TPTP Problem File: SLH0749^1.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 : Actuarial_Mathematics/0000_Preliminaries/prob_00025_000742__12805580_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1400 ( 679 unt; 131 typ; 0 def)
% Number of atoms : 3349 (1316 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10859 ( 292 ~; 64 |; 226 &;9030 @)
% ( 0 <=>;1247 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 872 ( 872 >; 0 *; 0 +; 0 <<)
% Number of symbols : 122 ( 119 usr; 17 con; 0-3 aty)
% Number of variables : 3906 ( 371 ^;3402 !; 133 ?;3906 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:06:43.088
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_I_Eo_J_J_J,type,
set_set_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (119)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
comple4436988014476444997_set_o: set_set_set_o > set_set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_Eo_M_Eo_J,type,
minus_minus_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_Eo_J,type,
ord_less_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
ord_less_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Set_OBex_001_Eo,type,
bex_o: set_o > ( $o > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Nat__Onat,type,
bex_nat: set_nat > ( nat > $o ) > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_Eo_J,type,
collect_set_o: ( set_o > $o ) > set_set_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001_062_I_Eo_M_Eo_J,type,
image_o_o_o: ( $o > $o > $o ) > set_o > set_o_o ).
thf(sy_c_Set_Oimage_001_Eo_001_062_It__Nat__Onat_M_Eo_J,type,
image_o_nat_o: ( $o > nat > $o ) > set_o > set_nat_o ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_o_set_set_nat: ( $o > set_set_nat ) > set_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J,type,
image_nat_nat_o: ( nat > nat > $o ) > set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
image_set_o_o_o: ( set_o > $o > $o ) > set_set_o > set_o_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_o_set_nat: ( set_o > set_nat ) > set_set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
image_5023573440332574309_set_o: ( set_o > set_set_o ) > set_set_o > set_set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7698617416147310703et_nat: ( set_o > set_set_nat ) > set_set_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
image_set_nat_nat_o: ( set_nat > nat > $o ) > set_set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
image_set_nat_set_o: ( set_nat > set_o ) > set_set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_Eo,type,
set_or7139685690850216873Than_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001_Eo,type,
set_ord_atLeast_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_I_Eo_J,type,
set_or8686861255860958915_set_o: set_o > set_set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Nat__Onat_J,type,
set_or1731685050470061051et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_Eo,type,
set_ord_atMost_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_I_Eo_J,type,
set_ord_atMost_set_o: set_o > set_set_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
set_ord_lessThan_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_A,type,
a: nat > set_nat ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1263)
thf(fact_0_assms_I1_J,axiom,
m != zero_zero_nat ).
% assms(1)
thf(fact_1_UN__I,axiom,
! [A: set_nat,A2: set_set_nat,B: nat,B2: set_nat > set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_2_UN__I,axiom,
! [A: nat,A2: set_nat,B: $o,B2: nat > set_o] :
( ( member_nat @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_3_UN__I,axiom,
! [A: $o,A2: set_o,B: $o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_4_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_5_UN__I,axiom,
! [A: $o,A2: set_o,B: nat,B2: $o > set_nat] :
( ( member_o @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_6_UN__iff,axiom,
! [B: nat,B2: $o > set_nat,A2: set_o] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( member_nat @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_7_UN__iff,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( member_nat @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_8_UN__iff,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( member_o @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_9_UN__iff,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( member_nat @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_10_SUP__identity__eq,axiom,
! [A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [X: set_o] : X
@ A2 ) )
= ( comple90263536869209701_set_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_11_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : X
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_12_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_13_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_14_UN__ball__bex__simps_I4_J,axiom,
! [B2: $o > set_nat,A2: set_o,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_15_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_16_UN__ball__bex__simps_I4_J,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ? [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: $o] :
( ( member_o @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_17_UN__ball__bex__simps_I4_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_18_UN__ball__bex__simps_I2_J,axiom,
! [B2: $o > set_nat,A2: set_o,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_19_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_20_UN__ball__bex__simps_I2_J,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_21_UN__ball__bex__simps_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_22_bex__UN,axiom,
! [B2: $o > set_nat,A2: set_o,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_23_bex__UN,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_24_bex__UN,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ? [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ? [Y: $o] :
( ( member_o @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_25_bex__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_26_ball__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_27_ball__UN,axiom,
! [B2: $o > set_nat,A2: set_o,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_28_ball__UN,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_29_ball__UN,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_30_SUP__UNION,axiom,
! [F: nat > $o,G: nat > set_nat,A2: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_31_SUP__UNION,axiom,
! [F: nat > $o,G: $o > set_nat,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y: $o] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_32_SUP__UNION,axiom,
! [F: $o > $o,G: $o > set_o,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_33_SUP__UNION,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_34_SUP__UNION,axiom,
! [F: nat > set_nat,G: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_35_SUP__UNION,axiom,
! [F: $o > set_nat,G: nat > set_o,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_36_SUP__UNION,axiom,
! [F: $o > set_nat,G: $o > set_o,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_37_SUP__UNION,axiom,
! [F: nat > $o,G: set_nat > set_nat,A2: set_set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_set_nat_o
@ ^ [Y: set_nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_38_SUP__UNION,axiom,
! [F: nat > set_o,G: nat > set_nat,A2: set_nat] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y: nat] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_39_SUP__UNION,axiom,
! [F: nat > set_o,G: $o > set_nat,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_40_image__ident,axiom,
! [Y2: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_41_image__ident,axiom,
! [Y2: set_o] :
( ( image_o_o
@ ^ [X: $o] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_42_image__ident,axiom,
! [Y2: set_set_nat] :
( ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_43_image__eqI,axiom,
! [B: set_nat,F: set_nat > set_nat,X2: set_nat,A2: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_44_image__eqI,axiom,
! [B: set_nat,F: $o > set_nat,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_o_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_45_image__eqI,axiom,
! [B: set_o,F: $o > set_o,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_46_image__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_47_image__eqI,axiom,
! [B: nat,F: $o > nat,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A2 )
=> ( member_nat @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_48_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_49_image__eqI,axiom,
! [B: $o,F: nat > $o,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_o @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_50_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_51_Union__iff,axiom,
! [A2: nat,C: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ C )
& ( member_nat @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_52_Union__iff,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X: set_o] :
( ( member_set_o @ X @ C )
& ( member_o @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_53_UnionI,axiom,
! [X3: set_nat,C: set_set_nat,A2: nat] :
( ( member_set_nat @ X3 @ C )
=> ( ( member_nat @ A2 @ X3 )
=> ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_54_UnionI,axiom,
! [X3: set_o,C: set_set_o,A2: $o] :
( ( member_set_o @ X3 @ C )
=> ( ( member_o @ A2 @ X3 )
=> ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_55_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( P @ X ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ X )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_56_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ A2 ) )
=> ( P @ X ) ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ! [Y: $o] :
( ( member_o @ Y @ X )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_57_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) )
& ( P @ X ) ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ? [Y: nat] :
( ( member_nat @ Y @ X )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_58_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ? [X: $o] :
( ( member_o @ X @ ( comple90263536869209701_set_o @ A2 ) )
& ( P @ X ) ) )
= ( ? [X: set_o] :
( ( member_set_o @ X @ A2 )
& ? [Y: $o] :
( ( member_o @ Y @ X )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_59_lessThan__eq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X2 = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_60_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_61_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_62_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_63_rev__image__eqI,axiom,
! [X2: $o,A2: set_o,B: set_nat,F: $o > set_nat] :
( ( member_o @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_o_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_64_rev__image__eqI,axiom,
! [X2: $o,A2: set_o,B: set_o,F: $o > set_o] :
( ( member_o @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_65_rev__image__eqI,axiom,
! [X2: $o,A2: set_o,B: $o,F: $o > $o] :
( ( member_o @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_66_rev__image__eqI,axiom,
! [X2: $o,A2: set_o,B: nat,F: $o > nat] :
( ( member_o @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_67_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_68_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_69_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_70_ball__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_71_ball__imageD,axiom,
! [F: $o > set_nat,A2: set_o,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_o_set_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_72_ball__imageD,axiom,
! [F: $o > set_o,A2: set_o,P: set_o > $o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ ( image_o_set_o @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_73_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_74_ball__imageD,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ ( image_o_o @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_75_ball__imageD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_76_image__cong,axiom,
! [M: set_set_nat,N: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( M = N )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_7916887816326733075et_nat @ F @ M )
= ( image_7916887816326733075et_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_77_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( M = N )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_o_set_nat @ F @ M )
= ( image_o_set_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_78_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > set_o,G: $o > set_o] :
( ( M = N )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_o_set_o @ F @ M )
= ( image_o_set_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_79_image__cong,axiom,
! [M: set_o,N: set_o,F: $o > $o,G: $o > $o] :
( ( M = N )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_o_o @ F @ M )
= ( image_o_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_80_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M = N )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_set_nat @ F @ M )
= ( image_nat_set_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_81_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_82_bex__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_83_bex__imageD,axiom,
! [F: $o > set_nat,A2: set_o,P: set_nat > $o] :
( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_o_set_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_84_bex__imageD,axiom,
! [F: $o > set_o,A2: set_o,P: set_o > $o] :
( ? [X5: set_o] :
( ( member_set_o @ X5 @ ( image_o_set_o @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_85_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_86_bex__imageD,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ? [X5: $o] :
( ( member_o @ X5 @ ( image_o_o @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_87_bex__imageD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_nat > $o] :
( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_88_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_89_image__iff,axiom,
! [Z: set_nat,F: $o > set_nat,A2: set_o] :
( ( member_set_nat @ Z @ ( image_o_set_nat @ F @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_90_image__iff,axiom,
! [Z: set_o,F: $o > set_o,A2: set_o] :
( ( member_set_o @ Z @ ( image_o_set_o @ F @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_91_image__iff,axiom,
! [Z: set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( member_set_nat @ Z @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_92_image__iff,axiom,
! [Z: $o,F: $o > $o,A2: set_o] :
( ( member_o @ Z @ ( image_o_o @ F @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_93_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_94_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_95_imageI,axiom,
! [X2: $o,A2: set_o,F: $o > set_nat] :
( ( member_o @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_o_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_96_imageI,axiom,
! [X2: $o,A2: set_o,F: $o > set_o] :
( ( member_o @ X2 @ A2 )
=> ( member_set_o @ ( F @ X2 ) @ ( image_o_set_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_97_imageI,axiom,
! [X2: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_98_imageI,axiom,
! [X2: $o,A2: set_o,F: $o > nat] :
( ( member_o @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_o_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_99_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_100_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_101_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_102_Sup_OSUP__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > set_nat,D: set_nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_7916887816326733075et_nat @ C @ A2 ) )
= ( Sup @ ( image_7916887816326733075et_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_103_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_nat,D: $o > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_o_set_nat @ C @ A2 ) )
= ( Sup @ ( image_o_set_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_104_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o,Sup: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_o_set_o @ C @ A2 ) )
= ( Sup @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_105_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o,Sup: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_o_o @ C @ A2 ) )
= ( Sup @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_106_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_107_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_108_Inf_OINF__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > set_nat,D: set_nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_7916887816326733075et_nat @ C @ A2 ) )
= ( Inf @ ( image_7916887816326733075et_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_109_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_nat,D: $o > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_o_set_nat @ C @ A2 ) )
= ( Inf @ ( image_o_set_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_110_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o,Inf: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_o_set_o @ C @ A2 ) )
= ( Inf @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_111_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o,Inf: set_o > $o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_o_o @ C @ A2 ) )
= ( Inf @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_112_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_113_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_114_UnionE,axiom,
! [A2: nat,C: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A2 @ X6 )
=> ~ ( member_set_nat @ X6 @ C ) ) ) ).
% UnionE
thf(fact_115_UnionE,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
=> ~ ! [X6: set_o] :
( ( member_o @ A2 @ X6 )
=> ~ ( member_set_o @ X6 @ C ) ) ) ).
% UnionE
thf(fact_116_Compr__image__eq,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( image_7916887816326733075et_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_7916887816326733075et_nat @ F
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_117_Compr__image__eq,axiom,
! [F: $o > set_nat,A2: set_o,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( image_o_set_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_set_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_118_Compr__image__eq,axiom,
! [F: $o > set_o,A2: set_o,P: set_o > $o] :
( ( collect_set_o
@ ^ [X: set_o] :
( ( member_set_o @ X @ ( image_o_set_o @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_set_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_119_Compr__image__eq,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_120_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_121_Compr__image__eq,axiom,
! [F: nat > $o,A2: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_122_Compr__image__eq,axiom,
! [F: $o > nat,A2: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_o_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_123_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_124_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_125_image__image,axiom,
! [F: $o > $o,G: $o > $o,A2: set_o] :
( ( image_o_o @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_126_image__image,axiom,
! [F: set_nat > nat,G: nat > set_nat,A2: set_nat] :
( ( image_set_nat_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_127_image__image,axiom,
! [F: set_nat > $o,G: $o > set_nat,A2: set_o] :
( ( image_set_nat_o @ F @ ( image_o_set_nat @ G @ A2 ) )
= ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_128_image__image,axiom,
! [F: set_o > $o,G: $o > set_o,A2: set_o] :
( ( image_set_o_o @ F @ ( image_o_set_o @ G @ A2 ) )
= ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_129_image__image,axiom,
! [F: nat > set_nat,G: $o > nat,A2: set_o] :
( ( image_nat_set_nat @ F @ ( image_o_nat @ G @ A2 ) )
= ( image_o_set_nat
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_130_image__image,axiom,
! [F: nat > set_nat,G: nat > nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_131_image__image,axiom,
! [F: $o > set_nat,G: nat > $o,A2: set_nat] :
( ( image_o_set_nat @ F @ ( image_nat_o @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_132_image__image,axiom,
! [F: $o > set_nat,G: $o > $o,A2: set_o] :
( ( image_o_set_nat @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_set_nat
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_133_image__image,axiom,
! [F: $o > set_o,G: $o > $o,A2: set_o] :
( ( image_o_set_o @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_set_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_134_imageE,axiom,
! [B: set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [X4: set_nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_set_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_135_imageE,axiom,
! [B: set_nat,F: $o > set_nat,A2: set_o] :
( ( member_set_nat @ B @ ( image_o_set_nat @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_136_imageE,axiom,
! [B: set_o,F: $o > set_o,A2: set_o] :
( ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_137_imageE,axiom,
! [B: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_138_imageE,axiom,
! [B: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_139_imageE,axiom,
! [B: $o,F: nat > $o,A2: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_140_imageE,axiom,
! [B: nat,F: $o > nat,A2: set_o] :
( ( member_nat @ B @ ( image_o_nat @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_141_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_142_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_143_Sup_OSUP__identity__eq,axiom,
! [Sup: set_o > $o,A2: set_o] :
( ( Sup
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_144_Sup_OSUP__identity__eq,axiom,
! [Sup: set_set_nat > set_nat,A2: set_set_nat] :
( ( Sup
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_145_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_146_Inf_OINF__identity__eq,axiom,
! [Inf: set_o > $o,A2: set_o] :
( ( Inf
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_147_Inf_OINF__identity__eq,axiom,
! [Inf: set_set_nat > set_nat,A2: set_set_nat] :
( ( Inf
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_148_SUP__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > set_nat,D: set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_149_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_nat,D: $o > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_150_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_151_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_152_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > $o,D: nat > $o] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_153_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > nat,D: $o > nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_154_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_155_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_156_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_o,D: nat > set_o] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_157_SUP__commute,axiom,
! [F: $o > $o > $o,B2: set_o,A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [J: $o] :
( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_158_SUP__commute,axiom,
! [F: nat > nat > set_nat,B2: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_159_SUP__commute,axiom,
! [F: nat > $o > set_nat,B2: set_o,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [J: $o] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_160_SUP__commute,axiom,
! [F: $o > nat > set_nat,B2: set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_161_SUP__commute,axiom,
! [F: $o > $o > set_nat,B2: set_o,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [J: $o] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_162_SUP__commute,axiom,
! [F: $o > $o > set_o,B2: set_o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [I: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [J: $o] :
( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_163_SUP__commute,axiom,
! [F: nat > set_nat > set_nat,B2: set_set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [J: set_nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_164_SUP__commute,axiom,
! [F: $o > set_nat > set_nat,B2: set_set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [J: set_nat] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_165_SUP__commute,axiom,
! [F: set_nat > nat > set_nat,B2: set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_166_SUP__commute,axiom,
! [F: set_nat > $o > set_nat,B2: set_o,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [J: $o] :
( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_167_image__Union,axiom,
! [F: set_nat > set_nat,S: set_set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( comple548664676211718543et_nat @ S ) )
= ( comple548664676211718543et_nat @ ( image_7884819252390400639et_nat @ ( image_7916887816326733075et_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_168_image__Union,axiom,
! [F: nat > set_nat,S: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_169_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_170_image__Union,axiom,
! [F: nat > $o,S: set_set_nat] :
( ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ ( image_nat_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_171_image__Union,axiom,
! [F: $o > set_nat,S: set_set_o] :
( ( image_o_set_nat @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple548664676211718543et_nat @ ( image_7698617416147310703et_nat @ ( image_o_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_172_image__Union,axiom,
! [F: $o > set_o,S: set_set_o] :
( ( image_o_set_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ ( image_o_set_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_173_image__Union,axiom,
! [F: $o > nat,S: set_set_o] :
( ( image_o_nat @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple7399068483239264473et_nat @ ( image_set_o_set_nat @ ( image_o_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_174_image__Union,axiom,
! [F: $o > $o,S: set_set_o] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_175_UN__UN__flatten,axiom,
! [C: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_176_UN__UN__flatten,axiom,
! [C: nat > set_nat,B2: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_177_UN__UN__flatten,axiom,
! [C: $o > set_nat,B2: nat > set_o,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_178_UN__UN__flatten,axiom,
! [C: $o > set_nat,B2: $o > set_o,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_179_UN__UN__flatten,axiom,
! [C: nat > set_o,B2: nat > set_nat,A2: set_nat] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y: nat] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_180_UN__UN__flatten,axiom,
! [C: nat > set_o,B2: $o > set_nat,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_181_UN__UN__flatten,axiom,
! [C: $o > set_o,B2: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_182_UN__UN__flatten,axiom,
! [C: set_nat > set_nat,B2: nat > set_set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_183_UN__UN__flatten,axiom,
! [C: set_nat > set_nat,B2: $o > set_set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_184_UN__UN__flatten,axiom,
! [C: nat > set_nat,B2: set_nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_185_UN__E,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_186_UN__E,axiom,
! [B: nat,B2: $o > set_nat,A2: set_o] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_187_UN__E,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_188_UN__E,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_189_UN__E,axiom,
! [B: $o,B2: nat > set_o,A2: set_nat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_190_UN__extend__simps_I8_J,axiom,
! [B2: set_nat > set_nat,A2: set_set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_5842784325960735177et_nat
@ ^ [Y: set_set_nat] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ Y ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ ( comple548664676211718543et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_191_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ Y ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_192_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_nat,A2: set_set_o] :
( ( comple7399068483239264473et_nat
@ ( image_set_o_set_nat
@ ^ [Y: set_o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ Y ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_193_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_o,A2: set_set_nat] :
( ( comple90263536869209701_set_o
@ ( image_set_nat_set_o
@ ^ [Y: set_nat] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ Y ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_194_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_o,A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [Y: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ Y ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_195_UN__extend__simps_I9_J,axiom,
! [C: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_196_UN__extend__simps_I9_J,axiom,
! [C: nat > set_nat,B2: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_197_UN__extend__simps_I9_J,axiom,
! [C: $o > set_nat,B2: nat > set_o,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_198_UN__extend__simps_I9_J,axiom,
! [C: $o > set_nat,B2: $o > set_o,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_199_UN__extend__simps_I9_J,axiom,
! [C: nat > set_o,B2: nat > set_nat,A2: set_nat] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X: nat] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_200_UN__extend__simps_I9_J,axiom,
! [C: nat > set_o,B2: $o > set_nat,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ C @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_201_UN__extend__simps_I9_J,axiom,
! [C: $o > set_o,B2: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_202_UN__extend__simps_I9_J,axiom,
! [C: set_nat > set_nat,B2: nat > set_set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_203_UN__extend__simps_I9_J,axiom,
! [C: set_nat > set_nat,B2: $o > set_set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_204_UN__extend__simps_I9_J,axiom,
! [C: nat > set_nat,B2: set_nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_205_image__UN,axiom,
! [F: nat > nat,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( image_nat_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_206_image__UN,axiom,
! [F: nat > nat,B2: $o > set_nat,A2: set_o] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( image_nat_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_207_image__UN,axiom,
! [F: nat > $o,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X: nat] : ( image_nat_o @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_208_image__UN,axiom,
! [F: nat > $o,B2: $o > set_nat,A2: set_o] :
( ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( image_nat_o @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_209_image__UN,axiom,
! [F: $o > nat,B2: nat > set_o,A2: set_nat] :
( ( image_o_nat @ F @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( image_o_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_210_image__UN,axiom,
! [F: $o > nat,B2: $o > set_o,A2: set_o] :
( ( image_o_nat @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( image_o_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_211_image__UN,axiom,
! [F: $o > $o,B2: $o > set_o,A2: set_o] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( image_o_o @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_212_image__UN,axiom,
! [F: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X: nat] : ( image_nat_set_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_213_image__UN,axiom,
! [F: nat > set_nat,B2: $o > set_nat,A2: set_o] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_o_set_set_nat
@ ^ [X: $o] : ( image_nat_set_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_214_image__UN,axiom,
! [F: nat > nat,B2: set_nat > set_nat,A2: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : ( image_nat_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_215_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_216_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_nat,F: nat > $o,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_217_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: $o > nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_218_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_nat,F: $o > $o,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_219_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_o,F: nat > nat,A2: set_nat] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_220_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_o,F: $o > $o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_221_UN__extend__simps_I10_J,axiom,
! [B2: set_nat > set_nat,F: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_222_UN__extend__simps_I10_J,axiom,
! [B2: set_o > set_nat,F: $o > set_o,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_set_o_set_nat @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_223_UN__extend__simps_I10_J,axiom,
! [B2: set_nat > set_nat,F: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A4: $o] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ ( image_o_set_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_224_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [A4: set_nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_225_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_226_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_227_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_228_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_229_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_230_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_231_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_232_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_233_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_234_mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_235_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_236_mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_237_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X2: real,B: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_238_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X2: real,Y3: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y3 ) )
= ( ( X2 = Y3 )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_239_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_240_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_241_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X2: real] :
( ( ( times_times_real @ A @ X2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_242_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_243_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_244_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_245_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_246_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_247_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_248_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_249_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_250_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_251_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_252_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_253_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_254_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_255_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_256_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_257_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X2: real,Y3: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y3 ) )
=> ( X2 = Y3 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_258_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_259_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_260_mult__left__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_261_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_262_mult__right__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_263_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_264_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_265_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_266_set__times__intro,axiom,
! [A: nat,C: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_267_set__times__intro,axiom,
! [A: real,C: set_real,B: real,D: set_real] :
( ( member_real @ A @ C )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_268_set__times__UNION__distrib_I2_J,axiom,
! [M: nat > set_nat,I2: set_nat,A2: set_nat] :
( ( times_times_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ I2 ) ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( times_times_set_nat @ ( M @ I ) @ A2 )
@ I2 ) ) ) ).
% set_times_UNION_distrib(2)
thf(fact_269_set__times__UNION__distrib_I2_J,axiom,
! [M: $o > set_nat,I2: set_o,A2: set_nat] :
( ( times_times_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ M @ I2 ) ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( times_times_set_nat @ ( M @ I ) @ A2 )
@ I2 ) ) ) ).
% set_times_UNION_distrib(2)
thf(fact_270_set__times__UNION__distrib_I2_J,axiom,
! [M: set_nat > set_nat,I2: set_set_nat,A2: set_nat] :
( ( times_times_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ M @ I2 ) ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( times_times_set_nat @ ( M @ I ) @ A2 )
@ I2 ) ) ) ).
% set_times_UNION_distrib(2)
thf(fact_271_set__times__UNION__distrib_I1_J,axiom,
! [A2: set_nat,M: nat > set_nat,I2: set_nat] :
( ( times_times_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ I2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( times_times_set_nat @ A2 @ ( M @ I ) )
@ I2 ) ) ) ).
% set_times_UNION_distrib(1)
thf(fact_272_set__times__UNION__distrib_I1_J,axiom,
! [A2: set_nat,M: $o > set_nat,I2: set_o] :
( ( times_times_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ M @ I2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( times_times_set_nat @ A2 @ ( M @ I ) )
@ I2 ) ) ) ).
% set_times_UNION_distrib(1)
thf(fact_273_set__times__UNION__distrib_I1_J,axiom,
! [A2: set_nat,M: set_nat > set_nat,I2: set_set_nat] :
( ( times_times_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ M @ I2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( times_times_set_nat @ A2 @ ( M @ I ) )
@ I2 ) ) ) ).
% set_times_UNION_distrib(1)
thf(fact_274_UN__constant__eq,axiom,
! [A: set_nat,A2: set_set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_275_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_nat,C2: set_nat] :
( ( member_o @ A @ A2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_276_UN__constant__eq,axiom,
! [A: nat,A2: set_nat,F: nat > set_nat,C2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_277_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_o,C2: set_o] :
( ( member_o @ A @ A2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_278_UN__constant__eq,axiom,
! [A: nat,A2: set_nat,F: nat > set_o,C2: set_o] :
( ( member_nat @ A @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_279_SUP__UN__eq,axiom,
! [R: nat > set_nat,S: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o
@ ^ [I: nat,X: nat] : ( member_nat @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_280_SUP__UN__eq,axiom,
! [R: $o > set_nat,S: set_o] :
( ( comple8317665133742190828_nat_o
@ ( image_o_nat_o
@ ^ [I: $o,X: nat] : ( member_nat @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_281_SUP__UN__eq,axiom,
! [R: set_nat > set_nat,S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I: set_nat,X: nat] : ( member_nat @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_282_SUP__UN__eq,axiom,
! [R: $o > set_o,S: set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_o_o_o
@ ^ [I: $o,X: $o] : ( member_o @ X @ ( R @ I ) )
@ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_set_o @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_283_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_284_mult__delta__left,axiom,
! [B: $o,X2: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y3 )
= ( times_times_nat @ X2 @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y3 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_285_mult__delta__left,axiom,
! [B: $o,X2: real,Y3: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y3 )
= ( times_times_real @ X2 @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y3 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_286_mult__delta__right,axiom,
! [B: $o,X2: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_287_mult__delta__right,axiom,
! [B: $o,X2: real,Y3: real] :
( ( B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= ( times_times_real @ X2 @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_288_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_289_SUP__Sup__eq,axiom,
! [S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I: set_nat,X: nat] : ( member_nat @ X @ I )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_290_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X: $o] : ( member_o @ X @ I )
@ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_291_UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_I
thf(fact_292_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_293_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_294_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_295_Sup__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Sup_UNIV
thf(fact_296_UNIV__witness,axiom,
? [X4: $o] : ( member_o @ X4 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_297_UNIV__witness,axiom,
? [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_298_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X4: $o] : ( member_o @ X4 @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_299_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X4: nat] : ( member_nat @ X4 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_300_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_301_Union__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Union_UNIV
thf(fact_302_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_303_range__eqI,axiom,
! [B: set_nat,F: $o > set_nat,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_o_set_nat @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_304_range__eqI,axiom,
! [B: set_o,F: $o > set_o,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_set_o @ B @ ( image_o_set_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_305_range__eqI,axiom,
! [B: set_nat,F: set_nat > set_nat,X2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) ) ) ).
% range_eqI
thf(fact_306_range__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_o_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_307_range__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_308_range__eqI,axiom,
! [B: $o,F: nat > $o,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_309_range__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_310_rangeI,axiom,
! [F: $o > set_nat,X2: $o] : ( member_set_nat @ ( F @ X2 ) @ ( image_o_set_nat @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_311_rangeI,axiom,
! [F: $o > set_o,X2: $o] : ( member_set_o @ ( F @ X2 ) @ ( image_o_set_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_312_rangeI,axiom,
! [F: set_nat > set_nat,X2: set_nat] : ( member_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) ) ).
% rangeI
thf(fact_313_rangeI,axiom,
! [F: $o > $o,X2: $o] : ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_314_rangeI,axiom,
! [F: nat > set_nat,X2: nat] : ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_315_rangeI,axiom,
! [F: nat > $o,X2: nat] : ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_316_rangeI,axiom,
! [F: nat > nat,X2: nat] : ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_317_range__composition,axiom,
! [F: $o > $o,G: $o > $o] :
( ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_o_o @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_318_range__composition,axiom,
! [F: $o > $o,G: nat > $o] :
( ( image_nat_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_o_o @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_319_range__composition,axiom,
! [F: nat > nat,G: nat > nat] :
( ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_320_range__composition,axiom,
! [F: nat > set_nat,G: $o > nat] :
( ( image_o_set_nat
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_nat_set_nat @ F @ ( image_o_nat @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_321_range__composition,axiom,
! [F: $o > set_nat,G: $o > $o] :
( ( image_o_set_nat
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_o_set_nat @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_322_range__composition,axiom,
! [F: $o > set_o,G: $o > $o] :
( ( image_o_set_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_o_set_o @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_323_range__composition,axiom,
! [F: set_nat > $o,G: $o > set_nat] :
( ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_set_nat_o @ F @ ( image_o_set_nat @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_324_range__composition,axiom,
! [F: set_o > $o,G: $o > set_o] :
( ( image_o_o
@ ^ [X: $o] : ( F @ ( G @ X ) )
@ top_top_set_o )
= ( image_set_o_o @ F @ ( image_o_set_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_325_range__composition,axiom,
! [F: $o > set_o,G: nat > $o] :
( ( image_nat_set_o
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_o_set_o @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_326_range__composition,axiom,
! [F: nat > set_nat,G: nat > nat] :
( ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_327_rangeE,axiom,
! [B: set_nat,F: $o > set_nat] :
( ( member_set_nat @ B @ ( image_o_set_nat @ F @ top_top_set_o ) )
=> ~ ! [X4: $o] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_328_rangeE,axiom,
! [B: set_o,F: $o > set_o] :
( ( member_set_o @ B @ ( image_o_set_o @ F @ top_top_set_o ) )
=> ~ ! [X4: $o] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_329_rangeE,axiom,
! [B: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) )
=> ~ ! [X4: set_nat] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_330_rangeE,axiom,
! [B: $o,F: $o > $o] :
( ( member_o @ B @ ( image_o_o @ F @ top_top_set_o ) )
=> ~ ! [X4: $o] :
( B
= ( ~ ( F @ X4 ) ) ) ) ).
% rangeE
thf(fact_331_rangeE,axiom,
! [B: set_nat,F: nat > set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ top_top_set_nat ) )
=> ~ ! [X4: nat] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_332_rangeE,axiom,
! [B: $o,F: nat > $o] :
( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X4: nat] :
( B
= ( ~ ( F @ X4 ) ) ) ) ).
% rangeE
thf(fact_333_rangeE,axiom,
! [B: nat,F: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X4: nat] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_334_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_335_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_336_set__times__elim,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X2 @ ( times_times_set_nat @ A2 @ B2 ) )
=> ~ ! [A5: nat,B3: nat] :
( ( X2
= ( times_times_nat @ A5 @ B3 ) )
=> ( ( member_nat @ A5 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_337_set__times__elim,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ( member_real @ X2 @ ( times_times_set_real @ A2 @ B2 ) )
=> ~ ! [A5: real,B3: real] :
( ( X2
= ( times_times_real @ A5 @ B3 ) )
=> ( ( member_real @ A5 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_338_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_339_mult_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_340_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_341_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_342_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_343_mult_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_344_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_345_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_346_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_347_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S2: set_o_o,X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_348_iso__tuple__UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_349_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_350_surj__def,axiom,
! [F: $o > set_nat] :
( ( ( image_o_set_nat @ F @ top_top_set_o )
= top_top_set_set_nat )
= ( ! [Y: set_nat] :
? [X: $o] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_351_surj__def,axiom,
! [F: $o > set_o] :
( ( ( image_o_set_o @ F @ top_top_set_o )
= top_top_set_set_o )
= ( ! [Y: set_o] :
? [X: $o] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_352_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y: $o] :
? [X: $o] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_353_surj__def,axiom,
! [F: set_nat > set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
= top_top_set_set_nat )
= ( ! [Y: set_nat] :
? [X: set_nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_354_surj__def,axiom,
! [F: nat > set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
= ( ! [Y: set_nat] :
? [X: nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_355_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X: nat] :
( Y
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_356_surjI,axiom,
! [G: $o > set_nat,F: set_nat > $o] :
( ! [X4: set_nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_o_set_nat @ G @ top_top_set_o )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_357_surjI,axiom,
! [G: $o > set_o,F: set_o > $o] :
( ! [X4: set_o] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_o_set_o @ G @ top_top_set_o )
= top_top_set_set_o ) ) ).
% surjI
thf(fact_358_surjI,axiom,
! [G: $o > $o,F: $o > $o] :
( ! [X4: $o] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_359_surjI,axiom,
! [G: set_nat > set_nat,F: set_nat > set_nat] :
( ! [X4: set_nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_7916887816326733075et_nat @ G @ top_top_set_set_nat )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_360_surjI,axiom,
! [G: nat > set_nat,F: set_nat > nat] :
( ! [X4: set_nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_nat_set_nat @ G @ top_top_set_nat )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_361_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_362_surjE,axiom,
! [F: $o > set_nat,Y3: set_nat] :
( ( ( image_o_set_nat @ F @ top_top_set_o )
= top_top_set_set_nat )
=> ~ ! [X4: $o] :
( Y3
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_363_surjE,axiom,
! [F: $o > set_o,Y3: set_o] :
( ( ( image_o_set_o @ F @ top_top_set_o )
= top_top_set_set_o )
=> ~ ! [X4: $o] :
( Y3
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_364_surjE,axiom,
! [F: $o > $o,Y3: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X4: $o] :
( Y3
= ( ~ ( F @ X4 ) ) ) ) ).
% surjE
thf(fact_365_surjE,axiom,
! [F: set_nat > set_nat,Y3: set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
= top_top_set_set_nat )
=> ~ ! [X4: set_nat] :
( Y3
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_366_surjE,axiom,
! [F: nat > set_nat,Y3: set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ~ ! [X4: nat] :
( Y3
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_367_surjE,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X4: nat] :
( Y3
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_368_surjD,axiom,
! [F: $o > set_nat,Y3: set_nat] :
( ( ( image_o_set_nat @ F @ top_top_set_o )
= top_top_set_set_nat )
=> ? [X4: $o] :
( Y3
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_369_surjD,axiom,
! [F: $o > set_o,Y3: set_o] :
( ( ( image_o_set_o @ F @ top_top_set_o )
= top_top_set_set_o )
=> ? [X4: $o] :
( Y3
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_370_surjD,axiom,
! [F: $o > $o,Y3: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X4: $o] :
( Y3
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_371_surjD,axiom,
! [F: set_nat > set_nat,Y3: set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
= top_top_set_set_nat )
=> ? [X4: set_nat] :
( Y3
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_372_surjD,axiom,
! [F: nat > set_nat,Y3: set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ? [X4: nat] :
( Y3
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_373_surjD,axiom,
! [F: nat > nat,Y3: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X4: nat] :
( Y3
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_374_UN__atLeast__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atLeast_UNIV
thf(fact_375_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_376_UN__atMost__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atMost_UNIV
thf(fact_377_atMost__eq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atMost_eq_iff
thf(fact_378_atLeast__eq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ( set_ord_atLeast_nat @ X2 )
= ( set_ord_atLeast_nat @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atLeast_eq_iff
thf(fact_379_Sup__atMost,axiom,
! [Y3: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y3 ) )
= Y3 ) ).
% Sup_atMost
thf(fact_380_Sup__atMost,axiom,
! [Y3: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ Y3 ) )
= Y3 ) ).
% Sup_atMost
thf(fact_381_Sup__atMost,axiom,
! [Y3: set_o] :
( ( comple90263536869209701_set_o @ ( set_ord_atMost_set_o @ Y3 ) )
= Y3 ) ).
% Sup_atMost
thf(fact_382_cSup__atMost,axiom,
! [X2: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ X2 ) )
= X2 ) ).
% cSup_atMost
thf(fact_383_cSup__atMost,axiom,
! [X2: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ X2 ) )
= X2 ) ).
% cSup_atMost
thf(fact_384_cSup__atMost,axiom,
! [X2: nat] :
( ( complete_Sup_Sup_nat @ ( set_ord_atMost_nat @ X2 ) )
= X2 ) ).
% cSup_atMost
thf(fact_385_cSup__atMost,axiom,
! [X2: set_o] :
( ( comple90263536869209701_set_o @ ( set_ord_atMost_set_o @ X2 ) )
= X2 ) ).
% cSup_atMost
thf(fact_386_Sup__atLeast,axiom,
! [X2: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or1731685050470061051et_nat @ X2 ) )
= top_top_set_nat ) ).
% Sup_atLeast
thf(fact_387_Sup__atLeast,axiom,
! [X2: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atLeast_o @ X2 ) )
= top_top_o ) ).
% Sup_atLeast
thf(fact_388_Sup__atLeast,axiom,
! [X2: set_o] :
( ( comple90263536869209701_set_o @ ( set_or8686861255860958915_set_o @ X2 ) )
= top_top_set_o ) ).
% Sup_atLeast
thf(fact_389_atLeast__0,axiom,
( ( set_ord_atLeast_nat @ zero_zero_nat )
= top_top_set_nat ) ).
% atLeast_0
thf(fact_390_not__Iic__eq__Ici,axiom,
! [H2: nat,L: nat] :
( ( set_ord_atMost_nat @ H2 )
!= ( set_ord_atLeast_nat @ L ) ) ).
% not_Iic_eq_Ici
thf(fact_391_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_392_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_393_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_394_not__UNIV__eq__Iic,axiom,
! [H3: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H3 ) ) ).
% not_UNIV_eq_Iic
thf(fact_395_atMost__eq__UNIV__iff,axiom,
! [X2: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X2 )
= top_top_set_set_nat )
= ( X2 = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_396_A__def,axiom,
( a
= ( ^ [I: nat] : ( set_or4665077453230672383an_nat @ ( times_times_nat @ I @ m ) @ ( times_times_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ m ) ) ) ) ).
% A_def
thf(fact_397_SUP__bool__eq,axiom,
( ( ^ [A3: set_o,F2: $o > $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F2 @ A3 ) ) )
= bex_o ) ).
% SUP_bool_eq
thf(fact_398_mult__commute__abs,axiom,
! [C2: nat] :
( ( ^ [X: nat] : ( times_times_nat @ X @ C2 ) )
= ( times_times_nat @ C2 ) ) ).
% mult_commute_abs
thf(fact_399_mult__commute__abs,axiom,
! [C2: real] :
( ( ^ [X: real] : ( times_times_real @ X @ C2 ) )
= ( times_times_real @ C2 ) ) ).
% mult_commute_abs
thf(fact_400_UN__UN__finite__eq,axiom,
! [A2: nat > set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [N3: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
@ top_top_set_nat ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) ) ) ).
% UN_UN_finite_eq
thf(fact_401_UN__UN__finite__eq,axiom,
! [A2: nat > set_o] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [N3: nat] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
@ top_top_set_nat ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ top_top_set_nat ) ) ) ).
% UN_UN_finite_eq
thf(fact_402_image__minus__const__AtMost,axiom,
! [C2: real,B: real] :
( ( image_real_real @ ( minus_minus_real @ C2 ) @ ( set_ord_atMost_real @ B ) )
= ( set_ord_atLeast_real @ ( minus_minus_real @ C2 @ B ) ) ) ).
% image_minus_const_AtMost
thf(fact_403_image__minus__const__atLeast,axiom,
! [C2: real,A: real] :
( ( image_real_real @ ( minus_minus_real @ C2 ) @ ( set_ord_atLeast_real @ A ) )
= ( set_ord_atMost_real @ ( minus_minus_real @ C2 @ A ) ) ) ).
% image_minus_const_atLeast
thf(fact_404_set__plus__intro,axiom,
! [A: nat,C: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_405_set__plus__intro,axiom,
! [A: real,C: set_real,B: real,D: set_real] :
( ( member_real @ A @ C )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_406_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_407_add__right__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_408_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_409_add__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_410_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_411_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_412_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_413_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_414_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_415_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_416_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_417_diff__diff__left,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_418_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_419_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_420_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_421_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_422_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_423_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_424_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_425_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_426_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_427_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_428_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_429_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_430_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_431_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_432_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_433_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_434_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_435_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_436_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_437_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_438_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_439_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_440_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_441_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_442_add__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_443_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_444_add__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_445_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_446_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_447_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_448_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_449_vector__space__over__itself_Oscale__one,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_450_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_451_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_452_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_453_add__diff__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_454_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_455_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_456_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_457_add__diff__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_458_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_459_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_460_lessThan__iff,axiom,
! [I3: $o,K: $o] :
( ( member_o @ I3 @ ( set_ord_lessThan_o @ K ) )
= ( ord_less_o @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_461_lessThan__iff,axiom,
! [I3: real,K: real] :
( ( member_real @ I3 @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_462_lessThan__iff,axiom,
! [I3: nat,K: nat] :
( ( member_nat @ I3 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I3 @ K ) ) ).
% lessThan_iff
thf(fact_463_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_464_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_465_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_466_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_467_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_468_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_469_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_470_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_471_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_472_lessThan__minus__lessThan,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( set_ord_lessThan_nat @ M2 ) )
= ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ).
% lessThan_minus_lessThan
thf(fact_473_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_474_True__in__image__Bex,axiom,
! [P: $o > $o,A2: set_o] :
( ( member_o @ $true @ ( image_o_o @ P @ A2 ) )
= ( ? [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ X ) ) ) ) ).
% True_in_image_Bex
thf(fact_475_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_476_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_477_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_478_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_479_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_480_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_481_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_482_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_483_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_484_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_485_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_486_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_487_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_488_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_489_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_490_image__add__0,axiom,
! [S: set_set_nat] :
( ( image_7916887816326733075et_nat @ ( plus_plus_set_nat @ zero_zero_set_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_491_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_492_image__add__0,axiom,
! [S: set_real] :
( ( image_real_real @ ( plus_plus_real @ zero_zero_real ) @ S )
= S ) ).
% image_add_0
thf(fact_493_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_494_image__add__atLeastLessThan,axiom,
! [K: real,I3: real,J2: real] :
( ( image_real_real @ ( plus_plus_real @ K ) @ ( set_or66887138388493659n_real @ I3 @ J2 ) )
= ( set_or66887138388493659n_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J2 @ K ) ) ) ).
% image_add_atLeastLessThan
thf(fact_495_image__add__atLeastLessThan,axiom,
! [K: nat,I3: nat,J2: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I3 @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% image_add_atLeastLessThan
thf(fact_496_image__add__atMost,axiom,
! [C2: real,A: real] :
( ( image_real_real @ ( plus_plus_real @ C2 ) @ ( set_ord_atMost_real @ A ) )
= ( set_ord_atMost_real @ ( plus_plus_real @ C2 @ A ) ) ) ).
% image_add_atMost
thf(fact_497_cSup__atLeastLessThan,axiom,
! [Y3: real,X2: real] :
( ( ord_less_real @ Y3 @ X2 )
=> ( ( comple1385675409528146559p_real @ ( set_or66887138388493659n_real @ Y3 @ X2 ) )
= X2 ) ) ).
% cSup_atLeastLessThan
thf(fact_498_image__add__atLeast,axiom,
! [K: real,I3: real] :
( ( image_real_real @ ( plus_plus_real @ K ) @ ( set_ord_atLeast_real @ I3 ) )
= ( set_ord_atLeast_real @ ( plus_plus_real @ K @ I3 ) ) ) ).
% image_add_atLeast
thf(fact_499_image__add__atLeast,axiom,
! [K: nat,I3: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_ord_atLeast_nat @ I3 ) )
= ( set_ord_atLeast_nat @ ( plus_plus_nat @ K @ I3 ) ) ) ).
% image_add_atLeast
thf(fact_500_surj__plus__right,axiom,
! [A: real] :
( ( image_real_real
@ ^ [B4: real] : ( plus_plus_real @ B4 @ A )
@ top_top_set_real )
= top_top_set_real ) ).
% surj_plus_right
thf(fact_501_surj__diff__right,axiom,
! [A: real] :
( ( image_real_real
@ ^ [X: real] : ( minus_minus_real @ X @ A )
@ top_top_set_real )
= top_top_set_real ) ).
% surj_diff_right
thf(fact_502_image__add__atLeastLessThan_H,axiom,
! [K: real,I3: real,J2: real] :
( ( image_real_real
@ ^ [N3: real] : ( plus_plus_real @ N3 @ K )
@ ( set_or66887138388493659n_real @ I3 @ J2 ) )
= ( set_or66887138388493659n_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J2 @ K ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_503_image__add__atLeastLessThan_H,axiom,
! [K: nat,I3: nat,J2: nat] :
( ( image_nat_nat
@ ^ [N3: nat] : ( plus_plus_nat @ N3 @ K )
@ ( set_or4665077453230672383an_nat @ I3 @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_504_translation__diff,axiom,
! [A: real,S3: set_real,T: set_real] :
( ( image_real_real @ ( plus_plus_real @ A ) @ ( minus_minus_set_real @ S3 @ T ) )
= ( minus_minus_set_real @ ( image_real_real @ ( plus_plus_real @ A ) @ S3 ) @ ( image_real_real @ ( plus_plus_real @ A ) @ T ) ) ) ).
% translation_diff
thf(fact_505_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_506_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_507_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_508_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_509_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_510_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_511_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_512_pos__add__strict,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_513_add__less__zeroD,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_514_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_515_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_516_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_517_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_518_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_519_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_520_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I3 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_521_order__less__imp__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_522_order__less__imp__not__less,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_523_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_524_order__less__imp__not__eq2,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_525_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_526_order__less__imp__not__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_527_linorder__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_528_linorder__less__linear,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_real @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_529_order__less__imp__triv,axiom,
! [X2: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_530_order__less__imp__triv,axiom,
! [X2: real,Y3: real,P: $o] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ( ord_less_real @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_531_order__less__not__sym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_532_order__less__not__sym,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_533_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_534_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_535_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_536_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_537_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_538_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_539_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_540_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_541_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_542_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_543_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_544_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_545_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_546_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_547_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_548_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_549_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_550_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_551_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_552_order__less__trans,axiom,
! [X2: real,Y3: real,Z: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_553_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_554_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_555_linorder__neq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_556_linorder__neq__iff,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
= ( ( ord_less_real @ X2 @ Y3 )
| ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_557_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_558_order__less__asym,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_559_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_560_linorder__neqE,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_561_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_562_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_563_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_564_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_565_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_566_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_567_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_568_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y3: real] :
( ( ~ ( ord_less_real @ X2 @ Y3 ) )
= ( ( ord_less_real @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_569_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_570_order_Ostrict__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_571_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_572_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B3: real] :
( ( ord_less_real @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B3: real] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_573_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_574_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_575_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_576_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_577_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_578_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_579_linorder__cases,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_580_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_581_antisym__conv3,axiom,
! [Y3: real,X2: real] :
( ~ ( ord_less_real @ Y3 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_582_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_583_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_584_ord__less__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_585_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_586_ord__eq__less__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_587_diff__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% diff_left_imp_eq
thf(fact_588_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_589_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_590_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_591_less__imp__neq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_592_dense,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ? [Z2: real] :
( ( ord_less_real @ X2 @ Z2 )
& ( ord_less_real @ Z2 @ Y3 ) ) ) ).
% dense
thf(fact_593_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_594_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_595_lt__ex,axiom,
! [X2: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% lt_ex
thf(fact_596_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 ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_597_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 ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_598_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_599_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_600_lessThan__strict__subset__iff,axiom,
! [M2: real,N2: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N2 ) )
= ( ord_less_real @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_601_lessThan__strict__subset__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_602_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_603_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_604_less__add__eq__less,axiom,
! [K: nat,L2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M2 @ L2 )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_605_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_606_trans__less__add2,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_607_trans__less__add1,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_608_less__diff__conv,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_609_add__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_610_not__add__less2,axiom,
! [J2: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_611_not__add__less1,axiom,
! [I3: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).
% not_add_less1
thf(fact_612_add__less__mono,axiom,
! [I3: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_613_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_614_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_615_add__lessD1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_616_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y3: real] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ Y3 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_617_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_618_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_619_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_620_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_621_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_622_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_623_atLeastLessThan__eq__iff,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C2 @ D3 ) )
= ( ( A = C2 )
& ( B = D3 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_624_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D3 ) )
= ( ( A = C2 )
& ( B = D3 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_625_Bex__def,axiom,
( bex_o
= ( ^ [A3: set_o,P3: $o > $o] :
? [X: $o] :
( ( member_o @ X @ A3 )
& ( P3 @ X ) ) ) ) ).
% Bex_def
thf(fact_626_Bex__def,axiom,
( bex_nat
= ( ^ [A3: set_nat,P3: nat > $o] :
? [X: nat] :
( ( member_nat @ X @ A3 )
& ( P3 @ X ) ) ) ) ).
% Bex_def
thf(fact_627_ex__gt__or__lt,axiom,
! [A: real] :
? [B3: real] :
( ( ord_less_real @ A @ B3 )
| ( ord_less_real @ B3 @ A ) ) ).
% ex_gt_or_lt
thf(fact_628_Ico__eq__Ico,axiom,
! [L2: real,H2: real,L: real,H3: real] :
( ( ( set_or66887138388493659n_real @ L2 @ H2 )
= ( set_or66887138388493659n_real @ L @ H3 ) )
= ( ( ( L2 = L )
& ( H2 = H3 ) )
| ( ~ ( ord_less_real @ L2 @ H2 )
& ~ ( ord_less_real @ L @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_629_Ico__eq__Ico,axiom,
! [L2: nat,H2: nat,L: nat,H3: nat] :
( ( ( set_or4665077453230672383an_nat @ L2 @ H2 )
= ( set_or4665077453230672383an_nat @ L @ H3 ) )
= ( ( ( L2 = L )
& ( H2 = H3 ) )
| ( ~ ( ord_less_nat @ L2 @ H2 )
& ~ ( ord_less_nat @ L @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_630_atLeastLessThan__inj_I1_J,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C2 @ D3 ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( A = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_631_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D3 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( A = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_632_atLeastLessThan__inj_I2_J,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C2 @ D3 ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( B = D3 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_633_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D3 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( B = D3 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_634_less__1__mult,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_635_less__1__mult,axiom,
! [M2: real,N2: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_636_square__diff__square__factored,axiom,
! [X2: real,Y3: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
= ( times_times_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ X2 @ Y3 ) ) ) ).
% square_diff_square_factored
thf(fact_637_eq__add__iff2,axiom,
! [A: real,E: real,C2: real,B: real,D3: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D3 ) )
= ( C2
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D3 ) ) ) ).
% eq_add_iff2
thf(fact_638_eq__add__iff1,axiom,
! [A: real,E: real,C2: real,B: real,D3: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D3 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C2 )
= D3 ) ) ).
% eq_add_iff1
thf(fact_639_set__plus__elim,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X2 @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A5: nat,B3: nat] :
( ( X2
= ( plus_plus_nat @ A5 @ B3 ) )
=> ( ( member_nat @ A5 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_640_set__plus__elim,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ( member_real @ X2 @ ( plus_plus_set_real @ A2 @ B2 ) )
=> ~ ! [A5: real,B3: real] :
( ( X2
= ( plus_plus_real @ A5 @ B3 ) )
=> ( ( member_real @ A5 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_641_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_642_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_643_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_644_diff__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_645_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_646_add__less__imp__less__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_647_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_648_add__less__imp__less__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_649_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_650_add__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_651_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_652_add__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_653_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_654_add__strict__mono,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D3 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_655_diff__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_656_diff__strict__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_657_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_658_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_659_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_660_add__implies__diff,axiom,
! [C2: real,B: real,A: real] :
( ( ( plus_plus_real @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_661_diff__eq__diff__less,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D3 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_662_diff__strict__mono,axiom,
! [A: real,B: real,D3: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D3 @ C2 )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_663_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_664_add__right__imp__eq,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_665_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_666_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_667_add__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_668_less__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_669_diff__less__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_670_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_671_add_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_672_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_673_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_674_add_Oright__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_675_diff__eq__diff__eq,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D3 ) )
=> ( ( A = B )
= ( C2 = D3 ) ) ) ).
% diff_eq_diff_eq
thf(fact_676_add_Oleft__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_677_diff__add__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_678_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_679_add_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_680_diff__diff__eq2,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_681_add__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_682_eq__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( A
= ( minus_minus_real @ C2 @ B ) )
= ( ( plus_plus_real @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_683_diff__eq__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( minus_minus_real @ A @ B )
= C2 )
= ( A
= ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_684_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_685_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_686_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_687_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_688_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_689_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J2: nat,K: nat,L2: nat] :
( ( ( I3 = J2 )
& ( K = L2 ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_690_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: real,J2: real,K: real,L2: real] :
( ( ( I3 = J2 )
& ( K = L2 ) )
=> ( ( plus_plus_real @ I3 @ K )
= ( plus_plus_real @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_691_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( K = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_692_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: real,J2: real,K: real,L2: real] :
( ( ( ord_less_real @ I3 @ J2 )
& ( K = L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_693_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J2: nat,K: nat,L2: nat] :
( ( ( I3 = J2 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_694_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: real,J2: real,K: real,L2: real] :
( ( ( I3 = J2 )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_695_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_696_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: real,J2: real,K: real,L2: real] :
( ( ( ord_less_real @ I3 @ J2 )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_697_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_698_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_699_less__add__iff1,axiom,
! [A: real,E: real,C2: real,B: real,D3: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D3 ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C2 ) @ D3 ) ) ).
% less_add_iff1
thf(fact_700_less__add__iff2,axiom,
! [A: real,E: real,C2: real,B: real,D3: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D3 ) )
= ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D3 ) ) ) ).
% less_add_iff2
thf(fact_701_square__diff__one__factored,axiom,
! [X2: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_702_set__plus__def,axiom,
( plus_plus_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [C4: nat] :
? [X: nat] :
( ( member_nat @ X @ A3 )
& ? [Y: nat] :
( ( member_nat @ Y @ B5 )
& ( C4
= ( plus_plus_nat @ X @ Y ) ) ) ) ) ) ) ).
% set_plus_def
thf(fact_703_set__plus__def,axiom,
( plus_plus_set_real
= ( ^ [A3: set_real,B5: set_real] :
( collect_real
@ ^ [C4: real] :
? [X: real] :
( ( member_real @ X @ A3 )
& ? [Y: real] :
( ( member_real @ Y @ B5 )
& ( C4
= ( plus_plus_real @ X @ Y ) ) ) ) ) ) ) ).
% set_plus_def
thf(fact_704_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_705_translation__subtract__diff,axiom,
! [A: real,S3: set_real,T: set_real] :
( ( image_real_real
@ ^ [X: real] : ( minus_minus_real @ X @ A )
@ ( minus_minus_set_real @ S3 @ T ) )
= ( minus_minus_set_real
@ ( image_real_real
@ ^ [X: real] : ( minus_minus_real @ X @ A )
@ S3 )
@ ( image_real_real
@ ^ [X: real] : ( minus_minus_real @ X @ A )
@ T ) ) ) ).
% translation_subtract_diff
thf(fact_706_not__sum__squares__lt__zero,axiom,
! [X2: real,Y3: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_707_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_708_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_709_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_710_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_711_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_712_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A: real,X2: real,Y3: real] :
( ( times_times_real @ A @ ( minus_minus_real @ X2 @ Y3 ) )
= ( minus_minus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ A @ Y3 ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_713_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ X2 )
= ( minus_minus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_714_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C2 ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_715_right__diff__distrib_H,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib'
thf(fact_716_left__diff__distrib_H,axiom,
! [B: nat,C2: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C2 ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_717_left__diff__distrib_H,axiom,
! [B: real,C2: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C2 ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C2 @ A ) ) ) ).
% left_diff_distrib'
thf(fact_718_right__diff__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% right_diff_distrib
thf(fact_719_left__diff__distrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% left_diff_distrib
thf(fact_720_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_721_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_722_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_723_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_724_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_725_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_726_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_727_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: real,X2: real,Y3: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X2 @ Y3 ) )
= ( plus_plus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ A @ Y3 ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_728_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X2 )
= ( plus_plus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_729_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_730_combine__common__factor,axiom,
! [A: real,E: real,B: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_731_distrib__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_732_distrib__right,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_733_distrib__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_734_distrib__left,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_735_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_736_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_737_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_738_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_739_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_740_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_741_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_742_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_743_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_744_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_745_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_746_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_747_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_748_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_749_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_750_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_751_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_752_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_753_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_754_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_755_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_756_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_757_diff__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_758_left__add__mult__distrib,axiom,
! [I3: nat,U: nat,J2: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_759_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_760_add__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_761_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_762_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_763_image__def,axiom,
( image_nat_set_nat
= ( ^ [F2: nat > set_nat,A3: set_nat] :
( collect_set_nat
@ ^ [Y: set_nat] :
? [X: nat] :
( ( member_nat @ X @ A3 )
& ( Y
= ( F2 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_764_image__def,axiom,
( image_o_set_nat
= ( ^ [F2: $o > set_nat,A3: set_o] :
( collect_set_nat
@ ^ [Y: set_nat] :
? [X: $o] :
( ( member_o @ X @ A3 )
& ( Y
= ( F2 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_765_image__def,axiom,
( image_o_set_o
= ( ^ [F2: $o > set_o,A3: set_o] :
( collect_set_o
@ ^ [Y: set_o] :
? [X: $o] :
( ( member_o @ X @ A3 )
& ( Y
= ( F2 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_766_image__def,axiom,
( image_nat_nat
= ( ^ [F2: nat > nat,A3: set_nat] :
( collect_nat
@ ^ [Y: nat] :
? [X: nat] :
( ( member_nat @ X @ A3 )
& ( Y
= ( F2 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_767_image__def,axiom,
( image_o_o
= ( ^ [F2: $o > $o,A3: set_o] :
( collect_o
@ ^ [Y: $o] :
? [X: $o] :
( ( member_o @ X @ A3 )
& ( Y
= ( F2 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_768_image__def,axiom,
( image_7916887816326733075et_nat
= ( ^ [F2: set_nat > set_nat,A3: set_set_nat] :
( collect_set_nat
@ ^ [Y: set_nat] :
? [X: set_nat] :
( ( member_set_nat @ X @ A3 )
& ( Y
= ( F2 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_769_Union__eq,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X: nat] :
? [Y: set_nat] :
( ( member_set_nat @ Y @ A3 )
& ( member_nat @ X @ Y ) ) ) ) ) ).
% Union_eq
thf(fact_770_Union__eq,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X: $o] :
? [Y: set_o] :
( ( member_set_o @ Y @ A3 )
& ( member_o @ X @ Y ) ) ) ) ) ).
% Union_eq
thf(fact_771_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_772_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_773_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_774_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_775_UN__le__add__shift__strict,axiom,
! [M: nat > set_nat,K: nat,N2: nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( M @ ( plus_plus_nat @ I @ K ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_776_UN__le__add__shift__strict,axiom,
! [M: nat > set_o,K: nat,N2: nat] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [I: nat] : ( M @ ( plus_plus_nat @ I @ K ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ M @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_777_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_778_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_779_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_780_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_781_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_782_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_783_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_784_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_785_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_786_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_787_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_788_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_789_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_790_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_791_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_792_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_793_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_794_mult__less__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_795_mult__less__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_796_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_797_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_798_mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_799_mult__less__cancel__left__disj,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_800_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_801_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_802_mult__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_803_mult__less__cancel__right__disj,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_804_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_805_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_806_UN__finite2__eq,axiom,
! [A2: nat > set_nat,B2: nat > set_nat,K: nat] :
( ! [N4: nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N4 @ K ) ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_eq
thf(fact_807_UN__finite2__eq,axiom,
! [A2: nat > set_o,B2: nat > set_o,K: nat] :
( ! [N4: nat] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N4 @ K ) ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ top_top_set_nat ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_eq
thf(fact_808_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_809_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_810_mult__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_811_mult__less__mono2,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_812_Collect__bex__eq,axiom,
! [A2: set_nat,P: nat > nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
? [Y: nat] :
( ( member_nat @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] :
( collect_nat
@ ^ [X: nat] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_813_Collect__bex__eq,axiom,
! [A2: set_o,P: nat > $o > $o] :
( ( collect_nat
@ ^ [X: nat] :
? [Y: $o] :
( ( member_o @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y: $o] :
( collect_nat
@ ^ [X: nat] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_814_Collect__bex__eq,axiom,
! [A2: set_set_nat,P: nat > set_nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
? [Y: set_nat] :
( ( member_set_nat @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y: set_nat] :
( collect_nat
@ ^ [X: nat] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_815_Collect__bex__eq,axiom,
! [A2: set_o,P: $o > $o > $o] :
( ( collect_o
@ ^ [X: $o] :
? [Y: $o] :
( ( member_o @ Y @ A2 )
& ( P @ X @ Y ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y: $o] :
( collect_o
@ ^ [X: $o] : ( P @ X @ Y ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_816_UNION__eq,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) )
= ( collect_nat
@ ^ [Y: nat] :
? [X: nat] :
( ( member_nat @ X @ A2 )
& ( member_nat @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_817_UNION__eq,axiom,
! [B2: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) )
= ( collect_nat
@ ^ [Y: nat] :
? [X: $o] :
( ( member_o @ X @ A2 )
& ( member_nat @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_818_UNION__eq,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) )
= ( collect_nat
@ ^ [Y: nat] :
? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( member_nat @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_819_UNION__eq,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) )
= ( collect_o
@ ^ [Y: $o] :
? [X: $o] :
( ( member_o @ X @ A2 )
& ( member_o @ Y @ ( B2 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_820_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_821_set__times__def,axiom,
( times_times_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [C4: nat] :
? [X: nat] :
( ( member_nat @ X @ A3 )
& ? [Y: nat] :
( ( member_nat @ Y @ B5 )
& ( C4
= ( times_times_nat @ X @ Y ) ) ) ) ) ) ) ).
% set_times_def
thf(fact_822_set__times__def,axiom,
( times_times_set_real
= ( ^ [A3: set_real,B5: set_real] :
( collect_real
@ ^ [C4: real] :
? [X: real] :
( ( member_real @ X @ A3 )
& ? [Y: real] :
( ( member_real @ Y @ B5 )
& ( C4
= ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% set_times_def
thf(fact_823_SUP__lessD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,Y3: set_nat,I3: set_nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ Y3 )
=> ( ( member_set_nat @ I3 @ A2 )
=> ( ord_less_set_nat @ ( F @ I3 ) @ Y3 ) ) ) ).
% SUP_lessD
thf(fact_824_SUP__lessD,axiom,
! [F: $o > set_nat,A2: set_o,Y3: set_nat,I3: $o] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ Y3 )
=> ( ( member_o @ I3 @ A2 )
=> ( ord_less_set_nat @ ( F @ I3 ) @ Y3 ) ) ) ).
% SUP_lessD
thf(fact_825_SUP__lessD,axiom,
! [F: nat > set_nat,A2: set_nat,Y3: set_nat,I3: nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ Y3 )
=> ( ( member_nat @ I3 @ A2 )
=> ( ord_less_set_nat @ ( F @ I3 ) @ Y3 ) ) ) ).
% SUP_lessD
thf(fact_826_SUP__lessD,axiom,
! [F: $o > $o,A2: set_o,Y3: $o,I3: $o] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ Y3 )
=> ( ( member_o @ I3 @ A2 )
=> ( ord_less_o @ ( F @ I3 ) @ Y3 ) ) ) ).
% SUP_lessD
thf(fact_827_SUP__lessD,axiom,
! [F: nat > $o,A2: set_nat,Y3: $o,I3: nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ Y3 )
=> ( ( member_nat @ I3 @ A2 )
=> ( ord_less_o @ ( F @ I3 ) @ Y3 ) ) ) ).
% SUP_lessD
thf(fact_828_SUP__lessD,axiom,
! [F: $o > set_o,A2: set_o,Y3: set_o,I3: $o] :
( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ Y3 )
=> ( ( member_o @ I3 @ A2 )
=> ( ord_less_set_o @ ( F @ I3 ) @ Y3 ) ) ) ).
% SUP_lessD
thf(fact_829_SUP__lessD,axiom,
! [F: nat > set_o,A2: set_nat,Y3: set_o,I3: nat] :
( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) ) @ Y3 )
=> ( ( member_nat @ I3 @ A2 )
=> ( ord_less_set_o @ ( F @ I3 ) @ Y3 ) ) ) ).
% SUP_lessD
thf(fact_830_UN__extend__simps_I6_J,axiom,
! [A2: nat > set_nat,C: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C ) ) @ B2 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( minus_minus_set_nat @ ( A2 @ X ) @ B2 )
@ C ) ) ) ).
% UN_extend_simps(6)
thf(fact_831_UN__extend__simps_I6_J,axiom,
! [A2: $o > set_nat,C: set_o,B2: set_nat] :
( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A2 @ C ) ) @ B2 )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( minus_minus_set_nat @ ( A2 @ X ) @ B2 )
@ C ) ) ) ).
% UN_extend_simps(6)
thf(fact_832_UN__extend__simps_I6_J,axiom,
! [A2: set_nat > set_nat,C: set_set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A2 @ C ) ) @ B2 )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : ( minus_minus_set_nat @ ( A2 @ X ) @ B2 )
@ C ) ) ) ).
% UN_extend_simps(6)
thf(fact_833_UN__extend__simps_I6_J,axiom,
! [A2: $o > set_o,C: set_o,B2: set_o] :
( ( minus_minus_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B2 )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( minus_minus_set_o @ ( A2 @ X ) @ B2 )
@ C ) ) ) ).
% UN_extend_simps(6)
thf(fact_834_range__diff,axiom,
! [A: real] :
( ( image_real_real @ ( minus_minus_real @ A ) @ top_top_set_real )
= top_top_set_real ) ).
% range_diff
thf(fact_835_range__add,axiom,
! [A: real] :
( ( image_real_real @ ( plus_plus_real @ A ) @ top_top_set_real )
= top_top_set_real ) ).
% range_add
thf(fact_836_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_837_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y3: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y3 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_838_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_839_sum__squares__gt__zero__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
= ( ( X2 != zero_zero_real )
| ( Y3 != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_840_mult__diff__mult,axiom,
! [X2: real,Y3: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ Y3 ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_841_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C2: nat,D3: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C2 != D3 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C2 ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D3 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_842_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C2: real,D3: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C2 != D3 ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R @ C2 ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D3 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_843_mult__if__delta,axiom,
! [P: $o,Q: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_844_mult__if__delta,axiom,
! [P: $o,Q: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_845_Diff__iff,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B2 ) )
= ( ( member_o @ C2 @ A2 )
& ~ ( member_o @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_846_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_847_DiffI,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ A2 )
=> ( ~ ( member_o @ C2 @ B2 )
=> ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_848_DiffI,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_849_less__set__def,axiom,
( ord_less_set_o
= ( ^ [A3: set_o,B5: set_o] :
( ord_less_o_o
@ ^ [X: $o] : ( member_o @ X @ A3 )
@ ^ [X: $o] : ( member_o @ X @ B5 ) ) ) ) ).
% less_set_def
thf(fact_850_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_set_def
thf(fact_851_minus__set__def,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B5: set_o] :
( collect_o
@ ( minus_minus_o_o
@ ^ [X: $o] : ( member_o @ X @ A3 )
@ ^ [X: $o] : ( member_o @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_852_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_853_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_854_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_855_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_856_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_857_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_858_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_859_less__not__refl3,axiom,
! [S3: nat,T: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( S3 != T ) ) ).
% less_not_refl3
thf(fact_860_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_861_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_862_diff__commute,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_863_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_864_psubset__imp__ex__mem,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_set_o @ A2 @ B2 )
=> ? [B3: $o] : ( member_o @ B3 @ ( minus_minus_set_o @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_865_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_866_psubsetD,axiom,
! [A2: set_o,B2: set_o,C2: $o] :
( ( ord_less_set_o @ A2 @ B2 )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_867_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_868_DiffD2,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B2 ) )
=> ~ ( member_o @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_869_DiffD2,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_870_DiffD1,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B2 ) )
=> ( member_o @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_871_DiffD1,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_872_DiffE,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( minus_minus_set_o @ A2 @ B2 ) )
=> ~ ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_873_DiffE,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_874_set__diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B5: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ~ ( member_o @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_875_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ~ ( member_nat @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_876_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_877_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_878_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_879_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_880_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_881_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_882_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_883_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_884_crossproduct__noteq,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ( A != B )
& ( C2 != D3 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D3 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D3 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_885_crossproduct__noteq,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( A != B )
& ( C2 != D3 ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D3 ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D3 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_886_crossproduct__eq,axiom,
! [W: nat,Y3: nat,X2: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X2 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X2 @ Y3 ) ) )
= ( ( W = X2 )
| ( Y3 = Z ) ) ) ).
% crossproduct_eq
thf(fact_887_crossproduct__eq,axiom,
! [W: real,Y3: real,X2: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X2 @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X2 @ Y3 ) ) )
= ( ( W = X2 )
| ( Y3 = Z ) ) ) ).
% crossproduct_eq
thf(fact_888_add__diff__add,axiom,
! [A: real,C2: real,B: real,D3: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C2 @ D3 ) ) ) ).
% add_diff_add
thf(fact_889_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_890_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_891_ex__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ( P @ M3 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_892_all__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_893_mult__less__iff1,axiom,
! [Z: real,X2: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
= ( ord_less_real @ X2 @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_894_affine__parallel__expl__aux,axiom,
! [S: set_real,A: real,T2: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
= ( member_real @ ( plus_plus_real @ A @ X4 ) @ T2 ) )
=> ( T2
= ( image_real_real @ ( plus_plus_real @ A ) @ S ) ) ) ).
% affine_parallel_expl_aux
thf(fact_895_translation__invert,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ( ( image_real_real @ ( plus_plus_real @ A ) @ A2 )
= ( image_real_real @ ( plus_plus_real @ A ) @ B2 ) )
=> ( A2 = B2 ) ) ).
% translation_invert
thf(fact_896_translation__assoc,axiom,
! [B: real,A: real,S: set_real] :
( ( image_real_real @ ( plus_plus_real @ B ) @ ( image_real_real @ ( plus_plus_real @ A ) @ S ) )
= ( image_real_real @ ( plus_plus_real @ ( plus_plus_real @ A @ B ) ) @ S ) ) ).
% translation_assoc
thf(fact_897_UN__finite2__subset,axiom,
! [A2: nat > set_nat,B2: nat > set_nat,K: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N4 @ K ) ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_subset
thf(fact_898_UN__finite2__subset,axiom,
! [A2: nat > set_o,B2: nat > set_o,K: nat] :
( ! [N4: nat] : ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N4 @ K ) ) ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ top_top_set_nat ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_subset
thf(fact_899_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_900_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_901_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_902_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_903_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_904_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ X4 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_905_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_906_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_907_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_908_add__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_909_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_910_add__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_911_atMost__iff,axiom,
! [I3: $o,K: $o] :
( ( member_o @ I3 @ ( set_ord_atMost_o @ K ) )
= ( ord_less_eq_o @ I3 @ K ) ) ).
% atMost_iff
thf(fact_912_atMost__iff,axiom,
! [I3: real,K: real] :
( ( member_real @ I3 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I3 @ K ) ) ).
% atMost_iff
thf(fact_913_atMost__iff,axiom,
! [I3: nat,K: nat] :
( ( member_nat @ I3 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I3 @ K ) ) ).
% atMost_iff
thf(fact_914_atLeast__iff,axiom,
! [I3: $o,K: $o] :
( ( member_o @ I3 @ ( set_ord_atLeast_o @ K ) )
= ( ord_less_eq_o @ K @ I3 ) ) ).
% atLeast_iff
thf(fact_915_atLeast__iff,axiom,
! [I3: real,K: real] :
( ( member_real @ I3 @ ( set_ord_atLeast_real @ K ) )
= ( ord_less_eq_real @ K @ I3 ) ) ).
% atLeast_iff
thf(fact_916_atLeast__iff,axiom,
! [I3: nat,K: nat] :
( ( member_nat @ I3 @ ( set_ord_atLeast_nat @ K ) )
= ( ord_less_eq_nat @ K @ I3 ) ) ).
% atLeast_iff
thf(fact_917_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_918_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_919_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_920_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_921_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_922_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_923_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_924_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_925_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_926_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_927_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_928_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_929_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_930_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_931_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_932_subset__translation__eq,axiom,
! [A: real,S3: set_real,T: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ ( plus_plus_real @ A ) @ S3 ) @ ( image_real_real @ ( plus_plus_real @ A ) @ T ) )
= ( ord_less_eq_set_real @ S3 @ T ) ) ).
% subset_translation_eq
thf(fact_933_atLeastLessThan__iff,axiom,
! [I3: $o,L2: $o,U: $o] :
( ( member_o @ I3 @ ( set_or7139685690850216873Than_o @ L2 @ U ) )
= ( ( ord_less_eq_o @ L2 @ I3 )
& ( ord_less_o @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_934_atLeastLessThan__iff,axiom,
! [I3: real,L2: real,U: real] :
( ( member_real @ I3 @ ( set_or66887138388493659n_real @ L2 @ U ) )
= ( ( ord_less_eq_real @ L2 @ I3 )
& ( ord_less_real @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_935_atLeastLessThan__iff,axiom,
! [I3: nat,L2: nat,U: nat] :
( ( member_nat @ I3 @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
= ( ( ord_less_eq_nat @ L2 @ I3 )
& ( ord_less_nat @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_936_lessThan__subset__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X2 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
= ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_937_lessThan__subset__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_938_ivl__subset,axiom,
! [I3: real,J2: real,M2: real,N2: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I3 @ J2 ) @ ( set_or66887138388493659n_real @ M2 @ N2 ) )
= ( ( ord_less_eq_real @ J2 @ I3 )
| ( ( ord_less_eq_real @ M2 @ I3 )
& ( ord_less_eq_real @ J2 @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_939_ivl__subset,axiom,
! [I3: nat,J2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ J2 ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ J2 @ I3 )
| ( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_eq_nat @ J2 @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_940_atMost__subset__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X2 ) @ ( set_ord_atMost_real @ Y3 ) )
= ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_941_atMost__subset__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_942_ivl__diff,axiom,
! [I3: real,N2: real,M2: real] :
( ( ord_less_eq_real @ I3 @ N2 )
=> ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I3 @ M2 ) @ ( set_or66887138388493659n_real @ I3 @ N2 ) )
= ( set_or66887138388493659n_real @ N2 @ M2 ) ) ) ).
% ivl_diff
thf(fact_943_ivl__diff,axiom,
! [I3: nat,N2: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ M2 ) @ ( set_or4665077453230672383an_nat @ I3 @ N2 ) )
= ( set_or4665077453230672383an_nat @ N2 @ M2 ) ) ) ).
% ivl_diff
thf(fact_944_atLeast__subset__iff,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ X2 ) @ ( set_ord_atLeast_real @ Y3 ) )
= ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_945_atLeast__subset__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X2 ) @ ( set_ord_atLeast_nat @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% atLeast_subset_iff
thf(fact_946_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_947_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_948_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D4: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D4 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D4 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_949_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: real] :
( ( ord_less_eq_real @ A @ C3 )
& ( ord_less_eq_real @ C3 @ B )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A @ X5 )
& ( ord_less_real @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D4: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D4 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D4 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_950_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_951_leD,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq_real @ Y3 @ X2 )
=> ~ ( ord_less_real @ X2 @ Y3 ) ) ).
% leD
thf(fact_952_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_953_leI,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% leI
thf(fact_954_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_955_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_956_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_957_antisym__conv1,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_958_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_959_antisym__conv2,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_960_dense__ge,axiom,
! [Z: real,Y3: real] :
( ! [X4: real] :
( ( ord_less_real @ Z @ X4 )
=> ( ord_less_eq_real @ Y3 @ X4 ) )
=> ( ord_less_eq_real @ Y3 @ Z ) ) ).
% dense_ge
thf(fact_961_dense__le,axiom,
! [Y3: real,Z: real] :
( ! [X4: real] :
( ( ord_less_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ( ord_less_eq_real @ Y3 @ Z ) ) ).
% dense_le
thf(fact_962_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_963_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_964_not__le__imp__less,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ord_less_nat @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_965_not__le__imp__less,axiom,
! [Y3: real,X2: real] :
( ~ ( ord_less_eq_real @ Y3 @ X2 )
=> ( ord_less_real @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_966_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_967_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_968_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_969_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_970_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_971_order_Ostrict__trans1,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_972_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_973_order_Ostrict__trans2,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_974_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_975_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_976_dense__ge__bounded,axiom,
! [Z: real,X2: real,Y3: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ! [W2: real] :
( ( ord_less_real @ Z @ W2 )
=> ( ( ord_less_real @ W2 @ X2 )
=> ( ord_less_eq_real @ Y3 @ W2 ) ) )
=> ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_977_dense__le__bounded,axiom,
! [X2: real,Y3: real,Z: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ! [W2: real] :
( ( ord_less_real @ X2 @ W2 )
=> ( ( ord_less_real @ W2 @ Y3 )
=> ( ord_less_eq_real @ W2 @ Z ) ) )
=> ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_978_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_979_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_980_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_981_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_982_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_983_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_984_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_985_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_986_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_987_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_988_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_989_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_990_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_991_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_992_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_993_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_994_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_995_order__less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_996_linorder__not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_997_linorder__not__le,axiom,
! [X2: real,Y3: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y3 ) )
= ( ord_less_real @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_998_linorder__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_999_linorder__not__less,axiom,
! [X2: real,Y3: real] :
( ( ~ ( ord_less_real @ X2 @ Y3 ) )
= ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_1000_order__less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1001_order__less__imp__le,axiom,
! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1002_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_1003_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1004_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_1005_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1006_order__le__less__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1007_order__le__less__trans,axiom,
! [X2: real,Y3: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1008_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1009_order__less__le__trans,axiom,
! [X2: real,Y3: real,Z: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1010_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1011_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1012_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1013_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1014_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1015_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1016_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1017_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1018_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1019_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1020_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1021_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1022_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1023_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1024_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1025_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
=> ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1026_linorder__le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1027_linorder__le__less__linear,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
| ( ord_less_real @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1028_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1029_order__le__imp__less__or__eq,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1030_diff__mono,axiom,
! [A: real,B: real,D3: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D3 @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D3 ) ) ) ) ).
% diff_mono
thf(fact_1031_diff__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_1032_diff__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1033_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D3 ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1034_in__mono,axiom,
! [A2: set_o,B2: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ X2 @ A2 )
=> ( member_o @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_1035_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_1036_subsetD,axiom,
! [A2: set_o,B2: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_1037_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_1038_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B5: set_o] :
! [X: $o] :
( ( member_o @ X @ A3 )
=> ( member_o @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1039_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1040_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B5: set_o] :
! [T3: $o] :
( ( member_o @ T3 @ A3 )
=> ( member_o @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1041_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A3 )
=> ( member_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1042_Union__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1043_Union__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_mono
thf(fact_1044_Sup__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_1045_Sup__subset__mono,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_1046_Sup__subset__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_1047_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_1048_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1049_le__cases3,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1050_le__cases3,axiom,
! [X2: real,Y3: real,Z: real] :
( ( ( ord_less_eq_real @ X2 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_real @ Y3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_real @ Z @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ Z )
=> ~ ( ord_less_eq_real @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1051_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1052_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1053_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1054_ord__eq__le__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1055_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_1056_ord__le__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_1057_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_1058_order__antisym,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_1059_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_1060_order_Otrans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% order.trans
thf(fact_1061_order__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_1062_order__trans,axiom,
! [X2: real,Y3: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z )
=> ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_1063_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_1064_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A5: real,B3: real] :
( ( ord_less_eq_real @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: real,B3: real] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_1065_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1066_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1067_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_1068_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1069_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_1070_dual__order_Otrans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_1071_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1072_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1073_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1074_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1075_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1076_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1077_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1078_order__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1079_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1080_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1081_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1082_order__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1083_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_1084_order__eq__refl,axiom,
! [X2: real,Y3: real] :
( ( X2 = Y3 )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_1085_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_1086_linorder__linear,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
| ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_1087_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1088_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1089_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1090_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1091_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1092_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1093_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1094_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1095_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_1096_linorder__le__cases,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_1097_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_1098_order__antisym__conv,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq_real @ Y3 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_1099_Union__subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ B2 )
& ( ord_less_eq_set_nat @ X4 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1100_Union__subsetI,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ? [Y5: set_o] :
( ( member_set_o @ Y5 @ B2 )
& ( ord_less_eq_set_o @ X4 @ Y5 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1101_Union__upper,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_1102_Union__upper,axiom,
! [B2: set_o,A2: set_set_o] :
( ( member_set_o @ B2 @ A2 )
=> ( ord_less_eq_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Union_upper
thf(fact_1103_Union__least,axiom,
! [A2: set_set_nat,C: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ X6 @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1104_Union__least,axiom,
! [A2: set_set_o,C: set_o] :
( ! [X6: set_o] :
( ( member_set_o @ X6 @ A2 )
=> ( ord_less_eq_set_o @ X6 @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ C ) ) ).
% Union_least
thf(fact_1105_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1106_subset__image__iff,axiom,
! [B2: set_set_nat,F: $o > set_nat,A2: set_o] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_o_set_nat @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1107_subset__image__iff,axiom,
! [B2: set_set_o,F: $o > set_o,A2: set_o] :
( ( ord_le4374716579403074808_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_set_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1108_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1109_subset__image__iff,axiom,
! [B2: set_o,F: $o > $o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1110_subset__image__iff,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1111_image__subset__iff,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1112_image__subset__iff,axiom,
! [F: $o > set_nat,A2: set_o,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A2 ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1113_image__subset__iff,axiom,
! [F: $o > set_o,A2: set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_set_o @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1114_image__subset__iff,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1115_image__subset__iff,axiom,
! [F: $o > $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1116_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1117_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1118_subset__imageE,axiom,
! [B2: set_set_nat,F: $o > set_nat,A2: set_o] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_o_set_nat @ F @ A2 ) )
=> ~ ! [C5: set_o] :
( ( ord_less_eq_set_o @ C5 @ A2 )
=> ( B2
!= ( image_o_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1119_subset__imageE,axiom,
! [B2: set_set_o,F: $o > set_o,A2: set_o] :
( ( ord_le4374716579403074808_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) )
=> ~ ! [C5: set_o] :
( ( ord_less_eq_set_o @ C5 @ A2 )
=> ( B2
!= ( image_o_set_o @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1120_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1121_subset__imageE,axiom,
! [B2: set_o,F: $o > $o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ ( image_o_o @ F @ A2 ) )
=> ~ ! [C5: set_o] :
( ( ord_less_eq_set_o @ C5 @ A2 )
=> ( B2
!= ( image_o_o @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1122_subset__imageE,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [C5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C5 @ A2 )
=> ( B2
!= ( image_7916887816326733075et_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1123_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1124_image__subsetI,axiom,
! [A2: set_o,F: $o > set_nat,B2: set_set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1125_image__subsetI,axiom,
! [A2: set_o,F: $o > set_o,B2: set_set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_set_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1126_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1127_image__subsetI,axiom,
! [A2: set_o,F: $o > nat,B2: set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1128_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1129_image__subsetI,axiom,
! [A2: set_nat,F: nat > $o,B2: set_o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1130_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1131_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1132_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A2 ) @ ( image_o_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1133_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ ( image_o_set_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1134_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1135_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ ( image_o_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1136_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1137_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_1138_SUP__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1139_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1140_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1141_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1142_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1143_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1144_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_o,G: nat > set_o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1145_cSup__eq,axiom,
! [X3: set_real,A: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X3 )
=> ( ord_less_eq_real @ X4 @ A ) )
=> ( ! [Y4: real] :
( ! [X5: real] :
( ( member_real @ X5 @ X3 )
=> ( ord_less_eq_real @ X5 @ Y4 ) )
=> ( ord_less_eq_real @ A @ Y4 ) )
=> ( ( comple1385675409528146559p_real @ X3 )
= A ) ) ) ).
% cSup_eq
thf(fact_1146_cSup__eq__maximum,axiom,
! [Z: real,X3: set_real] :
( ( member_real @ Z @ X3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ X3 )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ( ( comple1385675409528146559p_real @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1147_cSup__eq__maximum,axiom,
! [Z: set_nat,X3: set_set_nat] :
( ( member_set_nat @ Z @ X3 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X3 )
=> ( ord_less_eq_set_nat @ X4 @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1148_cSup__eq__maximum,axiom,
! [Z: $o,X3: set_o] :
( ( member_o @ Z @ X3 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X3 )
=> ( ord_less_eq_o @ X4 @ Z ) )
=> ( ( complete_Sup_Sup_o @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1149_cSup__eq__maximum,axiom,
! [Z: nat,X3: set_nat] :
( ( member_nat @ Z @ X3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X3 )
=> ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1150_cSup__eq__maximum,axiom,
! [Z: set_o,X3: set_set_o] :
( ( member_set_o @ Z @ X3 )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ X3 )
=> ( ord_less_eq_set_o @ X4 @ Z ) )
=> ( ( comple90263536869209701_set_o @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1151_Sup__upper2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1152_Sup__upper2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1153_Sup__upper2,axiom,
! [U: set_o,A2: set_set_o,V: set_o] :
( ( member_set_o @ U @ A2 )
=> ( ( ord_less_eq_set_o @ V @ U )
=> ( ord_less_eq_set_o @ V @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1154_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1155_Sup__le__iff,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1156_Sup__le__iff,axiom,
! [A2: set_set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ B )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A2 )
=> ( ord_less_eq_set_o @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1157_Sup__upper,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_1158_Sup__upper,axiom,
! [X2: $o,A2: set_o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ X2 @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_1159_Sup__upper,axiom,
! [X2: set_o,A2: set_set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ X2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_1160_Sup__least,axiom,
! [A2: set_set_nat,Z: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ X4 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_1161_Sup__least,axiom,
! [A2: set_o,Z: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ X4 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_1162_Sup__least,axiom,
! [A2: set_set_o,Z: set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ X4 @ Z ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_1163_Sup__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B2 )
& ( ord_less_eq_set_nat @ A5 @ X5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1164_Sup__mono,axiom,
! [A2: set_o,B2: set_o] :
( ! [A5: $o] :
( ( member_o @ A5 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B2 )
& ( ord_less_eq_o @ A5 @ X5 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_1165_Sup__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [A5: set_o] :
( ( member_set_o @ A5 @ A2 )
=> ? [X5: set_o] :
( ( member_set_o @ X5 @ B2 )
& ( ord_less_eq_set_o @ A5 @ X5 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_1166_Sup__eqI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ A2 )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( ! [Y4: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A2 )
=> ( ord_less_eq_set_nat @ Z4 @ Y4 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1167_Sup__eqI,axiom,
! [A2: set_o,X2: $o] :
( ! [Y4: $o] :
( ( member_o @ Y4 @ A2 )
=> ( ord_less_eq_o @ Y4 @ X2 ) )
=> ( ! [Y4: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A2 )
=> ( ord_less_eq_o @ Z4 @ Y4 ) )
=> ( ord_less_eq_o @ X2 @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1168_Sup__eqI,axiom,
! [A2: set_set_o,X2: set_o] :
( ! [Y4: set_o] :
( ( member_set_o @ Y4 @ A2 )
=> ( ord_less_eq_set_o @ Y4 @ X2 ) )
=> ( ! [Y4: set_o] :
( ! [Z4: set_o] :
( ( member_set_o @ Z4 @ A2 )
=> ( ord_less_eq_set_o @ Z4 @ Y4 ) )
=> ( ord_less_eq_set_o @ X2 @ Y4 ) )
=> ( ( comple90263536869209701_set_o @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1169_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1170_add__le__imp__le__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1171_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1172_add__le__imp__le__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1173_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_1174_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1175_add__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1176_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_1177_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1178_add__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1179_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_1180_add__mono,axiom,
! [A: real,B: real,C2: real,D3: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D3 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_1181_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1182_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1183_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1184_diff__diff__cancel,axiom,
! [I3: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1185_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1186_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1187_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1188_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1189_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1190_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1191_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1192_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1193_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1194_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1195_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1196_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1197_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1198_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1199_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1200_diff__le__mono,axiom,
! [M2: nat,N2: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N2 @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1201_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1202_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1203_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1204_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1205_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1206_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1207_mult__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1208_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1209_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1210_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1211_trans__le__add2,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1212_trans__le__add1,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1213_add__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1214_add__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1215_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N4: nat] :
( L2
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1216_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1217_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1218_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1219_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1220_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1221_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1222_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_1223_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_1224_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1225_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1226_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1227_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1228_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1229_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I3 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1230_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1231_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1232_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1233_le__diff__conv,axiom,
! [J2: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1234_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1235_less__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1236_nat__eq__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M2 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1237_nat__eq__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1238_nat__le__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1239_nat__le__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1240_nat__diff__add__eq1,axiom,
! [J2: nat,I3: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1241_nat__diff__add__eq2,axiom,
! [I3: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1242_nat__less__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1243_nat__less__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N2 ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1244_kuhn__lemma,axiom,
! [P4: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P4 )
=> ( ! [X4: nat > nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I5 ) @ P4 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ( ( Label @ X4 @ I4 )
= zero_zero_nat )
| ( ( Label @ X4 @ I4 )
= one_one_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I5 ) @ P4 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ( ( X4 @ I4 )
= zero_zero_nat )
=> ( ( Label @ X4 @ I4 )
= zero_zero_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I5 ) @ P4 ) )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ( ( X4 @ I4 )
= P4 )
=> ( ( Label @ X4 @ I4 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ( ord_less_nat @ ( Q2 @ I5 ) @ P4 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
=> ? [R2: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J4 ) @ ( R2 @ J4 ) )
& ( ord_less_eq_nat @ ( R2 @ J4 ) @ ( plus_plus_nat @ ( Q2 @ J4 ) @ one_one_nat ) ) ) )
& ? [S4: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J4 ) @ ( S4 @ J4 ) )
& ( ord_less_eq_nat @ ( S4 @ J4 ) @ ( plus_plus_nat @ ( Q2 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I5 )
!= ( Label @ S4 @ I5 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1245_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1246_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1247_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_1248_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_1249_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_1250_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_1251_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1252_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q3: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I4: nat] :
( ( Q3 @ I4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I4 ) )
& ( ord_less_eq_real @ ( X4 @ I4 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I5: nat] : ( ord_less_eq_nat @ ( L3 @ X5 @ I5 ) @ one_one_nat )
& ! [X5: nat > real,I5: nat] :
( ( ( P @ X5 )
& ( Q3 @ I5 )
& ( ( X5 @ I5 )
= zero_zero_real ) )
=> ( ( L3 @ X5 @ I5 )
= zero_zero_nat ) )
& ! [X5: nat > real,I5: nat] :
( ( ( P @ X5 )
& ( Q3 @ I5 )
& ( ( X5 @ I5 )
= one_one_real ) )
=> ( ( L3 @ X5 @ I5 )
= one_one_nat ) )
& ! [X5: nat > real,I5: nat] :
( ( ( P @ X5 )
& ( Q3 @ I5 )
& ( ( L3 @ X5 @ I5 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I5 ) @ ( F @ X5 @ I5 ) ) )
& ! [X5: nat > real,I5: nat] :
( ( ( P @ X5 )
& ( Q3 @ I5 )
& ( ( L3 @ X5 @ I5 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I5 ) @ ( X5 @ I5 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1253_segment__bound__lemma,axiom,
! [B2: real,X2: real,Y3: real,U: real] :
( ( ord_less_eq_real @ B2 @ X2 )
=> ( ( ord_less_eq_real @ B2 @ Y3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X2 ) @ ( times_times_real @ U @ Y3 ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_1254_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A5: real,B3: real,C3: real] :
( ( P @ A5 @ B3 )
=> ( ( P @ B3 @ C3 )
=> ( ( ord_less_eq_real @ A5 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C3 )
=> ( P @ A5 @ C3 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A5: real,B3: real] :
( ( ( ord_less_eq_real @ A5 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A5 ) @ D4 ) )
=> ( P @ A5 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1255_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_1256_sum__le__prod1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1257_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1258_square__bound__lemma,axiom,
! [X2: real] : ( ord_less_real @ X2 @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ).
% square_bound_lemma
thf(fact_1259_seq__mono__lemma,axiom,
! [M2: nat,D3: nat > real,E: nat > real] :
( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_real @ ( D3 @ N4 ) @ ( E @ N4 ) ) )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ord_less_eq_real @ ( E @ N4 ) @ ( E @ M2 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M2 @ N5 )
=> ( ord_less_real @ ( D3 @ N5 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1260_eq__diff__eq_H,axiom,
! [X2: real,Y3: real,Z: real] :
( ( X2
= ( minus_minus_real @ Y3 @ Z ) )
= ( Y3
= ( plus_plus_real @ X2 @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_1261_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1262_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ a @ ( set_ord_lessThan_nat @ n ) ) )
= ( set_ord_lessThan_nat @ ( times_times_nat @ n @ m ) ) ) ).
%------------------------------------------------------------------------------