TPTP Problem File: SLH0735^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_03752_157313__16028828_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1612 ( 599 unt; 333 typ; 0 def)
% Number of atoms : 3480 (1268 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10472 ( 183 ~; 8 |; 344 &;8561 @)
% ( 0 <=>;1376 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 56 ( 55 usr)
% Number of type conns : 1922 (1922 >; 0 *; 0 +; 0 <<)
% Number of symbols : 281 ( 278 usr; 39 con; 0-5 aty)
% Number of variables : 3884 ( 351 ^;3389 !; 144 ?;3884 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:12:16.034
%------------------------------------------------------------------------------
% Could-be-implicit typings (55)
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thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
image_list_a_nat: ( list_a > nat ) > set_list_a > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_4532641577586004224_a_nat: ( list_a > sum_sum_a_nat ) > set_list_a > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_6262589752765146990_a_nat: ( nat > list_Sum_sum_a_nat ) > set_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
image_nat_list_a: ( nat > list_a ) > set_nat > set_list_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7293268710728258664_a_nat: ( nat > sum_sum_a_nat ) > set_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001_Eo,type,
image_6095136190293192542_nat_o: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2473878607534554506at_nat: ( sum_sum_a_nat > nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7142520692256960453_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001tf__a,type,
image_6322530041254294468_nat_a: ( sum_sum_a_nat > a ) > set_Sum_sum_a_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_7897140031816760844_a_nat: ( a > list_Sum_sum_a_nat ) > set_a > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7873763678140191238_a_nat: ( a > sum_sum_a_nat ) > set_a > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inl_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Nat__Onat,type,
sum_Inl_a_nat: a > sum_sum_a_nat ).
thf(sy_c_Sum__Type_Osum_Oisl_001tf__a_001t__Nat__Onat,type,
sum_isl_a_nat: sum_sum_a_nat > $o ).
thf(sy_c_Sum__Type_Osum_Oprojl_001tf__a_001t__Nat__Onat,type,
sum_projl_a_nat: sum_sum_a_nat > a ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member8690443509505302927_a_nat: ( nat > sum_sum_a_nat ) > set_na3699693778330250182_a_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
member5553968465346197646_a_nat: set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member8098812455498974984_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v__092_060phi_062,type,
phi: fo_fmla_a_b ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_vs____,type,
vs: list_Sum_sum_a_nat ).
% Relevant facts (1277)
thf(fact_0_local_Owf,axiom,
wf_fo_intp_a_b @ phi @ i ).
% local.wf
thf(fact_1_list_Oset__map,axiom,
! [F: nat > sum_sum_a_nat,V: list_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ V ) )
= ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_2_list_Oset__map,axiom,
! [F: a > sum_sum_a_nat,V: list_a] :
( ( set_Sum_sum_a_nat2 @ ( map_a_Sum_sum_a_nat @ F @ V ) )
= ( image_7873763678140191238_a_nat @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_3_list_Oset__map,axiom,
! [F: list_a > list_Sum_sum_a_nat,V: list_list_a] :
( ( set_li2392974972034027290_a_nat @ ( map_li6996171348340886213_a_nat @ F @ V ) )
= ( image_674313660629153798_a_nat @ F @ ( set_list_a2 @ V ) ) ) ).
% list.set_map
thf(fact_4_list_Oset__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,V: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ V ) )
= ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_5_list_Oset__map,axiom,
! [F: a > $o,V: list_a] :
( ( set_o2 @ ( map_a_o @ F @ V ) )
= ( image_a_o @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_6_list_Oset__map,axiom,
! [F: nat > $o,V: list_nat] :
( ( set_o2 @ ( map_nat_o @ F @ V ) )
= ( image_nat_o @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_7_list_Oset__map,axiom,
! [F: a > a,V: list_a] :
( ( set_a2 @ ( map_a_a @ F @ V ) )
= ( image_a_a @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_8_list_Oset__map,axiom,
! [F: nat > a,V: list_nat] :
( ( set_a2 @ ( map_nat_a @ F @ V ) )
= ( image_nat_a @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_9_list_Oset__map,axiom,
! [F: a > nat,V: list_a] :
( ( set_nat2 @ ( map_a_nat @ F @ V ) )
= ( image_a_nat @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_10_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_11_map__eq__conv,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Xs ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_12_map__eq__conv,axiom,
! [F: list_a > list_Sum_sum_a_nat,Xs: list_list_a,G: list_a > list_Sum_sum_a_nat] :
( ( ( map_li6996171348340886213_a_nat @ F @ Xs )
= ( map_li6996171348340886213_a_nat @ G @ Xs ) )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_13_map__eq__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_14_map__eq__conv,axiom,
! [F: a > sum_sum_a_nat,Xs: list_a,G: a > sum_sum_a_nat] :
( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
= ( map_a_Sum_sum_a_nat @ G @ Xs ) )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_15_image__set,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_16_image__set,axiom,
! [F: a > sum_sum_a_nat,Xs: list_a] :
( ( image_7873763678140191238_a_nat @ F @ ( set_a2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_a_Sum_sum_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_17_image__set,axiom,
! [F: list_a > list_Sum_sum_a_nat,Xs: list_list_a] :
( ( image_674313660629153798_a_nat @ F @ ( set_list_a2 @ Xs ) )
= ( set_li2392974972034027290_a_nat @ ( map_li6996171348340886213_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_18_image__set,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_19_image__set,axiom,
! [F: a > $o,Xs: list_a] :
( ( image_a_o @ F @ ( set_a2 @ Xs ) )
= ( set_o2 @ ( map_a_o @ F @ Xs ) ) ) ).
% image_set
thf(fact_20_image__set,axiom,
! [F: a > a,Xs: list_a] :
( ( image_a_a @ F @ ( set_a2 @ Xs ) )
= ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_21_image__set,axiom,
! [F: a > nat,Xs: list_a] :
( ( image_a_nat @ F @ ( set_a2 @ Xs ) )
= ( set_nat2 @ ( map_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_22_image__set,axiom,
! [F: nat > $o,Xs: list_nat] :
( ( image_nat_o @ F @ ( set_nat2 @ Xs ) )
= ( set_o2 @ ( map_nat_o @ F @ Xs ) ) ) ).
% image_set
thf(fact_23_image__set,axiom,
! [F: nat > a,Xs: list_nat] :
( ( image_nat_a @ F @ ( set_nat2 @ Xs ) )
= ( set_a2 @ ( map_nat_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_24_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_25_range__subsetD,axiom,
! [F: a > $o,B: set_o,I: a] :
( ( ord_less_eq_set_o @ ( image_a_o @ F @ top_top_set_a ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_26_range__subsetD,axiom,
! [F: nat > $o,B: set_o,I: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ top_top_set_nat ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_27_range__subsetD,axiom,
! [F: list_a > $o,B: set_o,I: list_a] :
( ( ord_less_eq_set_o @ ( image_list_a_o @ F @ top_top_set_list_a ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_28_range__subsetD,axiom,
! [F: sum_sum_a_nat > $o,B: set_o,I: sum_sum_a_nat] :
( ( ord_less_eq_set_o @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) @ B )
=> ( member_o @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_29_range__subsetD,axiom,
! [F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat,I: a] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) @ B )
=> ( member_Sum_sum_a_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_30_range__subsetD,axiom,
! [F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,I: nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) @ B )
=> ( member_Sum_sum_a_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_31_range__subsetD,axiom,
! [F: a > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,I: a] :
( ( ord_le1147066620699065093_a_nat @ ( image_7897140031816760844_a_nat @ F @ top_top_set_a ) @ B )
=> ( member408289922725080238_a_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_32_range__subsetD,axiom,
! [F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,I: nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ top_top_set_nat ) @ B )
=> ( member408289922725080238_a_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_33_range__subsetD,axiom,
! [F: list_a > sum_sum_a_nat,B: set_Sum_sum_a_nat,I: list_a] :
( ( ord_le1325389633284124927_a_nat @ ( image_4532641577586004224_a_nat @ F @ top_top_set_list_a ) @ B )
=> ( member_Sum_sum_a_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_34_range__subsetD,axiom,
! [F: list_a > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,I: list_a] :
( ( ord_le1147066620699065093_a_nat @ ( image_674313660629153798_a_nat @ F @ top_top_set_list_a ) @ B )
=> ( member408289922725080238_a_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_35__092_060open_062vs_A_092_060in_062_Aproj__fmla_A_092_060phi_062_A_123_092_060sigma_062_O_Aesat_A_092_060phi_062_AI_A_092_060sigma_062_AUNIV_125_A_092_060inter_062_Arange_A_Imap_AInl_J_092_060close_062,axiom,
( member408289922725080238_a_nat @ vs
@ ( inf_in3249246906714053971_a_nat
@ ( proj_f6037546931041448027_a_nat @ phi
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ phi @ i @ Sigma @ top_to795618464972521135_a_nat ) ) )
@ ( image_674313660629153798_a_nat @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat ) @ top_top_set_list_a ) ) ) ).
% \<open>vs \<in> proj_fmla \<phi> {\<sigma>. esat \<phi> I \<sigma> UNIV} \<inter> range (map Inl)\<close>
thf(fact_36_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_Inl_a_nat @ X1 )
= ( sum_Inl_a_nat @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_37_old_Osum_Oinject_I1_J,axiom,
! [A: a,A2: a] :
( ( ( sum_Inl_a_nat @ A )
= ( sum_Inl_a_nat @ A2 ) )
= ( A = A2 ) ) ).
% old.sum.inject(1)
thf(fact_38_subsetI,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A3 )
=> ( member408289922725080238_a_nat @ X2 @ B ) )
=> ( ord_le1147066620699065093_a_nat @ A3 @ B ) ) ).
% subsetI
thf(fact_39_subsetI,axiom,
! [A3: set_o,B: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ X2 @ B ) )
=> ( ord_less_eq_set_o @ A3 @ B ) ) ).
% subsetI
thf(fact_40_subsetI,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A3 )
=> ( member_Sum_sum_a_nat @ X2 @ B ) )
=> ( ord_le1325389633284124927_a_nat @ A3 @ B ) ) ).
% subsetI
thf(fact_41_subset__antisym,axiom,
! [A3: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ( ord_less_eq_set_o @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_42_subset__antisym,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ( ( ord_le1147066620699065093_a_nat @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_43_subset__antisym,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_44_image__eqI,axiom,
! [B2: $o,F: nat > $o,X3: nat,A3: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A3 )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_45_image__eqI,axiom,
! [B2: $o,F: a > $o,X3: a,A3: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A3 )
=> ( member_o @ B2 @ ( image_a_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_46_image__eqI,axiom,
! [B2: $o,F: $o > $o,X3: $o,A3: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A3 )
=> ( member_o @ B2 @ ( image_o_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_47_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,X3: a,A3: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A3 )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_48_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,X3: nat,A3: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A3 )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_49_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: $o > sum_sum_a_nat,X3: $o,A3: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A3 )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_4139480514073730540_a_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_50_image__eqI,axiom,
! [B2: $o,F: sum_sum_a_nat > $o,X3: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_Sum_sum_a_nat @ X3 @ A3 )
=> ( member_o @ B2 @ ( image_6095136190293192542_nat_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_51_image__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,X3: nat,A3: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A3 )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_52_image__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: a > list_Sum_sum_a_nat,X3: a,A3: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A3 )
=> ( member408289922725080238_a_nat @ B2 @ ( image_7897140031816760844_a_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_53_image__eqI,axiom,
! [B2: $o,F: list_Sum_sum_a_nat > $o,X3: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A3 )
=> ( member_o @ B2 @ ( image_3984946558445957976_nat_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_54_UNIV__I,axiom,
! [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ top_to6433055325616222389_a_nat ) ).
% UNIV_I
thf(fact_55_UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_I
thf(fact_56_UNIV__I,axiom,
! [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ top_to795618464972521135_a_nat ) ).
% UNIV_I
thf(fact_57_UNIV__I,axiom,
! [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_I
thf(fact_58_UNIV__I,axiom,
! [X3: list_a] : ( member_list_a @ X3 @ top_top_set_list_a ) ).
% UNIV_I
thf(fact_59_UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_60_iso__tuple__UNIV__I,axiom,
! [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ top_to6433055325616222389_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_61_iso__tuple__UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_62_iso__tuple__UNIV__I,axiom,
! [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ top_to795618464972521135_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_63_iso__tuple__UNIV__I,axiom,
! [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_64_iso__tuple__UNIV__I,axiom,
! [X3: list_a] : ( member_list_a @ X3 @ top_top_set_list_a ) ).
% iso_tuple_UNIV_I
thf(fact_65_iso__tuple__UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_66_IntI,axiom,
! [C: $o,A3: set_o,B: set_o] :
( ( member_o @ C @ A3 )
=> ( ( member_o @ C @ B )
=> ( member_o @ C @ ( inf_inf_set_o @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_67_IntI,axiom,
! [C: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A3 )
=> ( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_68_IntI,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A3 )
=> ( ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_69_Int__iff,axiom,
! [C: $o,A3: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B ) )
= ( ( member_o @ C @ A3 )
& ( member_o @ C @ B ) ) ) ).
% Int_iff
thf(fact_70_Int__iff,axiom,
! [C: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A3 )
& ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_71_Int__iff,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) )
= ( ( member408289922725080238_a_nat @ C @ A3 )
& ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_72_map__ident,axiom,
( ( map_Su2790769393171190532_a_nat
@ ^ [X: sum_sum_a_nat] : X )
= ( ^ [Xs2: list_Sum_sum_a_nat] : Xs2 ) ) ).
% map_ident
thf(fact_73_Int__UNIV,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A3 @ B )
= top_to6433055325616222389_a_nat )
= ( ( A3 = top_to6433055325616222389_a_nat )
& ( B = top_to6433055325616222389_a_nat ) ) ) ).
% Int_UNIV
thf(fact_74_Int__UNIV,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A3 @ B )
= top_to795618464972521135_a_nat )
= ( ( A3 = top_to795618464972521135_a_nat )
& ( B = top_to795618464972521135_a_nat ) ) ) ).
% Int_UNIV
thf(fact_75_Int__UNIV,axiom,
! [A3: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A3 @ B )
= top_top_set_a )
= ( ( A3 = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_76_Int__UNIV,axiom,
! [A3: set_list_a,B: set_list_a] :
( ( ( inf_inf_set_list_a @ A3 @ B )
= top_top_set_list_a )
= ( ( A3 = top_top_set_list_a )
& ( B = top_top_set_list_a ) ) ) ).
% Int_UNIV
thf(fact_77_Int__UNIV,axiom,
! [A3: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B )
= top_top_set_nat )
= ( ( A3 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_78_Int__subset__iff,axiom,
! [C2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) )
= ( ( ord_le1325389633284124927_a_nat @ C2 @ A3 )
& ( ord_le1325389633284124927_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_79_Int__subset__iff,axiom,
! [C2: set_o,A3: set_o,B: set_o] :
( ( ord_less_eq_set_o @ C2 @ ( inf_inf_set_o @ A3 @ B ) )
= ( ( ord_less_eq_set_o @ C2 @ A3 )
& ( ord_less_eq_set_o @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_80_Int__subset__iff,axiom,
! [C2: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) )
= ( ( ord_le1147066620699065093_a_nat @ C2 @ A3 )
& ( ord_le1147066620699065093_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_81_list_Omap__ident,axiom,
! [T: list_Sum_sum_a_nat] :
( ( map_Su2790769393171190532_a_nat
@ ^ [X: sum_sum_a_nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_82_IntE,axiom,
! [C: $o,A3: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B ) )
=> ~ ( ( member_o @ C @ A3 )
=> ~ ( member_o @ C @ B ) ) ) ).
% IntE
thf(fact_83_IntE,axiom,
! [C: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) )
=> ~ ( ( member_Sum_sum_a_nat @ C @ A3 )
=> ~ ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% IntE
thf(fact_84_IntE,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) )
=> ~ ( ( member408289922725080238_a_nat @ C @ A3 )
=> ~ ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% IntE
thf(fact_85_IntD1,axiom,
! [C: $o,A3: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B ) )
=> ( member_o @ C @ A3 ) ) ).
% IntD1
thf(fact_86_IntD1,axiom,
! [C: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) )
=> ( member_Sum_sum_a_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_87_IntD1,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) )
=> ( member408289922725080238_a_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_88_IntD2,axiom,
! [C: $o,A3: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B ) )
=> ( member_o @ C @ B ) ) ).
% IntD2
thf(fact_89_IntD2,axiom,
! [C: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ).
% IntD2
thf(fact_90_IntD2,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) )
=> ( member408289922725080238_a_nat @ C @ B ) ) ).
% IntD2
thf(fact_91_imageE,axiom,
! [B2: $o,F: nat > $o,A3: set_nat] :
( ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) )
=> ~ ! [X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_92_imageE,axiom,
! [B2: $o,F: a > $o,A3: set_a] :
( ( member_o @ B2 @ ( image_a_o @ F @ A3 ) )
=> ~ ! [X2: a] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_93_imageE,axiom,
! [B2: $o,F: $o > $o,A3: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A3 ) )
=> ~ ! [X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_o @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_94_imageE,axiom,
! [B2: $o,F: sum_sum_a_nat > $o,A3: set_Sum_sum_a_nat] :
( ( member_o @ B2 @ ( image_6095136190293192542_nat_o @ F @ A3 ) )
=> ~ ! [X2: sum_sum_a_nat] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_Sum_sum_a_nat @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_95_imageE,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,A3: set_a] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
=> ~ ! [X2: a] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_96_imageE,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,A3: set_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
=> ~ ! [X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_97_imageE,axiom,
! [B2: sum_sum_a_nat,F: $o > sum_sum_a_nat,A3: set_o] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_4139480514073730540_a_nat @ F @ A3 ) )
=> ~ ! [X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_o @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_98_imageE,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,A3: set_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
=> ~ ! [X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_99_imageE,axiom,
! [B2: list_Sum_sum_a_nat,F: a > list_Sum_sum_a_nat,A3: set_a] :
( ( member408289922725080238_a_nat @ B2 @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
=> ~ ! [X2: a] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_100_imageE,axiom,
! [B2: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat,A3: set_o] :
( ( member408289922725080238_a_nat @ B2 @ ( image_733135354807394034_a_nat @ F @ A3 ) )
=> ~ ! [X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_o @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_101_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B3: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
& ( member_o @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_102_Int__def,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A4 )
& ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_103_Int__def,axiom,
( inf_in8399021836546144180_a_nat
= ( ^ [A4: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A4 )
& ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_104_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( member_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_105_Int__def,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A4 )
& ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_106_UNIV__def,axiom,
( top_to9106040778512017686_a_nat
= ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : $true ) ) ).
% UNIV_def
thf(fact_107_UNIV__def,axiom,
( top_to795618464972521135_a_nat
= ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] : $true ) ) ).
% UNIV_def
thf(fact_108_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X: a] : $true ) ) ).
% UNIV_def
thf(fact_109_UNIV__def,axiom,
( top_top_set_list_a
= ( collect_list_a
@ ^ [X: list_a] : $true ) ) ).
% UNIV_def
thf(fact_110_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_111_Int__assoc,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) @ C2 )
= ( inf_in3249246906714053971_a_nat @ A3 @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_112_Int__absorb,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_113_Int__Collect,axiom,
! [X3: $o,A3: set_o,P: $o > $o] :
( ( member_o @ X3 @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
= ( ( member_o @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_114_Int__Collect,axiom,
! [X3: sum_sum_a_nat,A3: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ A3 @ ( collec7073057861543223018_a_nat @ P ) ) )
= ( ( member_Sum_sum_a_nat @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_115_Int__Collect,axiom,
! [X3: nat > sum_sum_a_nat,A3: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X3 @ ( inf_in8399021836546144180_a_nat @ A3 @ ( collec5629555741568564177_a_nat @ P ) ) )
= ( ( member8690443509505302927_a_nat @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_116_Int__Collect,axiom,
! [X3: nat,A3: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_117_Int__Collect,axiom,
! [X3: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ A3 @ ( collec7555443234367654128_a_nat @ P ) ) )
= ( ( member408289922725080238_a_nat @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_118_Int__commute,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_119_image__image,axiom,
! [F: $o > $o,G: nat > $o,A3: set_nat] :
( ( image_o_o @ F @ ( image_nat_o @ G @ A3 ) )
= ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_120_image__image,axiom,
! [F: $o > $o,G: a > $o,A3: set_a] :
( ( image_o_o @ F @ ( image_a_o @ G @ A3 ) )
= ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_121_image__image,axiom,
! [F: nat > $o,G: nat > nat,A3: set_nat] :
( ( image_nat_o @ F @ ( image_nat_nat @ G @ A3 ) )
= ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_122_image__image,axiom,
! [F: nat > $o,G: a > nat,A3: set_a] :
( ( image_nat_o @ F @ ( image_a_nat @ G @ A3 ) )
= ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_123_image__image,axiom,
! [F: a > $o,G: nat > a,A3: set_nat] :
( ( image_a_o @ F @ ( image_nat_a @ G @ A3 ) )
= ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_124_image__image,axiom,
! [F: a > $o,G: a > a,A3: set_a] :
( ( image_a_o @ F @ ( image_a_a @ G @ A3 ) )
= ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_125_image__image,axiom,
! [F: sum_sum_a_nat > $o,G: a > sum_sum_a_nat,A3: set_a] :
( ( image_6095136190293192542_nat_o @ F @ ( image_7873763678140191238_a_nat @ G @ A3 ) )
= ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_126_image__image,axiom,
! [F: sum_sum_a_nat > $o,G: nat > sum_sum_a_nat,A3: set_nat] :
( ( image_6095136190293192542_nat_o @ F @ ( image_7293268710728258664_a_nat @ G @ A3 ) )
= ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_127_image__image,axiom,
! [F: $o > sum_sum_a_nat,G: nat > $o,A3: set_nat] :
( ( image_4139480514073730540_a_nat @ F @ ( image_nat_o @ G @ A3 ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_128_image__image,axiom,
! [F: $o > sum_sum_a_nat,G: a > $o,A3: set_a] :
( ( image_4139480514073730540_a_nat @ F @ ( image_a_o @ G @ A3 ) )
= ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_129_Collect__subset,axiom,
! [A3: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_130_Collect__subset,axiom,
! [A3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_131_Collect__subset,axiom,
! [A3: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_132_Collect__subset,axiom,
! [A3: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_133_Collect__subset,axiom,
! [A3: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_134_Compr__image__eq,axiom,
! [F: a > $o,A3: set_a,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_a_o @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_a_o @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_135_Compr__image__eq,axiom,
! [F: $o > $o,A3: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_136_Compr__image__eq,axiom,
! [F: nat > $o,A3: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_137_Compr__image__eq,axiom,
! [F: $o > nat,A3: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_o_nat @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_138_Compr__image__eq,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_139_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > $o,A3: set_Sum_sum_a_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_6095136190293192542_nat_o @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_6095136190293192542_nat_o @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_140_Compr__image__eq,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_7873763678140191238_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_141_Compr__image__eq,axiom,
! [F: $o > sum_sum_a_nat,A3: set_o,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_4139480514073730540_a_nat @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_4139480514073730540_a_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_142_Compr__image__eq,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_7293268710728258664_a_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_143_Compr__image__eq,axiom,
! [F: sum_sum_a_nat > nat,A3: set_Sum_sum_a_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_2473878607534554506at_nat @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_2473878607534554506at_nat @ F
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_144_Collect__conj__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_in8399021836546144180_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_145_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_146_Collect__conj__eq,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_in3249246906714053971_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_147_Int__left__absorb,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A3 @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) )
= ( inf_in3249246906714053971_a_nat @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_148_Int__left__commute,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A3 @ ( inf_in3249246906714053971_a_nat @ B @ C2 ) )
= ( inf_in3249246906714053971_a_nat @ B @ ( inf_in3249246906714053971_a_nat @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_149_proj__fmla__map,axiom,
( proj_f6037546931041448027_a_nat
= ( ^ [Phi: fo_fmla_a_b] :
( image_6721470456781115300_a_nat
@ ^ [Tau: nat > sum_sum_a_nat] : ( map_na823391071729141993_a_nat @ Tau @ ( fv_fo_fmla_list_a_b @ Phi ) ) ) ) ) ).
% proj_fmla_map
thf(fact_150_range__composition,axiom,
! [F: $o > $o,G: a > $o] :
( ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_o_o @ F @ ( image_a_o @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_151_range__composition,axiom,
! [F: nat > $o,G: a > nat] :
( ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_nat_o @ F @ ( image_a_nat @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_152_range__composition,axiom,
! [F: a > $o,G: a > a] :
( ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_a_o @ F @ ( image_a_a @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_153_range__composition,axiom,
! [F: $o > $o,G: nat > $o] :
( ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_o_o @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_154_range__composition,axiom,
! [F: nat > $o,G: nat > nat] :
( ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_o @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_155_range__composition,axiom,
! [F: a > $o,G: nat > a] :
( ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_a_o @ F @ ( image_nat_a @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_156_range__composition,axiom,
! [F: nat > $o,G: list_a > nat] :
( ( image_list_a_o
@ ^ [X: list_a] : ( F @ ( G @ X ) )
@ top_top_set_list_a )
= ( image_nat_o @ F @ ( image_list_a_nat @ G @ top_top_set_list_a ) ) ) ).
% range_composition
thf(fact_157_range__composition,axiom,
! [F: a > $o,G: list_a > a] :
( ( image_list_a_o
@ ^ [X: list_a] : ( F @ ( G @ X ) )
@ top_top_set_list_a )
= ( image_a_o @ F @ ( image_list_a_a @ G @ top_top_set_list_a ) ) ) ).
% range_composition
thf(fact_158_range__composition,axiom,
! [F: nat > $o,G: sum_sum_a_nat > nat] :
( ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ top_to795618464972521135_a_nat )
= ( image_nat_o @ F @ ( image_2473878607534554506at_nat @ G @ top_to795618464972521135_a_nat ) ) ) ).
% range_composition
thf(fact_159_range__composition,axiom,
! [F: a > $o,G: sum_sum_a_nat > a] :
( ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ top_to795618464972521135_a_nat )
= ( image_a_o @ F @ ( image_6322530041254294468_nat_a @ G @ top_to795618464972521135_a_nat ) ) ) ).
% range_composition
thf(fact_160_rangeE,axiom,
! [B2: $o,F: a > $o] :
( ( member_o @ B2 @ ( image_a_o @ F @ top_top_set_a ) )
=> ~ ! [X2: a] :
( B2
= ( ~ ( F @ X2 ) ) ) ) ).
% rangeE
thf(fact_161_rangeE,axiom,
! [B2: $o,F: nat > $o] :
( ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B2
= ( ~ ( F @ X2 ) ) ) ) ).
% rangeE
thf(fact_162_rangeE,axiom,
! [B2: $o,F: list_a > $o] :
( ( member_o @ B2 @ ( image_list_a_o @ F @ top_top_set_list_a ) )
=> ~ ! [X2: list_a] :
( B2
= ( ~ ( F @ X2 ) ) ) ) ).
% rangeE
thf(fact_163_rangeE,axiom,
! [B2: $o,F: sum_sum_a_nat > $o] :
( ( member_o @ B2 @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) )
=> ~ ! [X2: sum_sum_a_nat] :
( B2
= ( ~ ( F @ X2 ) ) ) ) ).
% rangeE
thf(fact_164_rangeE,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) )
=> ~ ! [X2: a] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_165_rangeE,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_166_rangeE,axiom,
! [B2: list_Sum_sum_a_nat,F: a > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( image_7897140031816760844_a_nat @ F @ top_top_set_a ) )
=> ~ ! [X2: a] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_167_rangeE,axiom,
! [B2: sum_sum_a_nat,F: list_a > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_4532641577586004224_a_nat @ F @ top_top_set_list_a ) )
=> ~ ! [X2: list_a] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_168_rangeE,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_169_rangeE,axiom,
! [B2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) )
=> ~ ! [X2: sum_sum_a_nat] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_170_Int__UNIV__right,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A3 @ top_to6433055325616222389_a_nat )
= A3 ) ).
% Int_UNIV_right
thf(fact_171_Int__UNIV__right,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A3 @ top_to795618464972521135_a_nat )
= A3 ) ).
% Int_UNIV_right
thf(fact_172_Int__UNIV__right,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ top_top_set_a )
= A3 ) ).
% Int_UNIV_right
thf(fact_173_Int__UNIV__right,axiom,
! [A3: set_list_a] :
( ( inf_inf_set_list_a @ A3 @ top_top_set_list_a )
= A3 ) ).
% Int_UNIV_right
thf(fact_174_Int__UNIV__right,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
= A3 ) ).
% Int_UNIV_right
thf(fact_175_Int__UNIV__left,axiom,
! [B: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ top_to6433055325616222389_a_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_176_Int__UNIV__left,axiom,
! [B: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_177_Int__UNIV__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_178_Int__UNIV__left,axiom,
! [B: set_list_a] :
( ( inf_inf_set_list_a @ top_top_set_list_a @ B )
= B ) ).
% Int_UNIV_left
thf(fact_179_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_180_Int__Collect__mono,axiom,
! [A3: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ A3 @ B )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le8108555184339247974_a_nat @ ( inf_in8399021836546144180_a_nat @ A3 @ ( collec5629555741568564177_a_nat @ P ) ) @ ( inf_in8399021836546144180_a_nat @ B @ ( collec5629555741568564177_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_181_Int__Collect__mono,axiom,
! [A3: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_182_Int__Collect__mono,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ ( collec7073057861543223018_a_nat @ P ) ) @ ( inf_in7084830621192376909_a_nat @ B @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_183_Int__Collect__mono,axiom,
! [A3: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_184_Int__Collect__mono,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A3 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A3 @ ( collec7555443234367654128_a_nat @ P ) ) @ ( inf_in3249246906714053971_a_nat @ B @ ( collec7555443234367654128_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_185_Int__greatest,axiom,
! [C2: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ C2 @ B )
=> ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_186_Int__greatest,axiom,
! [C2: set_o,A3: set_o,B: set_o] :
( ( ord_less_eq_set_o @ C2 @ A3 )
=> ( ( ord_less_eq_set_o @ C2 @ B )
=> ( ord_less_eq_set_o @ C2 @ ( inf_inf_set_o @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_187_Int__greatest,axiom,
! [C2: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C2 @ A3 )
=> ( ( ord_le1147066620699065093_a_nat @ C2 @ B )
=> ( ord_le1147066620699065093_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_188_Int__absorb2,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( ( inf_in7084830621192376909_a_nat @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_189_Int__absorb2,axiom,
! [A3: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ( inf_inf_set_o @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_190_Int__absorb2,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ( ( inf_in3249246906714053971_a_nat @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_191_Int__absorb1,axiom,
! [B: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A3 )
=> ( ( inf_in7084830621192376909_a_nat @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_192_Int__absorb1,axiom,
! [B: set_o,A3: set_o] :
( ( ord_less_eq_set_o @ B @ A3 )
=> ( ( inf_inf_set_o @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_193_Int__absorb1,axiom,
! [B: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ A3 )
=> ( ( inf_in3249246906714053971_a_nat @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_194_mem__Collect__eq,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_195_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_196_mem__Collect__eq,axiom,
! [A: sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_197_mem__Collect__eq,axiom,
! [A: nat > sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ A @ ( collec5629555741568564177_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_198_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_199_Collect__mem__eq,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_200_Collect__mem__eq,axiom,
! [A3: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_201_Collect__mem__eq,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
! [A3: set_na3699693778330250182_a_nat] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_204_Collect__cong,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ! [X2: nat > sum_sum_a_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec5629555741568564177_a_nat @ P )
= ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_205_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_206_Int__lower2,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_207_Int__lower2,axiom,
! [A3: set_o,B: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_208_Int__lower2,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_209_Int__lower1,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_210_Int__lower1,axiom,
! [A3: set_o,B: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_211_Int__lower1,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_212_Int__mono,axiom,
! [A3: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A3 @ B ) @ ( inf_in7084830621192376909_a_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_213_Int__mono,axiom,
! [A3: set_o,C2: set_o,B: set_o,D: set_o] :
( ( ord_less_eq_set_o @ A3 @ C2 )
=> ( ( ord_less_eq_set_o @ B @ D )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ B ) @ ( inf_inf_set_o @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_214_Int__mono,axiom,
! [A3: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ D )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) @ ( inf_in3249246906714053971_a_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_215_UNIV__witness,axiom,
? [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ top_to6433055325616222389_a_nat ) ).
% UNIV_witness
thf(fact_216_UNIV__witness,axiom,
? [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_217_UNIV__witness,axiom,
? [X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X2 @ top_to795618464972521135_a_nat ) ).
% UNIV_witness
thf(fact_218_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_219_UNIV__witness,axiom,
? [X2: list_a] : ( member_list_a @ X2 @ top_top_set_list_a ) ).
% UNIV_witness
thf(fact_220_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_221_UNIV__eq__I,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ! [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ A3 )
=> ( top_to6433055325616222389_a_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_222_UNIV__eq__I,axiom,
! [A3: set_o] :
( ! [X2: $o] : ( member_o @ X2 @ A3 )
=> ( top_top_set_o = A3 ) ) ).
% UNIV_eq_I
thf(fact_223_UNIV__eq__I,axiom,
! [A3: set_Sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X2 @ A3 )
=> ( top_to795618464972521135_a_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_224_UNIV__eq__I,axiom,
! [A3: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A3 )
=> ( top_top_set_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_225_UNIV__eq__I,axiom,
! [A3: set_list_a] :
( ! [X2: list_a] : ( member_list_a @ X2 @ A3 )
=> ( top_top_set_list_a = A3 ) ) ).
% UNIV_eq_I
thf(fact_226_UNIV__eq__I,axiom,
! [A3: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A3 )
=> ( top_top_set_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_227_rev__image__eqI,axiom,
! [X3: nat,A3: set_nat,B2: $o,F: nat > $o] :
( ( member_nat @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_228_rev__image__eqI,axiom,
! [X3: a,A3: set_a,B2: $o,F: a > $o] :
( ( member_a @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_a_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_229_rev__image__eqI,axiom,
! [X3: $o,A3: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_230_rev__image__eqI,axiom,
! [X3: a,A3: set_a,B2: sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( member_a @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_231_rev__image__eqI,axiom,
! [X3: nat,A3: set_nat,B2: sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_232_rev__image__eqI,axiom,
! [X3: $o,A3: set_o,B2: sum_sum_a_nat,F: $o > sum_sum_a_nat] :
( ( member_o @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_4139480514073730540_a_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_233_rev__image__eqI,axiom,
! [X3: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B2: $o,F: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_6095136190293192542_nat_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_234_rev__image__eqI,axiom,
! [X3: nat,A3: set_nat,B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_235_rev__image__eqI,axiom,
! [X3: a,A3: set_a,B2: list_Sum_sum_a_nat,F: a > list_Sum_sum_a_nat] :
( ( member_a @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_7897140031816760844_a_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_236_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B2: $o,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X3 @ A3 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_3984946558445957976_nat_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_237_ball__imageD,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,P: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_238_ball__imageD,axiom,
! [F: list_a > list_Sum_sum_a_nat,A3: set_list_a,P: list_Sum_sum_a_nat > $o] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( image_674313660629153798_a_nat @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_239_ball__imageD,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,P: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_240_ball__imageD,axiom,
! [F: nat > list_Sum_sum_a_nat,A3: set_nat,P: list_Sum_sum_a_nat > $o] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_241_ball__imageD,axiom,
! [F: nat > $o,A3: set_nat,P: $o > $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_242_ball__imageD,axiom,
! [F: a > list_Sum_sum_a_nat,A3: set_a,P: list_Sum_sum_a_nat > $o] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_243_ball__imageD,axiom,
! [F: a > $o,A3: set_a,P: $o > $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ ( image_a_o @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_244_image__cong,axiom,
! [M: set_a,N: set_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7873763678140191238_a_nat @ F @ M )
= ( image_7873763678140191238_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_245_image__cong,axiom,
! [M: set_list_a,N: set_list_a,F: list_a > list_Sum_sum_a_nat,G: list_a > list_Sum_sum_a_nat] :
( ( M = N )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_674313660629153798_a_nat @ F @ M )
= ( image_674313660629153798_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_246_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7293268710728258664_a_nat @ F @ M )
= ( image_7293268710728258664_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_247_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > list_Sum_sum_a_nat,G: nat > list_Sum_sum_a_nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_6262589752765146990_a_nat @ F @ M )
= ( image_6262589752765146990_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_248_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > $o,G: nat > $o] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_o @ F @ M )
= ( image_nat_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_249_image__cong,axiom,
! [M: set_a,N: set_a,F: a > list_Sum_sum_a_nat,G: a > list_Sum_sum_a_nat] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7897140031816760844_a_nat @ F @ M )
= ( image_7897140031816760844_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_250_image__cong,axiom,
! [M: set_a,N: set_a,F: a > $o,G: a > $o] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_o @ F @ M )
= ( image_a_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_251_bex__imageD,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,P: sum_sum_a_nat > $o] :
( ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_252_bex__imageD,axiom,
! [F: list_a > list_Sum_sum_a_nat,A3: set_list_a,P: list_Sum_sum_a_nat > $o] :
( ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_674313660629153798_a_nat @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_253_bex__imageD,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,P: sum_sum_a_nat > $o] :
( ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_254_bex__imageD,axiom,
! [F: nat > list_Sum_sum_a_nat,A3: set_nat,P: list_Sum_sum_a_nat > $o] :
( ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_255_bex__imageD,axiom,
! [F: nat > $o,A3: set_nat,P: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_nat_o @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_256_bex__imageD,axiom,
! [F: a > list_Sum_sum_a_nat,A3: set_a,P: list_Sum_sum_a_nat > $o] :
( ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_257_bex__imageD,axiom,
! [F: a > $o,A3: set_a,P: $o > $o] :
( ? [X4: $o] :
( ( member_o @ X4 @ ( image_a_o @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_258_image__iff,axiom,
! [Z: list_Sum_sum_a_nat,F: list_a > list_Sum_sum_a_nat,A3: set_list_a] :
( ( member408289922725080238_a_nat @ Z @ ( image_674313660629153798_a_nat @ F @ A3 ) )
= ( ? [X: list_a] :
( ( member_list_a @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_259_image__iff,axiom,
! [Z: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,A3: set_nat] :
( ( member408289922725080238_a_nat @ Z @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_260_image__iff,axiom,
! [Z: list_Sum_sum_a_nat,F: a > list_Sum_sum_a_nat,A3: set_a] :
( ( member408289922725080238_a_nat @ Z @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
= ( ? [X: a] :
( ( member_a @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_261_image__iff,axiom,
! [Z: $o,F: nat > $o,A3: set_nat] :
( ( member_o @ Z @ ( image_nat_o @ F @ A3 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_262_image__iff,axiom,
! [Z: $o,F: a > $o,A3: set_a] :
( ( member_o @ Z @ ( image_a_o @ F @ A3 ) )
= ( ? [X: a] :
( ( member_a @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_263_image__iff,axiom,
! [Z: sum_sum_a_nat,F: a > sum_sum_a_nat,A3: set_a] :
( ( member_Sum_sum_a_nat @ Z @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
= ( ? [X: a] :
( ( member_a @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_264_image__iff,axiom,
! [Z: sum_sum_a_nat,F: nat > sum_sum_a_nat,A3: set_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_265_imageI,axiom,
! [X3: nat,A3: set_nat,F: nat > $o] :
( ( member_nat @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_266_imageI,axiom,
! [X3: a,A3: set_a,F: a > $o] :
( ( member_a @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ ( image_a_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_267_imageI,axiom,
! [X3: $o,A3: set_o,F: $o > $o] :
( ( member_o @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_268_imageI,axiom,
! [X3: a,A3: set_a,F: a > sum_sum_a_nat] :
( ( member_a @ X3 @ A3 )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7873763678140191238_a_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_269_imageI,axiom,
! [X3: nat,A3: set_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7293268710728258664_a_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_270_imageI,axiom,
! [X3: $o,A3: set_o,F: $o > sum_sum_a_nat] :
( ( member_o @ X3 @ A3 )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_4139480514073730540_a_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_271_imageI,axiom,
! [X3: sum_sum_a_nat,A3: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ ( image_6095136190293192542_nat_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_272_imageI,axiom,
! [X3: nat,A3: set_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X3 @ A3 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_6262589752765146990_a_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_273_imageI,axiom,
! [X3: a,A3: set_a,F: a > list_Sum_sum_a_nat] :
( ( member_a @ X3 @ A3 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_7897140031816760844_a_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_274_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ ( image_3984946558445957976_nat_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_275_Collect__mono__iff,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) )
= ( ! [X: nat > sum_sum_a_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_276_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_277_Collect__mono__iff,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) )
= ( ! [X: sum_sum_a_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_278_Collect__mono__iff,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) )
= ( ! [X: $o] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_279_Collect__mono__iff,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_280_set__eq__subset,axiom,
( ( ^ [Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y = Z2 ) )
= ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A4 @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_281_set__eq__subset,axiom,
( ( ^ [Y: set_o,Z2: set_o] : ( Y = Z2 ) )
= ( ^ [A4: set_o,B3: set_o] :
( ( ord_less_eq_set_o @ A4 @ B3 )
& ( ord_less_eq_set_o @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_282_set__eq__subset,axiom,
( ( ^ [Y: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] : ( Y = Z2 ) )
= ( ^ [A4: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A4 @ B3 )
& ( ord_le1147066620699065093_a_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_283_subset__trans,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ord_le1325389633284124927_a_nat @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_284_subset__trans,axiom,
! [A3: set_o,B: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ord_less_eq_set_o @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_285_subset__trans,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C2 )
=> ( ord_le1147066620699065093_a_nat @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_286_Collect__mono,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ! [X2: nat > sum_sum_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le8108555184339247974_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_287_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_288_Collect__mono,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_289_Collect__mono,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X2: $o] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_290_Collect__mono,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ! [X2: list_Sum_sum_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_291_subset__refl,axiom,
! [A3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_292_subset__refl,axiom,
! [A3: set_o] : ( ord_less_eq_set_o @ A3 @ A3 ) ).
% subset_refl
thf(fact_293_subset__refl,axiom,
! [A3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_294_subset__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
! [T2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ T2 @ A4 )
=> ( member_Sum_sum_a_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_295_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B3: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A4 )
=> ( member_o @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_296_subset__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
! [T2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ T2 @ A4 )
=> ( member408289922725080238_a_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_297_Set_OequalityD2,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A3 = B )
=> ( ord_le1325389633284124927_a_nat @ B @ A3 ) ) ).
% Set.equalityD2
thf(fact_298_Set_OequalityD2,axiom,
! [A3: set_o,B: set_o] :
( ( A3 = B )
=> ( ord_less_eq_set_o @ B @ A3 ) ) ).
% Set.equalityD2
thf(fact_299_Set_OequalityD2,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( A3 = B )
=> ( ord_le1147066620699065093_a_nat @ B @ A3 ) ) ).
% Set.equalityD2
thf(fact_300_equalityD1,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A3 = B )
=> ( ord_le1325389633284124927_a_nat @ A3 @ B ) ) ).
% equalityD1
thf(fact_301_equalityD1,axiom,
! [A3: set_o,B: set_o] :
( ( A3 = B )
=> ( ord_less_eq_set_o @ A3 @ B ) ) ).
% equalityD1
thf(fact_302_equalityD1,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( A3 = B )
=> ( ord_le1147066620699065093_a_nat @ A3 @ B ) ) ).
% equalityD1
thf(fact_303_subset__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A4 )
=> ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_304_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B3: set_o] :
! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_o @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_305_subset__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A4 )
=> ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_306_equalityE,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A3 = B )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ~ ( ord_le1325389633284124927_a_nat @ B @ A3 ) ) ) ).
% equalityE
thf(fact_307_equalityE,axiom,
! [A3: set_o,B: set_o] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_o @ A3 @ B )
=> ~ ( ord_less_eq_set_o @ B @ A3 ) ) ) ).
% equalityE
thf(fact_308_equalityE,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( A3 = B )
=> ~ ( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ~ ( ord_le1147066620699065093_a_nat @ B @ A3 ) ) ) ).
% equalityE
thf(fact_309_subsetD,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( ( member_Sum_sum_a_nat @ C @ A3 )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_310_subsetD,axiom,
! [A3: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ( member_o @ C @ A3 )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_311_subsetD,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ( ( member408289922725080238_a_nat @ C @ A3 )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_312_in__mono,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( ( member_Sum_sum_a_nat @ X3 @ A3 )
=> ( member_Sum_sum_a_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_313_in__mono,axiom,
! [A3: set_o,B: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ( member_o @ X3 @ A3 )
=> ( member_o @ X3 @ B ) ) ) ).
% in_mono
thf(fact_314_in__mono,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ( ( member408289922725080238_a_nat @ X3 @ A3 )
=> ( member408289922725080238_a_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_315_image__Int__subset,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,B: set_a] : ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( inf_inf_set_a @ A3 @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_7873763678140191238_a_nat @ F @ A3 ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_316_image__Int__subset,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,B: set_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( inf_inf_set_nat @ A3 @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_7293268710728258664_a_nat @ F @ A3 ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_317_image__Int__subset,axiom,
! [F: list_Sum_sum_a_nat > sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_6705138127771971519_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_6705138127771971519_a_nat @ F @ A3 ) @ ( image_6705138127771971519_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_318_image__Int__subset,axiom,
! [F: nat > $o,A3: set_nat,B: set_nat] : ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( inf_inf_set_nat @ A3 @ B ) ) @ ( inf_inf_set_o @ ( image_nat_o @ F @ A3 ) @ ( image_nat_o @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_319_image__Int__subset,axiom,
! [F: a > $o,A3: set_a,B: set_a] : ( ord_less_eq_set_o @ ( image_a_o @ F @ ( inf_inf_set_a @ A3 @ B ) ) @ ( inf_inf_set_o @ ( image_a_o @ F @ A3 ) @ ( image_a_o @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_320_image__Int__subset,axiom,
! [F: list_Sum_sum_a_nat > $o,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_less_eq_set_o @ ( image_3984946558445957976_nat_o @ F @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) ) @ ( inf_inf_set_o @ ( image_3984946558445957976_nat_o @ F @ A3 ) @ ( image_3984946558445957976_nat_o @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_321_image__Int__subset,axiom,
! [F: list_a > list_Sum_sum_a_nat,A3: set_list_a,B: set_list_a] : ( ord_le1147066620699065093_a_nat @ ( image_674313660629153798_a_nat @ F @ ( inf_inf_set_list_a @ A3 @ B ) ) @ ( inf_in3249246906714053971_a_nat @ ( image_674313660629153798_a_nat @ F @ A3 ) @ ( image_674313660629153798_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_322_image__Int__subset,axiom,
! [F: nat > list_Sum_sum_a_nat,A3: set_nat,B: set_nat] : ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ ( inf_inf_set_nat @ A3 @ B ) ) @ ( inf_in3249246906714053971_a_nat @ ( image_6262589752765146990_a_nat @ F @ A3 ) @ ( image_6262589752765146990_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_323_image__Int__subset,axiom,
! [F: a > list_Sum_sum_a_nat,A3: set_a,B: set_a] : ( ord_le1147066620699065093_a_nat @ ( image_7897140031816760844_a_nat @ F @ ( inf_inf_set_a @ A3 @ B ) ) @ ( inf_in3249246906714053971_a_nat @ ( image_7897140031816760844_a_nat @ F @ A3 ) @ ( image_7897140031816760844_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_324_image__Int__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ A3 @ B ) ) @ ( inf_in3249246906714053971_a_nat @ ( image_5081948215111134021_a_nat @ F @ A3 ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_325_Inl__inject,axiom,
! [X3: a,Y2: a] :
( ( ( sum_Inl_a_nat @ X3 )
= ( sum_Inl_a_nat @ Y2 ) )
=> ( X3 = Y2 ) ) ).
% Inl_inject
thf(fact_326_range__eqI,axiom,
! [B2: $o,F: a > $o,X3: a] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_a_o @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_327_range__eqI,axiom,
! [B2: $o,F: nat > $o,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_328_range__eqI,axiom,
! [B2: $o,F: list_a > $o,X3: list_a] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_list_a_o @ F @ top_top_set_list_a ) ) ) ).
% range_eqI
thf(fact_329_range__eqI,axiom,
! [B2: $o,F: sum_sum_a_nat > $o,X3: sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) ) ) ).
% range_eqI
thf(fact_330_range__eqI,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,X3: a] :
( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_331_range__eqI,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_332_range__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: a > list_Sum_sum_a_nat,X3: a] :
( ( B2
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_7897140031816760844_a_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_333_range__eqI,axiom,
! [B2: sum_sum_a_nat,F: list_a > sum_sum_a_nat,X3: list_a] :
( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_4532641577586004224_a_nat @ F @ top_top_set_list_a ) ) ) ).
% range_eqI
thf(fact_334_range__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_335_range__eqI,axiom,
! [B2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) ) ) ).
% range_eqI
thf(fact_336_rangeI,axiom,
! [F: a > $o,X3: a] : ( member_o @ ( F @ X3 ) @ ( image_a_o @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_337_rangeI,axiom,
! [F: nat > $o,X3: nat] : ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_338_rangeI,axiom,
! [F: list_a > $o,X3: list_a] : ( member_o @ ( F @ X3 ) @ ( image_list_a_o @ F @ top_top_set_list_a ) ) ).
% rangeI
thf(fact_339_rangeI,axiom,
! [F: sum_sum_a_nat > $o,X3: sum_sum_a_nat] : ( member_o @ ( F @ X3 ) @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) ) ).
% rangeI
thf(fact_340_rangeI,axiom,
! [F: a > sum_sum_a_nat,X3: a] : ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_341_rangeI,axiom,
! [F: nat > sum_sum_a_nat,X3: nat] : ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_342_rangeI,axiom,
! [F: a > list_Sum_sum_a_nat,X3: a] : ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_7897140031816760844_a_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_343_rangeI,axiom,
! [F: list_a > sum_sum_a_nat,X3: list_a] : ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_4532641577586004224_a_nat @ F @ top_top_set_list_a ) ) ).
% rangeI
thf(fact_344_rangeI,axiom,
! [F: nat > list_Sum_sum_a_nat,X3: nat] : ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_6262589752765146990_a_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_345_rangeI,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) ) ).
% rangeI
thf(fact_346_subset__UNIV,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_347_subset__UNIV,axiom,
! [A3: set_list_a] : ( ord_le8861187494160871172list_a @ A3 @ top_top_set_list_a ) ).
% subset_UNIV
thf(fact_348_subset__UNIV,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_349_subset__UNIV,axiom,
! [A3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A3 @ top_to795618464972521135_a_nat ) ).
% subset_UNIV
thf(fact_350_subset__UNIV,axiom,
! [A3: set_o] : ( ord_less_eq_set_o @ A3 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_351_subset__UNIV,axiom,
! [A3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A3 @ top_to6433055325616222389_a_nat ) ).
% subset_UNIV
thf(fact_352_subset__image__iff,axiom,
! [B: set_o,F: nat > $o,A3: set_nat] :
( ( ord_less_eq_set_o @ B @ ( image_nat_o @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B
= ( image_nat_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_353_subset__image__iff,axiom,
! [B: set_o,F: a > $o,A3: set_a] :
( ( ord_less_eq_set_o @ B @ ( image_a_o @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B
= ( image_a_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_354_subset__image__iff,axiom,
! [B: set_o,F: $o > $o,A3: set_o] :
( ( ord_less_eq_set_o @ B @ ( image_o_o @ F @ A3 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A3 )
& ( B
= ( image_o_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_355_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A3: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B
= ( image_7873763678140191238_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_356_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A3: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B
= ( image_7293268710728258664_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_357_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: $o > sum_sum_a_nat,A3: set_o] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_4139480514073730540_a_nat @ F @ A3 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A3 )
& ( B
= ( image_4139480514073730540_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_358_subset__image__iff,axiom,
! [B: set_o,F: sum_sum_a_nat > $o,A3: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ B @ ( image_6095136190293192542_nat_o @ F @ A3 ) )
= ( ? [AA: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ AA @ A3 )
& ( B
= ( image_6095136190293192542_nat_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_359_subset__image__iff,axiom,
! [B: set_o,F: list_Sum_sum_a_nat > $o,A3: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_o @ B @ ( image_3984946558445957976_nat_o @ F @ A3 ) )
= ( ? [AA: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ AA @ A3 )
& ( B
= ( image_3984946558445957976_nat_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_360_subset__image__iff,axiom,
! [B: set_li6526943997496501093_a_nat,F: nat > list_Sum_sum_a_nat,A3: set_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B
= ( image_6262589752765146990_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_361_subset__image__iff,axiom,
! [B: set_li6526943997496501093_a_nat,F: a > list_Sum_sum_a_nat,A3: set_a] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A3 )
& ( B
= ( image_7897140031816760844_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_362_image__subset__iff,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A3 ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_363_image__subset__iff,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A3 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_364_image__subset__iff,axiom,
! [F: nat > $o,A3: set_nat,B: set_o] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ A3 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_o @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_365_image__subset__iff,axiom,
! [F: a > $o,A3: set_a,B: set_o] :
( ( ord_less_eq_set_o @ ( image_a_o @ F @ A3 ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_o @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_366_image__subset__iff,axiom,
! [F: list_a > list_Sum_sum_a_nat,A3: set_list_a,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_674313660629153798_a_nat @ F @ A3 ) @ B )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A3 )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_367_image__subset__iff,axiom,
! [F: nat > list_Sum_sum_a_nat,A3: set_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ A3 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_368_image__subset__iff,axiom,
! [F: a > list_Sum_sum_a_nat,A3: set_a,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_7897140031816760844_a_nat @ F @ A3 ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ A3 )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_369_subset__imageE,axiom,
! [B: set_o,F: nat > $o,A3: set_nat] :
( ( ord_less_eq_set_o @ B @ ( image_nat_o @ F @ A3 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
=> ( B
!= ( image_nat_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_370_subset__imageE,axiom,
! [B: set_o,F: a > $o,A3: set_a] :
( ( ord_less_eq_set_o @ B @ ( image_a_o @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B
!= ( image_a_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_371_subset__imageE,axiom,
! [B: set_o,F: $o > $o,A3: set_o] :
( ( ord_less_eq_set_o @ B @ ( image_o_o @ F @ A3 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A3 )
=> ( B
!= ( image_o_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_372_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A3: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B
!= ( image_7873763678140191238_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_373_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A3: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
=> ( B
!= ( image_7293268710728258664_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_374_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: $o > sum_sum_a_nat,A3: set_o] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_4139480514073730540_a_nat @ F @ A3 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A3 )
=> ( B
!= ( image_4139480514073730540_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_375_subset__imageE,axiom,
! [B: set_o,F: sum_sum_a_nat > $o,A3: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ B @ ( image_6095136190293192542_nat_o @ F @ A3 ) )
=> ~ ! [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A3 )
=> ( B
!= ( image_6095136190293192542_nat_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_376_subset__imageE,axiom,
! [B: set_o,F: list_Sum_sum_a_nat > $o,A3: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_o @ B @ ( image_3984946558445957976_nat_o @ F @ A3 ) )
=> ~ ! [C3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C3 @ A3 )
=> ( B
!= ( image_3984946558445957976_nat_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_377_subset__imageE,axiom,
! [B: set_li6526943997496501093_a_nat,F: nat > list_Sum_sum_a_nat,A3: set_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
=> ( B
!= ( image_6262589752765146990_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_378_subset__imageE,axiom,
! [B: set_li6526943997496501093_a_nat,F: a > list_Sum_sum_a_nat,A3: set_a] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( B
!= ( image_7897140031816760844_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_379_image__subsetI,axiom,
! [A3: set_nat,F: nat > $o,B: set_o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_380_image__subsetI,axiom,
! [A3: set_a,F: a > $o,B: set_o] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_381_image__subsetI,axiom,
! [A3: set_o,F: $o > $o,B: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_382_image__subsetI,axiom,
! [A3: set_a,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_383_image__subsetI,axiom,
! [A3: set_nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_384_image__subsetI,axiom,
! [A3: set_o,F: $o > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_4139480514073730540_a_nat @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_385_image__subsetI,axiom,
! [A3: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o,B: set_o] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_6095136190293192542_nat_o @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_386_image__subsetI,axiom,
! [A3: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o,B: set_o] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A3 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_3984946558445957976_nat_o @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_387_image__subsetI,axiom,
! [A3: set_nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_388_image__subsetI,axiom,
! [A3: set_a,F: a > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_7897140031816760844_a_nat @ F @ A3 ) @ B ) ) ).
% image_subsetI
thf(fact_389_image__mono,axiom,
! [A3: set_nat,B: set_nat,F: nat > $o] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A3 ) @ ( image_nat_o @ F @ B ) ) ) ).
% image_mono
thf(fact_390_image__mono,axiom,
! [A3: set_a,B: set_a,F: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ A3 ) @ ( image_a_o @ F @ B ) ) ) ).
% image_mono
thf(fact_391_image__mono,axiom,
! [A3: set_o,B: set_o,F: $o > $o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A3 ) @ ( image_o_o @ F @ B ) ) ) ).
% image_mono
thf(fact_392_image__mono,axiom,
! [A3: set_a,B: set_a,F: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A3 ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_393_image__mono,axiom,
! [A3: set_nat,B: set_nat,F: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A3 ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_394_image__mono,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( ord_less_eq_set_o @ ( image_6095136190293192542_nat_o @ F @ A3 ) @ ( image_6095136190293192542_nat_o @ F @ B ) ) ) ).
% image_mono
thf(fact_395_image__mono,axiom,
! [A3: set_o,B: set_o,F: $o > sum_sum_a_nat] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_4139480514073730540_a_nat @ F @ A3 ) @ ( image_4139480514073730540_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_396_image__mono,axiom,
! [A3: set_nat,B: set_nat,F: nat > list_Sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ A3 ) @ ( image_6262589752765146990_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_397_image__mono,axiom,
! [A3: set_a,B: set_a,F: a > list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A3 @ B )
=> ( ord_le1147066620699065093_a_nat @ ( image_7897140031816760844_a_nat @ F @ A3 ) @ ( image_7897140031816760844_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_398_image__mono,axiom,
! [A3: set_o,B: set_o,F: $o > list_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ord_le1147066620699065093_a_nat @ ( image_733135354807394034_a_nat @ F @ A3 ) @ ( image_733135354807394034_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_399_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( member_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_400_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_401_subset__code_I1_J,axiom,
! [Xs: list_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_402_subset__code_I1_J,axiom,
! [Xs: list_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ B )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( member_Sum_sum_a_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_403_subset__code_I1_J,axiom,
! [Xs: list_o,B: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B )
= ( ! [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( member_o @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_404_subset__code_I1_J,axiom,
! [Xs: list_l4703314356710769291_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ B )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( member408289922725080238_a_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_405_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F @ Xs2 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y3: nat] :
( X
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_406_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_Sum_sum_a_nat @ F @ Xs2 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y3: a] :
( X
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_407_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ? [Xs2: list_Sum_sum_a_nat] :
( Ys
= ( map_Su2790769393171190532_a_nat @ F @ Xs2 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y3: sum_sum_a_nat] :
( X
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_408_ex__map__conv,axiom,
! [Ys: list_l4703314356710769291_a_nat,F: list_a > list_Sum_sum_a_nat] :
( ( ? [Xs2: list_list_a] :
( Ys
= ( map_li6996171348340886213_a_nat @ F @ Xs2 ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Ys ) )
=> ? [Y3: list_a] :
( X
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_409_map__cong,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_410_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_a_Sum_sum_a_nat @ F @ Xs )
= ( map_a_Sum_sum_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_411_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_412_map__cong,axiom,
! [Xs: list_list_a,Ys: list_list_a,F: list_a > list_Sum_sum_a_nat,G: list_a > list_Sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_li6996171348340886213_a_nat @ F @ Xs )
= ( map_li6996171348340886213_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_413_map__idI,axiom,
! [Xs: list_o,F: $o > $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ ( set_o2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_o_o @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_414_map__idI,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_415_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_416_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_417_map__idI,axiom,
! [Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_418_map__idI,axiom,
! [Xs: list_list_a,F: list_a > list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_list_a_list_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_419_map__ext,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_420_map__ext,axiom,
! [Xs: list_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_a_Sum_sum_a_nat @ F @ Xs )
= ( map_a_Sum_sum_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_421_map__ext,axiom,
! [Xs: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_422_map__ext,axiom,
! [Xs: list_list_a,F: list_a > list_Sum_sum_a_nat,G: list_a > list_Sum_sum_a_nat] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_li6996171348340886213_a_nat @ F @ Xs )
= ( map_li6996171348340886213_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_423_list_Omap__ident__strong,axiom,
! [T: list_o,F: $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( set_o2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_o_o @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_424_list_Omap__ident__strong,axiom,
! [T: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z3 @ ( set_Sum_sum_a_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_425_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_a2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_426_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_427_list_Omap__ident__strong,axiom,
! [T: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z3 @ ( set_li2392974972034027290_a_nat @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_428_list_Omap__ident__strong,axiom,
! [T: list_list_a,F: list_a > list_a] :
( ! [Z3: list_a] :
( ( member_list_a @ Z3 @ ( set_list_a2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_list_a_list_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_429_list_Oinj__map__strong,axiom,
! [X3: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,Fa: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z3: sum_sum_a_nat,Za: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z3 @ ( set_Sum_sum_a_nat2 @ X3 ) )
=> ( ( member_Sum_sum_a_nat @ Za @ ( set_Sum_sum_a_nat2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_Su2790769393171190532_a_nat @ F @ X3 )
= ( map_Su2790769393171190532_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_430_list_Oinj__map__strong,axiom,
! [X3: list_a,Xa: list_a,F: a > sum_sum_a_nat,Fa: a > sum_sum_a_nat] :
( ! [Z3: a,Za: a] :
( ( member_a @ Z3 @ ( set_a2 @ X3 ) )
=> ( ( member_a @ Za @ ( set_a2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_a_Sum_sum_a_nat @ F @ X3 )
= ( map_a_Sum_sum_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_431_list_Oinj__map__strong,axiom,
! [X3: list_nat,Xa: list_nat,F: nat > sum_sum_a_nat,Fa: nat > sum_sum_a_nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X3 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F @ X3 )
= ( map_na823391071729141993_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_432_list_Oinj__map__strong,axiom,
! [X3: list_list_a,Xa: list_list_a,F: list_a > list_Sum_sum_a_nat,Fa: list_a > list_Sum_sum_a_nat] :
( ! [Z3: list_a,Za: list_a] :
( ( member_list_a @ Z3 @ ( set_list_a2 @ X3 ) )
=> ( ( member_list_a @ Za @ ( set_list_a2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_li6996171348340886213_a_nat @ F @ X3 )
= ( map_li6996171348340886213_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_433_list_Omap__cong0,axiom,
! [X3: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z3 @ ( set_Sum_sum_a_nat2 @ X3 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ X3 )
= ( map_Su2790769393171190532_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_434_list_Omap__cong0,axiom,
! [X3: list_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_a2 @ X3 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_a_Sum_sum_a_nat @ F @ X3 )
= ( map_a_Sum_sum_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_435_list_Omap__cong0,axiom,
! [X3: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X3 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X3 )
= ( map_na823391071729141993_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_436_list_Omap__cong0,axiom,
! [X3: list_list_a,F: list_a > list_Sum_sum_a_nat,G: list_a > list_Sum_sum_a_nat] :
( ! [Z3: list_a] :
( ( member_list_a @ Z3 @ ( set_list_a2 @ X3 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_li6996171348340886213_a_nat @ F @ X3 )
= ( map_li6996171348340886213_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_437_list_Omap__cong,axiom,
! [X3: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z3 @ ( set_Sum_sum_a_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ X3 )
= ( map_Su2790769393171190532_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_438_list_Omap__cong,axiom,
! [X3: list_a,Ya: list_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_a_Sum_sum_a_nat @ F @ X3 )
= ( map_a_Sum_sum_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_439_list_Omap__cong,axiom,
! [X3: list_nat,Ya: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X3 )
= ( map_na823391071729141993_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_440_list_Omap__cong,axiom,
! [X3: list_list_a,Ya: list_list_a,F: list_a > list_Sum_sum_a_nat,G: list_a > list_Sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z3: list_a] :
( ( member_list_a @ Z3 @ ( set_list_a2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_li6996171348340886213_a_nat @ F @ X3 )
= ( map_li6996171348340886213_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_441_inf__top_Oright__neutral,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ top_to6433055325616222389_a_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_442_inf__top_Oright__neutral,axiom,
! [A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ top_to795618464972521135_a_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_443_inf__top_Oright__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% inf_top.right_neutral
thf(fact_444_inf__top_Oright__neutral,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ top_top_set_list_a )
= A ) ).
% inf_top.right_neutral
thf(fact_445_inf__top_Oright__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_446_inf__top_Oneutr__eq__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( top_to6433055325616222389_a_nat
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) )
= ( ( A = top_to6433055325616222389_a_nat )
& ( B2 = top_to6433055325616222389_a_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_447_inf__top_Oneutr__eq__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( top_to795618464972521135_a_nat
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
= ( ( A = top_to795618464972521135_a_nat )
& ( B2 = top_to795618464972521135_a_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_448_inf__top_Oneutr__eq__iff,axiom,
! [A: set_a,B2: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A @ B2 ) )
= ( ( A = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_449_inf__top_Oneutr__eq__iff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( top_top_set_list_a
= ( inf_inf_set_list_a @ A @ B2 ) )
= ( ( A = top_top_set_list_a )
& ( B2 = top_top_set_list_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_450_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A @ B2 ) )
= ( ( A = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_451_inf__top_Oleft__neutral,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ top_to6433055325616222389_a_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_452_inf__top_Oleft__neutral,axiom,
! [A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_453_inf__top_Oleft__neutral,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_454_inf__top_Oleft__neutral,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ top_top_set_list_a @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_455_inf__top_Oleft__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_456_inf__top_Oeq__neutr__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A @ B2 )
= top_to6433055325616222389_a_nat )
= ( ( A = top_to6433055325616222389_a_nat )
& ( B2 = top_to6433055325616222389_a_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_457_inf__top_Oeq__neutr__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= top_to795618464972521135_a_nat )
= ( ( A = top_to795618464972521135_a_nat )
& ( B2 = top_to795618464972521135_a_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_458_inf__top_Oeq__neutr__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_459_inf__top_Oeq__neutr__iff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= top_top_set_list_a )
= ( ( A = top_top_set_list_a )
& ( B2 = top_top_set_list_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_460_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_461_top__eq__inf__iff,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( top_to6433055325616222389_a_nat
= ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) )
= ( ( X3 = top_to6433055325616222389_a_nat )
& ( Y2 = top_to6433055325616222389_a_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_462_top__eq__inf__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( top_to795618464972521135_a_nat
= ( inf_in7084830621192376909_a_nat @ X3 @ Y2 ) )
= ( ( X3 = top_to795618464972521135_a_nat )
& ( Y2 = top_to795618464972521135_a_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_463_top__eq__inf__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X3 @ Y2 ) )
= ( ( X3 = top_top_set_a )
& ( Y2 = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_464_top__eq__inf__iff,axiom,
! [X3: set_list_a,Y2: set_list_a] :
( ( top_top_set_list_a
= ( inf_inf_set_list_a @ X3 @ Y2 ) )
= ( ( X3 = top_top_set_list_a )
& ( Y2 = top_top_set_list_a ) ) ) ).
% top_eq_inf_iff
thf(fact_465_top__eq__inf__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X3 @ Y2 ) )
= ( ( X3 = top_top_set_nat )
& ( Y2 = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_466_inf__eq__top__iff,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ X3 @ Y2 )
= top_to6433055325616222389_a_nat )
= ( ( X3 = top_to6433055325616222389_a_nat )
& ( Y2 = top_to6433055325616222389_a_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_467_inf__eq__top__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ X3 @ Y2 )
= top_to795618464972521135_a_nat )
= ( ( X3 = top_to795618464972521135_a_nat )
& ( Y2 = top_to795618464972521135_a_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_468_inf__eq__top__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ( inf_inf_set_a @ X3 @ Y2 )
= top_top_set_a )
= ( ( X3 = top_top_set_a )
& ( Y2 = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_469_inf__eq__top__iff,axiom,
! [X3: set_list_a,Y2: set_list_a] :
( ( ( inf_inf_set_list_a @ X3 @ Y2 )
= top_top_set_list_a )
= ( ( X3 = top_top_set_list_a )
& ( Y2 = top_top_set_list_a ) ) ) ).
% inf_eq_top_iff
thf(fact_470_inf__eq__top__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ( inf_inf_set_nat @ X3 @ Y2 )
= top_top_set_nat )
= ( ( X3 = top_top_set_nat )
& ( Y2 = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_471_inf__top__right,axiom,
! [X3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X3 @ top_to6433055325616222389_a_nat )
= X3 ) ).
% inf_top_right
thf(fact_472_inf__top__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ top_to795618464972521135_a_nat )
= X3 ) ).
% inf_top_right
thf(fact_473_inf__top__right,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ top_top_set_a )
= X3 ) ).
% inf_top_right
thf(fact_474_inf__top__right,axiom,
! [X3: set_list_a] :
( ( inf_inf_set_list_a @ X3 @ top_top_set_list_a )
= X3 ) ).
% inf_top_right
thf(fact_475_inf__top__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ top_top_set_nat )
= X3 ) ).
% inf_top_right
thf(fact_476_inf__top__left,axiom,
! [X3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ top_to6433055325616222389_a_nat @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_477_inf__top__left,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_478_inf__top__left,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_479_inf__top__left,axiom,
! [X3: set_list_a] :
( ( inf_inf_set_list_a @ top_top_set_list_a @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_480_inf__top__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_481_inf_Obounded__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
= ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
& ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_482_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_483_inf_Obounded__iff,axiom,
! [A: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ ( inf_inf_set_o @ B2 @ C ) )
= ( ( ord_less_eq_set_o @ A @ B2 )
& ( ord_less_eq_set_o @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_484_inf_Obounded__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) )
= ( ( ord_le1147066620699065093_a_nat @ A @ B2 )
& ( ord_le1147066620699065093_a_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_485_le__inf__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y2 @ Z ) )
= ( ( ord_le1325389633284124927_a_nat @ X3 @ Y2 )
& ( ord_le1325389633284124927_a_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_486_le__inf__iff,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_487_le__inf__iff,axiom,
! [X3: set_o,Y2: set_o,Z: set_o] :
( ( ord_less_eq_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ Z ) )
= ( ( ord_less_eq_set_o @ X3 @ Y2 )
& ( ord_less_eq_set_o @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_488_le__inf__iff,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ Y2 @ Z ) )
= ( ( ord_le1147066620699065093_a_nat @ X3 @ Y2 )
& ( ord_le1147066620699065093_a_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_489_inf_Oidem,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_490_inf__idem,axiom,
! [X3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_491_inf__right__idem,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) @ Y2 )
= ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) ) ).
% inf_right_idem
thf(fact_492_inf_Oright__idem,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ B2 )
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_493_inf__left__idem,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) )
= ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) ) ).
% inf_left_idem
thf(fact_494_inf_Oleft__idem,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_495_top__set__def,axiom,
( top_to9106040778512017686_a_nat
= ( collec5629555741568564177_a_nat @ top_to660882137183040839_nat_o ) ) ).
% top_set_def
thf(fact_496_top__set__def,axiom,
( top_to795618464972521135_a_nat
= ( collec7073057861543223018_a_nat @ top_to1565196397637005550_nat_o ) ) ).
% top_set_def
thf(fact_497_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_498_top__set__def,axiom,
( top_top_set_list_a
= ( collect_list_a @ top_top_list_a_o ) ) ).
% top_set_def
thf(fact_499_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_500_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B3: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X: $o] : ( member_o @ X @ A4 )
@ ^ [X: $o] : ( member_o @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_501_inf__set__def,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( inf_in5242522483218605776_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A4 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_502_inf__set__def,axiom,
( inf_in8399021836546144180_a_nat
= ( ^ [A4: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( inf_in48813113380384809_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A4 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_503_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_504_inf__set__def,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( inf_in954358986474102090_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A4 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_505_less__eq__set__def,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ord_le1477630214076318366_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A4 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_506_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ A4 )
@ ^ [X: $o] : ( member_o @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_507_less__eq__set__def,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ord_le8737610411969296920_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A4 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_508_inf__left__commute,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ Y2 @ Z ) )
= ( inf_in3249246906714053971_a_nat @ Y2 @ ( inf_in3249246906714053971_a_nat @ X3 @ Z ) ) ) ).
% inf_left_commute
thf(fact_509_inf_Oleft__commute,axiom,
! [B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ B2 @ ( inf_in3249246906714053971_a_nat @ A @ C ) )
= ( inf_in3249246906714053971_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_510_inf__commute,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [X: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ Y3 @ X ) ) ) ).
% inf_commute
thf(fact_511_inf_Ocommute,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A5: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ B4 @ A5 ) ) ) ).
% inf.commute
thf(fact_512_inf__assoc,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) @ Z )
= ( inf_in3249246906714053971_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_513_inf_Oassoc,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ C )
= ( inf_in3249246906714053971_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_514_inf__sup__aci_I1_J,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [X: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ Y3 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_515_inf__sup__aci_I2_J,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) @ Z )
= ( inf_in3249246906714053971_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_516_inf__sup__aci_I3_J,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ Y2 @ Z ) )
= ( inf_in3249246906714053971_a_nat @ Y2 @ ( inf_in3249246906714053971_a_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_517_inf__sup__aci_I4_J,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) )
= ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_518_inf__sup__ord_I2_J,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_519_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_520_inf__sup__ord_I2_J,axiom,
! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_521_inf__sup__ord_I2_J,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_522_inf__sup__ord_I1_J,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_523_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_524_inf__sup__ord_I1_J,axiom,
! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_525_inf__sup__ord_I1_J,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_526_inf__le1,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_527_inf__le1,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_528_inf__le1,axiom,
! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_529_inf__le1,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_530_inf__le2,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_531_inf__le2,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_532_inf__le2,axiom,
! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_533_inf__le2,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_534_le__infE,axiom,
! [X3: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
=> ~ ( ( ord_le1325389633284124927_a_nat @ X3 @ A )
=> ~ ( ord_le1325389633284124927_a_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_535_le__infE,axiom,
! [X3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A )
=> ~ ( ord_less_eq_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_536_le__infE,axiom,
! [X3: set_o,A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ X3 @ ( inf_inf_set_o @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_o @ X3 @ A )
=> ~ ( ord_less_eq_set_o @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_537_le__infE,axiom,
! [X3: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) )
=> ~ ( ( ord_le1147066620699065093_a_nat @ X3 @ A )
=> ~ ( ord_le1147066620699065093_a_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_538_le__infI,axiom,
! [X3: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ X3 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_539_le__infI,axiom,
! [X3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A )
=> ( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_540_le__infI,axiom,
! [X3: set_o,A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ X3 @ A )
=> ( ( ord_less_eq_set_o @ X3 @ B2 )
=> ( ord_less_eq_set_o @ X3 @ ( inf_inf_set_o @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_541_le__infI,axiom,
! [X3: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ X3 @ B2 )
=> ( ord_le1147066620699065093_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_542_inf__mono,axiom,
! [A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ ( inf_in7084830621192376909_a_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_543_inf__mono,axiom,
! [A: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_544_inf__mono,axiom,
! [A: set_o,C: set_o,B2: set_o,D2: set_o] :
( ( ord_less_eq_set_o @ A @ C )
=> ( ( ord_less_eq_set_o @ B2 @ D2 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B2 ) @ ( inf_inf_set_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_545_inf__mono,axiom,
! [A: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,D2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ C )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ D2 )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ ( inf_in3249246906714053971_a_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_546_le__infI1,axiom,
! [A: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_547_le__infI1,axiom,
! [A: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_548_le__infI1,axiom,
! [A: set_o,X3: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A @ X3 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_549_le__infI1,axiom,
! [A: set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ X3 )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_550_le__infI2,axiom,
! [B2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_551_le__infI2,axiom,
! [B2: nat,X3: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_552_le__infI2,axiom,
! [B2: set_o,X3: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B2 @ X3 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_553_le__infI2,axiom,
! [B2: set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ X3 )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_554_inf_OorderE,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( A
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_555_inf_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( inf_inf_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_556_inf_OorderE,axiom,
! [A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( A
= ( inf_inf_set_o @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_557_inf_OorderE,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( A
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_558_inf_OorderI,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_559_inf_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_560_inf_OorderI,axiom,
! [A: set_o,B2: set_o] :
( ( A
= ( inf_inf_set_o @ A @ B2 ) )
=> ( ord_less_eq_set_o @ A @ B2 ) ) ).
% inf.orderI
thf(fact_561_inf_OorderI,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( A
= ( inf_in3249246906714053971_a_nat @ A @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_562_inf__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ( ord_le1325389633284124927_a_nat @ X2 @ Z3 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_563_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y2: nat] :
( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_564_inf__unique,axiom,
! [F: set_o > set_o > set_o,X3: set_o,Y2: set_o] :
( ! [X2: set_o,Y4: set_o] : ( ord_less_eq_set_o @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_o,Y4: set_o] : ( ord_less_eq_set_o @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_o,Y4: set_o,Z3: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ( ord_less_eq_set_o @ X2 @ Z3 )
=> ( ord_less_eq_set_o @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_o @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_565_inf__unique,axiom,
! [F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ! [X2: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat,Z3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X2 @ Y4 )
=> ( ( ord_le1147066620699065093_a_nat @ X2 @ Z3 )
=> ( ord_le1147066620699065093_a_nat @ X2 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in3249246906714053971_a_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_566_le__iff__inf,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ Y3 )
= X ) ) ) ).
% le_iff_inf
thf(fact_567_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y3: nat] :
( ( inf_inf_nat @ X @ Y3 )
= X ) ) ) ).
% le_iff_inf
thf(fact_568_le__iff__inf,axiom,
( ord_less_eq_set_o
= ( ^ [X: set_o,Y3: set_o] :
( ( inf_inf_set_o @ X @ Y3 )
= X ) ) ) ).
% le_iff_inf
thf(fact_569_le__iff__inf,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [X: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X @ Y3 )
= X ) ) ) ).
% le_iff_inf
thf(fact_570_inf_Oabsorb1,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_571_inf_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( inf_inf_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_572_inf_Oabsorb1,axiom,
! [A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( inf_inf_set_o @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_573_inf_Oabsorb1,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_574_inf_Oabsorb2,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_575_inf_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_576_inf_Oabsorb2,axiom,
! [B2: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( ( inf_inf_set_o @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_577_inf_Oabsorb2,axiom,
! [B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_578_inf__absorb1,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y2 )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_579_inf__absorb1,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_580_inf__absorb1,axiom,
! [X3: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y2 )
=> ( ( inf_inf_set_o @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_581_inf__absorb1,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y2 )
=> ( ( inf_in3249246906714053971_a_nat @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_582_inf__absorb2,axiom,
! [Y2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y2 @ X3 )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_583_inf__absorb2,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_584_inf__absorb2,axiom,
! [Y2: set_o,X3: set_o] :
( ( ord_less_eq_set_o @ Y2 @ X3 )
=> ( ( inf_inf_set_o @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_585_inf__absorb2,axiom,
! [Y2: set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y2 @ X3 )
=> ( ( inf_in3249246906714053971_a_nat @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_586_inf_OboundedE,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ~ ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_587_inf_OboundedE,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_588_inf_OboundedE,axiom,
! [A: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ ( inf_inf_set_o @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_o @ A @ B2 )
=> ~ ( ord_less_eq_set_o @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_589_inf_OboundedE,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) )
=> ~ ( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ~ ( ord_le1147066620699065093_a_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_590_inf_OboundedI,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ A @ C )
=> ( ord_le1325389633284124927_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_591_inf_OboundedI,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_592_inf_OboundedI,axiom,
! [A: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ord_less_eq_set_o @ A @ C )
=> ( ord_less_eq_set_o @ A @ ( inf_inf_set_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_593_inf_OboundedI,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ A @ C )
=> ( ord_le1147066620699065093_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_594_inf__greatest,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y2 )
=> ( ( ord_le1325389633284124927_a_nat @ X3 @ Z )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_595_inf__greatest,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_596_inf__greatest,axiom,
! [X3: set_o,Y2: set_o,Z: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y2 )
=> ( ( ord_less_eq_set_o @ X3 @ Z )
=> ( ord_less_eq_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_597_inf__greatest,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y2 )
=> ( ( ord_le1147066620699065093_a_nat @ X3 @ Z )
=> ( ord_le1147066620699065093_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_598_inf_Oorder__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( A5
= ( inf_in7084830621192376909_a_nat @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_599_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( A5
= ( inf_inf_nat @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_600_inf_Oorder__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B4: set_o] :
( A5
= ( inf_inf_set_o @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_601_inf_Oorder__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A5: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( A5
= ( inf_in3249246906714053971_a_nat @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_602_inf_Ocobounded1,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_603_inf_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_604_inf_Ocobounded1,axiom,
! [A: set_o,B2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_605_inf_Ocobounded1,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_606_inf_Ocobounded2,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_607_inf_Ocobounded2,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_608_inf_Ocobounded2,axiom,
! [A: set_o,B2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_609_inf_Ocobounded2,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_610_inf_Oabsorb__iff1,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_611_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( inf_inf_nat @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_612_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_o
= ( ^ [A5: set_o,B4: set_o] :
( ( inf_inf_set_o @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_613_inf_Oabsorb__iff1,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A5: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_614_inf_Oabsorb__iff2,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B4: set_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_615_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_616_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_o
= ( ^ [B4: set_o,A5: set_o] :
( ( inf_inf_set_o @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_617_inf_Oabsorb__iff2,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [B4: set_li6526943997496501093_a_nat,A5: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_618_inf_OcoboundedI1,axiom,
! [A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_619_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_620_inf_OcoboundedI1,axiom,
! [A: set_o,C: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A @ C )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_621_inf_OcoboundedI1,axiom,
! [A: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ C )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_622_inf_OcoboundedI2,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_623_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_624_inf_OcoboundedI2,axiom,
! [B2: set_o,C: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B2 @ C )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_625_inf_OcoboundedI2,axiom,
! [B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_626_proj__vals__def,axiom,
( proj_v3643391342904276326_a_nat
= ( ^ [R: set_na3699693778330250182_a_nat,Ns: list_nat] :
( image_6721470456781115300_a_nat
@ ^ [Tau: nat > sum_sum_a_nat] : ( map_na823391071729141993_a_nat @ Tau @ Ns )
@ R ) ) ) ).
% proj_vals_def
thf(fact_627_proj__fmla__def,axiom,
( proj_f6037546931041448027_a_nat
= ( ^ [Phi: fo_fmla_a_b,R: set_na3699693778330250182_a_nat] : ( proj_v3643391342904276326_a_nat @ R @ ( fv_fo_fmla_list_a_b @ Phi ) ) ) ) ).
% proj_fmla_def
thf(fact_628_order__refl,axiom,
! [X3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_629_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_630_order__refl,axiom,
! [X3: set_o] : ( ord_less_eq_set_o @ X3 @ X3 ) ).
% order_refl
thf(fact_631_order__refl,axiom,
! [X3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_632_dual__order_Orefl,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_633_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_634_dual__order_Orefl,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ A ) ).
% dual_order.refl
thf(fact_635_dual__order_Orefl,axiom,
! [A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_636_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_637_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: nat > sum_sum_a_nat] :
( ( P @ X2 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_1617820922473287582_a_nat @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_638_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_639_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > $o,B: set_o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_640_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > $o,B: set_o] :
( ! [X2: nat > sum_sum_a_nat] :
( ( P @ X2 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_2376713081370839351_nat_o @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_641_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > $o,B: set_o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_642_image__Collect__subsetI,axiom,
! [P: list_a > $o,F: list_a > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X2: list_a] :
( ( P @ X2 )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_674313660629153798_a_nat @ F @ ( collect_list_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_643_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_7897140031816760844_a_nat @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_644_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X2: nat > sum_sum_a_nat] :
( ( P @ X2 )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6721470456781115300_a_nat @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_645_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_646__092_060open_062_092_060And_062_092_060sigma_062_O_Asat_A_092_060phi_062_AI_A_092_060sigma_062_A_061_Aesat_A_092_060phi_062_AI_A_IInl_A_092_060circ_062_A_092_060sigma_062_J_AUNIV_092_060close_062,axiom,
! [Sigma2: nat > a] :
( ( sat_a_b @ phi @ i @ Sigma2 )
= ( esat_a_b @ phi @ i @ ( comp_a4874482103519161268at_nat @ sum_Inl_a_nat @ Sigma2 ) @ top_to795618464972521135_a_nat ) ) ).
% \<open>\<And>\<sigma>. sat \<phi> I \<sigma> = esat \<phi> I (Inl \<circ> \<sigma>) UNIV\<close>
thf(fact_647_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X3 @ top_to6433055325616222389_a_nat )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_648_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ top_to795618464972521135_a_nat )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_649_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ top_top_set_a )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_650_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_list_a] :
( ( inf_inf_set_list_a @ X3 @ top_top_set_list_a )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_651_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ top_top_set_nat )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_652_all__subset__image,axiom,
! [F: nat > $o,A3: set_nat,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_nat_o @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( P @ ( image_nat_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_653_all__subset__image,axiom,
! [F: a > $o,A3: set_a,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_a_o @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( P @ ( image_a_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_654_all__subset__image,axiom,
! [F: $o > $o,A3: set_o,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_o_o @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ A3 )
=> ( P @ ( image_o_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_655_all__subset__image,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_656_all__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_657_all__subset__image,axiom,
! [F: $o > sum_sum_a_nat,A3: set_o,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_4139480514073730540_a_nat @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ A3 )
=> ( P @ ( image_4139480514073730540_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_658_all__subset__image,axiom,
! [F: sum_sum_a_nat > $o,A3: set_Sum_sum_a_nat,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_6095136190293192542_nat_o @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ A3 )
=> ( P @ ( image_6095136190293192542_nat_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_659_all__subset__image,axiom,
! [F: list_Sum_sum_a_nat > $o,A3: set_li6526943997496501093_a_nat,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ord_less_eq_set_o @ B3 @ ( image_3984946558445957976_nat_o @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ A3 )
=> ( P @ ( image_3984946558445957976_nat_o @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_660_all__subset__image,axiom,
! [F: nat > list_Sum_sum_a_nat,A3: set_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( P @ ( image_6262589752765146990_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_661_all__subset__image,axiom,
! [F: a > list_Sum_sum_a_nat,A3: set_a,P: set_li6526943997496501093_a_nat > $o] :
( ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( P @ ( image_7897140031816760844_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_662_map__comp__map,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat] :
( ( comp_l4895937012117054562st_nat @ ( map_na823391071729141993_a_nat @ F ) @ ( map_nat_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_663_map__comp__map,axiom,
! [F: nat > sum_sum_a_nat,G: a > nat] :
( ( comp_l205733458383234690list_a @ ( map_na823391071729141993_a_nat @ F ) @ ( map_a_nat @ G ) )
= ( map_a_Sum_sum_a_nat @ ( comp_n4273600690000831346_nat_a @ F @ G ) ) ) ).
% map_comp_map
thf(fact_664_map__comp__map,axiom,
! [F: a > sum_sum_a_nat,G: nat > a] :
( ( comp_l1119906300942755268st_nat @ ( map_a_Sum_sum_a_nat @ F ) @ ( map_nat_a @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_665_map__comp__map,axiom,
! [F: a > sum_sum_a_nat,G: a > a] :
( ( comp_l3066312113933711776list_a @ ( map_a_Sum_sum_a_nat @ F ) @ ( map_a_a @ G ) )
= ( map_a_Sum_sum_a_nat @ ( comp_a5877636538679507994_nat_a @ F @ G ) ) ) ).
% map_comp_map
thf(fact_666_map__comp__map,axiom,
! [F: nat > sum_sum_a_nat,G: sum_sum_a_nat > nat] :
( ( comp_l3874044329958624643_a_nat @ ( map_na823391071729141993_a_nat @ F ) @ ( map_Su5227373005390213643at_nat @ G ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_n241567065475451507_a_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_667_map__comp__map,axiom,
! [F: a > sum_sum_a_nat,G: sum_sum_a_nat > a] :
( ( comp_l8031942592446849953_a_nat @ ( map_a_Sum_sum_a_nat @ F ) @ ( map_Sum_sum_a_nat_a @ G ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_a7301103983715706651_a_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_668_map__comp__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( comp_l2590854360456707st_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_na823391071729141993_a_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_669_map__comp__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( comp_l275772281349575457list_a @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_a_Sum_sum_a_nat @ G ) )
= ( map_a_Sum_sum_a_nat @ ( comp_S1182857061046812571_nat_a @ F @ G ) ) ) ).
% map_comp_map
thf(fact_670_map__comp__map,axiom,
! [F: list_a > list_Sum_sum_a_nat,G: list_a > list_a] :
( ( comp_l3640254568318768294list_a @ ( map_li6996171348340886213_a_nat @ F ) @ ( map_list_a_list_a @ G ) )
= ( map_li6996171348340886213_a_nat @ ( comp_l3066312113933711776list_a @ F @ G ) ) ) ).
% map_comp_map
thf(fact_671_map__comp__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( comp_l3689981812903912354_a_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_Su2790769393171190532_a_nat @ G ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_S4540598949873307932_a_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_672_List_Omap_Ocomp,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat] :
( ( comp_l4895937012117054562st_nat @ ( map_na823391071729141993_a_nat @ F ) @ ( map_nat_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_673_List_Omap_Ocomp,axiom,
! [F: nat > sum_sum_a_nat,G: a > nat] :
( ( comp_l205733458383234690list_a @ ( map_na823391071729141993_a_nat @ F ) @ ( map_a_nat @ G ) )
= ( map_a_Sum_sum_a_nat @ ( comp_n4273600690000831346_nat_a @ F @ G ) ) ) ).
% List.map.comp
thf(fact_674_List_Omap_Ocomp,axiom,
! [F: a > sum_sum_a_nat,G: nat > a] :
( ( comp_l1119906300942755268st_nat @ ( map_a_Sum_sum_a_nat @ F ) @ ( map_nat_a @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_675_List_Omap_Ocomp,axiom,
! [F: a > sum_sum_a_nat,G: a > a] :
( ( comp_l3066312113933711776list_a @ ( map_a_Sum_sum_a_nat @ F ) @ ( map_a_a @ G ) )
= ( map_a_Sum_sum_a_nat @ ( comp_a5877636538679507994_nat_a @ F @ G ) ) ) ).
% List.map.comp
thf(fact_676_List_Omap_Ocomp,axiom,
! [F: nat > sum_sum_a_nat,G: sum_sum_a_nat > nat] :
( ( comp_l3874044329958624643_a_nat @ ( map_na823391071729141993_a_nat @ F ) @ ( map_Su5227373005390213643at_nat @ G ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_n241567065475451507_a_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_677_List_Omap_Ocomp,axiom,
! [F: a > sum_sum_a_nat,G: sum_sum_a_nat > a] :
( ( comp_l8031942592446849953_a_nat @ ( map_a_Sum_sum_a_nat @ F ) @ ( map_Sum_sum_a_nat_a @ G ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_a7301103983715706651_a_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_678_List_Omap_Ocomp,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( comp_l2590854360456707st_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_na823391071729141993_a_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_679_List_Omap_Ocomp,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( comp_l275772281349575457list_a @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_a_Sum_sum_a_nat @ G ) )
= ( map_a_Sum_sum_a_nat @ ( comp_S1182857061046812571_nat_a @ F @ G ) ) ) ).
% List.map.comp
thf(fact_680_List_Omap_Ocomp,axiom,
! [F: list_a > list_Sum_sum_a_nat,G: list_a > list_a] :
( ( comp_l3640254568318768294list_a @ ( map_li6996171348340886213_a_nat @ F ) @ ( map_list_a_list_a @ G ) )
= ( map_li6996171348340886213_a_nat @ ( comp_l3066312113933711776list_a @ F @ G ) ) ) ).
% List.map.comp
thf(fact_681_List_Omap_Ocomp,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( comp_l3689981812903912354_a_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_Su2790769393171190532_a_nat @ G ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_S4540598949873307932_a_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_682_map__map,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,Xs: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( map_nat_nat @ G @ Xs ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_683_map__map,axiom,
! [F: nat > sum_sum_a_nat,G: a > nat,Xs: list_a] :
( ( map_na823391071729141993_a_nat @ F @ ( map_a_nat @ G @ Xs ) )
= ( map_a_Sum_sum_a_nat @ ( comp_n4273600690000831346_nat_a @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_684_map__map,axiom,
! [F: a > sum_sum_a_nat,G: nat > a,Xs: list_nat] :
( ( map_a_Sum_sum_a_nat @ F @ ( map_nat_a @ G @ Xs ) )
= ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_685_map__map,axiom,
! [F: a > sum_sum_a_nat,G: a > a,Xs: list_a] :
( ( map_a_Sum_sum_a_nat @ F @ ( map_a_a @ G @ Xs ) )
= ( map_a_Sum_sum_a_nat @ ( comp_a5877636538679507994_nat_a @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_686_map__map,axiom,
! [F: nat > sum_sum_a_nat,G: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( map_Su5227373005390213643at_nat @ G @ Xs ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_n241567065475451507_a_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_687_map__map,axiom,
! [F: a > sum_sum_a_nat,G: sum_sum_a_nat > a,Xs: list_Sum_sum_a_nat] :
( ( map_a_Sum_sum_a_nat @ F @ ( map_Sum_sum_a_nat_a @ G @ Xs ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_a7301103983715706651_a_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_688_map__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,Xs: list_nat] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_na823391071729141993_a_nat @ G @ Xs ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_689_map__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat,Xs: list_a] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_a_Sum_sum_a_nat @ G @ Xs ) )
= ( map_a_Sum_sum_a_nat @ ( comp_S1182857061046812571_nat_a @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_690_map__map,axiom,
! [F: list_a > list_Sum_sum_a_nat,G: list_a > list_a,Xs: list_list_a] :
( ( map_li6996171348340886213_a_nat @ F @ ( map_list_a_list_a @ G @ Xs ) )
= ( map_li6996171348340886213_a_nat @ ( comp_l3066312113933711776list_a @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_691_map__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_Su2790769393171190532_a_nat @ G @ Xs ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_S4540598949873307932_a_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_692_List_Omap_Ocompositionality,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,List: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( map_nat_nat @ G @ List ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_693_List_Omap_Ocompositionality,axiom,
! [F: nat > sum_sum_a_nat,G: a > nat,List: list_a] :
( ( map_na823391071729141993_a_nat @ F @ ( map_a_nat @ G @ List ) )
= ( map_a_Sum_sum_a_nat @ ( comp_n4273600690000831346_nat_a @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_694_List_Omap_Ocompositionality,axiom,
! [F: a > sum_sum_a_nat,G: nat > a,List: list_nat] :
( ( map_a_Sum_sum_a_nat @ F @ ( map_nat_a @ G @ List ) )
= ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_695_List_Omap_Ocompositionality,axiom,
! [F: a > sum_sum_a_nat,G: a > a,List: list_a] :
( ( map_a_Sum_sum_a_nat @ F @ ( map_a_a @ G @ List ) )
= ( map_a_Sum_sum_a_nat @ ( comp_a5877636538679507994_nat_a @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_696_List_Omap_Ocompositionality,axiom,
! [F: nat > sum_sum_a_nat,G: sum_sum_a_nat > nat,List: list_Sum_sum_a_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( map_Su5227373005390213643at_nat @ G @ List ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_n241567065475451507_a_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_697_List_Omap_Ocompositionality,axiom,
! [F: a > sum_sum_a_nat,G: sum_sum_a_nat > a,List: list_Sum_sum_a_nat] :
( ( map_a_Sum_sum_a_nat @ F @ ( map_Sum_sum_a_nat_a @ G @ List ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_a7301103983715706651_a_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_698_List_Omap_Ocompositionality,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,List: list_nat] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_na823391071729141993_a_nat @ G @ List ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_699_List_Omap_Ocompositionality,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat,List: list_a] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_a_Sum_sum_a_nat @ G @ List ) )
= ( map_a_Sum_sum_a_nat @ ( comp_S1182857061046812571_nat_a @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_700_List_Omap_Ocompositionality,axiom,
! [F: list_a > list_Sum_sum_a_nat,G: list_a > list_a,List: list_list_a] :
( ( map_li6996171348340886213_a_nat @ F @ ( map_list_a_list_a @ G @ List ) )
= ( map_li6996171348340886213_a_nat @ ( comp_l3066312113933711776list_a @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_701_List_Omap_Ocompositionality,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,List: list_Sum_sum_a_nat] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_Su2790769393171190532_a_nat @ G @ List ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_S4540598949873307932_a_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_702_list_Omap__comp,axiom,
! [G: nat > sum_sum_a_nat,F: nat > nat,V: list_nat] :
( ( map_na823391071729141993_a_nat @ G @ ( map_nat_nat @ F @ V ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_703_list_Omap__comp,axiom,
! [G: nat > sum_sum_a_nat,F: a > nat,V: list_a] :
( ( map_na823391071729141993_a_nat @ G @ ( map_a_nat @ F @ V ) )
= ( map_a_Sum_sum_a_nat @ ( comp_n4273600690000831346_nat_a @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_704_list_Omap__comp,axiom,
! [G: a > sum_sum_a_nat,F: nat > a,V: list_nat] :
( ( map_a_Sum_sum_a_nat @ G @ ( map_nat_a @ F @ V ) )
= ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_705_list_Omap__comp,axiom,
! [G: a > sum_sum_a_nat,F: a > a,V: list_a] :
( ( map_a_Sum_sum_a_nat @ G @ ( map_a_a @ F @ V ) )
= ( map_a_Sum_sum_a_nat @ ( comp_a5877636538679507994_nat_a @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_706_list_Omap__comp,axiom,
! [G: nat > sum_sum_a_nat,F: sum_sum_a_nat > nat,V: list_Sum_sum_a_nat] :
( ( map_na823391071729141993_a_nat @ G @ ( map_Su5227373005390213643at_nat @ F @ V ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_n241567065475451507_a_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_707_list_Omap__comp,axiom,
! [G: a > sum_sum_a_nat,F: sum_sum_a_nat > a,V: list_Sum_sum_a_nat] :
( ( map_a_Sum_sum_a_nat @ G @ ( map_Sum_sum_a_nat_a @ F @ V ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_a7301103983715706651_a_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_708_list_Omap__comp,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: nat > sum_sum_a_nat,V: list_nat] :
( ( map_Su2790769393171190532_a_nat @ G @ ( map_na823391071729141993_a_nat @ F @ V ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_709_list_Omap__comp,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: a > sum_sum_a_nat,V: list_a] :
( ( map_Su2790769393171190532_a_nat @ G @ ( map_a_Sum_sum_a_nat @ F @ V ) )
= ( map_a_Sum_sum_a_nat @ ( comp_S1182857061046812571_nat_a @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_710_list_Omap__comp,axiom,
! [G: list_a > list_Sum_sum_a_nat,F: list_a > list_a,V: list_list_a] :
( ( map_li6996171348340886213_a_nat @ G @ ( map_list_a_list_a @ F @ V ) )
= ( map_li6996171348340886213_a_nat @ ( comp_l3066312113933711776list_a @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_711_list_Omap__comp,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,V: list_Sum_sum_a_nat] :
( ( map_Su2790769393171190532_a_nat @ G @ ( map_Su2790769393171190532_a_nat @ F @ V ) )
= ( map_Su2790769393171190532_a_nat @ ( comp_S4540598949873307932_a_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_712_K__record__comp,axiom,
! [C: sum_sum_a_nat,F: nat > a] :
( ( comp_a4874482103519161268at_nat
@ ^ [X: a] : C
@ F )
= ( ^ [X: nat] : C ) ) ).
% K_record_comp
thf(fact_713_surj__fun__eq,axiom,
! [F: nat > a,X5: set_nat,G1: a > sum_sum_a_nat,G2: a > sum_sum_a_nat] :
( ( ( image_nat_a @ F @ X5 )
= top_top_set_a )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ( comp_a4874482103519161268at_nat @ G1 @ F @ X2 )
= ( comp_a4874482103519161268at_nat @ G2 @ F @ X2 ) ) )
=> ( G1 = G2 ) ) ) ).
% surj_fun_eq
thf(fact_714_sat__esat__conv,axiom,
! [Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
( ( wf_fo_intp_a_b @ Phi2 @ I2 )
=> ( ( sat_a_b @ Phi2 @ I2 @ Sigma2 )
= ( esat_a_b @ Phi2 @ I2 @ ( comp_a4874482103519161268at_nat @ sum_Inl_a_nat @ Sigma2 ) @ top_to795618464972521135_a_nat ) ) ) ).
% sat_esat_conv
thf(fact_715_order__antisym__conv,axiom,
! [Y2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y2 @ X3 )
=> ( ( ord_le1325389633284124927_a_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_716_order__antisym__conv,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_717_order__antisym__conv,axiom,
! [Y2: set_o,X3: set_o] :
( ( ord_less_eq_set_o @ Y2 @ X3 )
=> ( ( ord_less_eq_set_o @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_718_order__antisym__conv,axiom,
! [Y2: set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ Y2 @ X3 )
=> ( ( ord_le1147066620699065093_a_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_719_linorder__le__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_720_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_721_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_o,C: set_o] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_722_ord__le__eq__subst,axiom,
! [A: set_o,B2: set_o,F: set_o > nat,C: nat] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_723_ord__le__eq__subst,axiom,
! [A: set_o,B2: set_o,F: set_o > set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_724_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_725_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_726_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_o,C: set_o] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_727_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_728_ord__le__eq__subst,axiom,
! [A: set_o,B2: set_o,F: set_o > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_729_ord__le__eq__subst,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > nat,C: nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_730_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_731_ord__eq__le__subst,axiom,
! [A: set_o,F: nat > set_o,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_732_ord__eq__le__subst,axiom,
! [A: nat,F: set_o > nat,B2: set_o,C: set_o] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_733_ord__eq__le__subst,axiom,
! [A: set_o,F: set_o > set_o,B2: set_o,C: set_o] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_734_ord__eq__le__subst,axiom,
! [A: nat,F: set_Sum_sum_a_nat > nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_735_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_a_nat,F: nat > set_Sum_sum_a_nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_736_ord__eq__le__subst,axiom,
! [A: set_o,F: set_Sum_sum_a_nat > set_o,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_737_ord__eq__le__subst,axiom,
! [A: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1147066620699065093_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_738_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_a_nat,F: set_o > set_Sum_sum_a_nat,B2: set_o,C: set_o] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_739_ord__eq__le__subst,axiom,
! [A: nat,F: set_li6526943997496501093_a_nat > nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ! [X2: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_740_linorder__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_741_order__eq__refl,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( X3 = Y2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_742_order__eq__refl,axiom,
! [X3: nat,Y2: nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_743_order__eq__refl,axiom,
! [X3: set_o,Y2: set_o] :
( ( X3 = Y2 )
=> ( ord_less_eq_set_o @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_744_order__eq__refl,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( X3 = Y2 )
=> ( ord_le1147066620699065093_a_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_745_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_746_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_o,C: set_o] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_o @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_747_order__subst2,axiom,
! [A: set_o,B2: set_o,F: set_o > nat,C: nat] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_748_order__subst2,axiom,
! [A: set_o,B2: set_o,F: set_o > set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ord_less_eq_set_o @ ( F @ B2 ) @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_749_order__subst2,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_750_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_751_order__subst2,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_o,C: set_o] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_less_eq_set_o @ ( F @ B2 ) @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_752_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_753_order__subst2,axiom,
! [A: set_o,B2: set_o,F: set_o > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_754_order__subst2,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat > nat,C: nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_755_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_756_order__subst1,axiom,
! [A: nat,F: set_o > nat,B2: set_o,C: set_o] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_757_order__subst1,axiom,
! [A: set_o,F: nat > set_o,B2: nat,C: nat] :
( ( ord_less_eq_set_o @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_758_order__subst1,axiom,
! [A: set_o,F: set_o > set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_759_order__subst1,axiom,
! [A: set_Sum_sum_a_nat,F: nat > set_Sum_sum_a_nat,B2: nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_760_order__subst1,axiom,
! [A: nat,F: set_Sum_sum_a_nat > nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_761_order__subst1,axiom,
! [A: set_Sum_sum_a_nat,F: set_o > set_Sum_sum_a_nat,B2: set_o,C: set_o] :
( ( ord_le1325389633284124927_a_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ! [X2: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_762_order__subst1,axiom,
! [A: nat,F: set_li6526943997496501093_a_nat > nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ! [X2: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_763_order__subst1,axiom,
! [A: set_o,F: set_Sum_sum_a_nat > set_o,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ A @ ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_764_order__subst1,axiom,
! [A: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat,B2: nat,C: nat] :
( ( ord_le1147066620699065093_a_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1147066620699065093_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_765_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y = Z2 ) )
= ( ^ [A5: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A5 @ B4 )
& ( ord_le1325389633284124927_a_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_766_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_767_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_o,Z2: set_o] : ( Y = Z2 ) )
= ( ^ [A5: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A5 @ B4 )
& ( ord_less_eq_set_o @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_768_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] : ( Y = Z2 ) )
= ( ^ [A5: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A5 @ B4 )
& ( ord_le1147066620699065093_a_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_769_antisym,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_770_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_771_antisym,axiom,
! [A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_772_antisym,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_773_dual__order_Otrans,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ C @ B2 )
=> ( ord_le1325389633284124927_a_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_774_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_775_dual__order_Otrans,axiom,
! [B2: set_o,A: set_o,C: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( ( ord_less_eq_set_o @ C @ B2 )
=> ( ord_less_eq_set_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_776_dual__order_Otrans,axiom,
! [B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ C @ B2 )
=> ( ord_le1147066620699065093_a_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_777_dual__order_Oantisym,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_778_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_779_dual__order_Oantisym,axiom,
! [B2: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( ( ord_less_eq_set_o @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_780_dual__order_Oantisym,axiom,
! [B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_781_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y = Z2 ) )
= ( ^ [A5: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B4 @ A5 )
& ( ord_le1325389633284124927_a_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_782_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_783_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_o,Z2: set_o] : ( Y = Z2 ) )
= ( ^ [A5: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ B4 @ A5 )
& ( ord_less_eq_set_o @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_784_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] : ( Y = Z2 ) )
= ( ^ [A5: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B4 @ A5 )
& ( ord_le1147066620699065093_a_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_785_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: nat,B5: nat] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_786_order__trans,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y2 )
=> ( ( ord_le1325389633284124927_a_nat @ Y2 @ Z )
=> ( ord_le1325389633284124927_a_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_787_order__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_788_order__trans,axiom,
! [X3: set_o,Y2: set_o,Z: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y2 )
=> ( ( ord_less_eq_set_o @ Y2 @ Z )
=> ( ord_less_eq_set_o @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_789_order__trans,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat,Z: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y2 )
=> ( ( ord_le1147066620699065093_a_nat @ Y2 @ Z )
=> ( ord_le1147066620699065093_a_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_790_order_Otrans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_791_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_792_order_Otrans,axiom,
! [A: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ord_less_eq_set_o @ A @ C ) ) ) ).
% order.trans
thf(fact_793_order_Otrans,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ord_le1147066620699065093_a_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_794_order__antisym,axiom,
! [X3: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y2 )
=> ( ( ord_le1325389633284124927_a_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_795_order__antisym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_796_order__antisym,axiom,
! [X3: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y2 )
=> ( ( ord_less_eq_set_o @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_797_order__antisym,axiom,
! [X3: set_li6526943997496501093_a_nat,Y2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X3 @ Y2 )
=> ( ( ord_le1147066620699065093_a_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_798_ord__le__eq__trans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_799_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_800_ord__le__eq__trans,axiom,
! [A: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_801_ord__le__eq__trans,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le1147066620699065093_a_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_802_ord__eq__le__trans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A = B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_803_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_804_ord__eq__le__trans,axiom,
! [A: set_o,B2: set_o,C: set_o] :
( ( A = B2 )
=> ( ( ord_less_eq_set_o @ B2 @ C )
=> ( ord_less_eq_set_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_805_ord__eq__le__trans,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( A = B2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C )
=> ( ord_le1147066620699065093_a_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_806_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y = Z2 ) )
= ( ^ [X: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ Y3 )
& ( ord_le1325389633284124927_a_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_807_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_808_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_o,Z2: set_o] : ( Y = Z2 ) )
= ( ^ [X: set_o,Y3: set_o] :
( ( ord_less_eq_set_o @ X @ Y3 )
& ( ord_less_eq_set_o @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_809_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_li6526943997496501093_a_nat,Z2: set_li6526943997496501093_a_nat] : ( Y = Z2 ) )
= ( ^ [X: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X @ Y3 )
& ( ord_le1147066620699065093_a_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_810_le__cases3,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_811_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_812_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_li6526943997496501093_a_nat,K: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( A3
= ( inf_in3249246906714053971_a_nat @ K @ A ) )
=> ( ( inf_in3249246906714053971_a_nat @ A3 @ B2 )
= ( inf_in3249246906714053971_a_nat @ K @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_813_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_li6526943997496501093_a_nat,K: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( B
= ( inf_in3249246906714053971_a_nat @ K @ B2 ) )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B )
= ( inf_in3249246906714053971_a_nat @ K @ ( inf_in3249246906714053971_a_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_814_prop__restrict,axiom,
! [X3: nat > sum_sum_a_nat,Z4: set_na3699693778330250182_a_nat,X5: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X3 @ Z4 )
=> ( ( ord_le8108555184339247974_a_nat @ Z4
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_815_prop__restrict,axiom,
! [X3: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_816_prop__restrict,axiom,
! [X3: sum_sum_a_nat,Z4: set_Sum_sum_a_nat,X5: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X3 @ Z4 )
=> ( ( ord_le1325389633284124927_a_nat @ Z4
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_817_prop__restrict,axiom,
! [X3: $o,Z4: set_o,X5: set_o,P: $o > $o] :
( ( member_o @ X3 @ Z4 )
=> ( ( ord_less_eq_set_o @ Z4
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_818_prop__restrict,axiom,
! [X3: list_Sum_sum_a_nat,Z4: set_li6526943997496501093_a_nat,X5: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X3 @ Z4 )
=> ( ( ord_le1147066620699065093_a_nat @ Z4
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_819_Collect__restrict,axiom,
! [X5: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_820_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_821_Collect__restrict,axiom,
! [X5: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_822_Collect__restrict,axiom,
! [X5: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_823_Collect__restrict,axiom,
! [X5: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_824_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_825_top_Oextremum__uniqueI,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ top_top_set_list_a @ A )
=> ( A = top_top_set_list_a ) ) ).
% top.extremum_uniqueI
thf(fact_826_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_827_top_Oextremum__uniqueI,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ top_to795618464972521135_a_nat @ A )
=> ( A = top_to795618464972521135_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_828_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_829_top_Oextremum__uniqueI,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ top_to6433055325616222389_a_nat @ A )
=> ( A = top_to6433055325616222389_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_830_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_831_top_Oextremum__unique,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ top_top_set_list_a @ A )
= ( A = top_top_set_list_a ) ) ).
% top.extremum_unique
thf(fact_832_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_833_top_Oextremum__unique,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ top_to795618464972521135_a_nat @ A )
= ( A = top_to795618464972521135_a_nat ) ) ).
% top.extremum_unique
thf(fact_834_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_835_top_Oextremum__unique,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ top_to6433055325616222389_a_nat @ A )
= ( A = top_to6433055325616222389_a_nat ) ) ).
% top.extremum_unique
thf(fact_836_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_837_top__greatest,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ top_top_set_list_a ) ).
% top_greatest
thf(fact_838_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_839_top__greatest,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ top_to795618464972521135_a_nat ) ).
% top_greatest
thf(fact_840_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_841_top__greatest,axiom,
! [A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ A @ top_to6433055325616222389_a_nat ) ).
% top_greatest
thf(fact_842_comp__surj,axiom,
! [F: a > $o,G: $o > $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
=> ( ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_a_o @ ( comp_o_o_a @ G @ F ) @ top_top_set_a )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_843_comp__surj,axiom,
! [F: a > $o,G: $o > a] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
=> ( ( ( image_o_a @ G @ top_top_set_o )
= top_top_set_a )
=> ( ( image_a_a @ ( comp_o_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_844_comp__surj,axiom,
! [F: a > $o,G: $o > nat] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
=> ( ( ( image_o_nat @ G @ top_top_set_o )
= top_top_set_nat )
=> ( ( image_a_nat @ ( comp_o_nat_a @ G @ F ) @ top_top_set_a )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_845_comp__surj,axiom,
! [F: a > a,G: a > $o] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_o @ G @ top_top_set_a )
= top_top_set_o )
=> ( ( image_a_o @ ( comp_a_o_a @ G @ F ) @ top_top_set_a )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_846_comp__surj,axiom,
! [F: a > a,G: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a @ ( comp_a_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_847_comp__surj,axiom,
! [F: a > a,G: a > nat] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat )
=> ( ( image_a_nat @ ( comp_a_nat_a @ G @ F ) @ top_top_set_a )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_848_comp__surj,axiom,
! [F: a > nat,G: nat > $o] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( ( image_nat_o @ G @ top_top_set_nat )
= top_top_set_o )
=> ( ( image_a_o @ ( comp_nat_o_a @ G @ F ) @ top_top_set_a )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_849_comp__surj,axiom,
! [F: a > nat,G: nat > a] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( ( image_nat_a @ G @ top_top_set_nat )
= top_top_set_a )
=> ( ( image_a_a @ ( comp_nat_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_850_comp__surj,axiom,
! [F: a > nat,G: nat > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_a_nat @ ( comp_nat_nat_a @ G @ F ) @ top_top_set_a )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_851_comp__surj,axiom,
! [F: nat > $o,G: $o > $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ( ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_nat_o @ ( comp_o_o_nat @ G @ F ) @ top_top_set_nat )
= top_top_set_o ) ) ) ).
% comp_surj
thf(fact_852_fun_Omap__ident__strong,axiom,
! [T: a > $o,F: $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_a_o @ T @ top_top_set_a ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_o_o_a @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_853_fun_Omap__ident__strong,axiom,
! [T: nat > $o,F: $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_nat_o @ T @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_o_o_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_854_fun_Omap__ident__strong,axiom,
! [T: list_a > $o,F: $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_list_a_o @ T @ top_top_set_list_a ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_o_o_list_a @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_855_fun_Omap__ident__strong,axiom,
! [T: sum_sum_a_nat > $o,F: $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_6095136190293192542_nat_o @ T @ top_to795618464972521135_a_nat ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_o835585038542465372_a_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_856_fun_Omap__ident__strong,axiom,
! [T: a > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z3 @ ( image_7873763678140191238_a_nat @ T @ top_top_set_a ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_S1182857061046812571_nat_a @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_857_fun_Omap__ident__strong,axiom,
! [T: nat > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z3 @ ( image_7293268710728258664_a_nat @ T @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_S2395038772458240499at_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_858_fun_Omap__ident__strong,axiom,
! [T: a > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z3 @ ( image_7897140031816760844_a_nat @ T @ top_top_set_a ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_l7232043298170841371_nat_a @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_859_fun_Omap__ident__strong,axiom,
! [T: list_a > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z3 @ ( image_4532641577586004224_a_nat @ T @ top_top_set_list_a ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_S8917754823762946465list_a @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_860_fun_Omap__ident__strong,axiom,
! [T: nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z3 @ ( image_6262589752765146990_a_nat @ T @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_l2581977384674930803at_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_861_fun_Omap__ident__strong,axiom,
! [T: sum_sum_a_nat > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z3 @ ( image_7142520692256960453_a_nat @ T @ top_to795618464972521135_a_nat ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( comp_S4540598949873307932_a_nat @ F @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_862_fun_Oinj__map__strong,axiom,
! [X3: nat > a,Xa: nat > a,F: a > sum_sum_a_nat,Fa: a > sum_sum_a_nat] :
( ! [Z3: a,Za: a] :
( ( member_a @ Z3 @ ( image_nat_a @ X3 @ top_top_set_nat ) )
=> ( ( member_a @ Za @ ( image_nat_a @ Xa @ top_top_set_nat ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp_a4874482103519161268at_nat @ F @ X3 )
= ( comp_a4874482103519161268at_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% fun.inj_map_strong
thf(fact_863_fun_Omap__cong0,axiom,
! [X3: nat > a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_nat_a @ X3 @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_a4874482103519161268at_nat @ F @ X3 )
= ( comp_a4874482103519161268at_nat @ G @ X3 ) ) ) ).
% fun.map_cong0
thf(fact_864_fun_Omap__cong,axiom,
! [X3: nat > a,Ya: nat > a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z3: a] :
( ( member_a @ Z3 @ ( image_nat_a @ Ya @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_a4874482103519161268at_nat @ F @ X3 )
= ( comp_a4874482103519161268at_nat @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_865_fun_Oset__map,axiom,
! [F: $o > $o,V: a > $o] :
( ( image_a_o @ ( comp_o_o_a @ F @ V ) @ top_top_set_a )
= ( image_o_o @ F @ ( image_a_o @ V @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_866_fun_Oset__map,axiom,
! [F: nat > $o,V: a > nat] :
( ( image_a_o @ ( comp_nat_o_a @ F @ V ) @ top_top_set_a )
= ( image_nat_o @ F @ ( image_a_nat @ V @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_867_fun_Oset__map,axiom,
! [F: a > $o,V: a > a] :
( ( image_a_o @ ( comp_a_o_a @ F @ V ) @ top_top_set_a )
= ( image_a_o @ F @ ( image_a_a @ V @ top_top_set_a ) ) ) ).
% fun.set_map
thf(fact_868_fun_Oset__map,axiom,
! [F: $o > $o,V: nat > $o] :
( ( image_nat_o @ ( comp_o_o_nat @ F @ V ) @ top_top_set_nat )
= ( image_o_o @ F @ ( image_nat_o @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_869_fun_Oset__map,axiom,
! [F: nat > $o,V: nat > nat] :
( ( image_nat_o @ ( comp_nat_o_nat @ F @ V ) @ top_top_set_nat )
= ( image_nat_o @ F @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_870_fun_Oset__map,axiom,
! [F: a > $o,V: nat > a] :
( ( image_nat_o @ ( comp_a_o_nat @ F @ V ) @ top_top_set_nat )
= ( image_a_o @ F @ ( image_nat_a @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_871_fun_Oset__map,axiom,
! [F: nat > $o,V: list_a > nat] :
( ( image_list_a_o @ ( comp_nat_o_list_a @ F @ V ) @ top_top_set_list_a )
= ( image_nat_o @ F @ ( image_list_a_nat @ V @ top_top_set_list_a ) ) ) ).
% fun.set_map
thf(fact_872_fun_Oset__map,axiom,
! [F: a > $o,V: list_a > a] :
( ( image_list_a_o @ ( comp_a_o_list_a @ F @ V ) @ top_top_set_list_a )
= ( image_a_o @ F @ ( image_list_a_a @ V @ top_top_set_list_a ) ) ) ).
% fun.set_map
thf(fact_873_fun_Oset__map,axiom,
! [F: nat > $o,V: sum_sum_a_nat > nat] :
( ( image_6095136190293192542_nat_o @ ( comp_n3088003301918311578_a_nat @ F @ V ) @ top_to795618464972521135_a_nat )
= ( image_nat_o @ F @ ( image_2473878607534554506at_nat @ V @ top_to795618464972521135_a_nat ) ) ) ).
% fun.set_map
thf(fact_874_fun_Oset__map,axiom,
! [F: a > $o,V: sum_sum_a_nat > a] :
( ( image_6095136190293192542_nat_o @ ( comp_a6548673106616709442_a_nat @ F @ V ) @ top_to795618464972521135_a_nat )
= ( image_a_o @ F @ ( image_6322530041254294468_nat_a @ V @ top_to795618464972521135_a_nat ) ) ) ).
% fun.set_map
thf(fact_875_inf__Int__eq,axiom,
! [R2: set_o,S: set_o] :
( ( inf_inf_o_o
@ ^ [X: $o] : ( member_o @ X @ R2 )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( inf_inf_set_o @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_876_inf__Int__eq,axiom,
! [R2: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( inf_in5242522483218605776_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R2 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_877_inf__Int__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( inf_in954358986474102090_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R2 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( inf_in3249246906714053971_a_nat @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_878_pred__subset__eq,axiom,
! [R2: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( ord_le1477630214076318366_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R2 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ord_le1325389633284124927_a_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_879_pred__subset__eq,axiom,
! [R2: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ R2 )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ord_less_eq_set_o @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_880_pred__subset__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( ord_le8737610411969296920_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R2 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ord_le1147066620699065093_a_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_881_comp__apply,axiom,
( comp_a4874482103519161268at_nat
= ( ^ [F2: a > sum_sum_a_nat,G3: nat > a,X: nat] : ( F2 @ ( G3 @ X ) ) ) ) ).
% comp_apply
thf(fact_882_comp__eq__dest__lhs,axiom,
! [A: a > sum_sum_a_nat,B2: nat > a,C: nat > sum_sum_a_nat,V: nat] :
( ( ( comp_a4874482103519161268at_nat @ A @ B2 )
= C )
=> ( ( A @ ( B2 @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_883_comp__eq__elim,axiom,
! [A: a > sum_sum_a_nat,B2: nat > a,C: a > sum_sum_a_nat,D2: nat > a] :
( ( ( comp_a4874482103519161268at_nat @ A @ B2 )
= ( comp_a4874482103519161268at_nat @ C @ D2 ) )
=> ! [V2: nat] :
( ( A @ ( B2 @ V2 ) )
= ( C @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_884_comp__eq__dest,axiom,
! [A: a > sum_sum_a_nat,B2: nat > a,C: a > sum_sum_a_nat,D2: nat > a,V: nat] :
( ( ( comp_a4874482103519161268at_nat @ A @ B2 )
= ( comp_a4874482103519161268at_nat @ C @ D2 ) )
=> ( ( A @ ( B2 @ V ) )
= ( C @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_885_comp__assoc,axiom,
! [F: a > sum_sum_a_nat,G: nat > a,H: nat > nat] :
( ( comp_n1522318729830440540at_nat @ ( comp_a4874482103519161268at_nat @ F @ G ) @ H )
= ( comp_a4874482103519161268at_nat @ F @ ( comp_nat_a_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_886_comp__assoc,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat,H: nat > a] :
( ( comp_a4874482103519161268at_nat @ ( comp_S1182857061046812571_nat_a @ F @ G ) @ H )
= ( comp_S2395038772458240499at_nat @ F @ ( comp_a4874482103519161268at_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_887_comp__assoc,axiom,
! [F: a > sum_sum_a_nat,G: a > a,H: nat > a] :
( ( comp_a4874482103519161268at_nat @ ( comp_a5877636538679507994_nat_a @ F @ G ) @ H )
= ( comp_a4874482103519161268at_nat @ F @ ( comp_a_a_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_888_comp__def,axiom,
( comp_a4874482103519161268at_nat
= ( ^ [F2: a > sum_sum_a_nat,G3: nat > a,X: nat] : ( F2 @ ( G3 @ X ) ) ) ) ).
% comp_def
thf(fact_889_fun_Omap__comp,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: a > sum_sum_a_nat,V: nat > a] :
( ( comp_S2395038772458240499at_nat @ G @ ( comp_a4874482103519161268at_nat @ F @ V ) )
= ( comp_a4874482103519161268at_nat @ ( comp_S1182857061046812571_nat_a @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_890_fun_Omap__comp,axiom,
! [G: a > sum_sum_a_nat,F: nat > a,V: nat > nat] :
( ( comp_a4874482103519161268at_nat @ G @ ( comp_nat_a_nat @ F @ V ) )
= ( comp_n1522318729830440540at_nat @ ( comp_a4874482103519161268at_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_891_fun_Omap__comp,axiom,
! [G: a > sum_sum_a_nat,F: a > a,V: nat > a] :
( ( comp_a4874482103519161268at_nat @ G @ ( comp_a_a_nat @ F @ V ) )
= ( comp_a4874482103519161268at_nat @ ( comp_a5877636538679507994_nat_a @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_892_surjD,axiom,
! [F: a > $o,Y2: $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
=> ? [X2: a] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_893_surjD,axiom,
! [F: a > a,Y2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X2: a] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_894_surjD,axiom,
! [F: a > nat,Y2: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ? [X2: a] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_895_surjD,axiom,
! [F: nat > $o,Y2: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_896_surjD,axiom,
! [F: nat > a,Y2: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_897_surjD,axiom,
! [F: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_898_surjD,axiom,
! [F: a > list_a,Y2: list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
=> ? [X2: a] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_899_surjD,axiom,
! [F: list_a > a,Y2: a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
=> ? [X2: list_a] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_900_surjD,axiom,
! [F: list_a > nat,Y2: nat] :
( ( ( image_list_a_nat @ F @ top_top_set_list_a )
= top_top_set_nat )
=> ? [X2: list_a] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_901_surjD,axiom,
! [F: nat > list_a,Y2: list_a] :
( ( ( image_nat_list_a @ F @ top_top_set_nat )
= top_top_set_list_a )
=> ? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_902_surjE,axiom,
! [F: a > $o,Y2: $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
=> ~ ! [X2: a] :
( Y2
= ( ~ ( F @ X2 ) ) ) ) ).
% surjE
thf(fact_903_surjE,axiom,
! [F: a > a,Y2: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X2: a] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_904_surjE,axiom,
! [F: a > nat,Y2: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ~ ! [X2: a] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_905_surjE,axiom,
! [F: nat > $o,Y2: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ~ ! [X2: nat] :
( Y2
= ( ~ ( F @ X2 ) ) ) ) ).
% surjE
thf(fact_906_surjE,axiom,
! [F: nat > a,Y2: a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ~ ! [X2: nat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_907_surjE,axiom,
! [F: nat > nat,Y2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_908_surjE,axiom,
! [F: a > list_a,Y2: list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
=> ~ ! [X2: a] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_909_surjE,axiom,
! [F: list_a > a,Y2: a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
=> ~ ! [X2: list_a] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_910_surjE,axiom,
! [F: list_a > nat,Y2: nat] :
( ( ( image_list_a_nat @ F @ top_top_set_list_a )
= top_top_set_nat )
=> ~ ! [X2: list_a] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_911_surjE,axiom,
! [F: nat > list_a,Y2: list_a] :
( ( ( image_nat_list_a @ F @ top_top_set_nat )
= top_top_set_list_a )
=> ~ ! [X2: nat] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_912_surjI,axiom,
! [G: a > $o,F: $o > a] :
( ! [X2: $o] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_o @ G @ top_top_set_a )
= top_top_set_o ) ) ).
% surjI
thf(fact_913_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X2: a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_914_surjI,axiom,
! [G: a > nat,F: nat > a] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat ) ) ).
% surjI
thf(fact_915_surjI,axiom,
! [G: nat > $o,F: $o > nat] :
( ! [X2: $o] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_o @ G @ top_top_set_nat )
= top_top_set_o ) ) ).
% surjI
thf(fact_916_surjI,axiom,
! [G: nat > a,F: a > nat] :
( ! [X2: a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_a @ G @ top_top_set_nat )
= top_top_set_a ) ) ).
% surjI
thf(fact_917_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_918_surjI,axiom,
! [G: a > list_a,F: list_a > a] :
( ! [X2: list_a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_list_a @ G @ top_top_set_a )
= top_top_set_list_a ) ) ).
% surjI
thf(fact_919_surjI,axiom,
! [G: list_a > a,F: a > list_a] :
( ! [X2: a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_list_a_a @ G @ top_top_set_list_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_920_surjI,axiom,
! [G: list_a > nat,F: nat > list_a] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_list_a_nat @ G @ top_top_set_list_a )
= top_top_set_nat ) ) ).
% surjI
thf(fact_921_surjI,axiom,
! [G: nat > list_a,F: list_a > nat] :
( ! [X2: list_a] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_list_a @ G @ top_top_set_nat )
= top_top_set_list_a ) ) ).
% surjI
thf(fact_922_surj__def,axiom,
! [F: a > $o] :
( ( ( image_a_o @ F @ top_top_set_a )
= top_top_set_o )
= ( ! [Y3: $o] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_923_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y3: a] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_924_surj__def,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_925_surj__def,axiom,
! [F: nat > $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
= ( ! [Y3: $o] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_926_surj__def,axiom,
! [F: nat > a] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
= ( ! [Y3: a] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_927_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_928_surj__def,axiom,
! [F: a > list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
= ( ! [Y3: list_a] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_929_surj__def,axiom,
! [F: list_a > a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
= ( ! [Y3: a] :
? [X: list_a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_930_surj__def,axiom,
! [F: list_a > nat] :
( ( ( image_list_a_nat @ F @ top_top_set_list_a )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X: list_a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_931_surj__def,axiom,
! [F: nat > list_a] :
( ( ( image_nat_list_a @ F @ top_top_set_nat )
= top_top_set_list_a )
= ( ! [Y3: list_a] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_932_image__eq__imp__comp,axiom,
! [F: nat > $o,A3: set_nat,G: nat > $o,B: set_nat,H: $o > $o] :
( ( ( image_nat_o @ F @ A3 )
= ( image_nat_o @ G @ B ) )
=> ( ( image_nat_o @ ( comp_o_o_nat @ H @ F ) @ A3 )
= ( image_nat_o @ ( comp_o_o_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_933_image__eq__imp__comp,axiom,
! [F: nat > $o,A3: set_nat,G: a > $o,B: set_a,H: $o > $o] :
( ( ( image_nat_o @ F @ A3 )
= ( image_a_o @ G @ B ) )
=> ( ( image_nat_o @ ( comp_o_o_nat @ H @ F ) @ A3 )
= ( image_a_o @ ( comp_o_o_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_934_image__eq__imp__comp,axiom,
! [F: a > $o,A3: set_a,G: nat > $o,B: set_nat,H: $o > $o] :
( ( ( image_a_o @ F @ A3 )
= ( image_nat_o @ G @ B ) )
=> ( ( image_a_o @ ( comp_o_o_a @ H @ F ) @ A3 )
= ( image_nat_o @ ( comp_o_o_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_935_image__eq__imp__comp,axiom,
! [F: a > $o,A3: set_a,G: a > $o,B: set_a,H: $o > $o] :
( ( ( image_a_o @ F @ A3 )
= ( image_a_o @ G @ B ) )
=> ( ( image_a_o @ ( comp_o_o_a @ H @ F ) @ A3 )
= ( image_a_o @ ( comp_o_o_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_936_image__eq__imp__comp,axiom,
! [F: a > a,A3: set_a,G: nat > a,B: set_nat,H: a > sum_sum_a_nat] :
( ( ( image_a_a @ F @ A3 )
= ( image_nat_a @ G @ B ) )
=> ( ( image_7873763678140191238_a_nat @ ( comp_a5877636538679507994_nat_a @ H @ F ) @ A3 )
= ( image_7293268710728258664_a_nat @ ( comp_a4874482103519161268at_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_937_image__eq__imp__comp,axiom,
! [F: nat > a,A3: set_nat,G: a > a,B: set_a,H: a > sum_sum_a_nat] :
( ( ( image_nat_a @ F @ A3 )
= ( image_a_a @ G @ B ) )
=> ( ( image_7293268710728258664_a_nat @ ( comp_a4874482103519161268at_nat @ H @ F ) @ A3 )
= ( image_7873763678140191238_a_nat @ ( comp_a5877636538679507994_nat_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_938_image__eq__imp__comp,axiom,
! [F: nat > a,A3: set_nat,G: nat > a,B: set_nat,H: a > sum_sum_a_nat] :
( ( ( image_nat_a @ F @ A3 )
= ( image_nat_a @ G @ B ) )
=> ( ( image_7293268710728258664_a_nat @ ( comp_a4874482103519161268at_nat @ H @ F ) @ A3 )
= ( image_7293268710728258664_a_nat @ ( comp_a4874482103519161268at_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_939_image__eq__imp__comp,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,G: a > sum_sum_a_nat,B: set_a,H: sum_sum_a_nat > $o] :
( ( ( image_7873763678140191238_a_nat @ F @ A3 )
= ( image_7873763678140191238_a_nat @ G @ B ) )
=> ( ( image_a_o @ ( comp_S6848359305846158914at_o_a @ H @ F ) @ A3 )
= ( image_a_o @ ( comp_S6848359305846158914at_o_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_940_image__eq__imp__comp,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,G: nat > sum_sum_a_nat,B: set_nat,H: sum_sum_a_nat > $o] :
( ( ( image_7873763678140191238_a_nat @ F @ A3 )
= ( image_7293268710728258664_a_nat @ G @ B ) )
=> ( ( image_a_o @ ( comp_S6848359305846158914at_o_a @ H @ F ) @ A3 )
= ( image_nat_o @ ( comp_S1344612384428341132_o_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_941_image__eq__imp__comp,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,G: a > sum_sum_a_nat,B: set_a,H: sum_sum_a_nat > $o] :
( ( ( image_7293268710728258664_a_nat @ F @ A3 )
= ( image_7873763678140191238_a_nat @ G @ B ) )
=> ( ( image_nat_o @ ( comp_S1344612384428341132_o_nat @ H @ F ) @ A3 )
= ( image_a_o @ ( comp_S6848359305846158914at_o_a @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_942_image__comp,axiom,
! [F: $o > $o,G: nat > $o,R3: set_nat] :
( ( image_o_o @ F @ ( image_nat_o @ G @ R3 ) )
= ( image_nat_o @ ( comp_o_o_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_943_image__comp,axiom,
! [F: $o > $o,G: a > $o,R3: set_a] :
( ( image_o_o @ F @ ( image_a_o @ G @ R3 ) )
= ( image_a_o @ ( comp_o_o_a @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_944_image__comp,axiom,
! [F: nat > $o,G: nat > nat,R3: set_nat] :
( ( image_nat_o @ F @ ( image_nat_nat @ G @ R3 ) )
= ( image_nat_o @ ( comp_nat_o_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_945_image__comp,axiom,
! [F: nat > $o,G: a > nat,R3: set_a] :
( ( image_nat_o @ F @ ( image_a_nat @ G @ R3 ) )
= ( image_a_o @ ( comp_nat_o_a @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_946_image__comp,axiom,
! [F: a > $o,G: nat > a,R3: set_nat] :
( ( image_a_o @ F @ ( image_nat_a @ G @ R3 ) )
= ( image_nat_o @ ( comp_a_o_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_947_image__comp,axiom,
! [F: a > $o,G: a > a,R3: set_a] :
( ( image_a_o @ F @ ( image_a_a @ G @ R3 ) )
= ( image_a_o @ ( comp_a_o_a @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_948_image__comp,axiom,
! [F: sum_sum_a_nat > $o,G: a > sum_sum_a_nat,R3: set_a] :
( ( image_6095136190293192542_nat_o @ F @ ( image_7873763678140191238_a_nat @ G @ R3 ) )
= ( image_a_o @ ( comp_S6848359305846158914at_o_a @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_949_image__comp,axiom,
! [F: sum_sum_a_nat > $o,G: nat > sum_sum_a_nat,R3: set_nat] :
( ( image_6095136190293192542_nat_o @ F @ ( image_7293268710728258664_a_nat @ G @ R3 ) )
= ( image_nat_o @ ( comp_S1344612384428341132_o_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_950_image__comp,axiom,
! [F: $o > sum_sum_a_nat,G: nat > $o,R3: set_nat] :
( ( image_4139480514073730540_a_nat @ F @ ( image_nat_o @ G @ R3 ) )
= ( image_7293268710728258664_a_nat @ ( comp_o753276326763720218at_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_951_image__comp,axiom,
! [F: $o > sum_sum_a_nat,G: a > $o,R3: set_a] :
( ( image_4139480514073730540_a_nat @ F @ ( image_a_o @ G @ R3 ) )
= ( image_7873763678140191238_a_nat @ ( comp_o4021350297272338228_nat_a @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_952_top__empty__eq,axiom,
( top_to7646693901462700136_nat_o
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ top_to6433055325616222389_a_nat ) ) ) ).
% top_empty_eq
thf(fact_953_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_954_top__empty__eq,axiom,
( top_to1565196397637005550_nat_o
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ top_to795618464972521135_a_nat ) ) ) ).
% top_empty_eq
thf(fact_955_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X: a] : ( member_a @ X @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_956_top__empty__eq,axiom,
( top_top_list_a_o
= ( ^ [X: list_a] : ( member_list_a @ X @ top_top_set_list_a ) ) ) ).
% top_empty_eq
thf(fact_957_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_958_proj__sat__vals,axiom,
( proj_sat_a_b
= ( ^ [Phi: fo_fmla_a_b,I3: product_prod_b_nat > set_list_a] : ( proj_vals_a @ ( collect_nat_a @ ( sat_a_b @ Phi @ I3 ) ) @ ( fv_fo_fmla_list_a_b @ Phi ) ) ) ) ).
% proj_sat_vals
thf(fact_959_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: $o > $o,F: nat > $o,A3: set_nat] :
( ( Sup @ ( image_o_o @ G @ ( image_nat_o @ F @ A3 ) ) )
= ( Sup @ ( image_nat_o @ ( comp_o_o_nat @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_960_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: $o > $o,F: a > $o,A3: set_a] :
( ( Sup @ ( image_o_o @ G @ ( image_a_o @ F @ A3 ) ) )
= ( Sup @ ( image_a_o @ ( comp_o_o_a @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_961_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: nat > $o,F: nat > nat,A3: set_nat] :
( ( Sup @ ( image_nat_o @ G @ ( image_nat_nat @ F @ A3 ) ) )
= ( Sup @ ( image_nat_o @ ( comp_nat_o_nat @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_962_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: nat > $o,F: a > nat,A3: set_a] :
( ( Sup @ ( image_nat_o @ G @ ( image_a_nat @ F @ A3 ) ) )
= ( Sup @ ( image_a_o @ ( comp_nat_o_a @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_963_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: a > $o,F: nat > a,A3: set_nat] :
( ( Sup @ ( image_a_o @ G @ ( image_nat_a @ F @ A3 ) ) )
= ( Sup @ ( image_nat_o @ ( comp_a_o_nat @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_964_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: a > $o,F: a > a,A3: set_a] :
( ( Sup @ ( image_a_o @ G @ ( image_a_a @ F @ A3 ) ) )
= ( Sup @ ( image_a_o @ ( comp_a_o_a @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_965_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: sum_sum_a_nat > $o,F: a > sum_sum_a_nat,A3: set_a] :
( ( Sup @ ( image_6095136190293192542_nat_o @ G @ ( image_7873763678140191238_a_nat @ F @ A3 ) ) )
= ( Sup @ ( image_a_o @ ( comp_S6848359305846158914at_o_a @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_966_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,A3: set_nat] :
( ( Sup @ ( image_6095136190293192542_nat_o @ G @ ( image_7293268710728258664_a_nat @ F @ A3 ) ) )
= ( Sup @ ( image_nat_o @ ( comp_S1344612384428341132_o_nat @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_967_Sup_OSUP__image,axiom,
! [Sup: set_Sum_sum_a_nat > sum_sum_a_nat,G: $o > sum_sum_a_nat,F: nat > $o,A3: set_nat] :
( ( Sup @ ( image_4139480514073730540_a_nat @ G @ ( image_nat_o @ F @ A3 ) ) )
= ( Sup @ ( image_7293268710728258664_a_nat @ ( comp_o753276326763720218at_nat @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_968_Sup_OSUP__image,axiom,
! [Sup: set_Sum_sum_a_nat > sum_sum_a_nat,G: $o > sum_sum_a_nat,F: a > $o,A3: set_a] :
( ( Sup @ ( image_4139480514073730540_a_nat @ G @ ( image_a_o @ F @ A3 ) ) )
= ( Sup @ ( image_7873763678140191238_a_nat @ ( comp_o4021350297272338228_nat_a @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_969_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: $o > $o,F: nat > $o,A3: set_nat] :
( ( Inf @ ( image_o_o @ G @ ( image_nat_o @ F @ A3 ) ) )
= ( Inf @ ( image_nat_o @ ( comp_o_o_nat @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_970_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: $o > $o,F: a > $o,A3: set_a] :
( ( Inf @ ( image_o_o @ G @ ( image_a_o @ F @ A3 ) ) )
= ( Inf @ ( image_a_o @ ( comp_o_o_a @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_971_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: nat > $o,F: nat > nat,A3: set_nat] :
( ( Inf @ ( image_nat_o @ G @ ( image_nat_nat @ F @ A3 ) ) )
= ( Inf @ ( image_nat_o @ ( comp_nat_o_nat @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_972_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: nat > $o,F: a > nat,A3: set_a] :
( ( Inf @ ( image_nat_o @ G @ ( image_a_nat @ F @ A3 ) ) )
= ( Inf @ ( image_a_o @ ( comp_nat_o_a @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_973_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: a > $o,F: nat > a,A3: set_nat] :
( ( Inf @ ( image_a_o @ G @ ( image_nat_a @ F @ A3 ) ) )
= ( Inf @ ( image_nat_o @ ( comp_a_o_nat @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_974_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: a > $o,F: a > a,A3: set_a] :
( ( Inf @ ( image_a_o @ G @ ( image_a_a @ F @ A3 ) ) )
= ( Inf @ ( image_a_o @ ( comp_a_o_a @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_975_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: sum_sum_a_nat > $o,F: a > sum_sum_a_nat,A3: set_a] :
( ( Inf @ ( image_6095136190293192542_nat_o @ G @ ( image_7873763678140191238_a_nat @ F @ A3 ) ) )
= ( Inf @ ( image_a_o @ ( comp_S6848359305846158914at_o_a @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_976_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,A3: set_nat] :
( ( Inf @ ( image_6095136190293192542_nat_o @ G @ ( image_7293268710728258664_a_nat @ F @ A3 ) ) )
= ( Inf @ ( image_nat_o @ ( comp_S1344612384428341132_o_nat @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_977_Inf_OINF__image,axiom,
! [Inf: set_Sum_sum_a_nat > sum_sum_a_nat,G: $o > sum_sum_a_nat,F: nat > $o,A3: set_nat] :
( ( Inf @ ( image_4139480514073730540_a_nat @ G @ ( image_nat_o @ F @ A3 ) ) )
= ( Inf @ ( image_7293268710728258664_a_nat @ ( comp_o753276326763720218at_nat @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_978_Inf_OINF__image,axiom,
! [Inf: set_Sum_sum_a_nat > sum_sum_a_nat,G: $o > sum_sum_a_nat,F: a > $o,A3: set_a] :
( ( Inf @ ( image_4139480514073730540_a_nat @ G @ ( image_a_o @ F @ A3 ) ) )
= ( Inf @ ( image_7873763678140191238_a_nat @ ( comp_o4021350297272338228_nat_a @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_979_proj__sat__def,axiom,
( proj_sat_a_b
= ( ^ [Phi: fo_fmla_a_b,I3: product_prod_b_nat > set_list_a] :
( image_nat_a_list_a
@ ^ [Sigma: nat > a] : ( map_nat_a @ Sigma @ ( fv_fo_fmla_list_a_b @ Phi ) )
@ ( collect_nat_a @ ( sat_a_b @ Phi @ I3 ) ) ) ) ) ).
% proj_sat_def
thf(fact_980_fo__nmlzd__all__AD,axiom,
! [Xs: list_Sum_sum_a_nat,AD: set_a] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( fo_nmlzd_a @ AD @ Xs ) ) ).
% fo_nmlzd_all_AD
thf(fact_981_sp__equiv__list__subset,axiom,
! [Ms: list_a,Ns2: list_a,Sigma2: a > sum_sum_a_nat,Sigma3: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ms ) @ ( set_a2 @ Ns2 ) )
=> ( ( sp_equiv_list_a_nat @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ns2 ) @ ( map_a_Sum_sum_a_nat @ Sigma3 @ Ns2 ) )
=> ( sp_equiv_list_a_nat @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ms ) @ ( map_a_Sum_sum_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_982_sp__equiv__list__subset,axiom,
! [Ms: list_nat,Ns2: list_nat,Sigma2: nat > sum_sum_a_nat,Sigma3: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns2 ) )
=> ( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns2 ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ns2 ) )
=> ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_983_sp__equiv__list__subset,axiom,
! [Ms: list_Sum_sum_a_nat,Ns2: list_Sum_sum_a_nat,Sigma2: sum_sum_a_nat > sum_sum_a_nat,Sigma3: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Ms ) @ ( set_Sum_sum_a_nat2 @ Ns2 ) )
=> ( ( sp_equiv_list_a_nat @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ns2 ) @ ( map_Su2790769393171190532_a_nat @ Sigma3 @ Ns2 ) )
=> ( sp_equiv_list_a_nat @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ms ) @ ( map_Su2790769393171190532_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_984_fo__nmlz__rec__set__rev,axiom,
! [I: nat,M2: sum_sum_a_nat > option_nat,AD: set_a,Xs: list_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ I @ M2 @ AD @ Xs ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) ) ) ).
% fo_nmlz_rec_set_rev
thf(fact_985_Sup_OSUP__cong,axiom,
! [A3: set_a,B: set_a,C2: a > sum_sum_a_nat,D: a > sum_sum_a_nat,Sup: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_7873763678140191238_a_nat @ C2 @ A3 ) )
= ( Sup @ ( image_7873763678140191238_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_986_Sup_OSUP__cong,axiom,
! [A3: set_list_a,B: set_list_a,C2: list_a > list_Sum_sum_a_nat,D: list_a > list_Sum_sum_a_nat,Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_674313660629153798_a_nat @ C2 @ A3 ) )
= ( Sup @ ( image_674313660629153798_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_987_Sup_OSUP__cong,axiom,
! [A3: set_nat,B: set_nat,C2: nat > sum_sum_a_nat,D: nat > sum_sum_a_nat,Sup: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_7293268710728258664_a_nat @ C2 @ A3 ) )
= ( Sup @ ( image_7293268710728258664_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_988_Sup_OSUP__cong,axiom,
! [A3: set_nat,B: set_nat,C2: nat > list_Sum_sum_a_nat,D: nat > list_Sum_sum_a_nat,Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_6262589752765146990_a_nat @ C2 @ A3 ) )
= ( Sup @ ( image_6262589752765146990_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_989_Sup_OSUP__cong,axiom,
! [A3: set_nat,B: set_nat,C2: nat > $o,D: nat > $o,Sup: set_o > $o] :
( ( A3 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_o @ C2 @ A3 ) )
= ( Sup @ ( image_nat_o @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_990_Sup_OSUP__cong,axiom,
! [A3: set_a,B: set_a,C2: a > list_Sum_sum_a_nat,D: a > list_Sum_sum_a_nat,Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_7897140031816760844_a_nat @ C2 @ A3 ) )
= ( Sup @ ( image_7897140031816760844_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_991_Sup_OSUP__cong,axiom,
! [A3: set_a,B: set_a,C2: a > $o,D: a > $o,Sup: set_o > $o] :
( ( A3 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_a_o @ C2 @ A3 ) )
= ( Sup @ ( image_a_o @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_992_Inf_OINF__cong,axiom,
! [A3: set_a,B: set_a,C2: a > sum_sum_a_nat,D: a > sum_sum_a_nat,Inf: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_7873763678140191238_a_nat @ C2 @ A3 ) )
= ( Inf @ ( image_7873763678140191238_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_993_Inf_OINF__cong,axiom,
! [A3: set_list_a,B: set_list_a,C2: list_a > list_Sum_sum_a_nat,D: list_a > list_Sum_sum_a_nat,Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_674313660629153798_a_nat @ C2 @ A3 ) )
= ( Inf @ ( image_674313660629153798_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_994_Inf_OINF__cong,axiom,
! [A3: set_nat,B: set_nat,C2: nat > sum_sum_a_nat,D: nat > sum_sum_a_nat,Inf: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_7293268710728258664_a_nat @ C2 @ A3 ) )
= ( Inf @ ( image_7293268710728258664_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_995_Inf_OINF__cong,axiom,
! [A3: set_nat,B: set_nat,C2: nat > list_Sum_sum_a_nat,D: nat > list_Sum_sum_a_nat,Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_6262589752765146990_a_nat @ C2 @ A3 ) )
= ( Inf @ ( image_6262589752765146990_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_996_Inf_OINF__cong,axiom,
! [A3: set_nat,B: set_nat,C2: nat > $o,D: nat > $o,Inf: set_o > $o] :
( ( A3 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_o @ C2 @ A3 ) )
= ( Inf @ ( image_nat_o @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_997_Inf_OINF__cong,axiom,
! [A3: set_a,B: set_a,C2: a > list_Sum_sum_a_nat,D: a > list_Sum_sum_a_nat,Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A3 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_7897140031816760844_a_nat @ C2 @ A3 ) )
= ( Inf @ ( image_7897140031816760844_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_998_Inf_OINF__cong,axiom,
! [A3: set_a,B: set_a,C2: a > $o,D: a > $o,Inf: set_o > $o] :
( ( A3 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_a_o @ C2 @ A3 ) )
= ( Inf @ ( image_a_o @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_999_sp__equiv__list__link,axiom,
! [Sigma2: nat > sum_sum_a_nat,Ns2: list_nat,Tau2: nat > sum_sum_a_nat] :
( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns2 ) @ ( map_na823391071729141993_a_nat @ Tau2 @ Ns2 ) )
= ( sp_equiv_a_nat @ Sigma2 @ Tau2 @ ( set_nat2 @ Ns2 ) ) ) ).
% sp_equiv_list_link
thf(fact_1000_ad__agr__list__eq,axiom,
! [Ys: list_a,AD: set_a,Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ys ) @ AD )
=> ( ( ad_agr_list_a_nat @ AD @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Xs ) @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% ad_agr_list_eq
thf(fact_1001_map__eq__map__tailrec,axiom,
map_na823391071729141993_a_nat = map_ta1136998156224711455_a_nat ).
% map_eq_map_tailrec
thf(fact_1002_map__eq__map__tailrec,axiom,
map_a_Sum_sum_a_nat = map_ta822586218839268111_a_nat ).
% map_eq_map_tailrec
thf(fact_1003_map__eq__map__tailrec,axiom,
map_Su2790769393171190532_a_nat = map_ta197872344851190606_a_nat ).
% map_eq_map_tailrec
thf(fact_1004_map__eq__map__tailrec,axiom,
map_li6996171348340886213_a_nat = map_ta8237305816130785295_a_nat ).
% map_eq_map_tailrec
thf(fact_1005_conj__subset__def,axiom,
! [A3: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ A3
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le8108555184339247974_a_nat @ A3 @ ( collec5629555741568564177_a_nat @ P ) )
& ( ord_le8108555184339247974_a_nat @ A3 @ ( collec5629555741568564177_a_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1006_conj__subset__def,axiom,
! [A3: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1007_conj__subset__def,axiom,
! [A3: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A3
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le1325389633284124927_a_nat @ A3 @ ( collec7073057861543223018_a_nat @ P ) )
& ( ord_le1325389633284124927_a_nat @ A3 @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1008_conj__subset__def,axiom,
! [A3: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A3
@ ( collect_o
@ ^ [X: $o] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_o @ A3 @ ( collect_o @ P ) )
& ( ord_less_eq_set_o @ A3 @ ( collect_o @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1009_conj__subset__def,axiom,
! [A3: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ A3
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le1147066620699065093_a_nat @ A3 @ ( collec7555443234367654128_a_nat @ P ) )
& ( ord_le1147066620699065093_a_nat @ A3 @ ( collec7555443234367654128_a_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1010_ad__agr__list__set,axiom,
! [X5: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y2: a] :
( ( ad_agr_list_a_nat @ X5 @ Xs @ Ys )
=> ( ( member_a @ Y2 @ X5 )
=> ( ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y2 ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y2 ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ) ).
% ad_agr_list_set
thf(fact_1011_rewriteR__comp__comp2,axiom,
! [G: nat > a,H: nat > nat,R1: a > a,R22: nat > a,F: a > sum_sum_a_nat,L: a > sum_sum_a_nat] :
( ( ( comp_nat_a_nat @ G @ H )
= ( comp_a_a_nat @ R1 @ R22 ) )
=> ( ( ( comp_a5877636538679507994_nat_a @ F @ R1 )
= L )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_a4874482103519161268at_nat @ F @ G ) @ H )
= ( comp_a4874482103519161268at_nat @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1012_rewriteR__comp__comp2,axiom,
! [G: a > a,H: nat > a,R1: nat > a,R22: nat > nat,F: a > sum_sum_a_nat,L: nat > sum_sum_a_nat] :
( ( ( comp_a_a_nat @ G @ H )
= ( comp_nat_a_nat @ R1 @ R22 ) )
=> ( ( ( comp_a4874482103519161268at_nat @ F @ R1 )
= L )
=> ( ( comp_a4874482103519161268at_nat @ ( comp_a5877636538679507994_nat_a @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1013_rewriteR__comp__comp2,axiom,
! [G: a > sum_sum_a_nat,H: nat > a,R1: a > sum_sum_a_nat,R22: nat > a,F: sum_sum_a_nat > sum_sum_a_nat,L: a > sum_sum_a_nat] :
( ( ( comp_a4874482103519161268at_nat @ G @ H )
= ( comp_a4874482103519161268at_nat @ R1 @ R22 ) )
=> ( ( ( comp_S1182857061046812571_nat_a @ F @ R1 )
= L )
=> ( ( comp_a4874482103519161268at_nat @ ( comp_S1182857061046812571_nat_a @ F @ G ) @ H )
= ( comp_a4874482103519161268at_nat @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_1014_rewriteL__comp__comp2,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat,L1: a > sum_sum_a_nat,L2: a > a,H: nat > a,R3: nat > a] :
( ( ( comp_S1182857061046812571_nat_a @ F @ G )
= ( comp_a5877636538679507994_nat_a @ L1 @ L2 ) )
=> ( ( ( comp_a_a_nat @ L2 @ H )
= R3 )
=> ( ( comp_S2395038772458240499at_nat @ F @ ( comp_a4874482103519161268at_nat @ G @ H ) )
= ( comp_a4874482103519161268at_nat @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1015_rewriteL__comp__comp2,axiom,
! [F: a > sum_sum_a_nat,G: a > a,L1: sum_sum_a_nat > sum_sum_a_nat,L2: a > sum_sum_a_nat,H: nat > a,R3: nat > sum_sum_a_nat] :
( ( ( comp_a5877636538679507994_nat_a @ F @ G )
= ( comp_S1182857061046812571_nat_a @ L1 @ L2 ) )
=> ( ( ( comp_a4874482103519161268at_nat @ L2 @ H )
= R3 )
=> ( ( comp_a4874482103519161268at_nat @ F @ ( comp_a_a_nat @ G @ H ) )
= ( comp_S2395038772458240499at_nat @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1016_rewriteL__comp__comp2,axiom,
! [F: a > sum_sum_a_nat,G: nat > a,L1: a > sum_sum_a_nat,L2: nat > a,H: nat > nat,R3: nat > a] :
( ( ( comp_a4874482103519161268at_nat @ F @ G )
= ( comp_a4874482103519161268at_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_a_nat @ L2 @ H )
= R3 )
=> ( ( comp_a4874482103519161268at_nat @ F @ ( comp_nat_a_nat @ G @ H ) )
= ( comp_a4874482103519161268at_nat @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_1017_rewriteR__comp__comp,axiom,
! [G: nat > a,H: nat > nat,R3: nat > a,F: a > sum_sum_a_nat] :
( ( ( comp_nat_a_nat @ G @ H )
= R3 )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_a4874482103519161268at_nat @ F @ G ) @ H )
= ( comp_a4874482103519161268at_nat @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_1018_rewriteR__comp__comp,axiom,
! [G: a > a,H: nat > a,R3: nat > a,F: a > sum_sum_a_nat] :
( ( ( comp_a_a_nat @ G @ H )
= R3 )
=> ( ( comp_a4874482103519161268at_nat @ ( comp_a5877636538679507994_nat_a @ F @ G ) @ H )
= ( comp_a4874482103519161268at_nat @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_1019_rewriteR__comp__comp,axiom,
! [G: a > sum_sum_a_nat,H: nat > a,R3: nat > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( comp_a4874482103519161268at_nat @ G @ H )
= R3 )
=> ( ( comp_a4874482103519161268at_nat @ ( comp_S1182857061046812571_nat_a @ F @ G ) @ H )
= ( comp_S2395038772458240499at_nat @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_1020_rewriteL__comp__comp,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat,L: a > sum_sum_a_nat,H: nat > a] :
( ( ( comp_S1182857061046812571_nat_a @ F @ G )
= L )
=> ( ( comp_S2395038772458240499at_nat @ F @ ( comp_a4874482103519161268at_nat @ G @ H ) )
= ( comp_a4874482103519161268at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_1021_rewriteL__comp__comp,axiom,
! [F: a > sum_sum_a_nat,G: a > a,L: a > sum_sum_a_nat,H: nat > a] :
( ( ( comp_a5877636538679507994_nat_a @ F @ G )
= L )
=> ( ( comp_a4874482103519161268at_nat @ F @ ( comp_a_a_nat @ G @ H ) )
= ( comp_a4874482103519161268at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_1022_rewriteL__comp__comp,axiom,
! [F: a > sum_sum_a_nat,G: nat > a,L: nat > sum_sum_a_nat,H: nat > nat] :
( ( ( comp_a4874482103519161268at_nat @ F @ G )
= L )
=> ( ( comp_a4874482103519161268at_nat @ F @ ( comp_nat_a_nat @ G @ H ) )
= ( comp_n1522318729830440540at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_1023_type__copy__map__cong0,axiom,
! [M: nat > a,G: nat > nat,X3: nat,N: a > a,H: nat > a,F: a > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N @ ( H @ X3 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_a4874482103519161268at_nat @ F @ M ) @ G @ X3 )
= ( comp_a4874482103519161268at_nat @ ( comp_a5877636538679507994_nat_a @ F @ N ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_1024_type__copy__map__cong0,axiom,
! [M: a > a,G: nat > a,X3: nat,N: nat > a,H: nat > nat,F: a > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N @ ( H @ X3 ) ) )
=> ( ( comp_a4874482103519161268at_nat @ ( comp_a5877636538679507994_nat_a @ F @ M ) @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ ( comp_a4874482103519161268at_nat @ F @ N ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_1025_ad__agr__list__subset,axiom,
! [Ms: list_a,Ns2: list_a,X5: set_a,Sigma2: a > sum_sum_a_nat,Sigma3: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ms ) @ ( set_a2 @ Ns2 ) )
=> ( ( ad_agr_list_a_nat @ X5 @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ns2 ) @ ( map_a_Sum_sum_a_nat @ Sigma3 @ Ns2 ) )
=> ( ad_agr_list_a_nat @ X5 @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ms ) @ ( map_a_Sum_sum_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_1026_ad__agr__list__subset,axiom,
! [Ms: list_nat,Ns2: list_nat,X5: set_a,Sigma2: nat > sum_sum_a_nat,Sigma3: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns2 ) )
=> ( ( ad_agr_list_a_nat @ X5 @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns2 ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ns2 ) )
=> ( ad_agr_list_a_nat @ X5 @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_1027_ad__agr__list__subset,axiom,
! [Ms: list_Sum_sum_a_nat,Ns2: list_Sum_sum_a_nat,X5: set_a,Sigma2: sum_sum_a_nat > sum_sum_a_nat,Sigma3: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Ms ) @ ( set_Sum_sum_a_nat2 @ Ns2 ) )
=> ( ( ad_agr_list_a_nat @ X5 @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ns2 ) @ ( map_Su2790769393171190532_a_nat @ Sigma3 @ Ns2 ) )
=> ( ad_agr_list_a_nat @ X5 @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ms ) @ ( map_Su2790769393171190532_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_1028_sat__ad__agr__list,axiom,
! [Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,AD: set_a,Sigma2: nat > a,Tau2: nat > a] :
( ( wf_fo_intp_a_b @ Phi2 @ I2 )
=> ( ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ sum_Inl_a_nat @ Sigma2 ) @ ( fv_fo_fmla_list_a_b @ Phi2 ) ) @ ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ sum_Inl_a_nat @ Tau2 ) @ ( fv_fo_fmla_list_a_b @ Phi2 ) ) )
=> ( ( ord_less_eq_set_a @ ( act_edom_a_b @ Phi2 @ I2 ) @ AD )
=> ( ( sat_a_b @ Phi2 @ I2 @ Sigma2 )
= ( sat_a_b @ Phi2 @ I2 @ Tau2 ) ) ) ) ) ).
% sat_ad_agr_list
thf(fact_1029_esat__UNIV__ad__agr__list,axiom,
! [AD: set_a,Sigma2: nat > sum_sum_a_nat,Phi2: fo_fmla_a_b,Tau2: nat > sum_sum_a_nat,I2: product_prod_b_nat > set_list_a] :
( ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma2 @ ( fv_fo_fmla_list_a_b @ Phi2 ) ) @ ( map_na823391071729141993_a_nat @ Tau2 @ ( fv_fo_fmla_list_a_b @ Phi2 ) ) )
=> ( ( ord_less_eq_set_a @ ( act_edom_a_b @ Phi2 @ I2 ) @ AD )
=> ( ( esat_a_b @ Phi2 @ I2 @ Sigma2 @ top_to795618464972521135_a_nat )
= ( esat_a_b @ Phi2 @ I2 @ Tau2 @ top_to795618464972521135_a_nat ) ) ) ) ).
% esat_UNIV_ad_agr_list
thf(fact_1030_ad__agr__list__link,axiom,
! [Ns2: list_nat,AD: set_a,Sigma2: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ Ns2 ) @ ( set_nat2 @ Ns2 ) @ AD @ Sigma2 @ Tau2 )
= ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns2 ) @ ( map_na823391071729141993_a_nat @ Tau2 @ Ns2 ) ) ) ).
% ad_agr_list_link
thf(fact_1031_proj__out__list,axiom,
! [AD: set_a,Sigma2: nat > sum_sum_a_nat,Ns2: list_nat] :
( ( finite_finite_a @ AD )
=> ? [Tau3: nat > a] :
( ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns2 ) @ ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ sum_Inl_a_nat @ Tau3 ) @ Ns2 ) )
& ! [J: nat,X4: a] :
( ( member_nat @ J @ ( set_nat2 @ Ns2 ) )
=> ( ( ( Sigma2 @ J )
= ( sum_Inl_a_nat @ X4 ) )
=> ( ( Tau3 @ J )
= X4 ) ) ) ) ) ).
% proj_out_list
thf(fact_1032_proj__out__list,axiom,
! [AD: set_nat,Sigma2: nat > sum_sum_nat_nat,Ns2: list_nat] :
( ( finite_finite_nat @ AD )
=> ? [Tau3: nat > nat] :
( ( ad_agr_list_nat_nat @ AD @ ( map_na7095077453091140501at_nat @ Sigma2 @ Ns2 ) @ ( map_na7095077453091140501at_nat @ ( comp_n5293421440010881668at_nat @ sum_Inl_nat_nat @ Tau3 ) @ Ns2 ) )
& ! [J: nat,X4: nat] :
( ( member_nat @ J @ ( set_nat2 @ Ns2 ) )
=> ( ( ( Sigma2 @ J )
= ( sum_Inl_nat_nat @ X4 ) )
=> ( ( Tau3 @ J )
= X4 ) ) ) ) ) ).
% proj_out_list
thf(fact_1033_subset__Collect__iff,axiom,
! [B: set_na3699693778330250182_a_nat,A3: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ B @ A3 )
=> ( ( ord_le8108555184339247974_a_nat @ B
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1034_subset__Collect__iff,axiom,
! [B: set_nat,A3: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1035_subset__Collect__iff,axiom,
! [B: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ B @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ B
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1036_subset__Collect__iff,axiom,
! [B: set_o,A3: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A3 )
=> ( ( ord_less_eq_set_o @ B
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1037_subset__Collect__iff,axiom,
! [B: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ B @ A3 )
=> ( ( ord_le1147066620699065093_a_nat @ B
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1038_subset__CollectI,axiom,
! [B: set_na3699693778330250182_a_nat,A3: set_na3699693778330250182_a_nat,Q: ( nat > sum_sum_a_nat ) > $o,P: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ B @ A3 )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1039_subset__CollectI,axiom,
! [B: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1040_subset__CollectI,axiom,
! [B: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat,Q: sum_sum_a_nat > $o,P: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ B @ A3 )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1041_subset__CollectI,axiom,
! [B: set_o,A3: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A3 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ B )
& ( Q @ X ) ) )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1042_subset__CollectI,axiom,
! [B: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat,Q: list_Sum_sum_a_nat > $o,P: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ B @ A3 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1043_finite__Collect__disjI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( finite785833390020136079_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite785833390020136079_a_nat @ ( collec5629555741568564177_a_nat @ P ) )
& ( finite785833390020136079_a_nat @ ( collec5629555741568564177_a_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1044_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1045_finite__Collect__conjI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ( finite785833390020136079_a_nat @ ( collec5629555741568564177_a_nat @ P ) )
| ( finite785833390020136079_a_nat @ ( collec5629555741568564177_a_nat @ Q ) ) )
=> ( finite785833390020136079_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1046_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1047_finite__imageI,axiom,
! [F3: set_a,H: a > sum_sum_a_nat] :
( ( finite_finite_a @ F3 )
=> ( finite502105017643426984_a_nat @ ( image_7873763678140191238_a_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_1048_finite__imageI,axiom,
! [F3: set_list_a,H: list_a > list_Sum_sum_a_nat] :
( ( finite_finite_list_a @ F3 )
=> ( finite1487985464145237934_a_nat @ ( image_674313660629153798_a_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_1049_finite__imageI,axiom,
! [F3: set_a,H: a > list_Sum_sum_a_nat] :
( ( finite_finite_a @ F3 )
=> ( finite1487985464145237934_a_nat @ ( image_7897140031816760844_a_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_1050_finite__imageI,axiom,
! [F3: set_a,H: a > $o] :
( ( finite_finite_a @ F3 )
=> ( finite_finite_o @ ( image_a_o @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_1051_finite__imageI,axiom,
! [F3: set_nat,H: nat > sum_sum_a_nat] :
( ( finite_finite_nat @ F3 )
=> ( finite502105017643426984_a_nat @ ( image_7293268710728258664_a_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_1052_finite__imageI,axiom,
! [F3: set_nat,H: nat > list_Sum_sum_a_nat] :
( ( finite_finite_nat @ F3 )
=> ( finite1487985464145237934_a_nat @ ( image_6262589752765146990_a_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_1053_finite__imageI,axiom,
! [F3: set_nat,H: nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_o @ ( image_nat_o @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_1054_finite__imageI,axiom,
! [F3: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_1055_List_Ofinite__set,axiom,
! [Xs: list_Sum_sum_a_nat] : ( finite502105017643426984_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_1056_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_1057_List_Ofinite__set,axiom,
! [Xs: list_l4703314356710769291_a_nat] : ( finite1487985464145237934_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) ) ).
% List.finite_set
thf(fact_1058_List_Ofinite__set,axiom,
! [Xs: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_1059_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_1060_finite__Int,axiom,
! [F3: set_nat,G4: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G4 ) ) ) ).
% finite_Int
thf(fact_1061_finite__Int,axiom,
! [F3: set_li6526943997496501093_a_nat,G4: set_li6526943997496501093_a_nat] :
( ( ( finite1487985464145237934_a_nat @ F3 )
| ( finite1487985464145237934_a_nat @ G4 ) )
=> ( finite1487985464145237934_a_nat @ ( inf_in3249246906714053971_a_nat @ F3 @ G4 ) ) ) ).
% finite_Int
thf(fact_1062_finite__Collect__not,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o] :
( ( finite785833390020136079_a_nat @ ( collec5629555741568564177_a_nat @ P ) )
=> ( ( finite785833390020136079_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
~ ( P @ X ) ) )
= ( finite785833390020136079_a_nat @ top_to9106040778512017686_a_nat ) ) ) ).
% finite_Collect_not
thf(fact_1063_finite__Collect__not,axiom,
! [P: sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat @ ( collec7073057861543223018_a_nat @ P ) )
=> ( ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
~ ( P @ X ) ) )
= ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat ) ) ) ).
% finite_Collect_not
thf(fact_1064_finite__Collect__not,axiom,
! [P: a > $o] :
( ( finite_finite_a @ ( collect_a @ P ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
~ ( P @ X ) ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Collect_not
thf(fact_1065_finite__Collect__not,axiom,
! [P: list_a > $o] :
( ( finite_finite_list_a @ ( collect_list_a @ P ) )
=> ( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
~ ( P @ X ) ) )
= ( finite_finite_list_a @ top_top_set_list_a ) ) ) ).
% finite_Collect_not
thf(fact_1066_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_1067_finite__Collect__subsets,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1068_finite__Collect__subsets,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A3 )
=> ( finite4842507993062306312_a_nat
@ ( collec4049389696321283146_a_nat
@ ^ [B3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1069_finite__Collect__subsets,axiom,
! [A3: set_o] :
( ( finite_finite_o @ A3 )
=> ( finite_finite_set_o
@ ( collect_set_o
@ ^ [B3: set_o] : ( ord_less_eq_set_o @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1070_finite__Collect__subsets,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A3 )
=> ( finite8331804442905961358_a_nat
@ ( collec7528627406912015568_a_nat
@ ^ [B3: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1071_finite__Plus__UNIV__iff,axiom,
( ( finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1072_finite__Plus__UNIV__iff,axiom,
( ( finite3740268481367103950_nat_a @ top_to54524901450547413_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1073_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1074_finite__Plus__UNIV__iff,axiom,
( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1075_finite__Plus__UNIV__iff,axiom,
( ( finite7003286066433605570list_a @ top_to4747967031102520593list_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_list_a @ top_top_set_list_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1076_finite__Plus__UNIV__iff,axiom,
( ( finite896696315882757558st_a_a @ top_to2416585553260192005st_a_a )
= ( ( finite_finite_list_a @ top_top_set_list_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1077_finite__Plus__UNIV__iff,axiom,
( ( finite9133776500316014638_a_nat @ top_to4012036164214923573_a_nat )
= ( ( finite_finite_list_a @ top_top_set_list_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1078_finite__Plus__UNIV__iff,axiom,
( ( finite7226927734900800724list_a @ top_to1634804448594879195list_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_list_a @ top_top_set_list_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1079_finite__Plus__UNIV__iff,axiom,
( ( finite7642659526839096315_nat_a @ top_to860545285262905546_nat_a )
= ( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1080_finite__Plus__UNIV__iff,axiom,
( ( finite3067912454409353897at_nat @ top_to8676068415865862704at_nat )
= ( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_1081_Finite__Set_Ofinite__set,axiom,
( ( finite4842507993062306312_a_nat @ top_to9085961846241471503_a_nat )
= ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat ) ) ).
% Finite_Set.finite_set
thf(fact_1082_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_a @ top_top_set_set_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_1083_Finite__Set_Ofinite__set,axiom,
( ( finite5282473924520328461list_a @ top_to7106483174946246804list_a )
= ( finite_finite_list_a @ top_top_set_list_a ) ) ).
% Finite_Set.finite_set
thf(fact_1084_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_1085_finite__prod,axiom,
( ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1086_finite__prod,axiom,
( ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1087_finite__prod,axiom,
( ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1088_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1089_finite__prod,axiom,
( ( finite1633364483105306710list_a @ top_to6251280003177294685list_a )
= ( ( finite_finite_a @ top_top_set_a )
& ( finite_finite_list_a @ top_top_set_list_a ) ) ) ).
% finite_prod
thf(fact_1090_finite__prod,axiom,
( ( finite4750146769409234506st_a_a @ top_to3919898525334966097st_a_a )
= ( ( finite_finite_list_a @ top_top_set_list_a )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1091_finite__prod,axiom,
( ( finite4330131977071940250_a_nat @ top_to103321375020513257_a_nat )
= ( ( finite_finite_list_a @ top_top_set_list_a )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1092_finite__prod,axiom,
( ( finite2423283211656726336list_a @ top_to6949461696255244687list_a )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_list_a @ top_top_set_list_a ) ) ) ).
% finite_prod
thf(fact_1093_finite__prod,axiom,
( ( finite2075841124386597007_nat_a @ top_to5880318473739666198_nat_a )
= ( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_1094_finite__prod,axiom,
( ( finite4631044910092953877at_nat @ top_to2955724835671772388at_nat )
= ( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_1095_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite6544458595007987280od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1096_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6644898363146130708_a_nat @ top_to3353692345378799459_a_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1097_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite659689790015031866_nat_a @ top_to2612598781856825737_nat_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1098_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1099_finite__Prod__UNIV,axiom,
( ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_list_a @ top_top_set_list_a )
=> ( finite1633364483105306710list_a @ top_to6251280003177294685list_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1100_finite__Prod__UNIV,axiom,
( ( finite_finite_list_a @ top_top_set_list_a )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite4750146769409234506st_a_a @ top_to3919898525334966097st_a_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1101_finite__Prod__UNIV,axiom,
( ( finite_finite_list_a @ top_top_set_list_a )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite4330131977071940250_a_nat @ top_to103321375020513257_a_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1102_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_list_a @ top_top_set_list_a )
=> ( finite2423283211656726336list_a @ top_to6949461696255244687list_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1103_finite__Prod__UNIV,axiom,
( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite2075841124386597007_nat_a @ top_to5880318473739666198_nat_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_1104_finite__Prod__UNIV,axiom,
( ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite4631044910092953877at_nat @ top_to2955724835671772388at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_1105_finite__act__edom,axiom,
! [Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a] :
( ( wf_fo_intp_a_b @ Phi2 @ I2 )
=> ( finite_finite_a @ ( act_edom_a_b @ Phi2 @ I2 ) ) ) ).
% finite_act_edom
thf(fact_1106_pigeonhole__infinite__rel,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_nat,R2: list_Sum_sum_a_nat > nat > $o] :
( ~ ( finite1487985464145237934_a_nat @ A3 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R2 @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite1487985464145237934_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [A5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A5 @ A3 )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1107_pigeonhole__infinite__rel,axiom,
! [A3: set_o,B: set_nat,R2: $o > nat > $o] :
( ~ ( finite_finite_o @ A3 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R2 @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A5: $o] :
( ( member_o @ A5 @ A3 )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1108_pigeonhole__infinite__rel,axiom,
! [A3: set_Sum_sum_a_nat,B: set_nat,R2: sum_sum_a_nat > nat > $o] :
( ~ ( finite502105017643426984_a_nat @ A3 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R2 @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [A5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A5 @ A3 )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1109_pigeonhole__infinite__rel,axiom,
! [A3: set_na3699693778330250182_a_nat,B: set_nat,R2: ( nat > sum_sum_a_nat ) > nat > $o] :
( ~ ( finite785833390020136079_a_nat @ A3 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R2 @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite785833390020136079_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [A5: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ A5 @ A3 )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1110_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R2 @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A3 )
& ( R2 @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1111_not__finite__existsD,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o] :
( ~ ( finite785833390020136079_a_nat @ ( collec5629555741568564177_a_nat @ P ) )
=> ? [X_1: nat > sum_sum_a_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1112_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1113_finite__has__minimal2,axiom,
! [A3: set_o,A: $o] :
( ( finite_finite_o @ A3 )
=> ( ( member_o @ A @ A3 )
=> ? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( ord_less_eq_o @ X2 @ A )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A3 )
=> ( ( ord_less_eq_o @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1114_finite__has__minimal2,axiom,
! [A3: set_se4904748513628223167_a_nat,A: set_Sum_sum_a_nat] :
( ( finite4842507993062306312_a_nat @ A3 )
=> ( ( member8098812455498974984_a_nat @ A @ A3 )
=> ? [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ A3 )
& ( ord_le1325389633284124927_a_nat @ X2 @ A )
& ! [Xa2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ Xa2 @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1115_finite__has__minimal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A3 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1116_finite__has__minimal2,axiom,
! [A3: set_set_o,A: set_o] :
( ( finite_finite_set_o @ A3 )
=> ( ( member_set_o @ A @ A3 )
=> ? [X2: set_o] :
( ( member_set_o @ X2 @ A3 )
& ( ord_less_eq_set_o @ X2 @ A )
& ! [Xa2: set_o] :
( ( member_set_o @ Xa2 @ A3 )
=> ( ( ord_less_eq_set_o @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1117_finite__has__minimal2,axiom,
! [A3: set_se4330304633200676677_a_nat,A: set_li6526943997496501093_a_nat] :
( ( finite8331804442905961358_a_nat @ A3 )
=> ( ( member5553968465346197646_a_nat @ A @ A3 )
=> ? [X2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X2 @ A3 )
& ( ord_le1147066620699065093_a_nat @ X2 @ A )
& ! [Xa2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Xa2 @ A3 )
=> ( ( ord_le1147066620699065093_a_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1118_finite__has__maximal2,axiom,
! [A3: set_o,A: $o] :
( ( finite_finite_o @ A3 )
=> ( ( member_o @ A @ A3 )
=> ? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( ord_less_eq_o @ A @ X2 )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A3 )
=> ( ( ord_less_eq_o @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1119_finite__has__maximal2,axiom,
! [A3: set_se4904748513628223167_a_nat,A: set_Sum_sum_a_nat] :
( ( finite4842507993062306312_a_nat @ A3 )
=> ( ( member8098812455498974984_a_nat @ A @ A3 )
=> ? [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ A3 )
& ( ord_le1325389633284124927_a_nat @ A @ X2 )
& ! [Xa2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ Xa2 @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1120_finite__has__maximal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A3 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1121_finite__has__maximal2,axiom,
! [A3: set_set_o,A: set_o] :
( ( finite_finite_set_o @ A3 )
=> ( ( member_set_o @ A @ A3 )
=> ? [X2: set_o] :
( ( member_set_o @ X2 @ A3 )
& ( ord_less_eq_set_o @ A @ X2 )
& ! [Xa2: set_o] :
( ( member_set_o @ Xa2 @ A3 )
=> ( ( ord_less_eq_set_o @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1122_finite__has__maximal2,axiom,
! [A3: set_se4330304633200676677_a_nat,A: set_li6526943997496501093_a_nat] :
( ( finite8331804442905961358_a_nat @ A3 )
=> ( ( member5553968465346197646_a_nat @ A @ A3 )
=> ? [X2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X2 @ A3 )
& ( ord_le1147066620699065093_a_nat @ A @ X2 )
& ! [Xa2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Xa2 @ A3 )
=> ( ( ord_le1147066620699065093_a_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1123_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_1124_ex__new__if__finite,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ~ ( finite1487985464145237934_a_nat @ top_to6433055325616222389_a_nat )
=> ( ( finite1487985464145237934_a_nat @ A3 )
=> ? [A6: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1125_ex__new__if__finite,axiom,
! [A3: set_o] :
( ~ ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ A3 )
=> ? [A6: $o] :
~ ( member_o @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1126_ex__new__if__finite,axiom,
! [A3: set_Sum_sum_a_nat] :
( ~ ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
=> ( ( finite502105017643426984_a_nat @ A3 )
=> ? [A6: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1127_ex__new__if__finite,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A3 )
=> ? [A6: a] :
~ ( member_a @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1128_ex__new__if__finite,axiom,
! [A3: set_list_a] :
( ~ ( finite_finite_list_a @ top_top_set_list_a )
=> ( ( finite_finite_list_a @ A3 )
=> ? [A6: list_a] :
~ ( member_list_a @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1129_ex__new__if__finite,axiom,
! [A3: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A3 )
=> ? [A6: nat] :
~ ( member_nat @ A6 @ A3 ) ) ) ).
% ex_new_if_finite
thf(fact_1130_rev__finite__subset,axiom,
! [B: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A3 @ B )
=> ( finite_finite_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_1131_rev__finite__subset,axiom,
! [B: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( finite502105017643426984_a_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_1132_rev__finite__subset,axiom,
! [B: set_o,A3: set_o] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ A3 @ B )
=> ( finite_finite_o @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_1133_rev__finite__subset,axiom,
! [B: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ B )
=> ( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ( finite1487985464145237934_a_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_1134_infinite__super,axiom,
! [S: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_1135_infinite__super,axiom,
! [S: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ S @ T3 )
=> ( ~ ( finite502105017643426984_a_nat @ S )
=> ~ ( finite502105017643426984_a_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_1136_infinite__super,axiom,
! [S: set_o,T3: set_o] :
( ( ord_less_eq_set_o @ S @ T3 )
=> ( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ T3 ) ) ) ).
% infinite_super
thf(fact_1137_infinite__super,axiom,
! [S: set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ S @ T3 )
=> ( ~ ( finite1487985464145237934_a_nat @ S )
=> ~ ( finite1487985464145237934_a_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_1138_finite__subset,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_1139_finite__subset,axiom,
! [A3: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( finite502105017643426984_a_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_1140_finite__subset,axiom,
! [A3: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A3 @ B )
=> ( ( finite_finite_o @ B )
=> ( finite_finite_o @ A3 ) ) ) ).
% finite_subset
thf(fact_1141_finite__subset,axiom,
! [A3: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A3 @ B )
=> ( ( finite1487985464145237934_a_nat @ B )
=> ( finite1487985464145237934_a_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_1142_finite__list,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A3 )
=> ? [Xs3: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_1143_finite__list,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ? [Xs3: list_a] :
( ( set_a2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_1144_finite__list,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A3 )
=> ? [Xs3: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_1145_finite__list,axiom,
! [A3: set_list_a] :
( ( finite_finite_list_a @ A3 )
=> ? [Xs3: list_list_a] :
( ( set_list_a2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_1146_finite__list,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_1147_infinite__UNIV__listI,axiom,
~ ( finite_finite_list_a @ top_top_set_list_a ) ).
% infinite_UNIV_listI
thf(fact_1148_pigeonhole__infinite,axiom,
! [A3: set_a,F: a > $o] :
( ~ ( finite_finite_a @ A3 )
=> ( ( finite_finite_o @ ( image_a_o @ F @ A3 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1149_pigeonhole__infinite,axiom,
! [A3: set_o,F: $o > nat] :
( ~ ( finite_finite_o @ A3 )
=> ( ( finite_finite_nat @ ( image_o_nat @ F @ A3 ) )
=> ? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A5: $o] :
( ( member_o @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1150_pigeonhole__infinite,axiom,
! [A3: set_nat,F: nat > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_finite_o @ ( image_nat_o @ F @ A3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1151_pigeonhole__infinite,axiom,
! [A3: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1152_pigeonhole__infinite,axiom,
! [A3: set_a,F: a > sum_sum_a_nat] :
( ~ ( finite_finite_a @ A3 )
=> ( ( finite502105017643426984_a_nat @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1153_pigeonhole__infinite,axiom,
! [A3: set_Sum_sum_a_nat,F: sum_sum_a_nat > nat] :
( ~ ( finite502105017643426984_a_nat @ A3 )
=> ( ( finite_finite_nat @ ( image_2473878607534554506at_nat @ F @ A3 ) )
=> ? [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A3 )
& ~ ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [A5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1154_pigeonhole__infinite,axiom,
! [A3: set_nat,F: nat > sum_sum_a_nat] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite502105017643426984_a_nat @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1155_pigeonhole__infinite,axiom,
! [A3: set_a,F: a > list_Sum_sum_a_nat] :
( ~ ( finite_finite_a @ A3 )
=> ( ( finite1487985464145237934_a_nat @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1156_pigeonhole__infinite,axiom,
! [A3: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
( ~ ( finite1487985464145237934_a_nat @ A3 )
=> ( ( finite_finite_nat @ ( image_2535339886381165584at_nat @ F @ A3 ) )
=> ? [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A3 )
& ~ ( finite1487985464145237934_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [A5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1157_pigeonhole__infinite,axiom,
! [A3: set_nat,F: nat > list_Sum_sum_a_nat] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite1487985464145237934_a_nat @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A3 )
& ( ( F @ A5 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1158_all__finite__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1159_all__finite__subset__image,axiom,
! [F: $o > nat,A3: set_o,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_o_nat @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ A3 ) )
=> ( P @ ( image_o_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1160_all__finite__subset__image,axiom,
! [F: a > $o,A3: set_a,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ ( image_a_o @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) )
=> ( P @ ( image_a_o @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1161_all__finite__subset__image,axiom,
! [F: nat > $o,A3: set_nat,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ ( image_nat_o @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) )
=> ( P @ ( image_nat_o @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1162_all__finite__subset__image,axiom,
! [F: $o > $o,A3: set_o,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ ( image_o_o @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ A3 ) )
=> ( P @ ( image_o_o @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1163_all__finite__subset__image,axiom,
! [F: sum_sum_a_nat > nat,A3: set_Sum_sum_a_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_2473878607534554506at_nat @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A3 ) )
=> ( P @ ( image_2473878607534554506at_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1164_all__finite__subset__image,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) )
=> ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1165_all__finite__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) )
=> ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1166_all__finite__subset__image,axiom,
! [F: $o > sum_sum_a_nat,A3: set_o,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_4139480514073730540_a_nat @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_o] :
( ( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ A3 ) )
=> ( P @ ( image_4139480514073730540_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1167_all__finite__subset__image,axiom,
! [F: sum_sum_a_nat > $o,A3: set_Sum_sum_a_nat,P: set_o > $o] :
( ( ! [B3: set_o] :
( ( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ ( image_6095136190293192542_nat_o @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A3 ) )
=> ( P @ ( image_6095136190293192542_nat_o @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1168_ex__finite__subset__image,axiom,
! [F: nat > nat,A3: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 )
& ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1169_ex__finite__subset__image,axiom,
! [F: $o > nat,A3: set_o,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_o_nat @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_o] :
( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ A3 )
& ( P @ ( image_o_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1170_ex__finite__subset__image,axiom,
! [F: a > $o,A3: set_a,P: set_o > $o] :
( ( ? [B3: set_o] :
( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ ( image_a_o @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 )
& ( P @ ( image_a_o @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1171_ex__finite__subset__image,axiom,
! [F: nat > $o,A3: set_nat,P: set_o > $o] :
( ( ? [B3: set_o] :
( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ ( image_nat_o @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 )
& ( P @ ( image_nat_o @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1172_ex__finite__subset__image,axiom,
! [F: $o > $o,A3: set_o,P: set_o > $o] :
( ( ? [B3: set_o] :
( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ ( image_o_o @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_o] :
( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ A3 )
& ( P @ ( image_o_o @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1173_ex__finite__subset__image,axiom,
! [F: sum_sum_a_nat > nat,A3: set_Sum_sum_a_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_2473878607534554506at_nat @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A3 )
& ( P @ ( image_2473878607534554506at_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1174_ex__finite__subset__image,axiom,
! [F: a > sum_sum_a_nat,A3: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 )
& ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1175_ex__finite__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 )
& ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1176_ex__finite__subset__image,axiom,
! [F: $o > sum_sum_a_nat,A3: set_o,P: set_Sum_sum_a_nat > $o] :
( ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_4139480514073730540_a_nat @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_o] :
( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ A3 )
& ( P @ ( image_4139480514073730540_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1177_ex__finite__subset__image,axiom,
! [F: sum_sum_a_nat > $o,A3: set_Sum_sum_a_nat,P: set_o > $o] :
( ( ? [B3: set_o] :
( ( finite_finite_o @ B3 )
& ( ord_less_eq_set_o @ B3 @ ( image_6095136190293192542_nat_o @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A3 )
& ( P @ ( image_6095136190293192542_nat_o @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1178_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A3: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A3 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1179_finite__subset__image,axiom,
! [B: set_nat,F: $o > nat,A3: set_o] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_o_nat @ F @ A3 ) )
=> ? [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A3 )
& ( finite_finite_o @ C3 )
& ( B
= ( image_o_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1180_finite__subset__image,axiom,
! [B: set_o,F: a > $o,A3: set_a] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ B @ ( image_a_o @ F @ A3 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_o @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1181_finite__subset__image,axiom,
! [B: set_o,F: nat > $o,A3: set_nat] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ B @ ( image_nat_o @ F @ A3 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_o @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1182_finite__subset__image,axiom,
! [B: set_o,F: $o > $o,A3: set_o] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ B @ ( image_o_o @ F @ A3 ) )
=> ? [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A3 )
& ( finite_finite_o @ C3 )
& ( B
= ( image_o_o @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1183_finite__subset__image,axiom,
! [B: set_nat,F: sum_sum_a_nat > nat,A3: set_Sum_sum_a_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_2473878607534554506at_nat @ F @ A3 ) )
=> ? [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A3 )
& ( finite502105017643426984_a_nat @ C3 )
& ( B
= ( image_2473878607534554506at_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1184_finite__subset__image,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A3: set_a] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
& ( finite_finite_a @ C3 )
& ( B
= ( image_7873763678140191238_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1185_finite__subset__image,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A3: set_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_7293268710728258664_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1186_finite__subset__image,axiom,
! [B: set_Sum_sum_a_nat,F: $o > sum_sum_a_nat,A3: set_o] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_4139480514073730540_a_nat @ F @ A3 ) )
=> ? [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A3 )
& ( finite_finite_o @ C3 )
& ( B
= ( image_4139480514073730540_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1187_finite__subset__image,axiom,
! [B: set_o,F: sum_sum_a_nat > $o,A3: set_Sum_sum_a_nat] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ B @ ( image_6095136190293192542_nat_o @ F @ A3 ) )
=> ? [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A3 )
& ( finite502105017643426984_a_nat @ C3 )
& ( B
= ( image_6095136190293192542_nat_o @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1188_finite__surj,axiom,
! [A3: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A3 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1189_finite__surj,axiom,
! [A3: set_a,B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( finite_finite_a @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A3 ) )
=> ( finite502105017643426984_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_1190_finite__surj,axiom,
! [A3: set_nat,B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
=> ( finite502105017643426984_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_1191_finite__surj,axiom,
! [A3: set_a,B: set_o,F: a > $o] :
( ( finite_finite_a @ A3 )
=> ( ( ord_less_eq_set_o @ B @ ( image_a_o @ F @ A3 ) )
=> ( finite_finite_o @ B ) ) ) ).
% finite_surj
thf(fact_1192_finite__surj,axiom,
! [A3: set_nat,B: set_o,F: nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_set_o @ B @ ( image_nat_o @ F @ A3 ) )
=> ( finite_finite_o @ B ) ) ) ).
% finite_surj
thf(fact_1193_finite__surj,axiom,
! [A3: set_list_a,B: set_li6526943997496501093_a_nat,F: list_a > list_Sum_sum_a_nat] :
( ( finite_finite_list_a @ A3 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ ( image_674313660629153798_a_nat @ F @ A3 ) )
=> ( finite1487985464145237934_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_1194_finite__surj,axiom,
! [A3: set_a,B: set_li6526943997496501093_a_nat,F: a > list_Sum_sum_a_nat] :
( ( finite_finite_a @ A3 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ ( image_7897140031816760844_a_nat @ F @ A3 ) )
=> ( finite1487985464145237934_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_1195_finite__surj,axiom,
! [A3: set_nat,B: set_li6526943997496501093_a_nat,F: nat > list_Sum_sum_a_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_le1147066620699065093_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A3 ) )
=> ( finite1487985464145237934_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_1196_finite__range__imageI,axiom,
! [G: a > $o,F: $o > $o] :
( ( finite_finite_o @ ( image_a_o @ G @ top_top_set_a ) )
=> ( finite_finite_o
@ ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1197_finite__range__imageI,axiom,
! [G: a > $o,F: $o > nat] :
( ( finite_finite_o @ ( image_a_o @ G @ top_top_set_a ) )
=> ( finite_finite_nat
@ ( image_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1198_finite__range__imageI,axiom,
! [G: a > nat,F: nat > $o] :
( ( finite_finite_nat @ ( image_a_nat @ G @ top_top_set_a ) )
=> ( finite_finite_o
@ ( image_a_o
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1199_finite__range__imageI,axiom,
! [G: a > nat,F: nat > nat] :
( ( finite_finite_nat @ ( image_a_nat @ G @ top_top_set_a ) )
=> ( finite_finite_nat
@ ( image_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a ) ) ) ).
% finite_range_imageI
thf(fact_1200_finite__range__imageI,axiom,
! [G: nat > $o,F: $o > $o] :
( ( finite_finite_o @ ( image_nat_o @ G @ top_top_set_nat ) )
=> ( finite_finite_o
@ ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_1201_finite__range__imageI,axiom,
! [G: nat > $o,F: $o > nat] :
( ( finite_finite_o @ ( image_nat_o @ G @ top_top_set_nat ) )
=> ( finite_finite_nat
@ ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_1202_finite__range__imageI,axiom,
! [G: nat > nat,F: nat > $o] :
( ( finite_finite_nat @ ( image_nat_nat @ G @ top_top_set_nat ) )
=> ( finite_finite_o
@ ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_1203_finite__range__imageI,axiom,
! [G: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ G @ top_top_set_nat ) )
=> ( finite_finite_nat
@ ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_1204_finite__range__imageI,axiom,
! [G: list_a > nat,F: nat > nat] :
( ( finite_finite_nat @ ( image_list_a_nat @ G @ top_top_set_list_a ) )
=> ( finite_finite_nat
@ ( image_list_a_nat
@ ^ [X: list_a] : ( F @ ( G @ X ) )
@ top_top_set_list_a ) ) ) ).
% finite_range_imageI
thf(fact_1205_finite__range__imageI,axiom,
! [G: sum_sum_a_nat > nat,F: nat > nat] :
( ( finite_finite_nat @ ( image_2473878607534554506at_nat @ G @ top_to795618464972521135_a_nat ) )
=> ( finite_finite_nat
@ ( image_2473878607534554506at_nat
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ top_to795618464972521135_a_nat ) ) ) ).
% finite_range_imageI
thf(fact_1206_finite__option__UNIV,axiom,
( ( finite1805530552687753326_a_nat @ top_to5660197770879767925_a_nat )
= ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat ) ) ).
% finite_option_UNIV
thf(fact_1207_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_1208_finite__option__UNIV,axiom,
( ( finite2357637324228559219list_a @ top_to1591815355649308666list_a )
= ( finite_finite_list_a @ top_top_set_list_a ) ) ).
% finite_option_UNIV
thf(fact_1209_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_1210_proj__out,axiom,
! [Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma2: nat > sum_sum_a_nat] :
( ( wf_fo_intp_a_b @ Phi2 @ I2 )
=> ( ( esat_a_b @ Phi2 @ I2 @ Sigma2 @ top_to795618464972521135_a_nat )
=> ? [Tau3: nat > a] :
( ( esat_a_b @ Phi2 @ I2 @ ( comp_a4874482103519161268at_nat @ sum_Inl_a_nat @ Tau3 ) @ top_to795618464972521135_a_nat )
& ! [I4: nat,X4: a] :
( ( ( member_nat @ I4 @ ( fv_fo_fmla_a_b @ Phi2 ) )
& ( ( Sigma2 @ I4 )
= ( sum_Inl_a_nat @ X4 ) ) )
=> ( ( Tau3 @ I4 )
= X4 ) )
& ( ad_agr_list_a_nat @ ( act_edom_a_b @ Phi2 @ I2 ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ ( fv_fo_fmla_list_a_b @ Phi2 ) ) @ ( map_na823391071729141993_a_nat @ ( comp_a4874482103519161268at_nat @ sum_Inl_a_nat @ Tau3 ) @ ( fv_fo_fmla_list_a_b @ Phi2 ) ) ) ) ) ) ).
% proj_out
thf(fact_1211_esat__UNIV__cong,axiom,
! [Phi2: fo_fmla_a_b,AD: set_a,Sigma2: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat,I2: product_prod_b_nat > set_list_a] :
( ( ad_agr_a_b_nat @ Phi2 @ AD @ Sigma2 @ Tau2 )
=> ( ( ord_less_eq_set_a @ ( act_edom_a_b @ Phi2 @ I2 ) @ AD )
=> ( ( esat_a_b @ Phi2 @ I2 @ Sigma2 @ top_to795618464972521135_a_nat )
= ( esat_a_b @ Phi2 @ I2 @ Tau2 @ top_to795618464972521135_a_nat ) ) ) ) ).
% esat_UNIV_cong
thf(fact_1212_infinite__UNIV,axiom,
~ ( finite_finite_a @ top_top_set_a ) ).
% infinite_UNIV
thf(fact_1213_infinite__UNIV,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV
thf(fact_1214_ad__agr__list__fo__nmlzd__isl,axiom,
! [X5: set_a,F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X5 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) @ ( map_Su2790769393171190532_a_nat @ G @ Xs ) )
=> ( ( fo_nmlzd_a @ X5 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
=> ( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( sum_isl_a_nat @ ( F @ X3 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_1215_ad__agr__list__fo__nmlzd__isl,axiom,
! [X5: set_a,F: a > sum_sum_a_nat,Xs: list_a,G: a > sum_sum_a_nat,X3: a] :
( ( ad_agr_list_a_nat @ X5 @ ( map_a_Sum_sum_a_nat @ F @ Xs ) @ ( map_a_Sum_sum_a_nat @ G @ Xs ) )
=> ( ( fo_nmlzd_a @ X5 @ ( map_a_Sum_sum_a_nat @ F @ Xs ) )
=> ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( sum_isl_a_nat @ ( F @ X3 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_1216_ad__agr__list__fo__nmlzd__isl,axiom,
! [X5: set_a,F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat,X3: nat] :
( ( ad_agr_list_a_nat @ X5 @ ( map_na823391071729141993_a_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ G @ Xs ) )
=> ( ( fo_nmlzd_a @ X5 @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
=> ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( sum_isl_a_nat @ ( F @ X3 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_1217_fv__fo__fmla__list__eq,axiom,
! [Phi2: fo_fmla_a_b,Psi: fo_fmla_a_b] :
( ( ( fv_fo_fmla_a_b @ Phi2 )
= ( fv_fo_fmla_a_b @ Psi ) )
=> ( ( fv_fo_fmla_list_a_b @ Phi2 )
= ( fv_fo_fmla_list_a_b @ Psi ) ) ) ).
% fv_fo_fmla_list_eq
thf(fact_1218_esat__fv__cong,axiom,
! [Phi2: fo_fmla_a_b,Sigma2: nat > sum_sum_a_nat,Sigma3: nat > sum_sum_a_nat,I2: product_prod_b_nat > set_list_a,X5: set_Sum_sum_a_nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ ( fv_fo_fmla_a_b @ Phi2 ) )
=> ( ( Sigma2 @ N2 )
= ( Sigma3 @ N2 ) ) )
=> ( ( esat_a_b @ Phi2 @ I2 @ Sigma2 @ X5 )
= ( esat_a_b @ Phi2 @ I2 @ Sigma3 @ X5 ) ) ) ).
% esat_fv_cong
thf(fact_1219_sat__fv__cong,axiom,
! [Phi2: fo_fmla_a_b,Sigma2: nat > a,Sigma3: nat > a,I2: product_prod_b_nat > set_list_a] :
( ! [N2: nat] :
( ( member_nat @ N2 @ ( fv_fo_fmla_a_b @ Phi2 ) )
=> ( ( Sigma2 @ N2 )
= ( Sigma3 @ N2 ) ) )
=> ( ( sat_a_b @ Phi2 @ I2 @ Sigma2 )
= ( sat_a_b @ Phi2 @ I2 @ Sigma3 ) ) ) ).
% sat_fv_cong
thf(fact_1220_sum_Odisc_I1_J,axiom,
! [X1: a] : ( sum_isl_a_nat @ ( sum_Inl_a_nat @ X1 ) ) ).
% sum.disc(1)
thf(fact_1221_sum_OdiscI_I1_J,axiom,
! [Sum: sum_sum_a_nat,X1: a] :
( ( Sum
= ( sum_Inl_a_nat @ X1 ) )
=> ( sum_isl_a_nat @ Sum ) ) ).
% sum.discI(1)
thf(fact_1222_isl__def,axiom,
( sum_isl_a_nat
= ( ^ [Sum2: sum_sum_a_nat] :
? [X12: a] :
( Sum2
= ( sum_Inl_a_nat @ X12 ) ) ) ) ).
% isl_def
thf(fact_1223_fv__fo__fmla__list__set,axiom,
! [Phi2: fo_fmla_a_b] :
( ( set_nat2 @ ( fv_fo_fmla_list_a_b @ Phi2 ) )
= ( fv_fo_fmla_a_b @ Phi2 ) ) ).
% fv_fo_fmla_list_set
thf(fact_1224_sum_Ocollapse_I1_J,axiom,
! [Sum: sum_sum_a_nat] :
( ( sum_isl_a_nat @ Sum )
=> ( ( sum_Inl_a_nat @ ( sum_projl_a_nat @ Sum ) )
= Sum ) ) ).
% sum.collapse(1)
thf(fact_1225_finite__lists__length__eq,axiom,
! [A3: set_a,N3: nat] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ A3 )
& ( ( size_size_list_a @ Xs2 )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1226_finite__lists__length__eq,axiom,
! [A3: set_list_a,N3: nat] :
( ( finite_finite_list_a @ A3 )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [Xs2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ A3 )
& ( ( size_s349497388124573686list_a @ Xs2 )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1227_finite__lists__length__eq,axiom,
! [A3: set_nat,N3: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
& ( ( size_size_list_nat @ Xs2 )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1228_finite__lists__length__eq,axiom,
! [A3: set_Sum_sum_a_nat,N3: nat] :
( ( finite502105017643426984_a_nat @ A3 )
=> ( finite1487985464145237934_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [Xs2: list_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ A3 )
& ( ( size_s5283204784079214577_a_nat @ Xs2 )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1229_finite__lists__length__eq,axiom,
! [A3: set_o,N3: nat] :
( ( finite_finite_o @ A3 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs2: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
& ( ( size_size_list_o @ Xs2 )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1230_finite__lists__length__eq,axiom,
! [A3: set_li6526943997496501093_a_nat,N3: nat] :
( ( finite1487985464145237934_a_nat @ A3 )
=> ( finite3264696653626932_a_nat
@ ( collec4000455149352298102_a_nat
@ ^ [Xs2: list_l4703314356710769291_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs2 ) @ A3 )
& ( ( size_s5212483967078203639_a_nat @ Xs2 )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1231_finite__lists__length__le,axiom,
! [A3: set_a,N3: nat] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ A3 )
& ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1232_finite__lists__length__le,axiom,
! [A3: set_list_a,N3: nat] :
( ( finite_finite_list_a @ A3 )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [Xs2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ A3 )
& ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs2 ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1233_finite__lists__length__le,axiom,
! [A3: set_nat,N3: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1234_finite__lists__length__le,axiom,
! [A3: set_Sum_sum_a_nat,N3: nat] :
( ( finite502105017643426984_a_nat @ A3 )
=> ( finite1487985464145237934_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [Xs2: list_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ A3 )
& ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs2 ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1235_finite__lists__length__le,axiom,
! [A3: set_o,N3: nat] :
( ( finite_finite_o @ A3 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs2: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
& ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1236_finite__lists__length__le,axiom,
! [A3: set_li6526943997496501093_a_nat,N3: nat] :
( ( finite1487985464145237934_a_nat @ A3 )
=> ( finite3264696653626932_a_nat
@ ( collec4000455149352298102_a_nat
@ ^ [Xs2: list_l4703314356710769291_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs2 ) @ A3 )
& ( ord_less_eq_nat @ ( size_s5212483967078203639_a_nat @ Xs2 ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1237_length__map,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_1238_length__map,axiom,
! [F: a > sum_sum_a_nat,Xs: list_a] :
( ( size_s5283204784079214577_a_nat @ ( map_a_Sum_sum_a_nat @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_1239_length__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_1240_length__map,axiom,
! [F: list_a > list_Sum_sum_a_nat,Xs: list_list_a] :
( ( size_s5212483967078203639_a_nat @ ( map_li6996171348340886213_a_nat @ F @ Xs ) )
= ( size_s349497388124573686list_a @ Xs ) ) ).
% length_map
thf(fact_1241_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1242_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1243_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1244_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: a > sum_sum_a_nat,Ys: list_a] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_a_Sum_sum_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1245_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1246_map__eq__imp__length__eq,axiom,
! [F: a > sum_sum_a_nat,Xs: list_a,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1247_map__eq__imp__length__eq,axiom,
! [F: a > sum_sum_a_nat,Xs: list_a,G: a > sum_sum_a_nat,Ys: list_a] :
( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
= ( map_a_Sum_sum_a_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1248_map__eq__imp__length__eq,axiom,
! [F: a > sum_sum_a_nat,Xs: list_a,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_a_Sum_sum_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1249_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1250_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: a > sum_sum_a_nat,Ys: list_a] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_a_Sum_sum_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1251_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1252_map__eq__imp__length__eq,axiom,
! [F: list_a > list_Sum_sum_a_nat,Xs: list_list_a,G: list_a > list_Sum_sum_a_nat,Ys: list_list_a] :
( ( ( map_li6996171348340886213_a_nat @ F @ Xs )
= ( map_li6996171348340886213_a_nat @ G @ Ys ) )
=> ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1253_sum_Osel_I1_J,axiom,
! [X1: a] :
( ( sum_projl_a_nat @ ( sum_Inl_a_nat @ X1 ) )
= X1 ) ).
% sum.sel(1)
thf(fact_1254_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1255_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1256_finite__lists__distinct__length__eq,axiom,
! [A3: set_nat,N3: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= N3 )
& ( distinct_nat @ Xs2 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1257_finite__lists__distinct__length__eq,axiom,
! [A3: set_Sum_sum_a_nat,N3: nat] :
( ( finite502105017643426984_a_nat @ A3 )
=> ( finite1487985464145237934_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [Xs2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= N3 )
& ( distin2701893636801681144_a_nat @ Xs2 )
& ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1258_finite__lists__distinct__length__eq,axiom,
! [A3: set_o,N3: nat] :
( ( finite_finite_o @ A3 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= N3 )
& ( distinct_o @ Xs2 )
& ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1259_finite__lists__distinct__length__eq,axiom,
! [A3: set_li6526943997496501093_a_nat,N3: nat] :
( ( finite1487985464145237934_a_nat @ A3 )
=> ( finite3264696653626932_a_nat
@ ( collec4000455149352298102_a_nat
@ ^ [Xs2: list_l4703314356710769291_a_nat] :
( ( ( size_s5212483967078203639_a_nat @ Xs2 )
= N3 )
& ( distin811021574259663358_a_nat @ Xs2 )
& ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs2 ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1260_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M: nat] :
( ( P @ X3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1261_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1262_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A4: set_o] :
~ ( member_o @ $false @ A4 ) ) ) ).
% Inf_bool_def
thf(fact_1263_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1264_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1265_nat__le__linear,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
| ( ord_less_eq_nat @ N3 @ M2 ) ) ).
% nat_le_linear
thf(fact_1266_le__antisym,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M2 )
=> ( M2 = N3 ) ) ) ).
% le_antisym
thf(fact_1267_eq__imp__le,axiom,
! [M2: nat,N3: nat] :
( ( M2 = N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% eq_imp_le
thf(fact_1268_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1269_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_1270_diff__le__mono2,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1271_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1272_diff__le__self,axiom,
! [M2: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ).
% diff_le_self
thf(fact_1273_diff__le__mono,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1274_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1275_le__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1276_eq__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M2 = N3 ) ) ) ) ).
% eq_diff_iff
% Conjectures (2)
thf(conj_0,hypothesis,
! [Sigma4: nat > sum_sum_a_nat] :
( ( vs
= ( map_na823391071729141993_a_nat @ Sigma4 @ ( fv_fo_fmla_list_a_b @ phi ) ) )
=> ( ( esat_a_b @ phi @ i @ Sigma4 @ top_to795618464972521135_a_nat )
=> ( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ vs ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ top_top_set_a ) )
=> thesis ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------