raw_term_a_b @ freche665377491rm_a_b @ ( collect_trm_a_b @ dfree_a_b ) ).
% type_definition_strm
thf(fact_99_type__definition__strm,axiom,
type_d46389773_sf_sz @ freche782854530_sf_sz @ freche1046279700_sf_sz @ ( collect_trm_sf_sz @ dfree_sf_sz ) ).
% type_definition_strm
thf(fact_100_good__interp__inverse,axiom,
! [Y: denota723907260t_unit] :
( ( member122424677t_unit @ Y @ ( collec944067751t_unit @ denota1086158987_a_c_b ) )
=> ( ( freche288321865_a_c_b @ ( freche153761090_a_c_b @ Y ) )
= Y ) ) ).
% good_interp_inverse
thf(fact_101_good__interp__inverse,axiom,
! [Y: denota610675952t_unit] :
( ( member754147591t_unit @ Y @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( ( freche1421597129_sc_sz @ ( freche1784963216_sc_sz @ Y ) )
= Y ) ) ).
% good_interp_inverse
thf(fact_102_raw__interp__induct,axiom,
! [Y: denota723907260t_unit,P: denota723907260t_unit > $o] :
( ( member122424677t_unit @ Y @ ( collec944067751t_unit @ denota1086158987_a_c_b ) )
=> ( ! [X3: freche1734198479_a_c_b] : ( P @ ( freche288321865_a_c_b @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_interp_induct
thf(fact_103_raw__interp__induct,axiom,
! [Y: denota610675952t_unit,P: denota610675952t_unit > $o] :
( ( member754147591t_unit @ Y @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( ! [X3: freche2095075217_sc_sz] : ( P @ ( freche1421597129_sc_sz @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_interp_induct
thf(fact_104_raw__interp__cases,axiom,
! [Y: denota723907260t_unit] :
( ( member122424677t_unit @ Y @ ( collec944067751t_unit @ denota1086158987_a_c_b ) )
=> ~ ! [X3: freche1734198479_a_c_b] :
( Y
!= ( freche288321865_a_c_b @ X3 ) ) ) ).
% raw_interp_cases
thf(fact_105_raw__interp__cases,axiom,
! [Y: denota610675952t_unit] :
( ( member754147591t_unit @ Y @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ~ ! [X3: freche2095075217_sc_sz] :
( Y
!= ( freche1421597129_sc_sz @ X3 ) ) ) ).
% raw_interp_cases
thf(fact_106_raw__interp,axiom,
! [X2: freche1734198479_a_c_b] : ( member122424677t_unit @ ( freche288321865_a_c_b @ X2 ) @ ( collec944067751t_unit @ denota1086158987_a_c_b ) ) ).
% raw_interp
thf(fact_107_raw__interp,axiom,
! [X2: freche2095075217_sc_sz] : ( member754147591t_unit @ ( freche1421597129_sc_sz @ X2 ) @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) ) ).
% raw_interp
thf(fact_108_good__interp__inject,axiom,
! [X2: denota723907260t_unit,Y: denota723907260t_unit] :
( ( member122424677t_unit @ X2 @ ( collec944067751t_unit @ denota1086158987_a_c_b ) )
=> ( ( member122424677t_unit @ Y @ ( collec944067751t_unit @ denota1086158987_a_c_b ) )
=> ( ( ( freche153761090_a_c_b @ X2 )
= ( freche153761090_a_c_b @ Y ) )
= ( X2 = Y ) ) ) ) ).
% good_interp_inject
thf(fact_109_good__interp__inject,axiom,
! [X2: denota610675952t_unit,Y: denota610675952t_unit] :
( ( member754147591t_unit @ X2 @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( ( member754147591t_unit @ Y @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( ( ( freche1784963216_sc_sz @ X2 )
= ( freche1784963216_sc_sz @ Y ) )
= ( X2 = Y ) ) ) ) ).
% good_interp_inject
thf(fact_110_good__interp__induct,axiom,
! [P: freche1734198479_a_c_b > $o,X2: freche1734198479_a_c_b] :
( ! [Y3: denota723907260t_unit] :
( ( member122424677t_unit @ Y3 @ ( collec944067751t_unit @ denota1086158987_a_c_b ) )
=> ( P @ ( freche153761090_a_c_b @ Y3 ) ) )
=> ( P @ X2 ) ) ).
% good_interp_induct
thf(fact_111_good__interp__induct,axiom,
! [P: freche2095075217_sc_sz > $o,X2: freche2095075217_sc_sz] :
( ! [Y3: denota610675952t_unit] :
( ( member754147591t_unit @ Y3 @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( P @ ( freche1784963216_sc_sz @ Y3 ) ) )
=> ( P @ X2 ) ) ).
% good_interp_induct
thf(fact_112_good__interp__cases,axiom,
! [X2: freche1734198479_a_c_b] :
~ ! [Y3: denota723907260t_unit] :
( ( X2
= ( freche153761090_a_c_b @ Y3 ) )
=> ~ ( member122424677t_unit @ Y3 @ ( collec944067751t_unit @ denota1086158987_a_c_b ) ) ) ).
% good_interp_cases
thf(fact_113_good__interp__cases,axiom,
! [X2: freche2095075217_sc_sz] :
~ ! [Y3: denota610675952t_unit] :
( ( X2
= ( freche1784963216_sc_sz @ Y3 ) )
=> ~ ( member754147591t_unit @ Y3 @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) ) ) ).
% good_interp_cases
thf(fact_114_subsetI,axiom,
! [A3: set_trm_sf_sz,B: set_trm_sf_sz] :
( ! [X3: trm_sf_sz] :
( ( member_trm_sf_sz @ X3 @ A3 )
=> ( member_trm_sf_sz @ X3 @ B ) )
=> ( ord_le1697056455_sf_sz @ A3 @ B ) ) ).
% subsetI
thf(fact_115_subsetI,axiom,
! [A3: set_trm_a_b,B: set_trm_a_b] :
( ! [X3: trm_a_b] :
( ( member_trm_a_b @ X3 @ A3 )
=> ( member_trm_a_b @ X3 @ B ) )
=> ( ord_le230080206rm_a_b @ A3 @ B ) ) ).
% subsetI
thf(fact_116_subsetI,axiom,
! [A3: set_Fi291318197eal_sz,B: set_Fi291318197eal_sz] :
( ! [X3: finite824932053eal_sz] :
( ( member1693951742eal_sz @ X3 @ A3 )
=> ( member1693951742eal_sz @ X3 @ B ) )
=> ( ord_le41263445eal_sz @ A3 @ B ) ) ).
% subsetI
thf(fact_117_subsetI,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ! [X3: sum_sum_b_b] :
( ( member_Sum_sum_b_b @ X3 @ A3 )
=> ( member_Sum_sum_b_b @ X3 @ B ) )
=> ( ord_le412705147um_b_b @ A3 @ B ) ) ).
% subsetI
thf(fact_118_subsetI,axiom,
! [A3: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A3 @ B ) ) ).
% subsetI
thf(fact_119_subsetI,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b] :
( ! [X3: sum_su1965225555um_c_b] :
( ( member973578748um_c_b @ X3 @ A3 )
=> ( member973578748um_c_b @ X3 @ B ) )
=> ( ord_le1404954451um_c_b @ A3 @ B ) ) ).
% subsetI
thf(fact_120_subset__antisym,axiom,
! [A3: set_Fi291318197eal_sz,B: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ A3 @ B )
=> ( ( ord_le41263445eal_sz @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_121_subset__antisym,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ A3 @ B )
=> ( ( ord_le412705147um_b_b @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_122_subset__antisym,axiom,
! [A3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( ord_less_eq_set_a @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_123_subset__antisym,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A3 @ B )
=> ( ( ord_le1404954451um_c_b @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_124_sup__apply,axiom,
( sup_su2024788734_b_b_o
= ( ^ [F2: sum_sum_b_b > $o,G2: sum_sum_b_b > $o,X: sum_sum_b_b] : ( sup_sup_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% sup_apply
thf(fact_125_sup__apply,axiom,
( sup_su1211201046_c_b_o
= ( ^ [F2: sum_su1965225555um_c_b > $o,G2: sum_su1965225555um_c_b > $o,X: sum_su1965225555um_c_b] : ( sup_sup_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% sup_apply
thf(fact_126_sup__apply,axiom,
( sup_sup_a_o
= ( ^ [F2: a > $o,G2: a > $o,X: a] : ( sup_sup_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% sup_apply
thf(fact_127_sup_Oidem,axiom,
! [A: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_128_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_129_sup_Oidem,axiom,
! [A: sum_sum_b_b > $o] :
( ( sup_su2024788734_b_b_o @ A @ A )
= A ) ).
% sup.idem
thf(fact_130_sup_Oidem,axiom,
! [A: sum_su1965225555um_c_b > $o] :
( ( sup_su1211201046_c_b_o @ A @ A )
= A ) ).
% sup.idem
thf(fact_131_sup_Oidem,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ A @ A )
= A ) ).
% sup.idem
thf(fact_132_sup_Oidem,axiom,
! [A: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A @ A )
= A ) ).
% sup.idem
thf(fact_133_sup__idem,axiom,
! [X2: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_134_sup__idem,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_135_sup__idem,axiom,
! [X2: sum_sum_b_b > $o] :
( ( sup_su2024788734_b_b_o @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_136_sup__idem,axiom,
! [X2: sum_su1965225555um_c_b > $o] :
( ( sup_su1211201046_c_b_o @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_137_sup__idem,axiom,
! [X2: a > $o] :
( ( sup_sup_a_o @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_138_sup__idem,axiom,
! [X2: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_139_sup_Oleft__idem,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ A @ ( sup_su296826759um_c_b @ A @ B2 ) )
= ( sup_su296826759um_c_b @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_140_sup_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_141_sup_Oleft__idem,axiom,
! [A: sum_sum_b_b > $o,B2: sum_sum_b_b > $o] :
( ( sup_su2024788734_b_b_o @ A @ ( sup_su2024788734_b_b_o @ A @ B2 ) )
= ( sup_su2024788734_b_b_o @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_142_sup_Oleft__idem,axiom,
! [A: sum_su1965225555um_c_b > $o,B2: sum_su1965225555um_c_b > $o] :
( ( sup_su1211201046_c_b_o @ A @ ( sup_su1211201046_c_b_o @ A @ B2 ) )
= ( sup_su1211201046_c_b_o @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_143_sup_Oleft__idem,axiom,
! [A: a > $o,B2: a > $o] :
( ( sup_sup_a_o @ A @ ( sup_sup_a_o @ A @ B2 ) )
= ( sup_sup_a_o @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_144_sup_Oleft__idem,axiom,
! [A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A @ ( sup_su2137030215um_b_b @ A @ B2 ) )
= ( sup_su2137030215um_b_b @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_145_sup__left__idem,axiom,
! [X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ X2 @ ( sup_su296826759um_c_b @ X2 @ Y ) )
= ( sup_su296826759um_c_b @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_146_sup__left__idem,axiom,
! [X2: set_a,Y: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) )
= ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_147_sup__left__idem,axiom,
! [X2: sum_sum_b_b > $o,Y: sum_sum_b_b > $o] :
( ( sup_su2024788734_b_b_o @ X2 @ ( sup_su2024788734_b_b_o @ X2 @ Y ) )
= ( sup_su2024788734_b_b_o @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_148_sup__left__idem,axiom,
! [X2: sum_su1965225555um_c_b > $o,Y: sum_su1965225555um_c_b > $o] :
( ( sup_su1211201046_c_b_o @ X2 @ ( sup_su1211201046_c_b_o @ X2 @ Y ) )
= ( sup_su1211201046_c_b_o @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_149_sup__left__idem,axiom,
! [X2: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X2 @ ( sup_sup_a_o @ X2 @ Y ) )
= ( sup_sup_a_o @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_150_sup__left__idem,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ X2 @ ( sup_su2137030215um_b_b @ X2 @ Y ) )
= ( sup_su2137030215um_b_b @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_151_sup_Oright__idem,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ ( sup_su296826759um_c_b @ A @ B2 ) @ B2 )
= ( sup_su296826759um_c_b @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_152_sup_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ B2 )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_153_sup_Oright__idem,axiom,
! [A: sum_sum_b_b > $o,B2: sum_sum_b_b > $o] :
( ( sup_su2024788734_b_b_o @ ( sup_su2024788734_b_b_o @ A @ B2 ) @ B2 )
= ( sup_su2024788734_b_b_o @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_154_sup_Oright__idem,axiom,
! [A: sum_su1965225555um_c_b > $o,B2: sum_su1965225555um_c_b > $o] :
( ( sup_su1211201046_c_b_o @ ( sup_su1211201046_c_b_o @ A @ B2 ) @ B2 )
= ( sup_su1211201046_c_b_o @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_155_sup_Oright__idem,axiom,
! [A: a > $o,B2: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B2 ) @ B2 )
= ( sup_sup_a_o @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_156_sup_Oright__idem,axiom,
! [A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ ( sup_su2137030215um_b_b @ A @ B2 ) @ B2 )
= ( sup_su2137030215um_b_b @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_157_UnCI,axiom,
! [C2: a,B: set_a,A3: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A3 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) ) ) ).
% UnCI
thf(fact_158_UnCI,axiom,
! [C2: sum_sum_b_b,B: set_Sum_sum_b_b,A3: set_Sum_sum_b_b] :
( ( ~ ( member_Sum_sum_b_b @ C2 @ B )
=> ( member_Sum_sum_b_b @ C2 @ A3 ) )
=> ( member_Sum_sum_b_b @ C2 @ ( sup_su2137030215um_b_b @ A3 @ B ) ) ) ).
% UnCI
thf(fact_159_Un__iff,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) )
= ( ( member_a @ C2 @ A3 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_160_Un__iff,axiom,
! [C2: sum_sum_b_b,A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( member_Sum_sum_b_b @ C2 @ ( sup_su2137030215um_b_b @ A3 @ B ) )
= ( ( member_Sum_sum_b_b @ C2 @ A3 )
| ( member_Sum_sum_b_b @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_161_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X: a] : ( member_a @ X @ A4 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_162_sup__set__def,axiom,
( sup_su296826759um_c_b
= ( ^ [A4: set_Su111624115um_c_b,B3: set_Su111624115um_c_b] :
( collec1166375742um_c_b
@ ( sup_su1211201046_c_b_o
@ ^ [X: sum_su1965225555um_c_b] : ( member973578748um_c_b @ X @ A4 )
@ ^ [X: sum_su1965225555um_c_b] : ( member973578748um_c_b @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_163_sup__set__def,axiom,
( sup_su2137030215um_b_b
= ( ^ [A4: set_Sum_sum_b_b,B3: set_Sum_sum_b_b] :
( collect_Sum_sum_b_b
@ ( sup_su2024788734_b_b_o
@ ^ [X: sum_sum_b_b] : ( member_Sum_sum_b_b @ X @ A4 )
@ ^ [X: sum_sum_b_b] : ( member_Sum_sum_b_b @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_164_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A4 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_165_less__eq__set__def,axiom,
( ord_le1404954451um_c_b
= ( ^ [A4: set_Su111624115um_c_b,B3: set_Su111624115um_c_b] :
( ord_le1534094666_c_b_o
@ ^ [X: sum_su1965225555um_c_b] : ( member973578748um_c_b @ X @ A4 )
@ ^ [X: sum_su1965225555um_c_b] : ( member973578748um_c_b @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_166_in__mono,axiom,
! [A3: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_167_in__mono,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b,X2: sum_su1965225555um_c_b] :
( ( ord_le1404954451um_c_b @ A3 @ B )
=> ( ( member973578748um_c_b @ X2 @ A3 )
=> ( member973578748um_c_b @ X2 @ B ) ) ) ).
% in_mono
thf(fact_168_subsetD,axiom,
! [A3: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_169_subsetD,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b,C2: sum_su1965225555um_c_b] :
( ( ord_le1404954451um_c_b @ A3 @ B )
=> ( ( member973578748um_c_b @ C2 @ A3 )
=> ( member973578748um_c_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_170_equalityE,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b] :
( ( A3 = B )
=> ~ ( ( ord_le1404954451um_c_b @ A3 @ B )
=> ~ ( ord_le1404954451um_c_b @ B @ A3 ) ) ) ).
% equalityE
thf(fact_171_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A4 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_172_subset__eq,axiom,
( ord_le1404954451um_c_b
= ( ^ [A4: set_Su111624115um_c_b,B3: set_Su111624115um_c_b] :
! [X: sum_su1965225555um_c_b] :
( ( member973578748um_c_b @ X @ A4 )
=> ( member973578748um_c_b @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_173_equalityD1,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b] :
( ( A3 = B )
=> ( ord_le1404954451um_c_b @ A3 @ B ) ) ).
% equalityD1
thf(fact_174_equalityD2,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b] :
( ( A3 = B )
=> ( ord_le1404954451um_c_b @ B @ A3 ) ) ).
% equalityD2
thf(fact_175_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_176_subset__iff,axiom,
( ord_le1404954451um_c_b
= ( ^ [A4: set_Su111624115um_c_b,B3: set_Su111624115um_c_b] :
! [T: sum_su1965225555um_c_b] :
( ( member973578748um_c_b @ T @ A4 )
=> ( member973578748um_c_b @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_177_subset__refl,axiom,
! [A3: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ A3 @ A3 ) ).
% subset_refl
thf(fact_178_Collect__mono,axiom,
! [P: sum_su1965225555um_c_b > $o,Q: sum_su1965225555um_c_b > $o] :
( ! [X3: sum_su1965225555um_c_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1404954451um_c_b @ ( collec1166375742um_c_b @ P ) @ ( collec1166375742um_c_b @ Q ) ) ) ).
% Collect_mono
thf(fact_179_subset__trans,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b,C: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A3 @ B )
=> ( ( ord_le1404954451um_c_b @ B @ C )
=> ( ord_le1404954451um_c_b @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_180_set__eq__subset,axiom,
( ( ^ [Y4: set_Su111624115um_c_b,Z2: set_Su111624115um_c_b] : Y4 = Z2 )
= ( ^ [A4: set_Su111624115um_c_b,B3: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A4 @ B3 )
& ( ord_le1404954451um_c_b @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_181_Collect__mono__iff,axiom,
! [P: sum_su1965225555um_c_b > $o,Q: sum_su1965225555um_c_b > $o] :
( ( ord_le1404954451um_c_b @ ( collec1166375742um_c_b @ P ) @ ( collec1166375742um_c_b @ Q ) )
= ( ! [X: sum_su1965225555um_c_b] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_182_inf__sup__aci_I8_J,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ X2 @ ( sup_su2137030215um_b_b @ X2 @ Y ) )
= ( sup_su2137030215um_b_b @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_183_inf__sup__aci_I7_J,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b,Z: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ X2 @ ( sup_su2137030215um_b_b @ Y @ Z ) )
= ( sup_su2137030215um_b_b @ Y @ ( sup_su2137030215um_b_b @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_184_inf__sup__aci_I6_J,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b,Z: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ ( sup_su2137030215um_b_b @ X2 @ Y ) @ Z )
= ( sup_su2137030215um_b_b @ X2 @ ( sup_su2137030215um_b_b @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_185_inf__sup__aci_I5_J,axiom,
( sup_su2137030215um_b_b
= ( ^ [X: set_Sum_sum_b_b,Y2: set_Sum_sum_b_b] : ( sup_su2137030215um_b_b @ Y2 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_186_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_Sum_sum_b_b,K: set_Sum_sum_b_b,A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( A3
= ( sup_su2137030215um_b_b @ K @ A ) )
=> ( ( sup_su2137030215um_b_b @ A3 @ B2 )
= ( sup_su2137030215um_b_b @ K @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_187_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Sum_sum_b_b,K: set_Sum_sum_b_b,B2: set_Sum_sum_b_b,A: set_Sum_sum_b_b] :
( ( B
= ( sup_su2137030215um_b_b @ K @ B2 ) )
=> ( ( sup_su2137030215um_b_b @ A @ B )
= ( sup_su2137030215um_b_b @ K @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_188_sup_Oassoc,axiom,
! [A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b,C2: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ ( sup_su2137030215um_b_b @ A @ B2 ) @ C2 )
= ( sup_su2137030215um_b_b @ A @ ( sup_su2137030215um_b_b @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_189_sup__assoc,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b,Z: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ ( sup_su2137030215um_b_b @ X2 @ Y ) @ Z )
= ( sup_su2137030215um_b_b @ X2 @ ( sup_su2137030215um_b_b @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_190_sup_Ocommute,axiom,
( sup_su2137030215um_b_b
= ( ^ [A5: set_Sum_sum_b_b,B4: set_Sum_sum_b_b] : ( sup_su2137030215um_b_b @ B4 @ A5 ) ) ) ).
% sup.commute
thf(fact_191_sup__commute,axiom,
( sup_su2137030215um_b_b
= ( ^ [X: set_Sum_sum_b_b,Y2: set_Sum_sum_b_b] : ( sup_su2137030215um_b_b @ Y2 @ X ) ) ) ).
% sup_commute
thf(fact_192_sup_Oleft__commute,axiom,
! [B2: set_Sum_sum_b_b,A: set_Sum_sum_b_b,C2: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ B2 @ ( sup_su2137030215um_b_b @ A @ C2 ) )
= ( sup_su2137030215um_b_b @ A @ ( sup_su2137030215um_b_b @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_193_sup__left__commute,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b,Z: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ X2 @ ( sup_su2137030215um_b_b @ Y @ Z ) )
= ( sup_su2137030215um_b_b @ Y @ ( sup_su2137030215um_b_b @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_194_UnE,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) )
=> ( ~ ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_195_UnE,axiom,
! [C2: sum_sum_b_b,A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( member_Sum_sum_b_b @ C2 @ ( sup_su2137030215um_b_b @ A3 @ B ) )
=> ( ~ ( member_Sum_sum_b_b @ C2 @ A3 )
=> ( member_Sum_sum_b_b @ C2 @ B ) ) ) ).
% UnE
thf(fact_196_UnI1,axiom,
! [C2: a,A3: set_a,B: set_a] :
( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) ) ) ).
% UnI1
thf(fact_197_UnI1,axiom,
! [C2: sum_sum_b_b,A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( member_Sum_sum_b_b @ C2 @ A3 )
=> ( member_Sum_sum_b_b @ C2 @ ( sup_su2137030215um_b_b @ A3 @ B ) ) ) ).
% UnI1
thf(fact_198_UnI2,axiom,
! [C2: a,B: set_a,A3: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B ) ) ) ).
% UnI2
thf(fact_199_UnI2,axiom,
! [C2: sum_sum_b_b,B: set_Sum_sum_b_b,A3: set_Sum_sum_b_b] :
( ( member_Sum_sum_b_b @ C2 @ B )
=> ( member_Sum_sum_b_b @ C2 @ ( sup_su2137030215um_b_b @ A3 @ B ) ) ) ).
% UnI2
thf(fact_200_bex__Un,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b,P: sum_sum_b_b > $o] :
( ( ? [X: sum_sum_b_b] :
( ( member_Sum_sum_b_b @ X @ ( sup_su2137030215um_b_b @ A3 @ B ) )
& ( P @ X ) ) )
= ( ? [X: sum_sum_b_b] :
( ( member_Sum_sum_b_b @ X @ A3 )
& ( P @ X ) )
| ? [X: sum_sum_b_b] :
( ( member_Sum_sum_b_b @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_201_ball__Un,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b,P: sum_sum_b_b > $o] :
( ( ! [X: sum_sum_b_b] :
( ( member_Sum_sum_b_b @ X @ ( sup_su2137030215um_b_b @ A3 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: sum_sum_b_b] :
( ( member_Sum_sum_b_b @ X @ A3 )
=> ( P @ X ) )
& ! [X: sum_sum_b_b] :
( ( member_Sum_sum_b_b @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_202_Un__assoc,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b,C: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ ( sup_su2137030215um_b_b @ A3 @ B ) @ C )
= ( sup_su2137030215um_b_b @ A3 @ ( sup_su2137030215um_b_b @ B @ C ) ) ) ).
% Un_assoc
thf(fact_203_Un__absorb,axiom,
! [A3: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_204_Un__commute,axiom,
( sup_su2137030215um_b_b
= ( ^ [A4: set_Sum_sum_b_b,B3: set_Sum_sum_b_b] : ( sup_su2137030215um_b_b @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_205_Un__left__absorb,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A3 @ ( sup_su2137030215um_b_b @ A3 @ B ) )
= ( sup_su2137030215um_b_b @ A3 @ B ) ) ).
% Un_left_absorb
thf(fact_206_Un__left__commute,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b,C: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A3 @ ( sup_su2137030215um_b_b @ B @ C ) )
= ( sup_su2137030215um_b_b @ B @ ( sup_su2137030215um_b_b @ A3 @ C ) ) ) ).
% Un_left_commute
thf(fact_207_Collect__subset,axiom,
! [A3: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_208_Collect__subset,axiom,
! [A3: set_Su111624115um_c_b,P: sum_su1965225555um_c_b > $o] :
( ord_le1404954451um_c_b
@ ( collec1166375742um_c_b
@ ^ [X: sum_su1965225555um_c_b] :
( ( member973578748um_c_b @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_209_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
| ( member_a @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_210_Un__def,axiom,
( sup_su296826759um_c_b
= ( ^ [A4: set_Su111624115um_c_b,B3: set_Su111624115um_c_b] :
( collec1166375742um_c_b
@ ^ [X: sum_su1965225555um_c_b] :
( ( member973578748um_c_b @ X @ A4 )
| ( member973578748um_c_b @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_211_Un__def,axiom,
( sup_su2137030215um_b_b
= ( ^ [A4: set_Sum_sum_b_b,B3: set_Sum_sum_b_b] :
( collect_Sum_sum_b_b
@ ^ [X: sum_sum_b_b] :
( ( member_Sum_sum_b_b @ X @ A4 )
| ( member_Sum_sum_b_b @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_212_Collect__disj__eq,axiom,
! [P: sum_su1965225555um_c_b > $o,Q: sum_su1965225555um_c_b > $o] :
( ( collec1166375742um_c_b
@ ^ [X: sum_su1965225555um_c_b] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su296826759um_c_b @ ( collec1166375742um_c_b @ P ) @ ( collec1166375742um_c_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_213_Collect__disj__eq,axiom,
! [P: sum_sum_b_b > $o,Q: sum_sum_b_b > $o] :
( ( collect_Sum_sum_b_b
@ ^ [X: sum_sum_b_b] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su2137030215um_b_b @ ( collect_Sum_sum_b_b @ P ) @ ( collect_Sum_sum_b_b @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_214_inf__sup__ord_I4_J,axiom,
! [Y: set_Sum_sum_b_b,X2: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ Y @ ( sup_su2137030215um_b_b @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_215_inf__sup__ord_I4_J,axiom,
! [Y: set_Su111624115um_c_b,X2: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ Y @ ( sup_su296826759um_c_b @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_216_inf__sup__ord_I3_J,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ X2 @ ( sup_su2137030215um_b_b @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_217_inf__sup__ord_I3_J,axiom,
! [X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ X2 @ ( sup_su296826759um_c_b @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_218_le__supE,axiom,
! [A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b,X2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ A @ B2 ) @ X2 )
=> ~ ( ( ord_le412705147um_b_b @ A @ X2 )
=> ~ ( ord_le412705147um_b_b @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_219_le__supE,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,X2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ A @ B2 ) @ X2 )
=> ~ ( ( ord_le1404954451um_c_b @ A @ X2 )
=> ~ ( ord_le1404954451um_c_b @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_220_le__supI,axiom,
! [A: set_Sum_sum_b_b,X2: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ A @ X2 )
=> ( ( ord_le412705147um_b_b @ B2 @ X2 )
=> ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_221_le__supI,axiom,
! [A: set_Su111624115um_c_b,X2: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ X2 )
=> ( ( ord_le1404954451um_c_b @ B2 @ X2 )
=> ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_222_sup__ge1,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ X2 @ ( sup_su2137030215um_b_b @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_223_sup__ge1,axiom,
! [X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ X2 @ ( sup_su296826759um_c_b @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_224_sup__ge2,axiom,
! [Y: set_Sum_sum_b_b,X2: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ Y @ ( sup_su2137030215um_b_b @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_225_sup__ge2,axiom,
! [Y: set_Su111624115um_c_b,X2: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ Y @ ( sup_su296826759um_c_b @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_226_le__supI1,axiom,
! [X2: set_Sum_sum_b_b,A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ X2 @ A )
=> ( ord_le412705147um_b_b @ X2 @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_227_le__supI1,axiom,
! [X2: set_Su111624115um_c_b,A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X2 @ A )
=> ( ord_le1404954451um_c_b @ X2 @ ( sup_su296826759um_c_b @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_228_le__supI2,axiom,
! [X2: set_Sum_sum_b_b,B2: set_Sum_sum_b_b,A: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ X2 @ B2 )
=> ( ord_le412705147um_b_b @ X2 @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_229_le__supI2,axiom,
! [X2: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X2 @ B2 )
=> ( ord_le1404954451um_c_b @ X2 @ ( sup_su296826759um_c_b @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_230_sup_Omono,axiom,
! [C2: set_Sum_sum_b_b,A: set_Sum_sum_b_b,D: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ C2 @ A )
=> ( ( ord_le412705147um_b_b @ D @ B2 )
=> ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ C2 @ D ) @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_231_sup_Omono,axiom,
! [C2: set_Su111624115um_c_b,A: set_Su111624115um_c_b,D: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ C2 @ A )
=> ( ( ord_le1404954451um_c_b @ D @ B2 )
=> ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ C2 @ D ) @ ( sup_su296826759um_c_b @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_232_sup__mono,axiom,
! [A: set_Sum_sum_b_b,C2: set_Sum_sum_b_b,B2: set_Sum_sum_b_b,D: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ A @ C2 )
=> ( ( ord_le412705147um_b_b @ B2 @ D )
=> ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ A @ B2 ) @ ( sup_su2137030215um_b_b @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_233_sup__mono,axiom,
! [A: set_Su111624115um_c_b,C2: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,D: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ C2 )
=> ( ( ord_le1404954451um_c_b @ B2 @ D )
=> ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ A @ B2 ) @ ( sup_su296826759um_c_b @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_234_sup__least,axiom,
! [Y: set_Sum_sum_b_b,X2: set_Sum_sum_b_b,Z: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ Y @ X2 )
=> ( ( ord_le412705147um_b_b @ Z @ X2 )
=> ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_235_sup__least,axiom,
! [Y: set_Su111624115um_c_b,X2: set_Su111624115um_c_b,Z: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ Y @ X2 )
=> ( ( ord_le1404954451um_c_b @ Z @ X2 )
=> ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_236_le__iff__sup,axiom,
( ord_le412705147um_b_b
= ( ^ [X: set_Sum_sum_b_b,Y2: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_237_le__iff__sup,axiom,
( ord_le1404954451um_c_b
= ( ^ [X: set_Su111624115um_c_b,Y2: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_238_sup_OorderE,axiom,
! [B2: set_Sum_sum_b_b,A: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ B2 @ A )
=> ( A
= ( sup_su2137030215um_b_b @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_239_sup_OorderE,axiom,
! [B2: set_Su111624115um_c_b,A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ B2 @ A )
=> ( A
= ( sup_su296826759um_c_b @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_240_sup_OorderI,axiom,
! [A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( A
= ( sup_su2137030215um_b_b @ A @ B2 ) )
=> ( ord_le412705147um_b_b @ B2 @ A ) ) ).
% sup.orderI
thf(fact_241_sup_OorderI,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( A
= ( sup_su296826759um_c_b @ A @ B2 ) )
=> ( ord_le1404954451um_c_b @ B2 @ A ) ) ).
% sup.orderI
thf(fact_242_sup__unique,axiom,
! [F: set_Sum_sum_b_b > set_Sum_sum_b_b > set_Sum_sum_b_b,X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b] :
( ! [X3: set_Sum_sum_b_b,Y3: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Sum_sum_b_b,Y3: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Sum_sum_b_b,Y3: set_Sum_sum_b_b,Z3: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ Y3 @ X3 )
=> ( ( ord_le412705147um_b_b @ Z3 @ X3 )
=> ( ord_le412705147um_b_b @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su2137030215um_b_b @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_243_sup__unique,axiom,
! [F: set_Su111624115um_c_b > set_Su111624115um_c_b > set_Su111624115um_c_b,X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b] :
( ! [X3: set_Su111624115um_c_b,Y3: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Su111624115um_c_b,Y3: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Su111624115um_c_b,Y3: set_Su111624115um_c_b,Z3: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ Y3 @ X3 )
=> ( ( ord_le1404954451um_c_b @ Z3 @ X3 )
=> ( ord_le1404954451um_c_b @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su296826759um_c_b @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_244_sup_Oabsorb1,axiom,
! [B2: set_Sum_sum_b_b,A: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ B2 @ A )
=> ( ( sup_su2137030215um_b_b @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_245_sup_Oabsorb1,axiom,
! [B2: set_Su111624115um_c_b,A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ B2 @ A )
=> ( ( sup_su296826759um_c_b @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_246_sup_Oabsorb2,axiom,
! [A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ A @ B2 )
=> ( ( sup_su2137030215um_b_b @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_247_sup_Oabsorb2,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ B2 )
=> ( ( sup_su296826759um_c_b @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_248_sup__absorb1,axiom,
! [Y: set_Sum_sum_b_b,X2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ Y @ X2 )
=> ( ( sup_su2137030215um_b_b @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_249_sup__absorb1,axiom,
! [Y: set_Su111624115um_c_b,X2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ Y @ X2 )
=> ( ( sup_su296826759um_c_b @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_250_sup__absorb2,axiom,
! [X2: set_Sum_sum_b_b,Y: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ X2 @ Y )
=> ( ( sup_su2137030215um_b_b @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_251_sup__absorb2,axiom,
! [X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X2 @ Y )
=> ( ( sup_su296826759um_c_b @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_252_sup_OboundedE,axiom,
! [B2: set_Sum_sum_b_b,C2: set_Sum_sum_b_b,A: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ B2 @ C2 ) @ A )
=> ~ ( ( ord_le412705147um_b_b @ B2 @ A )
=> ~ ( ord_le412705147um_b_b @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_253_sup_OboundedE,axiom,
! [B2: set_Su111624115um_c_b,C2: set_Su111624115um_c_b,A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ B2 @ C2 ) @ A )
=> ~ ( ( ord_le1404954451um_c_b @ B2 @ A )
=> ~ ( ord_le1404954451um_c_b @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_254_sup_OboundedI,axiom,
! [B2: set_Sum_sum_b_b,A: set_Sum_sum_b_b,C2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ B2 @ A )
=> ( ( ord_le412705147um_b_b @ C2 @ A )
=> ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_255_sup_OboundedI,axiom,
! [B2: set_Su111624115um_c_b,A: set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ B2 @ A )
=> ( ( ord_le1404954451um_c_b @ C2 @ A )
=> ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_256_sup_Oorder__iff,axiom,
( ord_le412705147um_b_b
= ( ^ [B4: set_Sum_sum_b_b,A5: set_Sum_sum_b_b] :
( A5
= ( sup_su2137030215um_b_b @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_257_sup_Oorder__iff,axiom,
( ord_le1404954451um_c_b
= ( ^ [B4: set_Su111624115um_c_b,A5: set_Su111624115um_c_b] :
( A5
= ( sup_su296826759um_c_b @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_258_sup_Ocobounded1,axiom,
! [A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ A @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_259_sup_Ocobounded1,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ A @ ( sup_su296826759um_c_b @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_260_sup_Ocobounded2,axiom,
! [B2: set_Sum_sum_b_b,A: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ B2 @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_261_sup_Ocobounded2,axiom,
! [B2: set_Su111624115um_c_b,A: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ B2 @ ( sup_su296826759um_c_b @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_262_sup_Oabsorb__iff1,axiom,
( ord_le412705147um_b_b
= ( ^ [B4: set_Sum_sum_b_b,A5: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_263_sup_Oabsorb__iff1,axiom,
( ord_le1404954451um_c_b
= ( ^ [B4: set_Su111624115um_c_b,A5: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_264_sup_Oabsorb__iff2,axiom,
( ord_le412705147um_b_b
= ( ^ [A5: set_Sum_sum_b_b,B4: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_265_sup_Oabsorb__iff2,axiom,
( ord_le1404954451um_c_b
= ( ^ [A5: set_Su111624115um_c_b,B4: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_266_sup_OcoboundedI1,axiom,
! [C2: set_Sum_sum_b_b,A: set_Sum_sum_b_b,B2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ C2 @ A )
=> ( ord_le412705147um_b_b @ C2 @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_267_sup_OcoboundedI1,axiom,
! [C2: set_Su111624115um_c_b,A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ C2 @ A )
=> ( ord_le1404954451um_c_b @ C2 @ ( sup_su296826759um_c_b @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_268_sup_OcoboundedI2,axiom,
! [C2: set_Sum_sum_b_b,B2: set_Sum_sum_b_b,A: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ C2 @ B2 )
=> ( ord_le412705147um_b_b @ C2 @ ( sup_su2137030215um_b_b @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_269_sup_OcoboundedI2,axiom,
! [C2: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ C2 @ B2 )
=> ( ord_le1404954451um_c_b @ C2 @ ( sup_su296826759um_c_b @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_270_fst__mono,axiom,
! [X2: produc133005365real_b,Y: produc133005365real_b] :
( ( ord_le793200597real_b @ X2 @ Y )
=> ( ord_le767445194real_b @ ( produc936751193real_b @ X2 ) @ ( produc936751193real_b @ Y ) ) ) ).
% fst_mono
thf(fact_271_Un__mono,axiom,
! [A3: set_Sum_sum_b_b,C: set_Sum_sum_b_b,B: set_Sum_sum_b_b,D2: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ A3 @ C )
=> ( ( ord_le412705147um_b_b @ B @ D2 )
=> ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ A3 @ B ) @ ( sup_su2137030215um_b_b @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_272_Un__mono,axiom,
! [A3: set_Su111624115um_c_b,C: set_Su111624115um_c_b,B: set_Su111624115um_c_b,D2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A3 @ C )
=> ( ( ord_le1404954451um_c_b @ B @ D2 )
=> ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ A3 @ B ) @ ( sup_su296826759um_c_b @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_273_Un__least,axiom,
! [A3: set_Sum_sum_b_b,C: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ A3 @ C )
=> ( ( ord_le412705147um_b_b @ B @ C )
=> ( ord_le412705147um_b_b @ ( sup_su2137030215um_b_b @ A3 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_274_Un__least,axiom,
! [A3: set_Su111624115um_c_b,C: set_Su111624115um_c_b,B: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A3 @ C )
=> ( ( ord_le1404954451um_c_b @ B @ C )
=> ( ord_le1404954451um_c_b @ ( sup_su296826759um_c_b @ A3 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_275_Un__upper1,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ A3 @ ( sup_su2137030215um_b_b @ A3 @ B ) ) ).
% Un_upper1
thf(fact_276_Un__upper1,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ A3 @ ( sup_su296826759um_c_b @ A3 @ B ) ) ).
% Un_upper1
thf(fact_277_Un__upper2,axiom,
! [B: set_Sum_sum_b_b,A3: set_Sum_sum_b_b] : ( ord_le412705147um_b_b @ B @ ( sup_su2137030215um_b_b @ A3 @ B ) ) ).
% Un_upper2
thf(fact_278_Un__upper2,axiom,
! [B: set_Su111624115um_c_b,A3: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ B @ ( sup_su296826759um_c_b @ A3 @ B ) ) ).
% Un_upper2
thf(fact_279_Un__absorb1,axiom,
! [A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ A3 @ B )
=> ( ( sup_su2137030215um_b_b @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_280_Un__absorb1,axiom,
! [A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A3 @ B )
=> ( ( sup_su296826759um_c_b @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_281_Un__absorb2,axiom,
! [B: set_Sum_sum_b_b,A3: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ B @ A3 )
=> ( ( sup_su2137030215um_b_b @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_282_Un__absorb2,axiom,
! [B: set_Su111624115um_c_b,A3: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ B @ A3 )
=> ( ( sup_su296826759um_c_b @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_283_subset__UnE,axiom,
! [C: set_Sum_sum_b_b,A3: set_Sum_sum_b_b,B: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ C @ ( sup_su2137030215um_b_b @ A3 @ B ) )
=> ~ ! [A6: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ A6 @ A3 )
=> ! [B5: set_Sum_sum_b_b] :
( ( ord_le412705147um_b_b @ B5 @ B )
=> ( C
!= ( sup_su2137030215um_b_b @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_284_subset__UnE,axiom,
! [C: set_Su111624115um_c_b,A3: set_Su111624115um_c_b,B: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ C @ ( sup_su296826759um_c_b @ A3 @ B ) )
=> ~ ! [A6: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A6 @ A3 )
=> ! [B5: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ B5 @ B )
=> ( C
!= ( sup_su296826759um_c_b @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_285_subset__Un__eq,axiom,
( ord_le412705147um_b_b
= ( ^ [A4: set_Sum_sum_b_b,B3: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_286_subset__Un__eq,axiom,
( ord_le1404954451um_c_b
= ( ^ [A4: set_Su111624115um_c_b,B3: set_Su111624115um_c_b] :
( ( sup_su296826759um_c_b @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_287_type__definition__good__interp,axiom,
type_d1092523951t_unit @ freche1421597129_sc_sz @ freche1784963216_sc_sz @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) ).
% type_definition_good_interp
thf(fact_288_order__refl,axiom,
! [X2: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ X2 @ X2 ) ).
% order_refl
thf(fact_289_sterm__continuous_H,axiom,
! [I: denota610675952t_unit,Theta: trm_sf_sz,S: set_Fi291318197eal_sz] :
( ( denota1579475975_sc_sz @ I )
=> ( ( dfree_sf_sz @ Theta )
=> ( topolo1348430467z_real @ S @ ( denota1179238309_sc_sz @ I @ Theta ) ) ) ) ).
% sterm_continuous'
thf(fact_290_sterm__continuous_H,axiom,
! [I: denota723907260t_unit,Theta: trm_a_b,S: set_Fi268318752real_b] :
( ( denota1086158987_a_c_b @ I )
=> ( ( dfree_a_b @ Theta )
=> ( topolo238266006b_real @ S @ ( denota722380397_a_c_b @ I @ Theta ) ) ) ) ).
% sterm_continuous'
thf(fact_291_good__interp_Odomain,axiom,
( ( domain413207655_sc_sz @ freche58918398_sc_sz )
= denota1579475975_sc_sz ) ).
% good_interp.domain
thf(fact_292_strm_Odomain,axiom,
( ( domain134596861rm_a_b @ frechet_cr_strm_a_b )
= dfree_a_b ) ).
% strm.domain
thf(fact_293_strm_Odomain,axiom,
( ( domain512309999_sf_sz @ freche1244000341_sf_sz )
= dfree_sf_sz ) ).
% strm.domain
thf(fact_294_dual__order_Oantisym,axiom,
! [B2: set_Su111624115um_c_b,A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ B2 @ A )
=> ( ( ord_le1404954451um_c_b @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_295_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Su111624115um_c_b,Z2: set_Su111624115um_c_b] : Y4 = Z2 )
= ( ^ [A5: set_Su111624115um_c_b,B4: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ B4 @ A5 )
& ( ord_le1404954451um_c_b @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_296_dual__order_Otrans,axiom,
! [B2: set_Su111624115um_c_b,A: set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ B2 @ A )
=> ( ( ord_le1404954451um_c_b @ C2 @ B2 )
=> ( ord_le1404954451um_c_b @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_297_dual__order_Orefl,axiom,
! [A: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ A @ A ) ).
% dual_order.refl
thf(fact_298_order__trans,axiom,
! [X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b,Z: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X2 @ Y )
=> ( ( ord_le1404954451um_c_b @ Y @ Z )
=> ( ord_le1404954451um_c_b @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_299_order__class_Oorder_Oantisym,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ B2 )
=> ( ( ord_le1404954451um_c_b @ B2 @ A )
=> ( A = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_300_ord__le__eq__trans,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le1404954451um_c_b @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_301_ord__eq__le__trans,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( A = B2 )
=> ( ( ord_le1404954451um_c_b @ B2 @ C2 )
=> ( ord_le1404954451um_c_b @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_302_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_Su111624115um_c_b,Z2: set_Su111624115um_c_b] : Y4 = Z2 )
= ( ^ [A5: set_Su111624115um_c_b,B4: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A5 @ B4 )
& ( ord_le1404954451um_c_b @ B4 @ A5 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_303_antisym__conv,axiom,
! [Y: set_Su111624115um_c_b,X2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ Y @ X2 )
=> ( ( ord_le1404954451um_c_b @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_304_order_Otrans,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ B2 )
=> ( ( ord_le1404954451um_c_b @ B2 @ C2 )
=> ( ord_le1404954451um_c_b @ A @ C2 ) ) ) ).
% order.trans
thf(fact_305_eq__refl,axiom,
! [X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b] :
( ( X2 = Y )
=> ( ord_le1404954451um_c_b @ X2 @ Y ) ) ).
% eq_refl
thf(fact_306_antisym,axiom,
! [X2: set_Su111624115um_c_b,Y: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X2 @ Y )
=> ( ( ord_le1404954451um_c_b @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_307_eq__iff,axiom,
( ( ^ [Y4: set_Su111624115um_c_b,Z2: set_Su111624115um_c_b] : Y4 = Z2 )
= ( ^ [X: set_Su111624115um_c_b,Y2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X @ Y2 )
& ( ord_le1404954451um_c_b @ Y2 @ X ) ) ) ) ).
% eq_iff
thf(fact_308_ord__le__eq__subst,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,F: set_Su111624115um_c_b > set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: set_Su111624115um_c_b,Y3: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X3 @ Y3 )
=> ( ord_le1404954451um_c_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1404954451um_c_b @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_309_ord__eq__le__subst,axiom,
! [A: set_Su111624115um_c_b,F: set_Su111624115um_c_b > set_Su111624115um_c_b,B2: set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1404954451um_c_b @ B2 @ C2 )
=> ( ! [X3: set_Su111624115um_c_b,Y3: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X3 @ Y3 )
=> ( ord_le1404954451um_c_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1404954451um_c_b @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_310_order__subst2,axiom,
! [A: set_Su111624115um_c_b,B2: set_Su111624115um_c_b,F: set_Su111624115um_c_b > set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ B2 )
=> ( ( ord_le1404954451um_c_b @ ( F @ B2 ) @ C2 )
=> ( ! [X3: set_Su111624115um_c_b,Y3: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X3 @ Y3 )
=> ( ord_le1404954451um_c_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1404954451um_c_b @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_311_order__subst1,axiom,
! [A: set_Su111624115um_c_b,F: set_Su111624115um_c_b > set_Su111624115um_c_b,B2: set_Su111624115um_c_b,C2: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ A @ ( F @ B2 ) )
=> ( ( ord_le1404954451um_c_b @ B2 @ C2 )
=> ( ! [X3: set_Su111624115um_c_b,Y3: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ X3 @ Y3 )
=> ( ord_le1404954451um_c_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1404954451um_c_b @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_312_sterm__continuous,axiom,
! [I: denota610675952t_unit,Theta: trm_sf_sz] :
( ( denota1579475975_sc_sz @ I )
=> ( ( dfree_sf_sz @ Theta )
=> ( topolo1348430467z_real @ top_to1873962757eal_sz @ ( denota1179238309_sc_sz @ I @ Theta ) ) ) ) ).
% sterm_continuous
thf(fact_313_sterm__continuous,axiom,
! [I: denota723907260t_unit,Theta: trm_a_b] :
( ( denota1086158987_a_c_b @ I )
=> ( ( dfree_a_b @ Theta )
=> ( topolo238266006b_real @ top_to502572752real_b @ ( denota722380397_a_c_b @ I @ Theta ) ) ) ) ).
% sterm_continuous
thf(fact_314_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X: a] : ( member_a @ X @ R )
@ ^ [X: a] : ( member_a @ X @ S ) )
= ( ^ [X: a] : ( member_a @ X @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_315_sup__Un__eq,axiom,
! [R: set_Sum_sum_b_b,S: set_Sum_sum_b_b] :
( ( sup_su2024788734_b_b_o
@ ^ [X: sum_sum_b_b] : ( member_Sum_sum_b_b @ X @ R )
@ ^ [X: sum_sum_b_b] : ( member_Sum_sum_b_b @ X @ S ) )
= ( ^ [X: sum_sum_b_b] : ( member_Sum_sum_b_b @ X @ ( sup_su2137030215um_b_b @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_316_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ R )
@ ^ [X: a] : ( member_a @ X @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_317_pred__subset__eq,axiom,
! [R: set_Su111624115um_c_b,S: set_Su111624115um_c_b] :
( ( ord_le1534094666_c_b_o
@ ^ [X: sum_su1965225555um_c_b] : ( member973578748um_c_b @ X @ R )
@ ^ [X: sum_su1965225555um_c_b] : ( member973578748um_c_b @ X @ S ) )
= ( ord_le1404954451um_c_b @ R @ S ) ) ).
% pred_subset_eq
thf(fact_318_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_319_UNIV__I,axiom,
! [X2: finite824932053eal_sz] : ( member1693951742eal_sz @ X2 @ top_to1873962757eal_sz ) ).
% UNIV_I
thf(fact_320_sup__top__right,axiom,
! [X2: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ X2 @ top_to301096907um_b_b )
= top_to301096907um_b_b ) ).
% sup_top_right
thf(fact_321_sup__top__right,axiom,
! [X2: set_Fi291318197eal_sz] :
( ( sup_su1321388937eal_sz @ X2 @ top_to1873962757eal_sz )
= top_to1873962757eal_sz ) ).
% sup_top_right
thf(fact_322_sup__top__left,axiom,
! [X2: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ top_to301096907um_b_b @ X2 )
= top_to301096907um_b_b ) ).
% sup_top_left
thf(fact_323_sup__top__left,axiom,
! [X2: set_Fi291318197eal_sz] :
( ( sup_su1321388937eal_sz @ top_to1873962757eal_sz @ X2 )
= top_to1873962757eal_sz ) ).
% sup_top_left
thf(fact_324_top_Oextremum__uniqueI,axiom,
! [A: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ top_to1873962757eal_sz @ A )
=> ( A = top_to1873962757eal_sz ) ) ).
% top.extremum_uniqueI
thf(fact_325_top_Oextremum__uniqueI,axiom,
! [A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ top_to999478531um_c_b @ A )
=> ( A = top_to999478531um_c_b ) ) ).
% top.extremum_uniqueI
thf(fact_326_top_Oextremum__unique,axiom,
! [A: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ top_to1873962757eal_sz @ A )
= ( A = top_to1873962757eal_sz ) ) ).
% top.extremum_unique
thf(fact_327_top_Oextremum__unique,axiom,
! [A: set_Su111624115um_c_b] :
( ( ord_le1404954451um_c_b @ top_to999478531um_c_b @ A )
= ( A = top_to999478531um_c_b ) ) ).
% top.extremum_unique
thf(fact_328_top__greatest,axiom,
! [A: set_Fi291318197eal_sz] : ( ord_le41263445eal_sz @ A @ top_to1873962757eal_sz ) ).
% top_greatest
thf(fact_329_top__greatest,axiom,
! [A: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ A @ top_to999478531um_c_b ) ).
% top_greatest
thf(fact_330_subset__UNIV,axiom,
! [A3: set_Fi291318197eal_sz] : ( ord_le41263445eal_sz @ A3 @ top_to1873962757eal_sz ) ).
% subset_UNIV
thf(fact_331_subset__UNIV,axiom,
! [A3: set_Su111624115um_c_b] : ( ord_le1404954451um_c_b @ A3 @ top_to999478531um_c_b ) ).
% subset_UNIV
thf(fact_332_Un__UNIV__left,axiom,
! [B: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ top_to301096907um_b_b @ B )
= top_to301096907um_b_b ) ).
% Un_UNIV_left
thf(fact_333_Un__UNIV__left,axiom,
! [B: set_Fi291318197eal_sz] :
( ( sup_su1321388937eal_sz @ top_to1873962757eal_sz @ B )
= top_to1873962757eal_sz ) ).
% Un_UNIV_left
thf(fact_334_Un__UNIV__right,axiom,
! [A3: set_Sum_sum_b_b] :
( ( sup_su2137030215um_b_b @ A3 @ top_to301096907um_b_b )
= top_to301096907um_b_b ) ).
% Un_UNIV_right
thf(fact_335_Un__UNIV__right,axiom,
! [A3: set_Fi291318197eal_sz] :
( ( sup_su1321388937eal_sz @ A3 @ top_to1873962757eal_sz )
= top_to1873962757eal_sz ) ).
% Un_UNIV_right
thf(fact_336_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_337_UNIV__witness,axiom,
? [X3: finite824932053eal_sz] : ( member1693951742eal_sz @ X3 @ top_to1873962757eal_sz ) ).
% UNIV_witness
thf(fact_338_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_339_UNIV__eq__I,axiom,
! [A3: set_Fi291318197eal_sz] :
( ! [X3: finite824932053eal_sz] : ( member1693951742eal_sz @ X3 @ A3 )
=> ( top_to1873962757eal_sz = A3 ) ) ).
% UNIV_eq_I
thf(fact_340_UNIV__def,axiom,
( top_to999478531um_c_b
= ( collec1166375742um_c_b
@ ^ [X: sum_su1965225555um_c_b] : $true ) ) ).
% UNIV_def
thf(fact_341_UNIV__def,axiom,
( top_to1873962757eal_sz
= ( collec1002602816eal_sz
@ ^ [X: finite824932053eal_sz] : $true ) ) ).
% UNIV_def
thf(fact_342_continuous__on__cases__le,axiom,
! [S2: set_Fi291318197eal_sz,H: finite824932053eal_sz > real,A: real,F: finite824932053eal_sz > bounde472938360z_real,G: finite824932053eal_sz > bounde472938360z_real] :
( ( topolo885751137z_real
@ ( collec1002602816eal_sz
@ ^ [T: finite824932053eal_sz] :
( ( member1693951742eal_sz @ T @ S2 )
& ( ord_less_eq_real @ ( H @ T ) @ A ) ) )
@ F )
=> ( ( topolo885751137z_real
@ ( collec1002602816eal_sz
@ ^ [T: finite824932053eal_sz] :
( ( member1693951742eal_sz @ T @ S2 )
& ( ord_less_eq_real @ A @ ( H @ T ) ) ) )
@ G )
=> ( ( topolo1348430467z_real @ S2 @ H )
=> ( ! [T3: finite824932053eal_sz] :
( ( member1693951742eal_sz @ T3 @ S2 )
=> ( ( ( H @ T3 )
= A )
=> ( ( F @ T3 )
= ( G @ T3 ) ) ) )
=> ( topolo885751137z_real @ S2
@ ^ [T: finite824932053eal_sz] : ( if_Bou1785791806z_real @ ( ord_less_eq_real @ ( H @ T ) @ A ) @ ( F @ T ) @ ( G @ T ) ) ) ) ) ) ) ).
% continuous_on_cases_le
thf(fact_343_agree__UNIV__eq,axiom,
! [Nu: produc133005365real_b,Omega: produc133005365real_b] :
( ( denota1997846517gree_b @ Nu @ Omega @ top_to301096907um_b_b )
=> ( Nu = Omega ) ) ).
% agree_UNIV_eq
thf(fact_344_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X: a] : ( member_a @ X @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_345_top__empty__eq,axiom,
( top_to646513176l_sz_o
= ( ^ [X: finite824932053eal_sz] : ( member1693951742eal_sz @ X @ top_to1873962757eal_sz ) ) ) ).
% top_empty_eq
thf(fact_346_top__set__def,axiom,
( top_to999478531um_c_b
= ( collec1166375742um_c_b @ top_to1849113498_c_b_o ) ) ).
% top_set_def
thf(fact_347_top__set__def,axiom,
( top_to1873962757eal_sz
= ( collec1002602816eal_sz @ top_to646513176l_sz_o ) ) ).
% top_set_def
thf(fact_348_frechet__continuous,axiom,
! [I: denota610675952t_unit,Theta: trm_sf_sz] :
( ( denota1579475975_sc_sz @ I )
=> ( ( dfree_sf_sz @ Theta )
=> ( topolo885751137z_real @ top_to1873962757eal_sz @ ( freche585307148_sc_sz @ ( freche1784963216_sc_sz @ I ) @ ( freche1046279700_sf_sz @ Theta ) ) ) ) ) ).
% frechet_continuous
thf(fact_349_iso__tuple__UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_350_iso__tuple__UNIV__I,axiom,
! [X2: finite824932053eal_sz] : ( member1693951742eal_sz @ X2 @ top_to1873962757eal_sz ) ).
% iso_tuple_UNIV_I
thf(fact_351_continuous__on__cong,axiom,
! [S2: set_Fi291318197eal_sz,T4: set_Fi291318197eal_sz,F: finite824932053eal_sz > bounde472938360z_real,G: finite824932053eal_sz > bounde472938360z_real] :
( ( S2 = T4 )
=> ( ! [X3: finite824932053eal_sz] :
( ( member1693951742eal_sz @ X3 @ T4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( topolo885751137z_real @ S2 @ F )
= ( topolo885751137z_real @ T4 @ G ) ) ) ) ).
% continuous_on_cong
thf(fact_352_continuous__on__const,axiom,
! [S2: set_Fi291318197eal_sz,C2: bounde472938360z_real] :
( topolo885751137z_real @ S2
@ ^ [X: finite824932053eal_sz] : C2 ) ).
% continuous_on_const
thf(fact_353_continuous__on__subset,axiom,
! [S2: set_Fi291318197eal_sz,F: finite824932053eal_sz > bounde472938360z_real,T4: set_Fi291318197eal_sz] :
( ( topolo885751137z_real @ S2 @ F )
=> ( ( ord_le41263445eal_sz @ T4 @ S2 )
=> ( topolo885751137z_real @ T4 @ F ) ) ) ).
% continuous_on_subset
thf(fact_354_conj__subset__def,axiom,
! [A3: set_Su111624115um_c_b,P: sum_su1965225555um_c_b > $o,Q: sum_su1965225555um_c_b > $o] :
( ( ord_le1404954451um_c_b @ A3
@ ( collec1166375742um_c_b
@ ^ [X: sum_su1965225555um_c_b] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le1404954451um_c_b @ A3 @ ( collec1166375742um_c_b @ P ) )
& ( ord_le1404954451um_c_b @ A3 @ ( collec1166375742um_c_b @ Q ) ) ) ) ).
% conj_subset_def
% Helper facts (3)
thf(help_If_3_1_If_001t__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J_T,axiom,
! [X2: bounde472938360z_real,Y: bounde472938360z_real] :
( ( if_Bou1785791806z_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J_T,axiom,
! [X2: bounde472938360z_real,Y: bounde472938360z_real] :
( ( if_Bou1785791806z_real @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( denota896875430_a_c_b @ i @ j
@ ( collec1166375742um_c_b
@ ^ [Uu: sum_su1965225555um_c_b] :
? [X: a] :
( ( Uu
= ( sum_In811702836um_c_b @ X ) )
& ( member_a @ X @ ( static_SIGT_a_b @ theta_1 ) ) ) ) ) ).
%------------------------------------------------------------------------------