TPTP Problem File: SLH0489^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : VYDRA_MDL/0007_MDL/prob_00297_012519__16208784_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1481 ( 511 unt; 200 typ; 0 def)
% Number of atoms : 4074 (1313 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 13316 ( 362 ~; 76 |; 245 &;10537 @)
% ( 0 <=>;2096 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 1141 (1141 >; 0 *; 0 +; 0 <<)
% Number of symbols : 176 ( 173 usr; 25 con; 0-4 aty)
% Number of variables : 4063 ( 186 ^;3781 !; 96 ?;4063 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:52:35.587
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
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% Explicit typings (173)
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thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
accp_P1267725715503270512at_nat: ( produc859450856879609959at_nat > produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
accp_P795512239754848240at_nat: ( produc349518998152878311at_nat > produc349518998152878311at_nat > $o ) > produc349518998152878311at_nat > $o ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
lex_pr8029265285556086080at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > set_Pr575275573428919693at_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
lex_pr4669217892513370978at_nat: set_Pr8693737435421807431at_nat > set_Pr1261947904930325089at_nat > set_Pr7116486347545156417at_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
lex_pr8801849515957261039at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr553994874890374343at_nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
measur8038558561449204169at_nat: ( product_prod_nat_nat > nat ) > set_Pr8693737435421807431at_nat ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Nat__Onat,type,
mlex_prod_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
mlex_p6366001652026297872at_nat: ( product_prod_nat_nat > nat ) > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__Nat__Onat,type,
same_fst_nat_nat: ( nat > $o ) > ( nat > set_Pr1261947904930325089at_nat ) > set_Pr8693737435421807431at_nat ).
thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
same_f4956014544515070124at_nat: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > set_Pr8693737435421807431at_nat ) > set_Pr553994874890374343at_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member7775084264787586014at_nat: produc4471711990508489141at_nat > set_Pr4668228411283691157at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2223272150424702269at_nat: produc7248412053542808358at_nat > set_Pr7717912310451564380at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member3782324328723991648et_nat: produc2400336064389900727et_nat > set_Pr400265656397884439et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member3348759134392003351at_nat: produc8373899037510109440at_nat > set_Pr2539167527615954998at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member3048279782668807382at_nat: produc6277219514840344877at_nat > set_Pr575275573428919693at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member1995966531042493578at_nat: produc5224906263214031073at_nat > set_Pr7116486347545156417at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
member8873588116083876704at_nat: produc7491599851749785783at_nat > set_Pr3601174868274201367at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v__092_060sigma_062,type,
sigma: trace_a_t ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_ja____,type,
ja: nat ).
thf(sy_v_r1____,type,
r1: regex_a_t ).
thf(sy_v_r2____,type,
r2: regex_a_t ).
% Relevant facts (1267)
thf(fact_0_Times_Oprems_I2_J,axiom,
ord_less_eq_nat @ ia @ ja ).
% Times.prems(2)
thf(fact_1_match__le,axiom,
! [I: nat,J: nat,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ R ) )
=> ( ord_less_eq_nat @ I @ J ) ) ).
% match_le
thf(fact_2_match__refl__eps,axiom,
! [I: nat,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_t @ sigma @ R ) )
=> ( eps_a_t @ R ) ) ).
% match_refl_eps
thf(fact_3_wf__regex__eps__match,axiom,
! [R: regex_a_t,I: nat] :
( ( wf_regex_a_t @ R )
=> ( ( eps_a_t @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_t @ sigma @ R ) ) ) ) ).
% wf_regex_eps_match
thf(fact_4__092_060open_062_092_060And_062i_O_A_Ii_M_ASuc_Aj_J_A_092_060in_062_Amatch_Ar2_A_092_060Longrightarrow_062_Ai_A_092_060le_062_ASuc_Aj_092_060close_062,axiom,
! [I: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( suc @ ja ) ) @ ( match_a_t @ sigma @ r2 ) )
=> ( ord_less_eq_nat @ I @ ( suc @ ja ) ) ) ).
% \<open>\<And>i. (i, Suc j) \<in> match r2 \<Longrightarrow> i \<le> Suc j\<close>
thf(fact_5__092_060open_062_ISuc_Aj_M_ASuc_Aj_J_A_092_060in_062_Amatch_Ar2_A_092_060Longrightarrow_062_Aeps_Ar2_092_060close_062,axiom,
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( suc @ ja ) @ ( suc @ ja ) ) @ ( match_a_t @ sigma @ r2 ) )
=> ( eps_a_t @ r2 ) ) ).
% \<open>(Suc j, Suc j) \<in> match r2 \<Longrightarrow> eps r2\<close>
thf(fact_6_Times_OIH_I2_J,axiom,
! [I: nat,J: nat] :
( ( wf_regex_a_t @ r2 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) @ ( match_a_t @ sigma @ r2 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ ( rderive_a_t @ r2 ) ) ) ) ) ) ).
% Times.IH(2)
thf(fact_7_Times_OIH_I1_J,axiom,
! [I: nat,J: nat] :
( ( wf_regex_a_t @ r1 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) @ ( match_a_t @ sigma @ r1 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ ( rderive_a_t @ r1 ) ) ) ) ) ) ).
% Times.IH(1)
thf(fact_8_Times_Oprems_I1_J,axiom,
wf_regex_a_t @ ( times_a_t @ r1 @ r2 ) ).
% Times.prems(1)
thf(fact_9_MDL_Omatch__le,axiom,
! [I: nat,J: nat,Sigma: trace_a_t,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ Sigma @ R ) )
=> ( ord_less_eq_nat @ I @ J ) ) ).
% MDL.match_le
thf(fact_10_MDL_Omatch_Ocong,axiom,
match_a_t = match_a_t ).
% MDL.match.cong
thf(fact_11_MDL_Omatch__refl__eps,axiom,
! [I: nat,Sigma: trace_a_t,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_t @ Sigma @ R ) )
=> ( eps_a_t @ R ) ) ).
% MDL.match_refl_eps
thf(fact_12_MDL_Owf__regex__eps__match,axiom,
! [R: regex_a_t,I: nat,Sigma: trace_a_t] :
( ( wf_regex_a_t @ R )
=> ( ( eps_a_t @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_t @ Sigma @ R ) ) ) ) ).
% MDL.wf_regex_eps_match
thf(fact_13_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_14_lift__Suc__mono__le,axiom,
! [F: nat > set_set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6893508408891458716et_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_15_lift__Suc__mono__le,axiom,
! [F: nat > product_prod_nat_nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le704812498762024988_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le704812498762024988_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_16_lift__Suc__mono__le,axiom,
! [F: nat > product_prod_nat_nat > product_prod_nat_nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le5604493270027003598_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le5604493270027003598_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_17_lift__Suc__mono__le,axiom,
! [F: nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_18_lift__Suc__mono__le,axiom,
! [F: nat > nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le2646555220125990790_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_19_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_20_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_21_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6893508408891458716et_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_22_lift__Suc__antimono__le,axiom,
! [F: nat > product_prod_nat_nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le704812498762024988_nat_o @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le704812498762024988_nat_o @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_23_lift__Suc__antimono__le,axiom,
! [F: nat > product_prod_nat_nat > product_prod_nat_nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le5604493270027003598_nat_o @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le5604493270027003598_nat_o @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_24_lift__Suc__antimono__le,axiom,
! [F: nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat_o @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat_o @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_25_lift__Suc__antimono__le,axiom,
! [F: nat > nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le2646555220125990790_nat_o @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le2646555220125990790_nat_o @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_26_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_27_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_28_match_Osimps_I4_J,axiom,
! [R: regex_a_t,S: regex_a_t] :
( ( match_a_t @ sigma @ ( times_a_t @ R @ S ) )
= ( relcomp_nat_nat_nat @ ( match_a_t @ sigma @ R ) @ ( match_a_t @ sigma @ S ) ) ) ).
% match.simps(4)
thf(fact_29_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_30_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_31_prod_Oinject,axiom,
! [X1: set_nat,X2: set_nat,Y1: set_nat,Y2: set_nat] :
( ( ( produc4532415448927165861et_nat @ X1 @ X2 )
= ( produc4532415448927165861et_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_32_prod_Oinject,axiom,
! [X1: set_nat,X2: nat,Y1: set_nat,Y2: nat] :
( ( ( produc641871753055645167at_nat @ X1 @ X2 )
= ( produc641871753055645167at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_33_prod_Oinject,axiom,
! [X1: nat,X2: product_prod_nat_nat,Y1: nat,Y2: product_prod_nat_nat] :
( ( ( produc487386426758144856at_nat @ X1 @ X2 )
= ( produc487386426758144856at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_34_prod_Oinject,axiom,
! [X1: nat,X2: set_nat,Y1: nat,Y2: set_nat] :
( ( ( produc4207506657711014383et_nat @ X1 @ X2 )
= ( produc4207506657711014383et_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_35_prod_Oinject,axiom,
! [X1: nat > nat > nat,X2: produc7248412053542808358at_nat,Y1: nat > nat > nat,Y2: produc7248412053542808358at_nat] :
( ( ( produc3209952032786966637at_nat @ X1 @ X2 )
= ( produc3209952032786966637at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_36_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_37_prod_Oinject,axiom,
! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
= ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_38_old_Oprod_Oinject,axiom,
! [A: set_nat,B: set_nat,A2: set_nat,B2: set_nat] :
( ( ( produc4532415448927165861et_nat @ A @ B )
= ( produc4532415448927165861et_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_39_old_Oprod_Oinject,axiom,
! [A: set_nat,B: nat,A2: set_nat,B2: nat] :
( ( ( produc641871753055645167at_nat @ A @ B )
= ( produc641871753055645167at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_40_old_Oprod_Oinject,axiom,
! [A: nat,B: product_prod_nat_nat,A2: nat,B2: product_prod_nat_nat] :
( ( ( produc487386426758144856at_nat @ A @ B )
= ( produc487386426758144856at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_41_old_Oprod_Oinject,axiom,
! [A: nat,B: set_nat,A2: nat,B2: set_nat] :
( ( ( produc4207506657711014383et_nat @ A @ B )
= ( produc4207506657711014383et_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_42_old_Oprod_Oinject,axiom,
! [A: nat > nat > nat,B: produc7248412053542808358at_nat,A2: nat > nat > nat,B2: produc7248412053542808358at_nat] :
( ( ( produc3209952032786966637at_nat @ A @ B )
= ( produc3209952032786966637at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_43_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_44_old_Oprod_Oinject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A @ B )
= ( produc6161850002892822231at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_45_order__refl,axiom,
! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).
% order_refl
thf(fact_46_order__refl,axiom,
! [X: product_prod_nat_nat > $o] : ( ord_le704812498762024988_nat_o @ X @ X ) ).
% order_refl
thf(fact_47_order__refl,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > $o] : ( ord_le5604493270027003598_nat_o @ X @ X ) ).
% order_refl
thf(fact_48_order__refl,axiom,
! [X: nat > $o] : ( ord_less_eq_nat_o @ X @ X ) ).
% order_refl
thf(fact_49_order__refl,axiom,
! [X: nat > nat > $o] : ( ord_le2646555220125990790_nat_o @ X @ X ) ).
% order_refl
thf(fact_50_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_51_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_52_dual__order_Orefl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% dual_order.refl
thf(fact_53_dual__order_Orefl,axiom,
! [A: product_prod_nat_nat > $o] : ( ord_le704812498762024988_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_54_dual__order_Orefl,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o] : ( ord_le5604493270027003598_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_55_dual__order_Orefl,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_56_dual__order_Orefl,axiom,
! [A: nat > nat > $o] : ( ord_le2646555220125990790_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_57_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_58_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_59_relcomp_Ocases,axiom,
! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R @ S ) )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_60_relcomp_Ocases,axiom,
! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( relcom7295539661566034944at_nat @ R @ S ) )
=> ~ ! [B3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ B3 ) @ R )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_61_relcomp_Ocases,axiom,
! [A1: nat,A22: nat,R: set_Pr400265656397884439et_nat,S: set_Pr3601174868274201367at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcom6344913834383254569at_nat @ R @ S ) )
=> ~ ! [B3: set_nat] :
( ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ A1 @ B3 ) @ R )
=> ~ ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_62_relcomp_Ocases,axiom,
! [A1: set_nat,A22: nat,R: set_Pr3601174868274201367at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ A1 @ A22 ) @ ( relcom4526255650228441129at_nat @ R @ S ) )
=> ~ ! [B3: nat] :
( ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ A1 @ B3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_63_relcomp_Ocases,axiom,
! [A1: nat,A22: set_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr400265656397884439et_nat] :
( ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ A1 @ A22 ) @ ( relcom687176702183847977et_nat @ R @ S ) )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ R )
=> ~ ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_64_relcomp_Ocases,axiom,
! [A1: nat,A22: nat,R: set_Pr7717912310451564380at_nat,S: set_Pr2539167527615954998at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcom6771037729206764224at_nat @ R @ S ) )
=> ~ ! [B3: product_prod_nat_nat] :
( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A1 @ B3 ) @ R )
=> ~ ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_65_relcomp_Ocases,axiom,
! [A1: set_nat,A22: set_nat,R: set_Pr3601174868274201367at_nat,S: set_Pr400265656397884439et_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( relcom4805083259087181791et_nat @ R @ S ) )
=> ~ ! [B3: nat] :
( ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ A1 @ B3 ) @ R )
=> ~ ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_66_relcomp_Ocases,axiom,
! [A1: set_nat,A22: nat,R: set_Pr5488025237498180813et_nat,S: set_Pr3601174868274201367at_nat] :
( ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ A1 @ A22 ) @ ( relcom1239448354431812575at_nat @ R @ S ) )
=> ~ ! [B3: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ B3 ) @ R )
=> ~ ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_67_relcomp_Ocases,axiom,
! [A1: nat,A22: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr7717912310451564380at_nat] :
( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A1 @ A22 ) @ ( relcom907713085394703518at_nat @ R @ S ) )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ R )
=> ~ ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_68_relcomp_Ocases,axiom,
! [A1: nat,A22: set_nat,R: set_Pr400265656397884439et_nat,S: set_Pr5488025237498180813et_nat] :
( ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ A1 @ A22 ) @ ( relcom8128622903297633247et_nat @ R @ S ) )
=> ~ ! [B3: set_nat] :
( ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ A1 @ B3 ) @ R )
=> ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_69_regex_Oinject_I4_J,axiom,
! [X41: regex_a_t,X42: regex_a_t,Y41: regex_a_t,Y42: regex_a_t] :
( ( ( times_a_t @ X41 @ X42 )
= ( times_a_t @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% regex.inject(4)
thf(fact_70_subrelI,axiom,
! [R: set_Pr553994874890374343at_nat,S: set_Pr553994874890374343at_nat] :
( ! [X3: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y ) @ R )
=> ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y ) @ S ) )
=> ( ord_le2286497330737841255at_nat @ R @ S ) ) ).
% subrelI
thf(fact_71_subrelI,axiom,
! [R: set_Pr7116486347545156417at_nat,S: set_Pr7116486347545156417at_nat] :
( ! [X3: produc8373899037510109440at_nat,Y: produc8373899037510109440at_nat] :
( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ X3 @ Y ) @ R )
=> ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ X3 @ Y ) @ S ) )
=> ( ord_le5962939600335684321at_nat @ R @ S ) ) ).
% subrelI
thf(fact_72_subrelI,axiom,
! [R: set_Pr575275573428919693at_nat,S: set_Pr575275573428919693at_nat] :
( ! [X3: produc7248412053542808358at_nat,Y: produc7248412053542808358at_nat] :
( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ X3 @ Y ) @ R )
=> ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ X3 @ Y ) @ S ) )
=> ( ord_le8645100863074223405at_nat @ R @ S ) ) ).
% subrelI
thf(fact_73_subrelI,axiom,
! [R: set_Pr5488025237498180813et_nat,S: set_Pr5488025237498180813et_nat] :
( ! [X3: set_nat,Y: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y ) @ R )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y ) @ S ) )
=> ( ord_le4402255868550298733et_nat @ R @ S ) ) ).
% subrelI
thf(fact_74_subrelI,axiom,
! [R: set_Pr3601174868274201367at_nat,S: set_Pr3601174868274201367at_nat] :
( ! [X3: set_nat,Y: nat] :
( ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ X3 @ Y ) @ R )
=> ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ X3 @ Y ) @ S ) )
=> ( ord_le7046853370993658551at_nat @ R @ S ) ) ).
% subrelI
thf(fact_75_subrelI,axiom,
! [R: set_Pr7717912310451564380at_nat,S: set_Pr7717912310451564380at_nat] :
( ! [X3: nat,Y: product_prod_nat_nat] :
( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X3 @ Y ) @ R )
=> ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X3 @ Y ) @ S ) )
=> ( ord_le3050193496395145148at_nat @ R @ S ) ) ).
% subrelI
thf(fact_76_subrelI,axiom,
! [R: set_Pr400265656397884439et_nat,S: set_Pr400265656397884439et_nat] :
( ! [X3: nat,Y: set_nat] :
( ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ X3 @ Y ) @ R )
=> ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ X3 @ Y ) @ S ) )
=> ( ord_le3845944159117341623et_nat @ R @ S ) ) ).
% subrelI
thf(fact_77_subrelI,axiom,
! [R: set_Pr4668228411283691157at_nat,S: set_Pr4668228411283691157at_nat] :
( ! [X3: nat > nat > nat,Y: produc7248412053542808358at_nat] :
( ( member7775084264787586014at_nat @ ( produc3209952032786966637at_nat @ X3 @ Y ) @ R )
=> ( member7775084264787586014at_nat @ ( produc3209952032786966637at_nat @ X3 @ Y ) @ S ) )
=> ( ord_le8125556312545103413at_nat @ R @ S ) ) ).
% subrelI
thf(fact_78_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_79_subrelI,axiom,
! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ S ) )
=> ( ord_le3000389064537975527at_nat @ R @ S ) ) ).
% subrelI
thf(fact_80_relcomp__mono,axiom,
! [R2: set_Pr8693737435421807431at_nat,R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ( ord_le3000389064537975527at_nat @ R2 @ R )
=> ( ( ord_le3000389064537975527at_nat @ S2 @ S )
=> ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ R2 @ S2 ) @ ( relcom7295539661566034944at_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_81_relcomp__mono,axiom,
! [R2: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ R )
=> ( ( ord_le3146513528884898305at_nat @ S2 @ S )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R2 @ S2 ) @ ( relcomp_nat_nat_nat @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_82_eps_Osimps_I4_J,axiom,
! [R: regex_a_t,S: regex_a_t] :
( ( eps_a_t @ ( times_a_t @ R @ S ) )
= ( ( eps_a_t @ R )
& ( eps_a_t @ S ) ) ) ).
% eps.simps(4)
thf(fact_83_wf__regex_Osimps_I4_J,axiom,
! [R: regex_a_t,S: regex_a_t] :
( ( wf_regex_a_t @ ( times_a_t @ R @ S ) )
= ( ( wf_regex_a_t @ S )
& ( ~ ( eps_a_t @ S )
| ( wf_regex_a_t @ R ) ) ) ) ).
% wf_regex.simps(4)
thf(fact_84_MDL_Omatch_Osimps_I4_J,axiom,
! [Sigma: trace_a_t,R: regex_a_t,S: regex_a_t] :
( ( match_a_t @ Sigma @ ( times_a_t @ R @ S ) )
= ( relcomp_nat_nat_nat @ ( match_a_t @ Sigma @ R ) @ ( match_a_t @ Sigma @ S ) ) ) ).
% MDL.match.simps(4)
thf(fact_85_order__antisym__conv,axiom,
! [Y3: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y3 @ X )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_86_order__antisym__conv,axiom,
! [Y3: product_prod_nat_nat > $o,X: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ Y3 @ X )
=> ( ( ord_le704812498762024988_nat_o @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_87_order__antisym__conv,axiom,
! [Y3: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ Y3 @ X )
=> ( ( ord_le5604493270027003598_nat_o @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_88_order__antisym__conv,axiom,
! [Y3: nat > $o,X: nat > $o] :
( ( ord_less_eq_nat_o @ Y3 @ X )
=> ( ( ord_less_eq_nat_o @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_89_order__antisym__conv,axiom,
! [Y3: nat > nat > $o,X: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ Y3 @ X )
=> ( ( ord_le2646555220125990790_nat_o @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_90_order__antisym__conv,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_91_order__antisym__conv,axiom,
! [Y3: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_92_linorder__le__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_93_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_94_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_95_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_96_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_98_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_ord__le__eq__subst,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_100_ord__le__eq__subst,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_101_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_105_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_107_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_108_ord__eq__le__subst,axiom,
! [A: nat > $o,F: nat > nat > $o,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_109_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_110_ord__eq__le__subst,axiom,
! [A: nat,F: ( nat > $o ) > nat,B: nat > $o,C: nat > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A: nat > $o,F: set_nat > nat > $o,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_linorder__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_114_order__eq__refl,axiom,
! [X: set_set_nat,Y3: set_set_nat] :
( ( X = Y3 )
=> ( ord_le6893508408891458716et_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_115_order__eq__refl,axiom,
! [X: product_prod_nat_nat > $o,Y3: product_prod_nat_nat > $o] :
( ( X = Y3 )
=> ( ord_le704812498762024988_nat_o @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_116_order__eq__refl,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( X = Y3 )
=> ( ord_le5604493270027003598_nat_o @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_117_order__eq__refl,axiom,
! [X: nat > $o,Y3: nat > $o] :
( ( X = Y3 )
=> ( ord_less_eq_nat_o @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_118_order__eq__refl,axiom,
! [X: nat > nat > $o,Y3: nat > nat > $o] :
( ( X = Y3 )
=> ( ord_le2646555220125990790_nat_o @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_119_order__eq__refl,axiom,
! [X: nat,Y3: nat] :
( ( X = Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_120_order__eq__refl,axiom,
! [X: set_nat,Y3: set_nat] :
( ( X = Y3 )
=> ( ord_less_eq_set_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_121_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_123_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_124_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_125_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_126_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat_o @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_127_order__subst2,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_128_order__subst2,axiom,
! [A: nat > $o,B: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_129_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_130_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat_o @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_131_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_132_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_133_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_134_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_135_order__subst1,axiom,
! [A: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_136_order__subst1,axiom,
! [A: nat,F: ( nat > $o ) > nat,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_137_order__subst1,axiom,
! [A: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le6893508408891458716et_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_138_order__subst1,axiom,
! [A: nat > $o,F: nat > nat > $o,B: nat,C: nat] :
( ( ord_less_eq_nat_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_139_order__subst1,axiom,
! [A: set_nat,F: set_set_nat > set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_140_order__subst1,axiom,
! [A: set_nat,F: ( nat > $o ) > set_nat,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ! [X3: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_141_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_142_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: product_prod_nat_nat > $o,Z: product_prod_nat_nat > $o] : ( Y4 = Z ) )
= ( ^ [A3: product_prod_nat_nat > $o,B4: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ A3 @ B4 )
& ( ord_le704812498762024988_nat_o @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_143_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: product_prod_nat_nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat > product_prod_nat_nat > $o] : ( Y4 = Z ) )
= ( ^ [A3: product_prod_nat_nat > product_prod_nat_nat > $o,B4: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ A3 @ B4 )
& ( ord_le5604493270027003598_nat_o @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_144_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > $o,Z: nat > $o] : ( Y4 = Z ) )
= ( ^ [A3: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ A3 @ B4 )
& ( ord_less_eq_nat_o @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_145_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > nat > $o,Z: nat > nat > $o] : ( Y4 = Z ) )
= ( ^ [A3: nat > nat > $o,B4: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A3 @ B4 )
& ( ord_le2646555220125990790_nat_o @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_146_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_147_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_148_le__fun__def,axiom,
( ord_le704812498762024988_nat_o
= ( ^ [F2: product_prod_nat_nat > $o,G: product_prod_nat_nat > $o] :
! [X4: product_prod_nat_nat] : ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_149_le__fun__def,axiom,
( ord_le5604493270027003598_nat_o
= ( ^ [F2: product_prod_nat_nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
! [X4: product_prod_nat_nat] : ( ord_le704812498762024988_nat_o @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_150_le__fun__def,axiom,
( ord_less_eq_nat_o
= ( ^ [F2: nat > $o,G: nat > $o] :
! [X4: nat] : ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_151_le__fun__def,axiom,
( ord_le2646555220125990790_nat_o
= ( ^ [F2: nat > nat > $o,G: nat > nat > $o] :
! [X4: nat] : ( ord_less_eq_nat_o @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_152_le__funI,axiom,
! [F: product_prod_nat_nat > $o,G2: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] : ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_le704812498762024988_nat_o @ F @ G2 ) ) ).
% le_funI
thf(fact_153_le__funI,axiom,
! [F: nat > $o,G2: nat > $o] :
( ! [X3: nat] : ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq_nat_o @ F @ G2 ) ) ).
% le_funI
thf(fact_154_le__funI,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o,G2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] : ( ord_le704812498762024988_nat_o @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_le5604493270027003598_nat_o @ F @ G2 ) ) ).
% le_funI
thf(fact_155_le__funI,axiom,
! [F: nat > nat > $o,G2: nat > nat > $o] :
( ! [X3: nat] : ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_le2646555220125990790_nat_o @ F @ G2 ) ) ).
% le_funI
thf(fact_156_le__funE,axiom,
! [F: product_prod_nat_nat > $o,G2: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ( ord_le704812498762024988_nat_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_157_le__funE,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o,G2: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ( ord_le5604493270027003598_nat_o @ F @ G2 )
=> ( ord_le704812498762024988_nat_o @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_158_le__funE,axiom,
! [F: nat > $o,G2: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_159_le__funE,axiom,
! [F: nat > nat > $o,G2: nat > nat > $o,X: nat] :
( ( ord_le2646555220125990790_nat_o @ F @ G2 )
=> ( ord_less_eq_nat_o @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funE
thf(fact_160_le__funD,axiom,
! [F: product_prod_nat_nat > $o,G2: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ( ord_le704812498762024988_nat_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_161_le__funD,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o,G2: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ( ord_le5604493270027003598_nat_o @ F @ G2 )
=> ( ord_le704812498762024988_nat_o @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_162_le__funD,axiom,
! [F: nat > $o,G2: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_163_le__funD,axiom,
! [F: nat > nat > $o,G2: nat > nat > $o,X: nat] :
( ( ord_le2646555220125990790_nat_o @ F @ G2 )
=> ( ord_less_eq_nat_o @ ( F @ X ) @ ( G2 @ X ) ) ) ).
% le_funD
thf(fact_164_mem__Collect__eq,axiom,
! [A: produc349518998152878311at_nat,P: produc349518998152878311at_nat > $o] :
( ( member8062223511168850704at_nat @ A @ ( collec7334067512558549330at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_165_mem__Collect__eq,axiom,
! [A: produc5224906263214031073at_nat,P: produc5224906263214031073at_nat > $o] :
( ( member1995966531042493578at_nat @ A @ ( collec2200730763347676620at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_166_mem__Collect__eq,axiom,
! [A: produc6277219514840344877at_nat,P: produc6277219514840344877at_nat > $o] :
( ( member3048279782668807382at_nat @ A @ ( collec3253044014973990424at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_167_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_168_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_169_mem__Collect__eq,axiom,
! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_170_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_171_Collect__mem__eq,axiom,
! [A4: set_Pr553994874890374343at_nat] :
( ( collec7334067512558549330at_nat
@ ^ [X4: produc349518998152878311at_nat] : ( member8062223511168850704at_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_172_Collect__mem__eq,axiom,
! [A4: set_Pr7116486347545156417at_nat] :
( ( collec2200730763347676620at_nat
@ ^ [X4: produc5224906263214031073at_nat] : ( member1995966531042493578at_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_173_Collect__mem__eq,axiom,
! [A4: set_Pr575275573428919693at_nat] :
( ( collec3253044014973990424at_nat
@ ^ [X4: produc6277219514840344877at_nat] : ( member3048279782668807382at_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_174_Collect__mem__eq,axiom,
! [A4: set_set_nat] :
( ( collect_set_nat
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_175_Collect__mem__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_176_Collect__mem__eq,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ( collec7088162979684241874at_nat
@ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_177_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_178_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_179_Collect__cong,axiom,
! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
( ! [X3: produc859450856879609959at_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec7088162979684241874at_nat @ P )
= ( collec7088162979684241874at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_180_Collect__cong,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec3392354462482085612at_nat @ P )
= ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_181_antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_182_antisym,axiom,
! [A: product_prod_nat_nat > $o,B: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ A @ B )
=> ( ( ord_le704812498762024988_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_183_antisym,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ A @ B )
=> ( ( ord_le5604493270027003598_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_184_antisym,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_eq_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_185_antisym,axiom,
! [A: nat > nat > $o,B: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A @ B )
=> ( ( ord_le2646555220125990790_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_186_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_187_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_188_dual__order_Otrans,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_189_dual__order_Otrans,axiom,
! [B: product_prod_nat_nat > $o,A: product_prod_nat_nat > $o,C: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ B @ A )
=> ( ( ord_le704812498762024988_nat_o @ C @ B )
=> ( ord_le704812498762024988_nat_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_190_dual__order_Otrans,axiom,
! [B: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat > product_prod_nat_nat > $o,C: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ B @ A )
=> ( ( ord_le5604493270027003598_nat_o @ C @ B )
=> ( ord_le5604493270027003598_nat_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_191_dual__order_Otrans,axiom,
! [B: nat > $o,A: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ B @ A )
=> ( ( ord_less_eq_nat_o @ C @ B )
=> ( ord_less_eq_nat_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_192_dual__order_Otrans,axiom,
! [B: nat > nat > $o,A: nat > nat > $o,C: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ B @ A )
=> ( ( ord_le2646555220125990790_nat_o @ C @ B )
=> ( ord_le2646555220125990790_nat_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_193_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_194_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_195_dual__order_Oantisym,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_196_dual__order_Oantisym,axiom,
! [B: product_prod_nat_nat > $o,A: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ B @ A )
=> ( ( ord_le704812498762024988_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_197_dual__order_Oantisym,axiom,
! [B: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ B @ A )
=> ( ( ord_le5604493270027003598_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_198_dual__order_Oantisym,axiom,
! [B: nat > $o,A: nat > $o] :
( ( ord_less_eq_nat_o @ B @ A )
=> ( ( ord_less_eq_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_199_dual__order_Oantisym,axiom,
! [B: nat > nat > $o,A: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ B @ A )
=> ( ( ord_le2646555220125990790_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_200_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_201_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_202_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A3 )
& ( ord_le6893508408891458716et_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_203_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: product_prod_nat_nat > $o,Z: product_prod_nat_nat > $o] : ( Y4 = Z ) )
= ( ^ [A3: product_prod_nat_nat > $o,B4: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ B4 @ A3 )
& ( ord_le704812498762024988_nat_o @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_204_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: product_prod_nat_nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat > product_prod_nat_nat > $o] : ( Y4 = Z ) )
= ( ^ [A3: product_prod_nat_nat > product_prod_nat_nat > $o,B4: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ B4 @ A3 )
& ( ord_le5604493270027003598_nat_o @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_205_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat > $o,Z: nat > $o] : ( Y4 = Z ) )
= ( ^ [A3: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ B4 @ A3 )
& ( ord_less_eq_nat_o @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_206_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat > nat > $o,Z: nat > nat > $o] : ( Y4 = Z ) )
= ( ^ [A3: nat > nat > $o,B4: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ B4 @ A3 )
& ( ord_le2646555220125990790_nat_o @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_207_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_208_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_209_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B3: nat] :
( ( ord_less_eq_nat @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: nat,B3: nat] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_210_order__trans,axiom,
! [X: set_set_nat,Y3: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ Y3 @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_211_order__trans,axiom,
! [X: product_prod_nat_nat > $o,Y3: product_prod_nat_nat > $o,Z2: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ X @ Y3 )
=> ( ( ord_le704812498762024988_nat_o @ Y3 @ Z2 )
=> ( ord_le704812498762024988_nat_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_212_order__trans,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: product_prod_nat_nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ X @ Y3 )
=> ( ( ord_le5604493270027003598_nat_o @ Y3 @ Z2 )
=> ( ord_le5604493270027003598_nat_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_213_order__trans,axiom,
! [X: nat > $o,Y3: nat > $o,Z2: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y3 )
=> ( ( ord_less_eq_nat_o @ Y3 @ Z2 )
=> ( ord_less_eq_nat_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_214_order__trans,axiom,
! [X: nat > nat > $o,Y3: nat > nat > $o,Z2: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ X @ Y3 )
=> ( ( ord_le2646555220125990790_nat_o @ Y3 @ Z2 )
=> ( ord_le2646555220125990790_nat_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_215_order__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_216_order__trans,axiom,
! [X: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_217_order_Otrans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_218_order_Otrans,axiom,
! [A: product_prod_nat_nat > $o,B: product_prod_nat_nat > $o,C: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ A @ B )
=> ( ( ord_le704812498762024988_nat_o @ B @ C )
=> ( ord_le704812498762024988_nat_o @ A @ C ) ) ) ).
% order.trans
thf(fact_219_order_Otrans,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B: product_prod_nat_nat > product_prod_nat_nat > $o,C: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ A @ B )
=> ( ( ord_le5604493270027003598_nat_o @ B @ C )
=> ( ord_le5604493270027003598_nat_o @ A @ C ) ) ) ).
% order.trans
thf(fact_220_order_Otrans,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ord_less_eq_nat_o @ A @ C ) ) ) ).
% order.trans
thf(fact_221_order_Otrans,axiom,
! [A: nat > nat > $o,B: nat > nat > $o,C: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A @ B )
=> ( ( ord_le2646555220125990790_nat_o @ B @ C )
=> ( ord_le2646555220125990790_nat_o @ A @ C ) ) ) ).
% order.trans
thf(fact_222_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_223_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_224_order__antisym,axiom,
! [X: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y3 )
=> ( ( ord_le6893508408891458716et_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_225_order__antisym,axiom,
! [X: product_prod_nat_nat > $o,Y3: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ X @ Y3 )
=> ( ( ord_le704812498762024988_nat_o @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_226_order__antisym,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ X @ Y3 )
=> ( ( ord_le5604493270027003598_nat_o @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_227_order__antisym,axiom,
! [X: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y3 )
=> ( ( ord_less_eq_nat_o @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_228_order__antisym,axiom,
! [X: nat > nat > $o,Y3: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ X @ Y3 )
=> ( ( ord_le2646555220125990790_nat_o @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_229_order__antisym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_230_order__antisym,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_231_ord__le__eq__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_232_ord__le__eq__trans,axiom,
! [A: product_prod_nat_nat > $o,B: product_prod_nat_nat > $o,C: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ A @ B )
=> ( ( B = C )
=> ( ord_le704812498762024988_nat_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_233_ord__le__eq__trans,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B: product_prod_nat_nat > product_prod_nat_nat > $o,C: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ A @ B )
=> ( ( B = C )
=> ( ord_le5604493270027003598_nat_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_234_ord__le__eq__trans,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_235_ord__le__eq__trans,axiom,
! [A: nat > nat > $o,B: nat > nat > $o,C: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A @ B )
=> ( ( B = C )
=> ( ord_le2646555220125990790_nat_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_236_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_237_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_238_ord__eq__le__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_239_ord__eq__le__trans,axiom,
! [A: product_prod_nat_nat > $o,B: product_prod_nat_nat > $o,C: product_prod_nat_nat > $o] :
( ( A = B )
=> ( ( ord_le704812498762024988_nat_o @ B @ C )
=> ( ord_le704812498762024988_nat_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_240_ord__eq__le__trans,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B: product_prod_nat_nat > product_prod_nat_nat > $o,C: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( A = B )
=> ( ( ord_le5604493270027003598_nat_o @ B @ C )
=> ( ord_le5604493270027003598_nat_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_241_ord__eq__le__trans,axiom,
! [A: nat > $o,B: nat > $o,C: nat > $o] :
( ( A = B )
=> ( ( ord_less_eq_nat_o @ B @ C )
=> ( ord_less_eq_nat_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_242_ord__eq__le__trans,axiom,
! [A: nat > nat > $o,B: nat > nat > $o,C: nat > nat > $o] :
( ( A = B )
=> ( ( ord_le2646555220125990790_nat_o @ B @ C )
=> ( ord_le2646555220125990790_nat_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_243_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_244_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_245_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > $o,Z: nat > $o] : ( Y4 = Z ) )
= ( ^ [X4: nat > $o,Y5: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y5 )
& ( ord_less_eq_nat_o @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_246_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > nat > $o,Z: nat > nat > $o] : ( Y4 = Z ) )
= ( ^ [X4: nat > nat > $o,Y5: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ X4 @ Y5 )
& ( ord_le2646555220125990790_nat_o @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_247_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_248_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_249_le__cases3,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_250_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_251_prod__induct3,axiom,
! [P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
( ! [A5: product_prod_nat_nat,B3: nat,C2: nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B3 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_252_prod__cases3,axiom,
! [Y3: produc859450856879609959at_nat] :
~ ! [A5: product_prod_nat_nat,B3: nat,C2: nat] :
( Y3
!= ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_253_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_254_Pair__inject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A @ B )
= ( produc6161850002892822231at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_255_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A5: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_256_prod__cases,axiom,
! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
( ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_257_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X3: nat,Y: nat] :
( P2
= ( product_Pair_nat_nat @ X3 @ Y ) ) ).
% surj_pair
thf(fact_258_surj__pair,axiom,
! [P2: produc859450856879609959at_nat] :
? [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( P2
= ( produc6161850002892822231at_nat @ X3 @ Y ) ) ).
% surj_pair
thf(fact_259_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [A5: nat,B3: nat] :
( Y3
!= ( product_Pair_nat_nat @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_260_old_Oprod_Oexhaust,axiom,
! [Y3: produc859450856879609959at_nat] :
~ ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat] :
( Y3
!= ( produc6161850002892822231at_nat @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_261_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_262_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_263_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_264_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_265_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_266_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_267_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_268_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_269_O__assoc,axiom,
! [R3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ ( relcomp_nat_nat_nat @ R3 @ S3 ) @ T )
= ( relcomp_nat_nat_nat @ R3 @ ( relcomp_nat_nat_nat @ S3 @ T ) ) ) ).
% O_assoc
thf(fact_270_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z3: nat] :
( ( R3 @ X3 @ Y )
=> ( ( R3 @ Y @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R3 @ N3 @ ( suc @ N3 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_271_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_272_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_273_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_274_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_275_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_276_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_277_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_278_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_279_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_280_relcompEpair,axiom,
! [A: nat,C: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( relcomp_nat_nat_nat @ R @ S ) )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_281_relcompEpair,axiom,
! [A: product_prod_nat_nat,C: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C ) @ ( relcom7295539661566034944at_nat @ R @ S ) )
=> ~ ! [B3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B3 ) @ R )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ C ) @ S ) ) ) ).
% relcompEpair
thf(fact_282_relcompE,axiom,
! [Xz: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R @ S ) )
=> ~ ! [X3: nat,Y: nat,Z3: nat] :
( ( Xz
= ( product_Pair_nat_nat @ X3 @ Z3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_283_relcompE,axiom,
! [Xz: produc859450856879609959at_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ Xz @ ( relcom7295539661566034944at_nat @ R @ S ) )
=> ~ ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( Xz
= ( produc6161850002892822231at_nat @ X3 @ Z3 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_284_relcomp_OrelcompI,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C: nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( relcomp_nat_nat_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_285_relcomp_OrelcompI,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,C: product_prod_nat_nat,S: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C ) @ S )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C ) @ ( relcom7295539661566034944at_nat @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_286_relcomp_Osimps,axiom,
! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R @ S ) )
= ( ? [A3: nat,B4: nat,C3: nat] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_287_relcomp_Osimps,axiom,
! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( relcom7295539661566034944at_nat @ R @ S ) )
= ( ? [A3: product_prod_nat_nat,B4: product_prod_nat_nat,C3: product_prod_nat_nat] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ B4 ) @ R )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_288_match__Times,axiom,
! [I: nat,N: nat,R: regex_a_t,S: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ N ) ) @ ( match_a_t @ sigma @ ( times_a_t @ R @ S ) ) )
= ( ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ K2 ) ) @ ( match_a_t @ sigma @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ I @ N ) ) @ ( match_a_t @ sigma @ S ) ) ) ) ) ).
% match_Times
thf(fact_289_MDL_Omatch__Times,axiom,
! [I: nat,N: nat,Sigma: trace_a_t,R: regex_a_t,S: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ N ) ) @ ( match_a_t @ Sigma @ ( times_a_t @ R @ S ) ) )
= ( ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ K2 ) ) @ ( match_a_t @ Sigma @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ I @ N ) ) @ ( match_a_t @ Sigma @ S ) ) ) ) ) ).
% MDL.match_Times
thf(fact_290_relChain__def,axiom,
( bNF_Ca968750328013420230at_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat,As: nat > nat] :
! [I2: nat,J2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R4 )
=> ( ord_less_eq_nat @ ( As @ I2 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_291_relChain__def,axiom,
( bNF_Ca8308629720386654381at_nat
= ( ^ [R4: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > nat] :
! [I2: product_prod_nat_nat,J2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R4 )
=> ( ord_less_eq_nat @ ( As @ I2 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_292_relChain__def,axiom,
( bNF_Ca4389690986151823996et_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat,As: nat > set_nat] :
! [I2: nat,J2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R4 )
=> ( ord_less_eq_set_nat @ ( As @ I2 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_293_relChain__def,axiom,
( bNF_Ca100218009143674979et_nat
= ( ^ [R4: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > set_nat] :
! [I2: product_prod_nat_nat,J2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R4 )
=> ( ord_less_eq_set_nat @ ( As @ I2 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_294_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K3: nat,M4: nat] :
( X
!= ( product_Pair_nat_nat @ K3 @ M4 ) ) ).
% prod_decode_aux.cases
thf(fact_295_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F3: nat > nat > nat,A5: nat,B3: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_296_Greatest__equality,axiom,
! [P: set_nat > $o,X: set_nat] :
( ( P @ X )
=> ( ! [Y: set_nat] :
( ( P @ Y )
=> ( ord_less_eq_set_nat @ Y @ X ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_297_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_298_GreatestI2__order,axiom,
! [P: set_nat > $o,X: set_nat,Q: set_nat > $o] :
( ( P @ X )
=> ( ! [Y: set_nat] :
( ( P @ Y )
=> ( ord_less_eq_set_nat @ Y @ X ) )
=> ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( ! [Y6: set_nat] :
( ( P @ Y6 )
=> ( ord_less_eq_set_nat @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_299_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_300_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_301_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_302_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_303_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_304_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_305_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_306_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_307_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_308_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_309_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_310_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_311_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_312_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_313_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_314_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_315_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_316_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M6 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_317_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_318_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_319_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_320_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_321_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_322_subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ X3 @ B5 ) )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_323_subsetI,axiom,
! [A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ A4 )
=> ( member8206827879206165904at_nat @ X3 @ B5 ) )
=> ( ord_le3000389064537975527at_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_324_subsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% subsetI
thf(fact_325_subset__antisym,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_326_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_327_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_328_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_329_plus__prod_Oelims,axiom,
! [X: produc859450856879609959at_nat,Xa: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
( ( ( plus_p4591053195553783070at_nat @ X @ Xa )
= Y3 )
=> ~ ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( X
= ( produc6161850002892822231at_nat @ A5 @ B3 ) )
=> ! [C2: product_prod_nat_nat,D: product_prod_nat_nat] :
( ( Xa
= ( produc6161850002892822231at_nat @ C2 @ D ) )
=> ( Y3
!= ( produc6161850002892822231at_nat @ ( plus_p9057090461656269880at_nat @ A5 @ C2 ) @ ( plus_p9057090461656269880at_nat @ B3 @ D ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_330_plus__prod_Oelims,axiom,
! [X: product_prod_nat_nat,Xa: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ X @ Xa )
= Y3 )
=> ~ ! [A5: nat,B3: nat] :
( ( X
= ( product_Pair_nat_nat @ A5 @ B3 ) )
=> ! [C2: nat,D: nat] :
( ( Xa
= ( product_Pair_nat_nat @ C2 @ D ) )
=> ( Y3
!= ( product_Pair_nat_nat @ ( plus_plus_nat @ A5 @ C2 ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ) ) ).
% plus_prod.elims
thf(fact_331_plus__prod_Osimps,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D2: product_prod_nat_nat] :
( ( plus_p4591053195553783070at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ C @ D2 ) )
= ( produc6161850002892822231at_nat @ ( plus_p9057090461656269880at_nat @ A @ C ) @ ( plus_p9057090461656269880at_nat @ B @ D2 ) ) ) ).
% plus_prod.simps
thf(fact_332_plus__prod_Osimps,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D2 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ).
% plus_prod.simps
thf(fact_333_timestamp__total,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
| ( ord_less_eq_nat @ B @ A ) ) ).
% timestamp_total
thf(fact_334_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_335_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_336_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_337_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_338_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_339_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B4: nat] : ( plus_plus_nat @ B4 @ A3 ) ) ) ).
% add.commute
thf(fact_340_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_341_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_342_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_343_plus__prod_Ocases,axiom,
! [X: produc349518998152878311at_nat] :
~ ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat,C2: product_prod_nat_nat,D: product_prod_nat_nat] :
( X
!= ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A5 @ B3 ) @ ( produc6161850002892822231at_nat @ C2 @ D ) ) ) ).
% plus_prod.cases
thf(fact_344_plus__prod_Ocases,axiom,
! [X: produc859450856879609959at_nat] :
~ ! [A5: nat,B3: nat,C2: nat,D: nat] :
( X
!= ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A5 @ B3 ) @ ( product_Pair_nat_nat @ C2 @ D ) ) ) ).
% plus_prod.cases
thf(fact_345_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_346_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_347_subset__trans,axiom,
! [A4: set_nat,B5: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ord_less_eq_set_nat @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_348_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_349_subset__refl,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_350_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [T2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T2 @ A6 )
=> ( member8440522571783428010at_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_351_subset__iff,axiom,
( ord_le3000389064537975527at_nat
= ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
! [T2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ T2 @ A6 )
=> ( member8206827879206165904at_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_352_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_353_equalityD2,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 = B5 )
=> ( ord_less_eq_set_nat @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_354_equalityD1,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 = B5 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_355_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A6 )
=> ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_356_subset__eq,axiom,
( ord_le3000389064537975527at_nat
= ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ A6 )
=> ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_357_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member_nat @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_358_equalityE,axiom,
! [A4: set_nat,B5: set_nat] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_359_subsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_360_subsetD,axiom,
! [A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
( ( ord_le3000389064537975527at_nat @ A4 @ B5 )
=> ( ( member8206827879206165904at_nat @ C @ A4 )
=> ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_361_subsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_362_in__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_363_in__mono,axiom,
! [A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat] :
( ( ord_le3000389064537975527at_nat @ A4 @ B5 )
=> ( ( member8206827879206165904at_nat @ X @ A4 )
=> ( member8206827879206165904at_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_364_in__mono,axiom,
! [A4: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_365_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_366_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_367_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_368_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_369_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_370_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_371_ordered__ab__semigroup__add__class_Oadd__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% ordered_ab_semigroup_add_class.add_mono
thf(fact_372_add__mono__comm,axiom,
! [C: nat,D2: nat,A: nat] :
( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ D2 @ A ) ) ) ).
% add_mono_comm
thf(fact_373_timestamp__class_Oadd__mono,axiom,
! [C: nat,D2: nat,A: nat] :
( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ A @ D2 ) ) ) ).
% timestamp_class.add_mono
thf(fact_374_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_375_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_376_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_377_match__Star,axiom,
! [I: nat,N: nat,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ ( suc @ N ) ) ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) )
= ( ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ K2 ) ) @ ( match_a_t @ sigma @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ K2 ) @ ( plus_plus_nat @ I @ ( suc @ N ) ) ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) ) ) ) ) ).
% match_Star
thf(fact_378__092_060iota_062__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( embed_nat_iota_nat @ I ) @ ( embed_nat_iota_nat @ J ) ) ) ).
% \<iota>_mono
thf(fact_379_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_380_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_381_le__prod__encode__2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_2
thf(fact_382_le__prod__encode__1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_1
thf(fact_383_plus__prod_Opelims,axiom,
! [X: produc859450856879609959at_nat,Xa: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
( ( ( plus_p4591053195553783070at_nat @ X @ Xa )
= Y3 )
=> ( ( accp_P795512239754848240at_nat @ plus_p4626873463684405630at_nat @ ( produc4662710985925991255at_nat @ X @ Xa ) )
=> ~ ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( X
= ( produc6161850002892822231at_nat @ A5 @ B3 ) )
=> ! [C2: product_prod_nat_nat,D: product_prod_nat_nat] :
( ( Xa
= ( produc6161850002892822231at_nat @ C2 @ D ) )
=> ( ( Y3
= ( produc6161850002892822231at_nat @ ( plus_p9057090461656269880at_nat @ A5 @ C2 ) @ ( plus_p9057090461656269880at_nat @ B3 @ D ) ) )
=> ~ ( accp_P795512239754848240at_nat @ plus_p4626873463684405630at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A5 @ B3 ) @ ( produc6161850002892822231at_nat @ C2 @ D ) ) ) ) ) ) ) ) ).
% plus_prod.pelims
thf(fact_384_plus__prod_Opelims,axiom,
! [X: product_prod_nat_nat,Xa: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ( plus_p9057090461656269880at_nat @ X @ Xa )
= Y3 )
=> ( ( accp_P1267725715503270512at_nat @ plus_p6814068510987285344at_nat @ ( produc6161850002892822231at_nat @ X @ Xa ) )
=> ~ ! [A5: nat,B3: nat] :
( ( X
= ( product_Pair_nat_nat @ A5 @ B3 ) )
=> ! [C2: nat,D: nat] :
( ( Xa
= ( product_Pair_nat_nat @ C2 @ D ) )
=> ( ( Y3
= ( product_Pair_nat_nat @ ( plus_plus_nat @ A5 @ C2 ) @ ( plus_plus_nat @ B3 @ D ) ) )
=> ~ ( accp_P1267725715503270512at_nat @ plus_p6814068510987285344at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A5 @ B3 ) @ ( product_Pair_nat_nat @ C2 @ D ) ) ) ) ) ) ) ) ).
% plus_prod.pelims
thf(fact_385_ssubst__Pair__rhs,axiom,
! [R: nat,S: nat,R3: set_Pr1261947904930325089at_nat,S2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_386_ssubst__Pair__rhs,axiom,
! [R: product_prod_nat_nat,S: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_387_regex_Oinject_I5_J,axiom,
! [X52: regex_a_t,Y52: regex_a_t] :
( ( ( star_a_t @ X52 )
= ( star_a_t @ Y52 ) )
= ( X52 = Y52 ) ) ).
% regex.inject(5)
thf(fact_388_prod__encode__eq,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ( nat_prod_encode @ X )
= ( nat_prod_encode @ Y3 ) )
= ( X = Y3 ) ) ).
% prod_encode_eq
thf(fact_389_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_390_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_391_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_392_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_393_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_394_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_395_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_396_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_397_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_398_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_399_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_400_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_401_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_402_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_403_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_404_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_405_regex_Odistinct_I19_J,axiom,
! [X41: regex_a_t,X42: regex_a_t,X52: regex_a_t] :
( ( times_a_t @ X41 @ X42 )
!= ( star_a_t @ X52 ) ) ).
% regex.distinct(19)
thf(fact_406_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_407_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_408_wf__regex_Osimps_I5_J,axiom,
! [R: regex_a_t] :
( ( wf_regex_a_t @ ( star_a_t @ R ) )
= ( wf_regex_a_t @ R ) ) ).
% wf_regex.simps(5)
thf(fact_409_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_410_eps_Osimps_I5_J,axiom,
! [R: regex_a_t] : ( eps_a_t @ ( star_a_t @ R ) ) ).
% eps.simps(5)
thf(fact_411_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_412_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_413_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_414_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_415_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_416_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_417_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_418_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_419_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_420_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_421_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_422_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_423_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_424_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_425_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_426_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_427_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_428_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_429_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_430_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_431_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_432_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_433_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_434_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_435_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_436_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_437_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_438_rderive_Osimps_I5_J,axiom,
! [R: regex_a_t] :
( ( rderive_a_t @ ( star_a_t @ R ) )
= ( times_a_t @ ( star_a_t @ R ) @ ( rderive_a_t @ R ) ) ) ).
% rderive.simps(5)
thf(fact_439_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_440_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_441_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_442_MDL_Omatch__Star,axiom,
! [I: nat,N: nat,Sigma: trace_a_t,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ ( suc @ N ) ) ) @ ( match_a_t @ Sigma @ ( star_a_t @ R ) ) )
= ( ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ K2 ) ) @ ( match_a_t @ Sigma @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ K2 ) @ ( plus_plus_nat @ I @ ( suc @ N ) ) ) @ ( match_a_t @ Sigma @ ( star_a_t @ R ) ) ) ) ) ) ).
% MDL.match_Star
thf(fact_443_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_444_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_445_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_446_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_447_match_Osimps_I5_J,axiom,
! [R: regex_a_t] :
( ( match_a_t @ sigma @ ( star_a_t @ R ) )
= ( transi2905341329935302413cl_nat @ ( match_a_t @ sigma @ R ) ) ) ).
% match.simps(5)
thf(fact_448_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y3: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y3 )
=> ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y3
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y3
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_449_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K2: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K2 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K2 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M6 @ ( suc @ K2 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_450_rtrancl__unfold_H,axiom,
! [X: product_prod_nat_nat,Z2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ ( transi8609417484261754244at_nat @ R3 ) )
=> ( ( X = Z2 )
| ? [Y: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transi8609417484261754244at_nat @ R3 ) )
& ( Y != Z2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ R3 ) ) ) ) ).
% rtrancl_unfold'
thf(fact_451_rtrancl__unfold_H,axiom,
! [X: nat,Z2: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( X = Z2 )
| ? [Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
& ( Y != Z2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ R3 ) ) ) ) ).
% rtrancl_unfold'
thf(fact_452_local_Ortrancl__unfold,axiom,
! [X: product_prod_nat_nat,Z2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ ( transi8609417484261754244at_nat @ R3 ) )
=> ( ( X = Z2 )
| ? [Y: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
& ( X != Y )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ ( transi8609417484261754244at_nat @ R3 ) ) ) ) ) ).
% local.rtrancl_unfold
thf(fact_453_local_Ortrancl__unfold,axiom,
! [X: nat,Z2: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( X = Z2 )
| ? [Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
& ( X != Y )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ) ).
% local.rtrancl_unfold
thf(fact_454_double__diff,axiom,
! [A4: set_nat,B5: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C4 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_455_Diff__subset,axiom,
! [A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ A4 ) ).
% Diff_subset
thf(fact_456_Diff__mono,axiom,
! [A4: set_nat,C4: set_nat,D3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ D3 @ B5 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ ( minus_minus_set_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_457_MDL_Omatch_Osimps_I5_J,axiom,
! [Sigma: trace_a_t,R: regex_a_t] :
( ( match_a_t @ Sigma @ ( star_a_t @ R ) )
= ( transi2905341329935302413cl_nat @ ( match_a_t @ Sigma @ R ) ) ) ).
% MDL.match.simps(5)
thf(fact_458_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
= ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% prod_encode_prod_decode_aux
thf(fact_459_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( transi2905341329935302413cl_nat @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_460_rtrancl__idemp,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( transi2905341329935302413cl_nat @ ( transi2905341329935302413cl_nat @ R ) )
= ( transi2905341329935302413cl_nat @ R ) ) ).
% rtrancl_idemp
thf(fact_461_r__into__rtrancl,axiom,
! [P2: produc859450856879609959at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ P2 @ R )
=> ( member8206827879206165904at_nat @ P2 @ ( transi8609417484261754244at_nat @ R ) ) ) ).
% r_into_rtrancl
thf(fact_462_r__into__rtrancl,axiom,
! [P2: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ P2 @ R )
=> ( member8440522571783428010at_nat @ P2 @ ( transi2905341329935302413cl_nat @ R ) ) ) ).
% r_into_rtrancl
thf(fact_463_r__comp__rtrancl__eq,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ R @ ( transi2905341329935302413cl_nat @ R ) )
= ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R ) @ R ) ) ).
% r_comp_rtrancl_eq
thf(fact_464_rtrancl__mono,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R @ S )
=> ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).
% rtrancl_mono
thf(fact_465_rtrancl__subset,axiom,
! [R3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ S3 )
=> ( ( ord_le3146513528884898305at_nat @ S3 @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( transi2905341329935302413cl_nat @ S3 )
= ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_466_rtrancl__subset__rtrancl,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R @ ( transi2905341329935302413cl_nat @ S ) )
=> ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_467_DiffI,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ C @ B5 )
=> ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_468_DiffI,axiom,
! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ A4 )
=> ( ~ ( member8206827879206165904at_nat @ C @ B5 )
=> ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_469_DiffI,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_470_Diff__iff,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
= ( ( member8440522571783428010at_nat @ C @ A4 )
& ~ ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_471_Diff__iff,axiom,
! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) )
= ( ( member8206827879206165904at_nat @ C @ A4 )
& ~ ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_472_Diff__iff,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( ( member_nat @ C @ A4 )
& ~ ( member_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_473_DiffE,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
=> ~ ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_474_DiffE,axiom,
! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) )
=> ~ ( ( member8206827879206165904at_nat @ C @ A4 )
=> ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_475_DiffE,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_476_DiffD1,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
=> ( member8440522571783428010at_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_477_DiffD1,axiom,
! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) )
=> ( member8206827879206165904at_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_478_DiffD1,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ( member_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_479_DiffD2,axiom,
! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) )
=> ~ ( member8440522571783428010at_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_480_DiffD2,axiom,
! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) )
=> ~ ( member8206827879206165904at_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_481_DiffD2,axiom,
! [C: nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B5 ) )
=> ~ ( member_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_482_converse__rtrancl__induct2,axiom,
! [Ax: product_prod_nat_nat,Ay: product_prod_nat_nat,Bx: product_prod_nat_nat,By: product_prod_nat_nat,R: set_Pr553994874890374343at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ Bx @ By ) ) @ ( transi5470879363895450730at_nat @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A5 @ B3 ) @ ( produc6161850002892822231at_nat @ Aa @ Ba ) ) @ R )
=> ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Aa @ Ba ) @ ( produc6161850002892822231at_nat @ Bx @ By ) ) @ ( transi5470879363895450730at_nat @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A5 @ B3 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_483_converse__rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A5: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A5 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A5 @ B3 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_484_converse__rtranclE2,axiom,
! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat,Za: product_prod_nat_nat,Zb: product_prod_nat_nat,R: set_Pr553994874890374343at_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Xa @ Xb ) @ ( produc6161850002892822231at_nat @ Za @ Zb ) ) @ ( transi5470879363895450730at_nat @ R ) )
=> ( ( ( produc6161850002892822231at_nat @ Xa @ Xb )
!= ( produc6161850002892822231at_nat @ Za @ Zb ) )
=> ~ ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Xa @ Xb ) @ ( produc6161850002892822231at_nat @ A5 @ B3 ) ) @ R )
=> ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A5 @ B3 ) @ ( produc6161850002892822231at_nat @ Za @ Zb ) ) @ ( transi5470879363895450730at_nat @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_485_converse__rtranclE2,axiom,
! [Xa: nat,Xb: nat,Za: nat,Zb: nat,R: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( ( product_Pair_nat_nat @ Xa @ Xb )
!= ( product_Pair_nat_nat @ Za @ Zb ) )
=> ~ ! [A5: nat,B3: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ A5 @ B3 ) ) @ R )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A5 @ B3 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_486_rtrancl__induct2,axiom,
! [Ax: product_prod_nat_nat,Ay: product_prod_nat_nat,Bx: product_prod_nat_nat,By: product_prod_nat_nat,R: set_Pr553994874890374343at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ Bx @ By ) ) @ ( transi5470879363895450730at_nat @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ A5 @ B3 ) ) @ ( transi5470879363895450730at_nat @ R ) )
=> ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A5 @ B3 ) @ ( produc6161850002892822231at_nat @ Aa @ Ba ) ) @ R )
=> ( ( P @ A5 @ B3 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_487_rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A5: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A5 @ B3 ) ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A5 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R )
=> ( ( P @ A5 @ B3 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_488_rtrancl_Ocases,axiom,
! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ B3 ) @ ( transi8609417484261754244at_nat @ R ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_489_rtrancl_Ocases,axiom,
! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_490_rtrancl_Osimps,axiom,
! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( transi8609417484261754244at_nat @ R ) )
= ( ? [A3: product_prod_nat_nat] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: product_prod_nat_nat,B4: product_prod_nat_nat,C3: product_prod_nat_nat] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ B4 ) @ ( transi8609417484261754244at_nat @ R ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ C3 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_491_rtrancl_Osimps,axiom,
! [A1: nat,A22: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R ) )
= ( ? [A3: nat] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: nat,B4: nat,C3: nat] :
( ( A1 = A3 )
& ( A22 = C3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B4 ) @ ( transi2905341329935302413cl_nat @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C3 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_492_rtrancl_Ortrancl__refl,axiom,
! [A: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( transi8609417484261754244at_nat @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_493_rtrancl_Ortrancl__refl,axiom,
! [A: nat,R: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( transi2905341329935302413cl_nat @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_494_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,C: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C ) @ R )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C ) @ ( transi8609417484261754244at_nat @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_495_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_496_rtranclE,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( A != B )
=> ~ ! [Y: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y ) @ ( transi8609417484261754244at_nat @ R ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_497_rtranclE,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( A != B )
=> ~ ! [Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_498_rtrancl__trans,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,Z2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z2 ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ ( transi8609417484261754244at_nat @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_499_rtrancl__trans,axiom,
! [X: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_500_rtrancl__induct,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( P @ A )
=> ( ! [Y: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z3 ) @ R )
=> ( ( P @ Y )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_501_rtrancl__induct,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( P @ A )
=> ( ! [Y: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ R )
=> ( ( P @ Y )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_502_converse__rtranclE,axiom,
! [X: product_prod_nat_nat,Z2: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ ( transi8609417484261754244at_nat @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_503_converse__rtranclE,axiom,
! [X: nat,Z2: nat,R: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_504_converse__rtrancl__induct,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( P @ B )
=> ( ! [Y: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z3 ) @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z3 @ B ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( ( P @ Z3 )
=> ( P @ Y ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_505_converse__rtrancl__induct,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( P @ B )
=> ( ! [Y: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( P @ Z3 )
=> ( P @ Y ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_506_converse__rtrancl__into__rtrancl,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,C: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C ) @ ( transi8609417484261754244at_nat @ R ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C ) @ ( transi8609417484261754244at_nat @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_507_converse__rtrancl__into__rtrancl,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_508_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_509_prod__decode__triangle__add,axiom,
! [K: nat,M: nat] :
( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
= ( nat_prod_decode_aux @ K @ M ) ) ).
% prod_decode_triangle_add
thf(fact_510_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_511_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y3: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y3 )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y3
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y3
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_512_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N4: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_513_prod__decode__eq,axiom,
! [X: nat,Y3: nat] :
( ( ( nat_prod_decode @ X )
= ( nat_prod_decode @ Y3 ) )
= ( X = Y3 ) ) ).
% prod_decode_eq
thf(fact_514_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_515_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_516_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_517_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_518_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_519_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_520_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_521_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_522_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_523_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_524_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_525_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_526_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_527_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_528_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_529_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_530_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_531_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_532_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_533_prod__encode__inverse,axiom,
! [X: product_prod_nat_nat] :
( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
= X ) ).
% prod_encode_inverse
thf(fact_534_prod__decode__inverse,axiom,
! [N: nat] :
( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
= N ) ).
% prod_decode_inverse
thf(fact_535_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_536_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_537_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_538_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_539_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_540_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_541_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_542_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ zero_zero_nat ) ) ).
% prod_decode_def
thf(fact_543_zero__prod__def,axiom,
( zero_z8332228408419305374at_nat
= ( produc6161850002892822231at_nat @ zero_z3979849011205770936at_nat @ zero_z3979849011205770936at_nat ) ) ).
% zero_prod_def
thf(fact_544_zero__prod__def,axiom,
( zero_z3979849011205770936at_nat
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_545_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_546_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_547_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_548_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_549_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_550_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_551_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_552_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_553_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_554_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_555_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_556_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X3: nat,Y: nat] :
( ( P @ X3 @ Y )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_557_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_558_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_559_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_560_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_561_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_562_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_563_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_564_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_565_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_566_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_567_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_568_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_569_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_570_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_571_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_572_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_573_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_574_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_575_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_576_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_577_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_578_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_579_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_580_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_581_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_582_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_583_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_584_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_585_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_586_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_587_accp_Ocases,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat] :
( ( accp_P4275260045618599050at_nat @ R @ A )
=> ! [Y6: product_prod_nat_nat] :
( ( R @ Y6 @ A )
=> ( accp_P4275260045618599050at_nat @ R @ Y6 ) ) ) ).
% accp.cases
thf(fact_588_accp_Ocases,axiom,
! [R: nat > nat > $o,A: nat] :
( ( accp_nat @ R @ A )
=> ! [Y6: nat] :
( ( R @ Y6 @ A )
=> ( accp_nat @ R @ Y6 ) ) ) ).
% accp.cases
thf(fact_589_accp_Osimps,axiom,
( accp_P4275260045618599050at_nat
= ( ^ [R4: product_prod_nat_nat > product_prod_nat_nat > $o,A3: product_prod_nat_nat] :
? [X4: product_prod_nat_nat] :
( ( A3 = X4 )
& ! [Y5: product_prod_nat_nat] :
( ( R4 @ Y5 @ X4 )
=> ( accp_P4275260045618599050at_nat @ R4 @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_590_accp_Osimps,axiom,
( accp_nat
= ( ^ [R4: nat > nat > $o,A3: nat] :
? [X4: nat] :
( ( A3 = X4 )
& ! [Y5: nat] :
( ( R4 @ Y5 @ X4 )
=> ( accp_nat @ R4 @ Y5 ) ) ) ) ) ).
% accp.simps
thf(fact_591_accpI,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [Y: product_prod_nat_nat] :
( ( R @ Y @ X )
=> ( accp_P4275260045618599050at_nat @ R @ Y ) )
=> ( accp_P4275260045618599050at_nat @ R @ X ) ) ).
% accpI
thf(fact_592_accpI,axiom,
! [R: nat > nat > $o,X: nat] :
( ! [Y: nat] :
( ( R @ Y @ X )
=> ( accp_nat @ R @ Y ) )
=> ( accp_nat @ R @ X ) ) ).
% accpI
thf(fact_593_accp__induct,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( accp_P4275260045618599050at_nat @ R @ A )
=> ( ! [X3: product_prod_nat_nat] :
( ( accp_P4275260045618599050at_nat @ R @ X3 )
=> ( ! [Y6: product_prod_nat_nat] :
( ( R @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_594_accp__induct,axiom,
! [R: nat > nat > $o,A: nat,P: nat > $o] :
( ( accp_nat @ R @ A )
=> ( ! [X3: nat] :
( ( accp_nat @ R @ X3 )
=> ( ! [Y6: nat] :
( ( R @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_595_accp__downward,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( accp_P4275260045618599050at_nat @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P4275260045618599050at_nat @ R @ A ) ) ) ).
% accp_downward
thf(fact_596_accp__downward,axiom,
! [R: nat > nat > $o,B: nat,A: nat] :
( ( accp_nat @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_nat @ R @ A ) ) ) ).
% accp_downward
thf(fact_597_not__accp__down,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ~ ( accp_P4275260045618599050at_nat @ R3 @ X )
=> ~ ! [Z3: product_prod_nat_nat] :
( ( R3 @ Z3 @ X )
=> ( accp_P4275260045618599050at_nat @ R3 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_598_not__accp__down,axiom,
! [R3: nat > nat > $o,X: nat] :
( ~ ( accp_nat @ R3 @ X )
=> ~ ! [Z3: nat] :
( ( R3 @ Z3 @ X )
=> ( accp_nat @ R3 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_599_accp__induct__rule,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( accp_P4275260045618599050at_nat @ R @ A )
=> ( ! [X3: product_prod_nat_nat] :
( ( accp_P4275260045618599050at_nat @ R @ X3 )
=> ( ! [Y6: product_prod_nat_nat] :
( ( R @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_600_accp__induct__rule,axiom,
! [R: nat > nat > $o,A: nat,P: nat > $o] :
( ( accp_nat @ R @ A )
=> ( ! [X3: nat] :
( ( accp_nat @ R @ X3 )
=> ( ! [Y6: nat] :
( ( R @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_601_accp__subset__induct,axiom,
! [D3: product_prod_nat_nat > $o,R3: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ D3 @ ( accp_P4275260045618599050at_nat @ R3 ) )
=> ( ! [X3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( D3 @ X3 )
=> ( ( R3 @ Z3 @ X3 )
=> ( D3 @ Z3 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: product_prod_nat_nat] :
( ( D3 @ X3 )
=> ( ! [Z4: product_prod_nat_nat] :
( ( R3 @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_602_accp__subset__induct,axiom,
! [D3: nat > $o,R3: nat > nat > $o,X: nat,P: nat > $o] :
( ( ord_less_eq_nat_o @ D3 @ ( accp_nat @ R3 ) )
=> ( ! [X3: nat,Z3: nat] :
( ( D3 @ X3 )
=> ( ( R3 @ Z3 @ X3 )
=> ( D3 @ Z3 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: nat] :
( ( D3 @ X3 )
=> ( ! [Z4: nat] :
( ( R3 @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_603_accp__subset,axiom,
! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
=> ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_604_accp__subset,axiom,
! [R1: nat > nat > $o,R22: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
=> ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_605_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B3: nat] :
( ( P @ A5 @ B3 )
= ( P @ B3 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B3: nat] :
( ( P @ A5 @ B3 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_606_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_607_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_608_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_609_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_610_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_611_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_612_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp_nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
=> ( P @ zero_zero_nat ) )
=> ( ! [N3: nat] :
( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
=> ( ! [X5: nat,Y6: nat] :
( ( ( product_Pair_nat_nat @ X5 @ Y6 )
= ( nat_prod_decode @ N3 ) )
=> ( P @ Y6 ) )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_613_regex_Osize_I10_J,axiom,
! [X25: regex_a_t] :
( ( size_size_regex_a_t @ ( star_a_t @ X25 ) )
= ( plus_plus_nat @ ( size_size_regex_a_t @ X25 ) @ ( suc @ zero_zero_nat ) ) ) ).
% regex.size(10)
thf(fact_614_regex_Osize_I9_J,axiom,
! [X241: regex_a_t,X242: regex_a_t] :
( ( size_size_regex_a_t @ ( times_a_t @ X241 @ X242 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_regex_a_t @ X241 ) @ ( size_size_regex_a_t @ X242 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% regex.size(9)
thf(fact_615_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_616_in__lex__prod,axiom,
! [A: nat,B: product_prod_nat_nat,A2: nat,B2: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr8693737435421807431at_nat] :
( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ A2 @ B2 ) ) @ ( lex_pr8029265285556086080at_nat @ R @ S ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_617_in__lex__prod,axiom,
! [A: product_prod_nat_nat,B: nat,A2: product_prod_nat_nat,B2: nat,R: set_Pr8693737435421807431at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ A2 @ B2 ) ) @ ( lex_pr4669217892513370978at_nat @ R @ S ) )
= ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_618_in__lex__prod,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ A2 @ B2 ) ) @ ( lex_pr8801849515957261039at_nat @ R @ S ) )
= ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_619_in__lex__prod,axiom,
! [A: nat,B: nat,A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) @ ( lex_prod_nat_nat @ R @ S ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
| ( ( A = A2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_620_mlex__leq,axiom,
! [F: nat > nat,X: nat,Y3: nat,R3: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( mlex_prod_nat @ F @ R3 ) ) ) ) ).
% mlex_leq
thf(fact_621_mlex__leq,axiom,
! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( mlex_p6366001652026297872at_nat @ F @ R3 ) ) ) ) ).
% mlex_leq
thf(fact_622_same__fstI,axiom,
! [P: nat > $o,X: nat,Y7: nat,Y3: nat,R3: nat > set_Pr1261947904930325089at_nat] :
( ( P @ X )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y3 ) @ ( R3 @ X ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y7 ) @ ( product_Pair_nat_nat @ X @ Y3 ) ) @ ( same_fst_nat_nat @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_623_same__fstI,axiom,
! [P: product_prod_nat_nat > $o,X: product_prod_nat_nat,Y7: product_prod_nat_nat,Y3: product_prod_nat_nat,R3: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
( ( P @ X )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ Y3 ) @ ( R3 @ X ) )
=> ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ X @ Y7 ) @ ( produc6161850002892822231at_nat @ X @ Y3 ) ) @ ( same_f4956014544515070124at_nat @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_624_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_625_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_626_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_627_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_628_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_629_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_630_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_631_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_632_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_633_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_634_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_635_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_636_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_637_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_638_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_639_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_640_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_641_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_642_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_643_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_644_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_645_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_646_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_647_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_648_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_649_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_650_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_651_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_652_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_653_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_654_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_655_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_656_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_657_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_658_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_659_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_660_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_661_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_662_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_663_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_664_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_665_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_666_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_667_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_668_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_669_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_670_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_671_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_nat @ M6 @ N4 )
| ( M6 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_672_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_673_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_eq_nat @ M6 @ N4 )
& ( M6 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_674_mlex__less,axiom,
! [F: nat > nat,X: nat,Y3: nat,R3: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( mlex_prod_nat @ F @ R3 ) ) ) ).
% mlex_less
thf(fact_675_mlex__less,axiom,
! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( mlex_p6366001652026297872at_nat @ F @ R3 ) ) ) ).
% mlex_less
thf(fact_676_mlex__iff,axiom,
! [X: nat,Y3: nat,F: nat > nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( mlex_prod_nat @ F @ R3 ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) )
| ( ( ( F @ X )
= ( F @ Y3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R3 ) ) ) ) ).
% mlex_iff
thf(fact_677_mlex__iff,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,F: product_prod_nat_nat > nat,R3: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( mlex_p6366001652026297872at_nat @ F @ R3 ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) )
| ( ( ( F @ X )
= ( F @ Y3 ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R3 ) ) ) ) ).
% mlex_iff
thf(fact_678_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_679_less__imp__neq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_680_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_681_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_682_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_683_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_684_antisym__conv3,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_nat @ Y3 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_685_linorder__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_686_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_687_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_688_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N4: nat] :
( ( P4 @ N4 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N4 )
=> ~ ( P4 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_689_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B3: nat] :
( ( ord_less_nat @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B3: nat] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_690_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_691_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_692_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_693_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_694_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_695_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_696_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_697_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_698_less__not__refl3,axiom,
! [S: nat,T3: nat] :
( ( ord_less_nat @ S @ T3 )
=> ( S != T3 ) ) ).
% less_not_refl3
thf(fact_699_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_700_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_701_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_702_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_703_linorder__neqE,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_704_order__less__asym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_705_linorder__neq__iff,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
= ( ( ord_less_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_706_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_707_order__less__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_708_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_709_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_710_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_711_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_712_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_713_order__less__not__sym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_714_order__less__imp__triv,axiom,
! [X: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_715_linorder__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_716_order__less__imp__not__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_717_order__less__imp__not__eq2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_718_order__less__imp__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_719_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_720_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_721_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_722_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_723_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_724_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_725_add__mono__strict,axiom,
! [C: nat,D2: nat,A: nat] :
( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ A @ D2 ) ) ) ).
% add_mono_strict
thf(fact_726_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_727_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_728_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_729_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_730_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_731_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_732_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_733_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_734_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_735_order__le__imp__less__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_736_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ( ord_less_set_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_737_linorder__le__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_738_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_739_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_740_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_741_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_742_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_743_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_744_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_745_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_746_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_747_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_748_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_749_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_750_order__less__le__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_751_order__less__le__trans,axiom,
! [X: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_752_order__le__less__trans,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_753_order__le__less__trans,axiom,
! [X: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_754_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_755_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_756_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_757_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_758_order__less__imp__le,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_759_order__less__imp__le,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_760_linorder__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_761_linorder__not__le,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y3 ) )
= ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_762_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_763_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_764_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_765_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_766_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_767_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_768_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_769_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_770_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_771_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_772_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_773_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_774_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_775_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_776_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_eq_nat @ B4 @ A3 )
& ( A3 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_777_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
& ( A3 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_778_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A3: nat] :
( ( ord_less_nat @ B4 @ A3 )
| ( A3 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_779_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B4 @ A3 )
| ( A3 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_780_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_781_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_782_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_783_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_784_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_785_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_786_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_787_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_788_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] :
( ( ord_less_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_789_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_790_not__le__imp__less,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X )
=> ( ord_less_nat @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_791_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_792_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_793_antisym__conv2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_794_antisym__conv2,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ( ~ ( ord_less_set_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_795_antisym__conv1,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_796_antisym__conv1,axiom,
! [X: set_nat,Y3: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y3 )
=> ( ( ord_less_eq_set_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_797_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_798_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_799_leI,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% leI
thf(fact_800_leD,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_nat @ X @ Y3 ) ) ).
% leD
thf(fact_801_leD,axiom,
! [Y3: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X )
=> ~ ( ord_less_set_nat @ X @ Y3 ) ) ).
% leD
thf(fact_802_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_803_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_804_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_805_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_806_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_807_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_808_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_809_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_810_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_811_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_812_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_813_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_814_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_815_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_816_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_817_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_818_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_819_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_820_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_821_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_822_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_823_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_824_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_825_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_826_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_827_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_828_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_829_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_830_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_831_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_832_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_833_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_834_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_835_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_836_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_837_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_838_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_839_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_840_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_841_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_842_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_843_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M6 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_844_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_845_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_846_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_847_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_848_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_849_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_850_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_851_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_852_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_853_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_854_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_855_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_856_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_857_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_858_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_859_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_860_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_861_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_862_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_863_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_864_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_865_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_866_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_867_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_868_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_869_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_870_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_871_less__prod__simp,axiom,
! [X1: product_prod_nat_nat,Y1: product_prod_nat_nat,X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ord_le9033551061567896339at_nat @ ( produc6161850002892822231at_nat @ X1 @ Y1 ) @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) )
= ( ( ord_le1203424502768444845at_nat @ X1 @ X2 )
| ( ( ord_le8460144461188290721at_nat @ X1 @ X2 )
& ( ord_le1203424502768444845at_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_872_less__prod__simp,axiom,
! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ X1 @ X2 )
| ( ( ord_less_eq_nat @ X1 @ X2 )
& ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_873_less__prod__simp,axiom,
! [X1: set_nat,Y1: nat,X2: set_nat,Y2: nat] :
( ( ord_le4872869340735563107at_nat @ ( produc641871753055645167at_nat @ X1 @ Y1 ) @ ( produc641871753055645167at_nat @ X2 @ Y2 ) )
= ( ( ord_less_set_nat @ X1 @ X2 )
| ( ( ord_less_eq_set_nat @ X1 @ X2 )
& ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_874_less__eq__prod__simp,axiom,
! [X1: product_prod_nat_nat,Y1: product_prod_nat_nat,X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ord_le6722121967309221895at_nat @ ( produc6161850002892822231at_nat @ X1 @ Y1 ) @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) )
= ( ( ord_le1203424502768444845at_nat @ X1 @ X2 )
| ( ( ord_le8460144461188290721at_nat @ X1 @ X2 )
& ( ord_le8460144461188290721at_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_875_less__eq__prod__simp,axiom,
! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ X1 @ X2 )
| ( ( ord_less_eq_nat @ X1 @ X2 )
& ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_876_less__eq__prod__simp,axiom,
! [X1: nat,Y1: set_nat,X2: nat,Y2: set_nat] :
( ( ord_le4284901688344473943et_nat @ ( produc4207506657711014383et_nat @ X1 @ Y1 ) @ ( produc4207506657711014383et_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ X1 @ X2 )
| ( ( ord_less_eq_nat @ X1 @ X2 )
& ( ord_less_eq_set_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_877_less__eq__prod__simp,axiom,
! [X1: set_nat,Y1: nat,X2: set_nat,Y2: nat] :
( ( ord_le152793438849583191at_nat @ ( produc641871753055645167at_nat @ X1 @ Y1 ) @ ( produc641871753055645167at_nat @ X2 @ Y2 ) )
= ( ( ord_less_set_nat @ X1 @ X2 )
| ( ( ord_less_eq_set_nat @ X1 @ X2 )
& ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_878_less__eq__prod__simp,axiom,
! [X1: set_nat,Y1: set_nat,X2: set_nat,Y2: set_nat] :
( ( ord_le2041963031926835469et_nat @ ( produc4532415448927165861et_nat @ X1 @ Y1 ) @ ( produc4532415448927165861et_nat @ X2 @ Y2 ) )
= ( ( ord_less_set_nat @ X1 @ X2 )
| ( ( ord_less_eq_set_nat @ X1 @ X2 )
& ( ord_less_eq_set_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_879_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A0: nat > nat > nat,A1: nat,A22: nat,A32: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A1 @ ( product_Pair_nat_nat @ A22 @ A32 ) ) ) )
=> ( ! [F3: nat > nat > nat,A5: nat,B3: nat,Acc: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) )
=> ( ( ~ ( ord_less_nat @ B3 @ A5 )
=> ( P @ F3 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B3 @ ( F3 @ A5 @ Acc ) ) )
=> ( P @ F3 @ A5 @ B3 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_880_match__Star__unfold,axiom,
! [I: nat,J: nat,R: regex_a_t] :
( ( ord_less_nat @ I @ J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ I @ J ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ X3 ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ J ) @ ( match_a_t @ sigma @ R ) ) ) ) ) ).
% match_Star_unfold
thf(fact_881_psubsetI,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less_set_nat @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_882_atLeastLessThan__iff,axiom,
! [I: product_prod_nat_nat,L: product_prod_nat_nat,U: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I @ ( set_or2842531625620587538at_nat @ L @ U ) )
= ( ( ord_le8460144461188290721at_nat @ L @ I )
& ( ord_le1203424502768444845at_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_883_atLeastLessThan__iff,axiom,
! [I: produc859450856879609959at_nat,L: produc859450856879609959at_nat,U: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ I @ ( set_or8929655749483318776at_nat @ L @ U ) )
= ( ( ord_le6722121967309221895at_nat @ L @ I )
& ( ord_le9033551061567896339at_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_884_atLeastLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_885_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_886_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_887_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_888_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_889_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_890_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_891_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( A = C )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_892_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_893_subset__psubset__trans,axiom,
! [A4: set_nat,B5: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( ord_less_set_nat @ B5 @ C4 )
=> ( ord_less_set_nat @ A4 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_894_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_895_psubset__subset__trans,axiom,
! [A4: set_nat,B5: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ord_less_set_nat @ A4 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_896_psubset__imp__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_897_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_898_psubsetE,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_899_psubset__imp__ex__mem,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
=> ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_900_psubset__imp__ex__mem,axiom,
! [A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( ord_le6428140832669894131at_nat @ A4 @ B5 )
=> ? [B3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_901_psubset__imp__ex__mem,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_902_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_903_MDL_Omatch__Star__unfold,axiom,
! [I: nat,J: nat,Sigma: trace_a_t,R: regex_a_t] :
( ( ord_less_nat @ I @ J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ Sigma @ ( star_a_t @ R ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ I @ J ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ X3 ) @ ( match_a_t @ Sigma @ ( star_a_t @ R ) ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ J ) @ ( match_a_t @ Sigma @ R ) ) ) ) ) ).
% MDL.match_Star_unfold
thf(fact_904_fold__atLeastAtMost__nat_Opsimps,axiom,
! [F: nat > nat > nat,A: nat,B: nat,Acc2: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc2 ) ) ) )
=> ( ( ( ord_less_nat @ B @ A )
=> ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less_nat @ B @ A )
=> ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
= ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_905_fold__atLeastAtMost__nat_Opelims,axiom,
! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y3: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
= Y3 )
=> ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
=> ( Y3 = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa )
=> ( Y3
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_906_in__measure,axiom,
! [X: nat,Y3: nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( measure_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) ) ).
% in_measure
thf(fact_907_in__measure,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ ( measur8038558561449204169at_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) ) ).
% in_measure
thf(fact_908_psubsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ C @ A4 )
=> ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_909_psubsetD,axiom,
! [A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
( ( ord_le6428140832669894131at_nat @ A4 @ B5 )
=> ( ( member8206827879206165904at_nat @ C @ A4 )
=> ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_910_psubsetD,axiom,
! [A4: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A4 @ B5 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_911_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_912_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D5: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D5 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_913_minf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T3 @ X5 ) ) ).
% minf(8)
thf(fact_914_minf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T3 ) ) ).
% minf(6)
thf(fact_915_pinf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T3 @ X5 ) ) ).
% pinf(8)
thf(fact_916_pinf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T3 ) ) ).
% pinf(6)
thf(fact_917_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_918_pair__lessI2,axiom,
! [A: nat,B: nat,S: nat,T3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S @ T3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_919_pair__less__iff1,axiom,
! [X: nat,Y3: nat,Z2: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( product_Pair_nat_nat @ X @ Z2 ) ) @ fun_pair_less )
= ( ord_less_nat @ Y3 @ Z2 ) ) ).
% pair_less_iff1
thf(fact_920_sum_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P2: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P2 )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ N @ P2 ) ) )
= ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ P2 ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_921_sum__shift__lb__Suc0__0__upt,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_922_sum_OatLeast0__lessThan__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_923_sum_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( plus_plus_nat @ ( G2 @ M ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_924_sum_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G2 @ B ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_925_pair__lessI1,axiom,
! [A: nat,B: nat,S: nat,T3: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_926_sum__nonpos,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_927_sum__nonpos,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1900718384385340925at_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_928_sum__nonpos,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_929_sum__nonneg,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups977919841031483927at_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_930_sum__nonneg,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1900718384385340925at_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_931_sum__nonneg,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) ) ) ).
% sum_nonneg
thf(fact_932_pair__leqI2,axiom,
! [A: nat,B: nat,S: nat,T3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S @ T3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_933_pair__leqI1,axiom,
! [A: nat,B: nat,S: nat,T3: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T3 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_934_sum__strict__mono2,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ( member8440522571783428010at_nat @ B @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_935_sum__strict__mono2,axiom,
! [B5: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat,F: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ B5 )
=> ( ( ord_le3000389064537975527at_nat @ A4 @ B5 )
=> ( ( member8206827879206165904at_nat @ B @ ( minus_8321449233255521966at_nat @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ ( groups1900718384385340925at_nat @ F @ A4 ) @ ( groups1900718384385340925at_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_936_sum__strict__mono2,axiom,
! [B5: set_nat,A4: set_nat,B: nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_937_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_938_finite__relcomp,axiom,
! [R3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ R3 )
=> ( ( finite6177210948735845034at_nat @ S3 )
=> ( finite6177210948735845034at_nat @ ( relcomp_nat_nat_nat @ R3 @ S3 ) ) ) ) ).
% finite_relcomp
thf(fact_939_sum__eq__Suc0__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( F @ X4 )
= ( suc @ zero_zero_nat ) )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
=> ( ( X4 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_940_sum__mono__inv,axiom,
! [F: product_prod_nat_nat > nat,I5: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,I: product_prod_nat_nat] :
( ( ( groups977919841031483927at_nat @ F @ I5 )
= ( groups977919841031483927at_nat @ G2 @ I5 ) )
=> ( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) )
=> ( ( member8440522571783428010at_nat @ I @ I5 )
=> ( ( finite6177210948735845034at_nat @ I5 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_941_sum__mono__inv,axiom,
! [F: produc859450856879609959at_nat > nat,I5: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,I: produc859450856879609959at_nat] :
( ( ( groups1900718384385340925at_nat @ F @ I5 )
= ( groups1900718384385340925at_nat @ G2 @ I5 ) )
=> ( ! [I3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) )
=> ( ( member8206827879206165904at_nat @ I @ I5 )
=> ( ( finite4392333629123659920at_nat @ I5 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_942_sum__mono__inv,axiom,
! [F: nat > nat,I5: set_nat,G2: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I5 )
= ( groups3542108847815614940at_nat @ G2 @ I5 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) )
=> ( ( member_nat @ I @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_943_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_944_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M6: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_nat @ X4 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_945_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M6: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_946_sum__eq__1__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= one_one_nat )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( F @ X4 )
= one_one_nat )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A4 )
=> ( ( X4 != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_947_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_948_sum__nonneg__eq__0__iff,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups977919841031483927at_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_949_sum__nonneg__eq__0__iff,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ A4 )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups1900718384385340925at_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ A4 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_950_sum__nonneg__eq__0__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A4 )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_951_sum__le__included,axiom,
! [S: set_nat,T3: set_nat,G2: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G2 @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ T3 )
& ( ( I @ Xa2 )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ Xa2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S ) @ ( groups3542108847815614940at_nat @ G2 @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_952_sum__strict__mono__ex1,axiom,
! [A4: set_nat,F: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G2 @ X5 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G2 @ A4 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_953_sum_Orelated,axiom,
! [R3: nat > nat > $o,S3: set_nat,H: nat > nat,G2: nat > nat] :
( ( R3 @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
( ( ( R3 @ X12 @ X22 )
& ( R3 @ Y12 @ Y22 ) )
=> ( R3 @ ( plus_plus_nat @ X12 @ Y12 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R3 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R3 @ ( groups3542108847815614940at_nat @ H @ S3 ) @ ( groups3542108847815614940at_nat @ G2 @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_954_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr1261947904930325089at_nat,T4: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,I: product_prod_nat_nat > product_prod_nat_nat,J: product_prod_nat_nat > product_prod_nat_nat,T: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ S4 )
=> ( ( finite6177210948735845034at_nat @ T4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( member8440522571783428010at_nat @ ( J @ A5 ) @ ( minus_1356011639430497352at_nat @ T @ T4 ) ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( member8440522571783428010at_nat @ ( I @ B3 ) @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups977919841031483927at_nat @ G2 @ S3 )
= ( groups977919841031483927at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_955_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr1261947904930325089at_nat,T4: set_Pr8693737435421807431at_nat,S3: set_Pr1261947904930325089at_nat,I: produc859450856879609959at_nat > product_prod_nat_nat,J: product_prod_nat_nat > produc859450856879609959at_nat,T: set_Pr8693737435421807431at_nat,G2: product_prod_nat_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite6177210948735845034at_nat @ S4 )
=> ( ( finite4392333629123659920at_nat @ T4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( member8206827879206165904at_nat @ ( J @ A5 ) @ ( minus_8321449233255521966at_nat @ T @ T4 ) ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( member8440522571783428010at_nat @ ( I @ B3 ) @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups977919841031483927at_nat @ G2 @ S3 )
= ( groups1900718384385340925at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_956_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr8693737435421807431at_nat,T4: set_Pr1261947904930325089at_nat,S3: set_Pr8693737435421807431at_nat,I: product_prod_nat_nat > produc859450856879609959at_nat,J: produc859450856879609959at_nat > product_prod_nat_nat,T: set_Pr1261947904930325089at_nat,G2: produc859450856879609959at_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite4392333629123659920at_nat @ S4 )
=> ( ( finite6177210948735845034at_nat @ T4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( member8440522571783428010at_nat @ ( J @ A5 ) @ ( minus_1356011639430497352at_nat @ T @ T4 ) ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( member8206827879206165904at_nat @ ( I @ B3 ) @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups1900718384385340925at_nat @ G2 @ S3 )
= ( groups977919841031483927at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_957_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr8693737435421807431at_nat,T4: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat,I: produc859450856879609959at_nat > produc859450856879609959at_nat,J: produc859450856879609959at_nat > produc859450856879609959at_nat,T: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ S4 )
=> ( ( finite4392333629123659920at_nat @ T4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( member8206827879206165904at_nat @ ( J @ A5 ) @ ( minus_8321449233255521966at_nat @ T @ T4 ) ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( member8206827879206165904at_nat @ ( I @ B3 ) @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups1900718384385340925at_nat @ G2 @ S3 )
= ( groups1900718384385340925at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_958_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr1261947904930325089at_nat,T4: set_nat,S3: set_Pr1261947904930325089at_nat,I: nat > product_prod_nat_nat,J: product_prod_nat_nat > nat,T: set_nat,G2: product_prod_nat_nat > nat,H: nat > nat] :
( ( finite6177210948735845034at_nat @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( member_nat @ ( J @ A5 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member8440522571783428010at_nat @ ( I @ B3 ) @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups977919841031483927at_nat @ G2 @ S3 )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_959_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr8693737435421807431at_nat,T4: set_nat,S3: set_Pr8693737435421807431at_nat,I: nat > produc859450856879609959at_nat,J: produc859450856879609959at_nat > nat,T: set_nat,G2: produc859450856879609959at_nat > nat,H: nat > nat] :
( ( finite4392333629123659920at_nat @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( member_nat @ ( J @ A5 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member8206827879206165904at_nat @ ( I @ B3 ) @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups1900718384385340925at_nat @ G2 @ S3 )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_960_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_Pr1261947904930325089at_nat,S3: set_nat,I: product_prod_nat_nat > nat,J: nat > product_prod_nat_nat,T: set_Pr1261947904930325089at_nat,G2: nat > nat,H: product_prod_nat_nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite6177210948735845034at_nat @ T4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member8440522571783428010at_nat @ ( J @ A5 ) @ ( minus_1356011639430497352at_nat @ T @ T4 ) ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S3 )
= ( groups977919841031483927at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_961_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_Pr8693737435421807431at_nat,S3: set_nat,I: produc859450856879609959at_nat > nat,J: nat > produc859450856879609959at_nat,T: set_Pr8693737435421807431at_nat,G2: nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite4392333629123659920at_nat @ T4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member8206827879206165904at_nat @ ( J @ A5 ) @ ( minus_8321449233255521966at_nat @ T @ T4 ) ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S3 )
= ( groups1900718384385340925at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_962_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_nat,S3: set_nat,I: nat > nat,J: nat > nat,T: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member_nat @ ( J @ A5 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S3 )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_963_sum__diff__nat,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_964_sum__pos2,axiom,
! [I5: set_Pr1261947904930325089at_nat,I: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ I5 )
=> ( ( member8440522571783428010at_nat @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups977919841031483927at_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_965_sum__pos2,axiom,
! [I5: set_Pr8693737435421807431at_nat,I: produc859450856879609959at_nat,F: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ I5 )
=> ( ( member8206827879206165904at_nat @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1900718384385340925at_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_966_sum__pos2,axiom,
! [I5: set_nat,I: nat,F: nat > nat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_967_sum_Osame__carrier,axiom,
! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ C4 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ C4 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ C4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups977919841031483927at_nat @ G2 @ A4 )
= ( groups977919841031483927at_nat @ H @ B5 ) )
= ( ( groups977919841031483927at_nat @ G2 @ C4 )
= ( groups977919841031483927at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_968_sum_Osame__carrier,axiom,
! [C4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ C4 )
=> ( ( ord_le3000389064537975527at_nat @ A4 @ C4 )
=> ( ( ord_le3000389064537975527at_nat @ B5 @ C4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups1900718384385340925at_nat @ G2 @ A4 )
= ( groups1900718384385340925at_nat @ H @ B5 ) )
= ( ( groups1900718384385340925at_nat @ G2 @ C4 )
= ( groups1900718384385340925at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_969_sum_Osame__carrier,axiom,
! [C4: set_nat,A4: set_nat,B5: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G2 @ A4 )
= ( groups3542108847815614940at_nat @ H @ B5 ) )
= ( ( groups3542108847815614940at_nat @ G2 @ C4 )
= ( groups3542108847815614940at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_970_sum_Osame__carrierI,axiom,
! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ C4 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ C4 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ C4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups977919841031483927at_nat @ G2 @ C4 )
= ( groups977919841031483927at_nat @ H @ C4 ) )
=> ( ( groups977919841031483927at_nat @ G2 @ A4 )
= ( groups977919841031483927at_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_971_sum_Osame__carrierI,axiom,
! [C4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ C4 )
=> ( ( ord_le3000389064537975527at_nat @ A4 @ C4 )
=> ( ( ord_le3000389064537975527at_nat @ B5 @ C4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups1900718384385340925at_nat @ G2 @ C4 )
= ( groups1900718384385340925at_nat @ H @ C4 ) )
=> ( ( groups1900718384385340925at_nat @ G2 @ A4 )
= ( groups1900718384385340925at_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_972_sum_Osame__carrierI,axiom,
! [C4: set_nat,A4: set_nat,B5: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G2 @ C4 )
= ( groups3542108847815614940at_nat @ H @ C4 ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ A4 )
= ( groups3542108847815614940at_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_973_sum_Omono__neutral__left,axiom,
! [T: set_nat,S3: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S3 )
= ( groups3542108847815614940at_nat @ G2 @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_974_sum_Omono__neutral__right,axiom,
! [T: set_nat,S3: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ T )
= ( groups3542108847815614940at_nat @ G2 @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_975_sum_Omono__neutral__cong__left,axiom,
! [T: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,H: product_prod_nat_nat > nat,G2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ T )
=> ( ( ord_le3146513528884898305at_nat @ S3 @ T )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T @ S3 ) )
=> ( ( H @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups977919841031483927at_nat @ G2 @ S3 )
= ( groups977919841031483927at_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_976_sum_Omono__neutral__cong__left,axiom,
! [T: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat,H: produc859450856879609959at_nat > nat,G2: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ T )
=> ( ( ord_le3000389064537975527at_nat @ S3 @ T )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ ( minus_8321449233255521966at_nat @ T @ S3 ) )
=> ( ( H @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1900718384385340925at_nat @ G2 @ S3 )
= ( groups1900718384385340925at_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_977_sum_Omono__neutral__cong__left,axiom,
! [T: set_nat,S3: set_nat,H: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S3 ) )
=> ( ( H @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S3 )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_978_sum_Omono__neutral__cong__right,axiom,
! [T: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ T )
=> ( ( ord_le3146513528884898305at_nat @ S3 @ T )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups977919841031483927at_nat @ G2 @ T )
= ( groups977919841031483927at_nat @ H @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_979_sum_Omono__neutral__cong__right,axiom,
! [T: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ T )
=> ( ( ord_le3000389064537975527at_nat @ S3 @ T )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ ( minus_8321449233255521966at_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1900718384385340925at_nat @ G2 @ T )
= ( groups1900718384385340925at_nat @ H @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_980_sum_Omono__neutral__cong__right,axiom,
! [T: set_nat,S3: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ T )
= ( groups3542108847815614940at_nat @ H @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_981_sum_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G2: nat > nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups3542108847815614940at_nat @ G2 @ A4 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3542108847815614940at_nat @ G2 @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_982_sum__mono2,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_983_sum__mono2,axiom,
! [B5: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ B5 )
=> ( ( ord_le3000389064537975527at_nat @ A4 @ B5 )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups1900718384385340925at_nat @ F @ A4 ) @ ( groups1900718384385340925at_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_984_sum__mono2,axiom,
! [B5: set_nat,A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_985_finite__Diff2,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_986_finite__Diff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% finite_Diff
thf(fact_987_Diff__infinite__finite,axiom,
! [T: set_nat,S3: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_988_finite__has__maximal2,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
& ( ord_le8460144461188290721at_nat @ A @ X3 )
& ! [Xa2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa2 @ A4 )
=> ( ( ord_le8460144461188290721at_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_989_finite__has__maximal2,axiom,
! [A4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
( ( finite4392333629123659920at_nat @ A4 )
=> ( ( member8206827879206165904at_nat @ A @ A4 )
=> ? [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ A4 )
& ( ord_le6722121967309221895at_nat @ A @ X3 )
& ! [Xa2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Xa2 @ A4 )
=> ( ( ord_le6722121967309221895at_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_990_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_991_finite__has__maximal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ( ord_less_eq_set_nat @ A @ X3 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_992_finite__has__minimal2,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ? [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
& ( ord_le8460144461188290721at_nat @ X3 @ A )
& ! [Xa2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa2 @ A4 )
=> ( ( ord_le8460144461188290721at_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_993_finite__has__minimal2,axiom,
! [A4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
( ( finite4392333629123659920at_nat @ A4 )
=> ( ( member8206827879206165904at_nat @ A @ A4 )
=> ? [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ A4 )
& ( ord_le6722121967309221895at_nat @ X3 @ A )
& ! [Xa2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Xa2 @ A4 )
=> ( ( ord_le6722121967309221895at_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_994_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A4 )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_995_finite__has__minimal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_996_rev__finite__subset,axiom,
! [B5: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_997_infinite__super,axiom,
! [S3: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_998_finite__subset,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_999_infinite__nat__iff__unbounded__le,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M6: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M6 @ N4 )
& ( member_nat @ N4 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1000_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_1001_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H )
= ( set_or4548717258645045905et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H )
& ~ ( ord_less_eq_set_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1002_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1003_atLeastAtMost__iff,axiom,
! [I: product_prod_nat_nat,L: product_prod_nat_nat,U: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I @ ( set_or5187384067897641014at_nat @ L @ U ) )
= ( ( ord_le8460144461188290721at_nat @ L @ I )
& ( ord_le8460144461188290721at_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_1004_atLeastAtMost__iff,axiom,
! [I: produc859450856879609959at_nat,L: produc859450856879609959at_nat,U: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ I @ ( set_or5255168418374126620at_nat @ L @ U ) )
= ( ( ord_le6722121967309221895at_nat @ L @ I )
& ( ord_le6722121967309221895at_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_1005_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_1006_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_1007_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1008_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D2 ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1009_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1010_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_1011_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D2 )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D2 ) ) ) )
& ( ord_less_eq_set_nat @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1012_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D2 ) ) ) )
& ( ord_less_eq_nat @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1013_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1014_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or3540276404033026485et_nat @ C @ D2 ) )
= ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_set_nat @ B @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1015_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_nat @ B @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1016_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1017_sum_OatLeast0__atMost__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_1018_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( G2 @ M ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_1019_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G2 @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_1020_sum_Olast__plus,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( G2 @ N ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% sum.last_plus
thf(fact_1021_sum_Ohead__if,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% sum.head_if
thf(fact_1022_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > nat,P2: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_1023_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_eq_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_1024_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_eq_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_1025_sum_OatLeastAtMost__shift__0,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G2 @ ( plus_plus_nat @ M ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_1026_prod__eq__1__iff,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( groups708209901874060359at_nat @ F @ A4 )
= one_one_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ( F @ X4 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_1027_prod_Oinfinite,axiom,
! [A4: set_nat,G2: nat > nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( groups708209901874060359at_nat @ G2 @ A4 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_1028_prod_OatLeastAtMost__shift__0,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G2 @ ( plus_plus_nat @ M ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_1029_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G2 @ suc ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_1030_prod_OatLeastAtMost__shift__bounds,axiom,
! [G2: nat > nat,M: nat,K: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G2 @ ( plus_plus_nat @ K ) ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_1031_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_1032_prod__ge__1,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_1033_prod__ge__1,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups7771183085424866898at_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_1034_prod__ge__1,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% prod_ge_1
thf(fact_1035_prod__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat,G2: product_prod_nat_nat > nat] :
( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) @ ( groups4077766827762148844at_nat @ G2 @ A4 ) ) ) ).
% prod_mono
thf(fact_1036_prod__mono,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat,G2: produc859450856879609959at_nat > nat] :
( ! [I3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups7771183085424866898at_nat @ F @ A4 ) @ ( groups7771183085424866898at_nat @ G2 @ A4 ) ) ) ).
% prod_mono
thf(fact_1037_prod__mono,axiom,
! [A4: set_nat,F: nat > nat,G2: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G2 @ A4 ) ) ) ).
% prod_mono
thf(fact_1038_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( groups4077766827762148844at_nat @ G2 @ A4 )
!= one_one_nat )
=> ~ ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ A4 )
=> ( ( G2 @ A5 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1039_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: produc859450856879609959at_nat > nat,A4: set_Pr8693737435421807431at_nat] :
( ( ( groups7771183085424866898at_nat @ G2 @ A4 )
!= one_one_nat )
=> ~ ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ A4 )
=> ( ( G2 @ A5 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1040_prod_Onot__neutral__contains__not__neutral,axiom,
! [G2: nat > nat,A4: set_nat] :
( ( ( groups708209901874060359at_nat @ G2 @ A4 )
!= one_one_nat )
=> ~ ! [A5: nat] :
( ( member_nat @ A5 @ A4 )
=> ( ( G2 @ A5 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1041_prod_OatLeastLessThan__shift__0,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G2 @ ( plus_plus_nat @ M ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_1042_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G2 @ suc ) @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_1043_prod_OatLeastLessThan__shift__bounds,axiom,
! [G2: nat > nat,M: nat,K: nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G2 @ ( plus_plus_nat @ K ) ) @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_1044_prod__le__1,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_1045_prod__le__1,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups7771183085424866898at_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_1046_prod__le__1,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_1047_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr1261947904930325089at_nat,T4: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,I: product_prod_nat_nat > product_prod_nat_nat,J: product_prod_nat_nat > product_prod_nat_nat,T: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ S4 )
=> ( ( finite6177210948735845034at_nat @ T4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( member8440522571783428010at_nat @ ( J @ A5 ) @ ( minus_1356011639430497352at_nat @ T @ T4 ) ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( member8440522571783428010at_nat @ ( I @ B3 ) @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups4077766827762148844at_nat @ G2 @ S3 )
= ( groups4077766827762148844at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1048_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr1261947904930325089at_nat,T4: set_Pr8693737435421807431at_nat,S3: set_Pr1261947904930325089at_nat,I: produc859450856879609959at_nat > product_prod_nat_nat,J: product_prod_nat_nat > produc859450856879609959at_nat,T: set_Pr8693737435421807431at_nat,G2: product_prod_nat_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite6177210948735845034at_nat @ S4 )
=> ( ( finite4392333629123659920at_nat @ T4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( member8206827879206165904at_nat @ ( J @ A5 ) @ ( minus_8321449233255521966at_nat @ T @ T4 ) ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( member8440522571783428010at_nat @ ( I @ B3 ) @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups4077766827762148844at_nat @ G2 @ S3 )
= ( groups7771183085424866898at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1049_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr8693737435421807431at_nat,T4: set_Pr1261947904930325089at_nat,S3: set_Pr8693737435421807431at_nat,I: product_prod_nat_nat > produc859450856879609959at_nat,J: produc859450856879609959at_nat > product_prod_nat_nat,T: set_Pr1261947904930325089at_nat,G2: produc859450856879609959at_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite4392333629123659920at_nat @ S4 )
=> ( ( finite6177210948735845034at_nat @ T4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( member8440522571783428010at_nat @ ( J @ A5 ) @ ( minus_1356011639430497352at_nat @ T @ T4 ) ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( member8206827879206165904at_nat @ ( I @ B3 ) @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups7771183085424866898at_nat @ G2 @ S3 )
= ( groups4077766827762148844at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1050_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr8693737435421807431at_nat,T4: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat,I: produc859450856879609959at_nat > produc859450856879609959at_nat,J: produc859450856879609959at_nat > produc859450856879609959at_nat,T: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ S4 )
=> ( ( finite4392333629123659920at_nat @ T4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( member8206827879206165904at_nat @ ( J @ A5 ) @ ( minus_8321449233255521966at_nat @ T @ T4 ) ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( member8206827879206165904at_nat @ ( I @ B3 ) @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups7771183085424866898at_nat @ G2 @ S3 )
= ( groups7771183085424866898at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1051_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr1261947904930325089at_nat,T4: set_nat,S3: set_Pr1261947904930325089at_nat,I: nat > product_prod_nat_nat,J: product_prod_nat_nat > nat,T: set_nat,G2: product_prod_nat_nat > nat,H: nat > nat] :
( ( finite6177210948735845034at_nat @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) )
=> ( member_nat @ ( J @ A5 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member8440522571783428010at_nat @ ( I @ B3 ) @ ( minus_1356011639430497352at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups4077766827762148844at_nat @ G2 @ S3 )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1052_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Pr8693737435421807431at_nat,T4: set_nat,S3: set_Pr8693737435421807431at_nat,I: nat > produc859450856879609959at_nat,J: produc859450856879609959at_nat > nat,T: set_nat,G2: produc859450856879609959at_nat > nat,H: nat > nat] :
( ( finite4392333629123659920at_nat @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) )
=> ( member_nat @ ( J @ A5 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member8206827879206165904at_nat @ ( I @ B3 ) @ ( minus_8321449233255521966at_nat @ S3 @ S4 ) ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups7771183085424866898at_nat @ G2 @ S3 )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1053_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_Pr1261947904930325089at_nat,S3: set_nat,I: product_prod_nat_nat > nat,J: nat > product_prod_nat_nat,T: set_Pr1261947904930325089at_nat,G2: nat > nat,H: product_prod_nat_nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite6177210948735845034at_nat @ T4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member8440522571783428010at_nat @ ( J @ A5 ) @ ( minus_1356011639430497352at_nat @ T @ T4 ) ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ T @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups708209901874060359at_nat @ G2 @ S3 )
= ( groups4077766827762148844at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1054_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_Pr8693737435421807431at_nat,S3: set_nat,I: produc859450856879609959at_nat > nat,J: nat > produc859450856879609959at_nat,T: set_Pr8693737435421807431at_nat,G2: nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite4392333629123659920at_nat @ T4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member8206827879206165904at_nat @ ( J @ A5 ) @ ( minus_8321449233255521966at_nat @ T @ T4 ) ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ T @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups708209901874060359at_nat @ G2 @ S3 )
= ( groups7771183085424866898at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1055_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T4: set_nat,S3: set_nat,I: nat > nat,J: nat > nat,T: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ S3 @ S4 ) )
=> ( member_nat @ ( J @ A5 ) @ ( minus_minus_set_nat @ T @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S3 @ S4 ) ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S4 )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S3 )
=> ( ( H @ ( J @ A5 ) )
= ( G2 @ A5 ) ) )
=> ( ( groups708209901874060359at_nat @ G2 @ S3 )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1056_sum__comp__morphism,axiom,
! [H: nat > nat,G2: nat > nat,A4: set_nat] :
( ( ( H @ zero_zero_nat )
= zero_zero_nat )
=> ( ! [X3: nat,Y: nat] :
( ( H @ ( plus_plus_nat @ X3 @ Y ) )
= ( plus_plus_nat @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ H @ G2 ) @ A4 )
= ( H @ ( groups3542108847815614940at_nat @ G2 @ A4 ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_1057_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G2 @ suc ) @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_1058_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G2 @ suc ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_1059_sum_OatLeastLessThan__shift__bounds,axiom,
! [G2: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G2 @ ( plus_plus_nat @ K ) ) @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_1060_sum_OatLeastAtMost__shift__bounds,axiom,
! [G2: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G2 @ ( plus_plus_nat @ K ) ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_1061_prod_Osame__carrier,axiom,
! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ C4 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ C4 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ C4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ( ( groups4077766827762148844at_nat @ G2 @ A4 )
= ( groups4077766827762148844at_nat @ H @ B5 ) )
= ( ( groups4077766827762148844at_nat @ G2 @ C4 )
= ( groups4077766827762148844at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1062_prod_Osame__carrier,axiom,
! [C4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ C4 )
=> ( ( ord_le3000389064537975527at_nat @ A4 @ C4 )
=> ( ( ord_le3000389064537975527at_nat @ B5 @ C4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ( ( groups7771183085424866898at_nat @ G2 @ A4 )
= ( groups7771183085424866898at_nat @ H @ B5 ) )
= ( ( groups7771183085424866898at_nat @ G2 @ C4 )
= ( groups7771183085424866898at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1063_prod_Osame__carrier,axiom,
! [C4: set_nat,A4: set_nat,B5: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ( ( groups708209901874060359at_nat @ G2 @ A4 )
= ( groups708209901874060359at_nat @ H @ B5 ) )
= ( ( groups708209901874060359at_nat @ G2 @ C4 )
= ( groups708209901874060359at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1064_prod_Osame__carrierI,axiom,
! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ C4 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ C4 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ C4 )
=> ( ! [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ ( minus_1356011639430497352at_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ( ( groups4077766827762148844at_nat @ G2 @ C4 )
= ( groups4077766827762148844at_nat @ H @ C4 ) )
=> ( ( groups4077766827762148844at_nat @ G2 @ A4 )
= ( groups4077766827762148844at_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1065_prod_Osame__carrierI,axiom,
! [C4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ C4 )
=> ( ( ord_le3000389064537975527at_nat @ A4 @ C4 )
=> ( ( ord_le3000389064537975527at_nat @ B5 @ C4 )
=> ( ! [A5: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ A5 @ ( minus_8321449233255521966at_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ( ( groups7771183085424866898at_nat @ G2 @ C4 )
= ( groups7771183085424866898at_nat @ H @ C4 ) )
=> ( ( groups7771183085424866898at_nat @ G2 @ A4 )
= ( groups7771183085424866898at_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1066_prod_Osame__carrierI,axiom,
! [C4: set_nat,A4: set_nat,B5: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ B5 @ C4 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A4 ) )
=> ( ( G2 @ A5 )
= one_one_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
=> ( ( H @ B3 )
= one_one_nat ) )
=> ( ( ( groups708209901874060359at_nat @ G2 @ C4 )
= ( groups708209901874060359at_nat @ H @ C4 ) )
=> ( ( groups708209901874060359at_nat @ G2 @ A4 )
= ( groups708209901874060359at_nat @ H @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1067_prod_Omono__neutral__left,axiom,
! [T: set_nat,S3: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G2 @ S3 )
= ( groups708209901874060359at_nat @ G2 @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_1068_prod_Omono__neutral__right,axiom,
! [T: set_nat,S3: set_nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G2 @ T )
= ( groups708209901874060359at_nat @ G2 @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_1069_prod_Omono__neutral__cong__left,axiom,
! [T: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,H: product_prod_nat_nat > nat,G2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ T )
=> ( ( ord_le3146513528884898305at_nat @ S3 @ T )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T @ S3 ) )
=> ( ( H @ X3 )
= one_one_nat ) )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups4077766827762148844at_nat @ G2 @ S3 )
= ( groups4077766827762148844at_nat @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1070_prod_Omono__neutral__cong__left,axiom,
! [T: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat,H: produc859450856879609959at_nat > nat,G2: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ T )
=> ( ( ord_le3000389064537975527at_nat @ S3 @ T )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ ( minus_8321449233255521966at_nat @ T @ S3 ) )
=> ( ( H @ X3 )
= one_one_nat ) )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7771183085424866898at_nat @ G2 @ S3 )
= ( groups7771183085424866898at_nat @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1071_prod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S3: set_nat,H: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S3 ) )
=> ( ( H @ X3 )
= one_one_nat ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups708209901874060359at_nat @ G2 @ S3 )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1072_prod_Omono__neutral__cong__right,axiom,
! [T: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,G2: product_prod_nat_nat > nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ T )
=> ( ( ord_le3146513528884898305at_nat @ S3 @ T )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( minus_1356011639430497352at_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups4077766827762148844at_nat @ G2 @ T )
= ( groups4077766827762148844at_nat @ H @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1073_prod_Omono__neutral__cong__right,axiom,
! [T: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat,G2: produc859450856879609959at_nat > nat,H: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ T )
=> ( ( ord_le3000389064537975527at_nat @ S3 @ T )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ ( minus_8321449233255521966at_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7771183085424866898at_nat @ G2 @ T )
= ( groups7771183085424866898at_nat @ H @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1074_prod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S3: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S3 @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T @ S3 ) )
=> ( ( G2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups708209901874060359at_nat @ G2 @ T )
= ( groups708209901874060359at_nat @ H @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1075_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G2 @ zero_zero_nat ) @ ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G2 @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_1076_sum_OatLeast0__atMost__Suc__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G2 @ zero_zero_nat ) @ ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G2 @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_1077_sum_OatLeastLessThan__shift__0,axiom,
! [G2: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G2 @ ( plus_plus_nat @ M ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_1078_prod__mono__strict,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat,G2: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ord_less_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) @ ( groups4077766827762148844at_nat @ G2 @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_1079_prod__mono__strict,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat,G2: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ A4 )
=> ( ! [I3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ( A4 != bot_bo5327735625951526323at_nat )
=> ( ord_less_nat @ ( groups7771183085424866898at_nat @ F @ A4 ) @ ( groups7771183085424866898at_nat @ G2 @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_1080_prod__mono__strict,axiom,
! [A4: set_nat,F: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ A4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_nat @ ( F @ I3 ) @ ( G2 @ I3 ) ) ) )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G2 @ A4 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_1081_less__than__iff,axiom,
! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ less_than )
= ( ord_less_nat @ X @ Y3 ) ) ).
% less_than_iff
thf(fact_1082_empty__iff,axiom,
! [C: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_1083_empty__iff,axiom,
! [C: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ C @ bot_bo5327735625951526323at_nat ) ).
% empty_iff
thf(fact_1084_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1085_all__not__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ! [X4: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X4 @ A4 ) )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_1086_all__not__in__conv,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ( ! [X4: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ X4 @ A4 ) )
= ( A4 = bot_bo5327735625951526323at_nat ) ) ).
% all_not_in_conv
thf(fact_1087_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1088_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_1089_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_1090_relcomp__empty2,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ R3 @ bot_bo2099793752762293965at_nat )
= bot_bo2099793752762293965at_nat ) ).
% relcomp_empty2
thf(fact_1091_relcomp__empty1,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ bot_bo2099793752762293965at_nat @ R3 )
= bot_bo2099793752762293965at_nat ) ).
% relcomp_empty1
thf(fact_1092_subset__empty,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1093_empty__subsetI,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_1094_Diff__cancel,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ A4 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1095_empty__Diff,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1096_Diff__empty,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Diff_empty
thf(fact_1097_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or4548717258645045905et_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1098_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1099_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1100_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1101_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_1102_atLeastLessThan__empty,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( set_or3540276404033026485et_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_1103_atLeastLessThan__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_1104_atLeastLessThan__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A @ B ) )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_1105_atLeastLessThan__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_1106_prod_Oempty,axiom,
! [G2: nat > nat] :
( ( groups708209901874060359at_nat @ G2 @ bot_bot_set_nat )
= one_one_nat ) ).
% prod.empty
thf(fact_1107_Diff__eq__empty__iff,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( minus_minus_set_nat @ A4 @ B5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_1108_bot__prod__def,axiom,
( bot_bo7480491830437098195at_nat
= ( produc6161850002892822231at_nat @ bot_bo2769642828321324397at_nat @ bot_bo2769642828321324397at_nat ) ) ).
% bot_prod_def
thf(fact_1109_bot__prod__def,axiom,
( bot_bo3047382831089536473et_nat
= ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_1110_bot__prod__def,axiom,
( bot_bo8329445699581754659at_nat
= ( produc641871753055645167at_nat @ bot_bot_set_nat @ bot_bot_nat ) ) ).
% bot_prod_def
thf(fact_1111_bot__prod__def,axiom,
( bot_bo3238181912221869603et_nat
= ( produc4207506657711014383et_nat @ bot_bot_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_1112_bot__prod__def,axiom,
( bot_bo2769642828321324397at_nat
= ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat ) ) ).
% bot_prod_def
thf(fact_1113_emptyE,axiom,
! [A: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_1114_emptyE,axiom,
! [A: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ A @ bot_bo5327735625951526323at_nat ) ).
% emptyE
thf(fact_1115_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1116_equals0D,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_1117_equals0D,axiom,
! [A4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
( ( A4 = bot_bo5327735625951526323at_nat )
=> ~ ( member8206827879206165904at_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_1118_equals0D,axiom,
! [A4: set_nat,A: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_1119_equals0I,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [Y: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y @ A4 )
=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_1120_equals0I,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ! [Y: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ Y @ A4 )
=> ( A4 = bot_bo5327735625951526323at_nat ) ) ).
% equals0I
thf(fact_1121_equals0I,axiom,
! [A4: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1122_ex__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ? [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
= ( A4 != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_1123_ex__in__conv,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ( ? [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A4 ) )
= ( A4 != bot_bo5327735625951526323at_nat ) ) ).
% ex_in_conv
thf(fact_1124_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1125_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1126_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1127_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1128_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1129_not__psubset__empty,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1130_infinite__growing,axiom,
! [X7: set_Pr1261947904930325089at_nat] :
( ( X7 != bot_bo2099793752762293965at_nat )
=> ( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ X7 )
=> ? [Xa2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Xa2 @ X7 )
& ( ord_le1203424502768444845at_nat @ X3 @ Xa2 ) ) )
=> ~ ( finite6177210948735845034at_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_1131_infinite__growing,axiom,
! [X7: set_Pr8693737435421807431at_nat] :
( ( X7 != bot_bo5327735625951526323at_nat )
=> ( ! [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ X7 )
=> ? [Xa2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Xa2 @ X7 )
& ( ord_le9033551061567896339at_nat @ X3 @ Xa2 ) ) )
=> ~ ( finite4392333629123659920at_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_1132_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ X7 )
& ( ord_less_nat @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_1133_ex__min__if__finite,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S3 )
& ~ ? [Xa2: nat] :
( ( member_nat @ Xa2 @ S3 )
& ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1134_subset__emptyI,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_1135_subset__emptyI,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ! [X3: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ X3 @ A4 )
=> ( ord_le3000389064537975527at_nat @ A4 @ bot_bo5327735625951526323at_nat ) ) ).
% subset_emptyI
thf(fact_1136_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1137_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1138_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1139_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1140_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1141_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1142_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_1143_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A4 )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1144_finite__has__maximal,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1145_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A4 )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1146_finite__has__minimal,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1147_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1148_diff__shunt__var,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1149_arg__min__if__finite_I2_J,axiom,
! [S3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S3 )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S3 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1150_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1151_bot__empty__eq,axiom,
( bot_bo482883023278783056_nat_o
= ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1152_bot__empty__eq,axiom,
( bot_bo7573314457883560170_nat_o
= ( ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ bot_bo5327735625951526323at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1153_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1154_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1155_arg__min__least,axiom,
! [S3: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ S3 )
=> ( ( S3 != bot_bo2099793752762293965at_nat )
=> ( ( member8440522571783428010at_nat @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic4984276347100956536at_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1156_arg__min__least,axiom,
! [S3: set_Pr8693737435421807431at_nat,Y3: produc859450856879609959at_nat,F: produc859450856879609959at_nat > nat] :
( ( finite4392333629123659920at_nat @ S3 )
=> ( ( S3 != bot_bo5327735625951526323at_nat )
=> ( ( member8206827879206165904at_nat @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic390166758595302878at_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1157_arg__min__least,axiom,
! [S3: set_nat,Y3: nat,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ( ( member_nat @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1158_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1159_prod_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_1160_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1161_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1162_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1163_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1164_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1165_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1166_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1167_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1168_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1169_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1170_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1171_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1172_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1173_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1174_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1175_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1176_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1177_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1178_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1179_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1180_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1181_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1182_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1183_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1184_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1185_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1186_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1187_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1188_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1189_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1190_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1191_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1192_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1193_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1194_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1195_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1196_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B4: nat] : ( times_times_nat @ B4 @ A3 ) ) ) ).
% mult.commute
thf(fact_1197_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1198_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1199_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1200_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1201_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1202_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1203_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1204_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1205_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1206_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_1207_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1208_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1209_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1210_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1211_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1212_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1213_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( A != B )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1214_crossproduct__eq,axiom,
! [W: nat,Y3: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y3 ) ) )
= ( ( W = X )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_1215_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1216_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1217_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D2: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1218_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1219_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1220_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1221_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1222_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1223_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1224_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1225_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1226_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1227_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1228_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1229_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1230_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1231_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1232_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1233_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1234_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1235_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1236_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1237_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1238_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1239_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1240_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1241_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1242_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1243_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1244_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1245_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1246_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1247_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1248_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1249_prod_Orelated,axiom,
! [R3: nat > nat > $o,S3: set_nat,H: nat > nat,G2: nat > nat] :
( ( R3 @ one_one_nat @ one_one_nat )
=> ( ! [X12: nat,Y12: nat,X22: nat,Y22: nat] :
( ( ( R3 @ X12 @ X22 )
& ( R3 @ Y12 @ Y22 ) )
=> ( R3 @ ( times_times_nat @ X12 @ Y12 ) @ ( times_times_nat @ X22 @ Y22 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R3 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R3 @ ( groups708209901874060359at_nat @ H @ S3 ) @ ( groups708209901874060359at_nat @ G2 @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_1250_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1251_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1252_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1253_prod_OatLeastLessThan__concat,axiom,
! [M: nat,N: nat,P2: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ P2 )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ N @ P2 ) ) )
= ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ P2 ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_1254_prod_Osubset__diff,axiom,
! [B5: set_nat,A4: set_nat,G2: nat > nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( groups708209901874060359at_nat @ G2 @ A4 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G2 @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_1255_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M6: nat,N4: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1256_prod_OatLeast0__lessThan__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G2 @ N ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_1257_prod_OatLeast0__atMost__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_1258_prod_OatLeast__Suc__lessThan,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( times_times_nat @ ( G2 @ M ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_1259_prod_OatLeastLessThan__Suc,axiom,
! [A: nat,B: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G2 @ B ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_1260_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( G2 @ M ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_1261_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( G2 @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_1262_prod_Olast__plus,axiom,
! [M: nat,N: nat,G2: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( G2 @ N ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).
% prod.last_plus
thf(fact_1263_prod_Ohead__if,axiom,
! [N: nat,M: nat,G2: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G2 @ N ) ) ) ) ) ).
% prod.head_if
thf(fact_1264_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G2: nat > nat,P2: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_1265_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G2 @ zero_zero_nat ) @ ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G2 @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_1266_prod_OatLeast0__atMost__Suc__shift,axiom,
! [G2: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G2 @ zero_zero_nat ) @ ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G2 @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y3 )
= X ) ).
% Conjectures (9)
thf(conj_0,hypothesis,
! [I4: nat,J4: nat] :
( ( ord_less_eq_nat @ I4 @ J4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ ( suc @ J4 ) ) @ ( match_a_t @ sigma @ r1 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ J4 ) @ ( match_a_t @ sigma @ ( rderive_a_t @ r1 ) ) ) ) ) ).
thf(conj_1,hypothesis,
! [I4: nat,J4: nat] :
( ( ord_less_eq_nat @ I4 @ J4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ ( suc @ J4 ) ) @ ( match_a_t @ sigma @ r2 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ J4 ) @ ( match_a_t @ sigma @ ( rderive_a_t @ r2 ) ) ) ) ) ).
thf(conj_2,hypothesis,
$true ).
thf(conj_3,hypothesis,
! [I4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ ( suc @ ja ) ) @ ( match_a_t @ sigma @ r2 ) )
=> ( ord_less_eq_nat @ I4 @ ( suc @ ja ) ) ) ).
thf(conj_4,hypothesis,
wf_regex_a_t @ r2 ).
thf(conj_5,hypothesis,
wf_regex_a_t @ r1 ).
thf(conj_6,hypothesis,
eps_a_t @ r2 ).
thf(conj_7,hypothesis,
member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ia @ ja ) @ ( match_a_t @ sigma @ ( rderive_a_t @ r1 ) ) ).
thf(conj_8,conjecture,
member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ia @ ( suc @ ja ) ) @ ( relcomp_nat_nat_nat @ ( match_a_t @ sigma @ r1 ) @ ( match_a_t @ sigma @ r2 ) ) ).
%------------------------------------------------------------------------------