TPTP Problem File: ITP156^1.p
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%------------------------------------------------------------------------------
% File : ITP156^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_474__6256716_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Preferences/prob_474__6256716_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 482 ( 221 unt; 128 typ; 0 def)
% Number of atoms : 879 ( 416 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2819 ( 33 ~; 15 |; 57 &;2451 @)
% ( 0 <=>; 263 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 158 ( 158 >; 0 *; 0 +; 0 <<)
% Number of symbols : 97 ( 96 usr; 21 con; 0-2 aty)
% Number of variables : 912 ( 22 ^; 888 !; 2 ?; 912 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:46:49.616
%------------------------------------------------------------------------------
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real_V1214981142real_a: real > product_prod_real_a > product_prod_real_a ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
real_V296206913od_a_a: real > produc1921647824od_a_a > produc1921647824od_a_a ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
real_V512797224a_real: real > product_prod_a_real > product_prod_a_real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
real_V543523736od_a_a: real > product_prod_a_a > product_prod_a_a ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V453051771R_real: real > real > real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001tf__a,type,
real_V1035702895aleR_a: real > a > a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
collec1300223524l_real: ( produc957004601l_real > $o ) > set_Pr147102617l_real ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
member384820540a_real: produc1826081171a_real > set_Pr1741234931a_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
member1068169442l_real: produc957004601l_real > set_Pr147102617l_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_relation,type,
relation: set_Product_prod_a_a ).
thf(sy_v_u,type,
u: real ).
thf(sy_v_v,type,
v: real ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (353)
thf(fact_0_assms_I1_J,axiom,
prefer529818233pref_a @ relation ).
% assms(1)
thf(fact_1_indifferent__imp__weak__pref_I2_J,axiom,
! [X: a,Y: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ).
% indifferent_imp_weak_pref(2)
thf(fact_2_indifferent__imp__weak__pref_I1_J,axiom,
! [X: a,Y: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ).
% indifferent_imp_weak_pref(1)
thf(fact_3_indiff__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% indiff_trans
thf(fact_4_strict__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
=> ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
& ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% strict_trans
thf(fact_5_assms_I2_J,axiom,
member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation ).
% assms(2)
thf(fact_6__092_060open_062u_A_092_060noteq_062_A0_A_092_060and_062_Au_A_092_060noteq_062_A1_A_092_060longrightarrow_062_Au_A_K_092_060_094sub_062R_Ax_A_L_Av_A_K_092_060_094sub_062R_Ay_A_092_060succeq_062_Ay_092_060close_062,axiom,
( ( ( u != zero_zero_real )
& ( u != one_one_real ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ) ).
% \<open>u \<noteq> 0 \<and> u \<noteq> 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<succeq> y\<close>
thf(fact_7_u__0,axiom,
( ( u = zero_zero_real )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ) ).
% u_0
thf(fact_8_assms_I4_J,axiom,
ord_less_eq_real @ zero_zero_real @ v ).
% assms(4)
thf(fact_9_assms_I5_J,axiom,
( ( plus_plus_real @ u @ v )
= one_one_real ) ).
% assms(5)
thf(fact_10_assms_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ u ).
% assms(3)
thf(fact_11_scaleR__Pair,axiom,
! [R: real,A: a,B: product_prod_a_a] :
( ( real_V296206913od_a_a @ R @ ( produc1299253312od_a_a @ A @ B ) )
= ( produc1299253312od_a_a @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_12_scaleR__Pair,axiom,
! [R: real,A: a,B: real] :
( ( real_V512797224a_real @ R @ ( product_Pair_a_real @ A @ B ) )
= ( product_Pair_a_real @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_13_scaleR__Pair,axiom,
! [R: real,A: product_prod_a_a,B: a] :
( ( real_V1538726831_a_a_a @ R @ ( produc2061588782_a_a_a @ A @ B ) )
= ( produc2061588782_a_a_a @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_14_scaleR__Pair,axiom,
! [R: real,A: product_prod_a_a,B: product_prod_a_a] :
( ( real_V368978776od_a_a @ R @ ( produc1474507607od_a_a @ A @ B ) )
= ( produc1474507607od_a_a @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_15_scaleR__Pair,axiom,
! [R: real,A: product_prod_a_a,B: real] :
( ( real_V195069393a_real @ R @ ( produc454274172a_real @ A @ B ) )
= ( produc454274172a_real @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_16_scaleR__Pair,axiom,
! [R: real,A: real,B: a] :
( ( real_V1214981142real_a @ R @ ( product_Pair_real_a @ A @ B ) )
= ( product_Pair_real_a @ ( real_V453051771R_real @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_17_scaleR__Pair,axiom,
! [R: real,A: real,B: product_prod_a_a] :
( ( real_V1943155903od_a_a @ R @ ( produc454550562od_a_a @ A @ B ) )
= ( produc454550562od_a_a @ ( real_V453051771R_real @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_18_scaleR__Pair,axiom,
! [R: real,A: real,B: real] :
( ( real_V1139189034l_real @ R @ ( produc705216881l_real @ A @ B ) )
= ( produc705216881l_real @ ( real_V453051771R_real @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_19_scaleR__Pair,axiom,
! [R: real,A: a,B: a] :
( ( real_V543523736od_a_a @ R @ ( product_Pair_a_a @ A @ B ) )
= ( product_Pair_a_a @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% scaleR_Pair
thf(fact_20_scale__prod,axiom,
! [X: real,A: a,B: product_prod_a_a] :
( ( real_V296206913od_a_a @ X @ ( produc1299253312od_a_a @ A @ B ) )
= ( produc1299253312od_a_a @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% scale_prod
thf(fact_21_scale__prod,axiom,
! [X: real,A: a,B: real] :
( ( real_V512797224a_real @ X @ ( product_Pair_a_real @ A @ B ) )
= ( product_Pair_a_real @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% scale_prod
thf(fact_22_scale__prod,axiom,
! [X: real,A: product_prod_a_a,B: a] :
( ( real_V1538726831_a_a_a @ X @ ( produc2061588782_a_a_a @ A @ B ) )
= ( produc2061588782_a_a_a @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% scale_prod
thf(fact_23_scale__prod,axiom,
! [X: real,A: product_prod_a_a,B: product_prod_a_a] :
( ( real_V368978776od_a_a @ X @ ( produc1474507607od_a_a @ A @ B ) )
= ( produc1474507607od_a_a @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% scale_prod
thf(fact_24_scale__prod,axiom,
! [X: real,A: product_prod_a_a,B: real] :
( ( real_V195069393a_real @ X @ ( produc454274172a_real @ A @ B ) )
= ( produc454274172a_real @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% scale_prod
thf(fact_25_scale__prod,axiom,
! [X: real,A: real,B: a] :
( ( real_V1214981142real_a @ X @ ( product_Pair_real_a @ A @ B ) )
= ( product_Pair_real_a @ ( real_V453051771R_real @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% scale_prod
thf(fact_26_scale__prod,axiom,
! [X: real,A: real,B: product_prod_a_a] :
( ( real_V1943155903od_a_a @ X @ ( produc454550562od_a_a @ A @ B ) )
= ( produc454550562od_a_a @ ( real_V453051771R_real @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% scale_prod
thf(fact_27_scale__prod,axiom,
! [X: real,A: real,B: real] :
( ( real_V1139189034l_real @ X @ ( produc705216881l_real @ A @ B ) )
= ( produc705216881l_real @ ( real_V453051771R_real @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% scale_prod
thf(fact_28_scale__prod,axiom,
! [X: real,A: a,B: a] :
( ( real_V543523736od_a_a @ X @ ( product_Pair_a_a @ A @ B ) )
= ( product_Pair_a_a @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% scale_prod
thf(fact_29_add__Pair,axiom,
! [A: a,B: a,C: a,D: a] :
( ( plus_p1505579230od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Pair_a_a @ C @ D ) )
= ( product_Pair_a_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% add_Pair
thf(fact_30_add__Pair,axiom,
! [A: a,B: real,C: a,D: real] :
( ( plus_p541014626a_real @ ( product_Pair_a_real @ A @ B ) @ ( product_Pair_a_real @ C @ D ) )
= ( product_Pair_a_real @ ( plus_plus_a @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% add_Pair
thf(fact_31_add__Pair,axiom,
! [A: real,B: a,C: real,D: a] :
( ( plus_p1243198544real_a @ ( product_Pair_real_a @ A @ B ) @ ( product_Pair_real_a @ C @ D ) )
= ( product_Pair_real_a @ ( plus_plus_real @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% add_Pair
thf(fact_32_add__Pair,axiom,
! [A: real,B: real,C: real,D: real] :
( ( plus_p77862768l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
= ( produc705216881l_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% add_Pair
thf(fact_33_add__Pair,axiom,
! [A: a,B: set_real,C: a,D: set_real] :
( ( plus_p1745648536t_real @ ( produc1391501065t_real @ A @ B ) @ ( produc1391501065t_real @ C @ D ) )
= ( produc1391501065t_real @ ( plus_plus_a @ A @ C ) @ ( plus_plus_set_real @ B @ D ) ) ) ).
% add_Pair
thf(fact_34_add__Pair,axiom,
! [A: a,B: set_a,C: a,D: set_a] :
( ( plus_p1265825342_set_a @ ( product_Pair_a_set_a @ A @ B ) @ ( product_Pair_a_set_a @ C @ D ) )
= ( product_Pair_a_set_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_set_a @ B @ D ) ) ) ).
% add_Pair
thf(fact_35_add__Pair,axiom,
! [A: real,B: set_real,C: real,D: set_real] :
( ( plus_p2005278886t_real @ ( produc247649703t_real @ A @ B ) @ ( produc247649703t_real @ C @ D ) )
= ( produc247649703t_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_set_real @ B @ D ) ) ) ).
% add_Pair
thf(fact_36_add__Pair,axiom,
! [A: real,B: set_a,C: real,D: set_a] :
( ( plus_p125342128_set_a @ ( produc1872317017_set_a @ A @ B ) @ ( produc1872317017_set_a @ C @ D ) )
= ( produc1872317017_set_a @ ( plus_plus_real @ A @ C ) @ ( plus_plus_set_a @ B @ D ) ) ) ).
% add_Pair
thf(fact_37_add__Pair,axiom,
! [A: set_real,B: a,C: set_real,D: a] :
( ( plus_p1582417306real_a @ ( produc617496131real_a @ A @ B ) @ ( produc617496131real_a @ C @ D ) )
= ( produc617496131real_a @ ( plus_plus_set_real @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% add_Pair
thf(fact_38_add__Pair,axiom,
! [A: set_real,B: real,C: set_real,D: real] :
( ( plus_p1952869542l_real @ ( produc1812461223l_real @ A @ B ) @ ( produc1812461223l_real @ C @ D ) )
= ( produc1812461223l_real @ ( plus_plus_set_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% add_Pair
thf(fact_39_prod_Oinject,axiom,
! [X1: real,X2: real,Y1: real,Y2: real] :
( ( ( produc705216881l_real @ X1 @ X2 )
= ( produc705216881l_real @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_40_prod_Oinject,axiom,
! [X1: real,X2: a,Y1: real,Y2: a] :
( ( ( product_Pair_real_a @ X1 @ X2 )
= ( product_Pair_real_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_41_prod_Oinject,axiom,
! [X1: a,X2: real,Y1: a,Y2: real] :
( ( ( product_Pair_a_real @ X1 @ X2 )
= ( product_Pair_a_real @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_42_prod_Oinject,axiom,
! [X1: a,X2: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X2 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_43_old_Oprod_Oinject,axiom,
! [A: real,B: real,A2: real,B2: real] :
( ( ( produc705216881l_real @ A @ B )
= ( produc705216881l_real @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_44_old_Oprod_Oinject,axiom,
! [A: real,B: a,A2: real,B2: a] :
( ( ( product_Pair_real_a @ A @ B )
= ( product_Pair_real_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_45_old_Oprod_Oinject,axiom,
! [A: a,B: real,A2: a,B2: real] :
( ( ( product_Pair_a_real @ A @ B )
= ( product_Pair_a_real @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_46_old_Oprod_Oinject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_47_set__plus__intro,axiom,
! [A: produc957004601l_real,C2: set_Pr147102617l_real,B: produc957004601l_real,D2: set_Pr147102617l_real] :
( ( member1068169442l_real @ A @ C2 )
=> ( ( member1068169442l_real @ B @ D2 )
=> ( member1068169442l_real @ ( plus_p77862768l_real @ A @ B ) @ ( plus_p1708760016l_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_48_set__plus__intro,axiom,
! [A: product_prod_a_real,C2: set_Pr1928503567a_real,B: product_prod_a_real,D2: set_Pr1928503567a_real] :
( ( member1103263856a_real @ A @ C2 )
=> ( ( member1103263856a_real @ B @ D2 )
=> ( member1103263856a_real @ ( plus_p541014626a_real @ A @ B ) @ ( plus_p130512152a_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_49_set__plus__intro,axiom,
! [A: set_Product_prod_a_a,C2: set_se1596668135od_a_a,B: set_Product_prod_a_a,D2: set_se1596668135od_a_a] :
( ( member1838126896od_a_a @ A @ C2 )
=> ( ( member1838126896od_a_a @ B @ D2 )
=> ( member1838126896od_a_a @ ( plus_p634297534od_a_a @ A @ B ) @ ( plus_p1613276318od_a_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_50_set__plus__intro,axiom,
! [A: set_real,C2: set_set_real,B: set_real,D2: set_set_real] :
( ( member_set_real @ A @ C2 )
=> ( ( member_set_real @ B @ D2 )
=> ( member_set_real @ ( plus_plus_set_real @ A @ B ) @ ( plus_p768704801t_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_51_set__plus__intro,axiom,
! [A: set_a,C2: set_set_a,B: set_a,D2: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( member_set_a @ B @ D2 )
=> ( member_set_a @ ( plus_plus_set_a @ A @ B ) @ ( plus_plus_set_set_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_52_set__plus__intro,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: product_prod_a_a,D2: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ C2 )
=> ( ( member449909584od_a_a @ B @ D2 )
=> ( member449909584od_a_a @ ( plus_p1505579230od_a_a @ A @ B ) @ ( plus_p634297534od_a_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_53_set__plus__intro,axiom,
! [A: a,C2: set_a,B: a,D2: set_a] :
( ( member_a @ A @ C2 )
=> ( ( member_a @ B @ D2 )
=> ( member_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_54_set__plus__intro,axiom,
! [A: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_55_add__left__cancel,axiom,
! [A: product_prod_a_real,B: product_prod_a_real,C: product_prod_a_real] :
( ( ( plus_p541014626a_real @ A @ B )
= ( plus_p541014626a_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_56_add__left__cancel,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a] :
( ( ( plus_p1505579230od_a_a @ A @ B )
= ( plus_p1505579230od_a_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_57_add__left__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_58_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_59_add__right__cancel,axiom,
! [B: product_prod_a_real,A: product_prod_a_real,C: product_prod_a_real] :
( ( ( plus_p541014626a_real @ B @ A )
= ( plus_p541014626a_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_60_add__right__cancel,axiom,
! [B: product_prod_a_a,A: product_prod_a_a,C: product_prod_a_a] :
( ( ( plus_p1505579230od_a_a @ B @ A )
= ( plus_p1505579230od_a_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_61_add__right__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_62_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_63_pth__9_I3_J,axiom,
! [C: real,X: real,W: real,D: real,Z: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ X ) @ Z ) )
= ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ ( plus_plus_real @ W @ Z ) ) ) ).
% pth_9(3)
thf(fact_64_add__le__cancel__right,axiom,
! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) )
= ( ord_le1342644953l_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_65_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_66_add__le__cancel__left,axiom,
! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) )
= ( ord_le1342644953l_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_67_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_68_add__cancel__right__right,axiom,
! [A: product_prod_a_real,B: product_prod_a_real] :
( ( A
= ( plus_p541014626a_real @ A @ B ) )
= ( B = zero_z705155042a_real ) ) ).
% add_cancel_right_right
thf(fact_69_add__cancel__right__right,axiom,
! [A: produc957004601l_real,B: produc957004601l_real] :
( ( A
= ( plus_p77862768l_real @ A @ B ) )
= ( B = zero_z659284464l_real ) ) ).
% add_cancel_right_right
thf(fact_70_add__cancel__right__right,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( A
= ( plus_p1505579230od_a_a @ A @ B ) )
= ( B = zero_z950819678od_a_a ) ) ).
% add_cancel_right_right
thf(fact_71_add__cancel__right__right,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ A @ B ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_right
thf(fact_72_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_73_add__cancel__right__left,axiom,
! [A: product_prod_a_real,B: product_prod_a_real] :
( ( A
= ( plus_p541014626a_real @ B @ A ) )
= ( B = zero_z705155042a_real ) ) ).
% add_cancel_right_left
thf(fact_74_add__cancel__right__left,axiom,
! [A: produc957004601l_real,B: produc957004601l_real] :
( ( A
= ( plus_p77862768l_real @ B @ A ) )
= ( B = zero_z659284464l_real ) ) ).
% add_cancel_right_left
thf(fact_75_add__cancel__right__left,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( A
= ( plus_p1505579230od_a_a @ B @ A ) )
= ( B = zero_z950819678od_a_a ) ) ).
% add_cancel_right_left
thf(fact_76_add__cancel__right__left,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ B @ A ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_left
thf(fact_77_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_78_add__cancel__left__right,axiom,
! [A: product_prod_a_real,B: product_prod_a_real] :
( ( ( plus_p541014626a_real @ A @ B )
= A )
= ( B = zero_z705155042a_real ) ) ).
% add_cancel_left_right
thf(fact_79_add__cancel__left__right,axiom,
! [A: produc957004601l_real,B: produc957004601l_real] :
( ( ( plus_p77862768l_real @ A @ B )
= A )
= ( B = zero_z659284464l_real ) ) ).
% add_cancel_left_right
thf(fact_80_add__cancel__left__right,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( ( plus_p1505579230od_a_a @ A @ B )
= A )
= ( B = zero_z950819678od_a_a ) ) ).
% add_cancel_left_right
thf(fact_81_add__cancel__left__right,axiom,
! [A: a,B: a] :
( ( ( plus_plus_a @ A @ B )
= A )
= ( B = zero_zero_a ) ) ).
% add_cancel_left_right
thf(fact_82_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_83_add__cancel__left__left,axiom,
! [B: product_prod_a_real,A: product_prod_a_real] :
( ( ( plus_p541014626a_real @ B @ A )
= A )
= ( B = zero_z705155042a_real ) ) ).
% add_cancel_left_left
thf(fact_84_add__cancel__left__left,axiom,
! [B: produc957004601l_real,A: produc957004601l_real] :
( ( ( plus_p77862768l_real @ B @ A )
= A )
= ( B = zero_z659284464l_real ) ) ).
% add_cancel_left_left
thf(fact_85_add__cancel__left__left,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( ( plus_p1505579230od_a_a @ B @ A )
= A )
= ( B = zero_z950819678od_a_a ) ) ).
% add_cancel_left_left
thf(fact_86_add__cancel__left__left,axiom,
! [B: a,A: a] :
( ( ( plus_plus_a @ B @ A )
= A )
= ( B = zero_zero_a ) ) ).
% add_cancel_left_left
thf(fact_87_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_88_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_89_double__zero,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_zero
thf(fact_90_add_Oright__neutral,axiom,
! [A: product_prod_a_real] :
( ( plus_p541014626a_real @ A @ zero_z705155042a_real )
= A ) ).
% add.right_neutral
thf(fact_91_add_Oright__neutral,axiom,
! [A: set_Product_prod_a_a] :
( ( plus_p634297534od_a_a @ A @ zero_z257140542od_a_a )
= A ) ).
% add.right_neutral
thf(fact_92_add_Oright__neutral,axiom,
! [A: set_real] :
( ( plus_plus_set_real @ A @ zero_zero_set_real )
= A ) ).
% add.right_neutral
thf(fact_93_add_Oright__neutral,axiom,
! [A: set_a] :
( ( plus_plus_set_a @ A @ zero_zero_set_a )
= A ) ).
% add.right_neutral
thf(fact_94_add_Oright__neutral,axiom,
! [A: produc957004601l_real] :
( ( plus_p77862768l_real @ A @ zero_z659284464l_real )
= A ) ).
% add.right_neutral
thf(fact_95_add_Oright__neutral,axiom,
! [A: product_prod_a_a] :
( ( plus_p1505579230od_a_a @ A @ zero_z950819678od_a_a )
= A ) ).
% add.right_neutral
thf(fact_96_add_Oright__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.right_neutral
thf(fact_97_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_98_add_Oleft__neutral,axiom,
! [A: product_prod_a_real] :
( ( plus_p541014626a_real @ zero_z705155042a_real @ A )
= A ) ).
% add.left_neutral
thf(fact_99_add_Oleft__neutral,axiom,
! [A: set_Product_prod_a_a] :
( ( plus_p634297534od_a_a @ zero_z257140542od_a_a @ A )
= A ) ).
% add.left_neutral
thf(fact_100_add_Oleft__neutral,axiom,
! [A: set_real] :
( ( plus_plus_set_real @ zero_zero_set_real @ A )
= A ) ).
% add.left_neutral
thf(fact_101_add_Oleft__neutral,axiom,
! [A: set_a] :
( ( plus_plus_set_a @ zero_zero_set_a @ A )
= A ) ).
% add.left_neutral
thf(fact_102_add_Oleft__neutral,axiom,
! [A: produc957004601l_real] :
( ( plus_p77862768l_real @ zero_z659284464l_real @ A )
= A ) ).
% add.left_neutral
thf(fact_103_add_Oleft__neutral,axiom,
! [A: product_prod_a_a] :
( ( plus_p1505579230od_a_a @ zero_z950819678od_a_a @ A )
= A ) ).
% add.left_neutral
thf(fact_104_add_Oleft__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.left_neutral
thf(fact_105_add_Oleft__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.left_neutral
thf(fact_106_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_107_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_108_le__add__same__cancel2,axiom,
! [A: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ A @ ( plus_p77862768l_real @ B @ A ) )
= ( ord_le1342644953l_real @ zero_z659284464l_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_109_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_110_le__add__same__cancel1,axiom,
! [A: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ A @ ( plus_p77862768l_real @ A @ B ) )
= ( ord_le1342644953l_real @ zero_z659284464l_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_111_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_112_add__le__same__cancel2,axiom,
! [A: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ B ) @ B )
= ( ord_le1342644953l_real @ A @ zero_z659284464l_real ) ) ).
% add_le_same_cancel2
thf(fact_113_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_114_add__le__same__cancel1,axiom,
! [B: produc957004601l_real,A: produc957004601l_real] :
( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ B @ A ) @ B )
= ( ord_le1342644953l_real @ A @ zero_z659284464l_real ) ) ).
% add_le_same_cancel1
thf(fact_115_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_116_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_117_zero__reorient,axiom,
! [X: a] :
( ( zero_zero_a = X )
= ( X = zero_zero_a ) ) ).
% zero_reorient
thf(fact_118_zero__reorient,axiom,
! [X: produc957004601l_real] :
( ( zero_z659284464l_real = X )
= ( X = zero_z659284464l_real ) ) ).
% zero_reorient
thf(fact_119_zero__reorient,axiom,
! [X: product_prod_a_a] :
( ( zero_z950819678od_a_a = X )
= ( X = zero_z950819678od_a_a ) ) ).
% zero_reorient
thf(fact_120_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_121_add__nonpos__eq__0__iff,axiom,
! [X: produc957004601l_real,Y: produc957004601l_real] :
( ( ord_le1342644953l_real @ X @ zero_z659284464l_real )
=> ( ( ord_le1342644953l_real @ Y @ zero_z659284464l_real )
=> ( ( ( plus_p77862768l_real @ X @ Y )
= zero_z659284464l_real )
= ( ( X = zero_z659284464l_real )
& ( Y = zero_z659284464l_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_122_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_123_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A: produc957004601l_real,P: produc957004601l_real > $o] :
( ( member1068169442l_real @ A @ ( collec1300223524l_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_127_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A3: set_Pr147102617l_real] :
( ( collec1300223524l_real
@ ^ [X3: produc957004601l_real] : ( member1068169442l_real @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A3: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_131_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X4: product_prod_a_a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec645855634od_a_a @ P )
= ( collec645855634od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_132_add__nonneg__eq__0__iff,axiom,
! [X: produc957004601l_real,Y: produc957004601l_real] :
( ( ord_le1342644953l_real @ zero_z659284464l_real @ X )
=> ( ( ord_le1342644953l_real @ zero_z659284464l_real @ Y )
=> ( ( ( plus_p77862768l_real @ X @ Y )
= zero_z659284464l_real )
= ( ( X = zero_z659284464l_real )
& ( Y = zero_z659284464l_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_133_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_134_add__nonpos__nonpos,axiom,
! [A: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ A @ zero_z659284464l_real )
=> ( ( ord_le1342644953l_real @ B @ zero_z659284464l_real )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ B ) @ zero_z659284464l_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_135_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_136_add__nonneg__nonneg,axiom,
! [A: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ zero_z659284464l_real @ A )
=> ( ( ord_le1342644953l_real @ zero_z659284464l_real @ B )
=> ( ord_le1342644953l_real @ zero_z659284464l_real @ ( plus_p77862768l_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_137_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_138_add__increasing2,axiom,
! [C: produc957004601l_real,B: produc957004601l_real,A: produc957004601l_real] :
( ( ord_le1342644953l_real @ zero_z659284464l_real @ C )
=> ( ( ord_le1342644953l_real @ B @ A )
=> ( ord_le1342644953l_real @ B @ ( plus_p77862768l_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_139_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_140_add__decreasing2,axiom,
! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ C @ zero_z659284464l_real )
=> ( ( ord_le1342644953l_real @ A @ B )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_141_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_142_add__increasing,axiom,
! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
( ( ord_le1342644953l_real @ zero_z659284464l_real @ A )
=> ( ( ord_le1342644953l_real @ B @ C )
=> ( ord_le1342644953l_real @ B @ ( plus_p77862768l_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_143_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_144_add__decreasing,axiom,
! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ A @ zero_z659284464l_real )
=> ( ( ord_le1342644953l_real @ C @ B )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_145_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_146_zero__prod__def,axiom,
( zero_z950819678od_a_a
= ( product_Pair_a_a @ zero_zero_a @ zero_zero_a ) ) ).
% zero_prod_def
thf(fact_147_zero__prod__def,axiom,
( zero_z659284464l_real
= ( produc705216881l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_148_zero__prod__def,axiom,
( zero_z1407338960real_a
= ( product_Pair_real_a @ zero_zero_real @ zero_zero_a ) ) ).
% zero_prod_def
thf(fact_149_zero__prod__def,axiom,
( zero_z705155042a_real
= ( product_Pair_a_real @ zero_zero_a @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_150_zero__prod__def,axiom,
( zero_z590476811l_real
= ( produc1926903988l_real @ zero_zero_real @ zero_z659284464l_real ) ) ).
% zero_prod_def
thf(fact_151_zero__prod__def,axiom,
( zero_z396988537od_a_a
= ( produc454550562od_a_a @ zero_zero_real @ zero_z950819678od_a_a ) ) ).
% zero_prod_def
thf(fact_152_zero__prod__def,axiom,
( zero_z2135370393l_real
= ( produc64206290l_real @ zero_zero_a @ zero_z659284464l_real ) ) ).
% zero_prod_def
thf(fact_153_zero__prod__def,axiom,
( zero_z53797895od_a_a
= ( produc1299253312od_a_a @ zero_zero_a @ zero_z950819678od_a_a ) ) ).
% zero_prod_def
thf(fact_154_zero__prod__def,axiom,
( zero_z342251165l_real
= ( produc1175086478l_real @ zero_z659284464l_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_155_zero__prod__def,axiom,
( zero_z1937883107real_a
= ( produc1430099868real_a @ zero_z659284464l_real @ zero_zero_a ) ) ).
% zero_prod_def
thf(fact_156_add__le__imp__le__right,axiom,
! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) )
=> ( ord_le1342644953l_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_157_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_158_add__le__imp__le__left,axiom,
! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) )
=> ( ord_le1342644953l_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_159_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_160_add__right__mono,axiom,
! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
( ( ord_le1342644953l_real @ A @ B )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_161_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_162_add__left__mono,axiom,
! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
( ( ord_le1342644953l_real @ A @ B )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_163_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_164_add__mono,axiom,
! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real,D: produc957004601l_real] :
( ( ord_le1342644953l_real @ A @ B )
=> ( ( ord_le1342644953l_real @ C @ D )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_165_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_166_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
( ( ( ord_le1342644953l_real @ I @ J )
& ( ord_le1342644953l_real @ K @ L ) )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_167_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_168_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
( ( ( I = J )
& ( ord_le1342644953l_real @ K @ L ) )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_169_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_170_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
( ( ( ord_le1342644953l_real @ I @ J )
& ( K = L ) )
=> ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_171_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_172_pth__4_I1_J,axiom,
! [X: produc957004601l_real] :
( ( real_V1139189034l_real @ zero_zero_real @ X )
= zero_z659284464l_real ) ).
% pth_4(1)
thf(fact_173_pth__4_I1_J,axiom,
! [X: real] :
( ( real_V453051771R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% pth_4(1)
thf(fact_174_convex__ge__imp__conved,axiom,
! [Pr: set_Pr1741234931a_real] :
( ! [X4: product_prod_a_real,Y3: product_prod_a_real] :
( ( member384820540a_real @ ( produc886678603a_real @ X4 @ Y3 ) @ Pr )
=> ! [Alpha: real,Beta: real] :
( ( ( ( plus_plus_real @ Alpha @ Beta )
= one_one_real )
& ( ord_less_eq_real @ zero_zero_real @ Alpha )
& ( ord_less_eq_real @ zero_zero_real @ Beta ) )
=> ( member384820540a_real @ ( produc886678603a_real @ ( plus_p541014626a_real @ ( real_V512797224a_real @ Alpha @ X4 ) @ ( real_V512797224a_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
=> ( prefer1113819806a_real @ Pr ) ) ).
% convex_ge_imp_conved
thf(fact_175_convex__ge__imp__conved,axiom,
! [Pr: set_Pr1948701895od_a_a] :
( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Pr )
=> ! [Alpha: real,Beta: real] :
( ( ( ( plus_plus_real @ Alpha @ Beta )
= one_one_real )
& ( ord_less_eq_real @ zero_zero_real @ Alpha )
& ( ord_less_eq_real @ zero_zero_real @ Beta ) )
=> ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( plus_p1505579230od_a_a @ ( real_V543523736od_a_a @ Alpha @ X4 ) @ ( real_V543523736od_a_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
=> ( prefer225425826od_a_a @ Pr ) ) ).
% convex_ge_imp_conved
thf(fact_176_convex__ge__imp__conved,axiom,
! [Pr: set_Pr147102617l_real] :
( ! [X4: real,Y3: real] :
( ( member1068169442l_real @ ( produc705216881l_real @ X4 @ Y3 ) @ Pr )
=> ! [Alpha: real,Beta: real] :
( ( ( ( plus_plus_real @ Alpha @ Beta )
= one_one_real )
& ( ord_less_eq_real @ zero_zero_real @ Alpha )
& ( ord_less_eq_real @ zero_zero_real @ Beta ) )
=> ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha @ X4 ) @ ( real_V453051771R_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
=> ( prefer1247792113f_real @ Pr ) ) ).
% convex_ge_imp_conved
thf(fact_177_convex__ge__imp__conved,axiom,
! [Pr: set_Product_prod_a_a] :
( ! [X4: a,Y3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Pr )
=> ! [Alpha: real,Beta: real] :
( ( ( ( plus_plus_real @ Alpha @ Beta )
= one_one_real )
& ( ord_less_eq_real @ zero_zero_real @ Alpha )
& ( ord_less_eq_real @ zero_zero_real @ Beta ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha @ X4 ) @ ( real_V1035702895aleR_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
=> ( prefer529818233pref_a @ Pr ) ) ).
% convex_ge_imp_conved
thf(fact_178_add_Ogroup__left__neutral,axiom,
! [A: product_prod_a_real] :
( ( plus_p541014626a_real @ zero_z705155042a_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_179_add_Ogroup__left__neutral,axiom,
! [A: produc957004601l_real] :
( ( plus_p77862768l_real @ zero_z659284464l_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_180_add_Ogroup__left__neutral,axiom,
! [A: product_prod_a_a] :
( ( plus_p1505579230od_a_a @ zero_z950819678od_a_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_181_add_Ogroup__left__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_182_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_183_add_Ocomm__neutral,axiom,
! [A: set_a] :
( ( plus_plus_set_a @ A @ zero_zero_set_a )
= A ) ).
% add.comm_neutral
thf(fact_184_add_Ocomm__neutral,axiom,
! [A: produc957004601l_real] :
( ( plus_p77862768l_real @ A @ zero_z659284464l_real )
= A ) ).
% add.comm_neutral
thf(fact_185_add_Ocomm__neutral,axiom,
! [A: product_prod_a_a] :
( ( plus_p1505579230od_a_a @ A @ zero_z950819678od_a_a )
= A ) ).
% add.comm_neutral
thf(fact_186_add_Ocomm__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.comm_neutral
thf(fact_187_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_188_comm__monoid__add__class_Oadd__0,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_189_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_190_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_191_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_192_pth__4_I2_J,axiom,
! [C: real] :
( ( real_V453051771R_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% pth_4(2)
thf(fact_193_add__right__imp__eq,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_194_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_195_add__left__imp__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_196_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_197_add_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_198_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_199_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A4: a,B3: a] : ( plus_plus_a @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_200_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_201_add_Oright__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_202_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_203_add_Oleft__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_204_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_205_add_Oassoc,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_206_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_207_set__plus__elim,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ A3 @ B4 ) )
=> ~ ! [A5: product_prod_a_a,B5: product_prod_a_a] :
( ( X
= ( plus_p1505579230od_a_a @ A5 @ B5 ) )
=> ( ( member449909584od_a_a @ A5 @ A3 )
=> ~ ( member449909584od_a_a @ B5 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_208_set__plus__elim,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( member_a @ X @ ( plus_plus_set_a @ A3 @ B4 ) )
=> ~ ! [A5: a,B5: a] :
( ( X
= ( plus_plus_a @ A5 @ B5 ) )
=> ( ( member_a @ A5 @ A3 )
=> ~ ( member_a @ B5 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_209_set__plus__elim,axiom,
! [X: real,A3: set_real,B4: set_real] :
( ( member_real @ X @ ( plus_plus_set_real @ A3 @ B4 ) )
=> ~ ! [A5: real,B5: real] :
( ( X
= ( plus_plus_real @ A5 @ B5 ) )
=> ( ( member_real @ A5 @ A3 )
=> ~ ( member_real @ B5 @ B4 ) ) ) ) ).
% set_plus_elim
thf(fact_210_group__cancel_Oadd2,axiom,
! [B4: a,K: a,B: a,A: a] :
( ( B4
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B4 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_211_group__cancel_Oadd2,axiom,
! [B4: real,K: real,B: real,A: real] :
( ( B4
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_212_group__cancel_Oadd1,axiom,
! [A3: a,K: a,A: a,B: a] :
( ( A3
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A3 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_213_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_214_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_215_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_216_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_217_old_Oprod_Oinducts,axiom,
! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
( ! [A5: a,B5: a] : ( P @ ( product_Pair_a_a @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_218_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A5: a,B5: a] :
( Y
!= ( product_Pair_a_a @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_219_Pair__inject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_220_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A5: a,B5: a] : ( P @ ( product_Pair_a_a @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_221_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X4: a,Y3: a] :
( P2
= ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_222_pth__a,axiom,
! [X: real,Y: real] :
( ( plus_plus_real @ ( real_V453051771R_real @ zero_zero_real @ X ) @ Y )
= Y ) ).
% pth_a
thf(fact_223_pth__8,axiom,
! [C: real,X: real,D: real] :
( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ X ) )
= ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) ) ).
% pth_8
thf(fact_224_pth__6,axiom,
! [C: real,X: real,Y: real] :
( ( real_V453051771R_real @ C @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ C @ Y ) ) ) ).
% pth_6
thf(fact_225_pth__c_I1_J,axiom,
! [C: real,X: real,D: real,Y: real] :
( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) )
= ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( real_V453051771R_real @ C @ X ) ) ) ).
% pth_c(1)
thf(fact_226_pth__c_I2_J,axiom,
! [C: real,X: real,Z: real,D: real,Y: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ Y ) )
= ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) ) ) ).
% pth_c(2)
thf(fact_227_pth__c_I3_J,axiom,
! [C: real,X: real,D: real,Y: real,Z: real] :
( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
= ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) ) ) ).
% pth_c(3)
thf(fact_228_pth__c_I4_J,axiom,
! [C: real,X: real,W: real,D: real,Y: real,Z: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
= ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ Z ) ) ) ).
% pth_c(4)
thf(fact_229_pth__b_I1_J,axiom,
! [C: real,X: real,D: real,Y: real] :
( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) )
= ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) ) ) ).
% pth_b(1)
thf(fact_230_pth__b_I2_J,axiom,
! [C: real,X: real,Z: real,D: real,Y: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ Y ) )
= ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ Z @ ( real_V453051771R_real @ D @ Y ) ) ) ) ).
% pth_b(2)
thf(fact_231_pth__b_I3_J,axiom,
! [C: real,X: real,D: real,Y: real,Z: real] :
( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
= ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) ) ) ).
% pth_b(3)
thf(fact_232_pth__b_I4_J,axiom,
! [C: real,X: real,W: real,D: real,Y: real,Z: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
= ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ W @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) ) ) ) ).
% pth_b(4)
thf(fact_233_pth__9_I1_J,axiom,
! [C: real,X: real,Z: real,D: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ X ) )
= ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ Z ) ) ).
% pth_9(1)
thf(fact_234_pth__9_I2_J,axiom,
! [C: real,X: real,D: real,Z: real] :
( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ X ) @ Z ) )
= ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ Z ) ) ).
% pth_9(2)
thf(fact_235_scaleR__eq__iff,axiom,
! [B: real,U: real,A: real] :
( ( ( plus_plus_real @ B @ ( real_V453051771R_real @ U @ A ) )
= ( plus_plus_real @ A @ ( real_V453051771R_real @ U @ B ) ) )
= ( ( A = B )
| ( U = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_236_scaleR__eq__iff,axiom,
! [B: a,U: real,A: a] :
( ( ( plus_plus_a @ B @ ( real_V1035702895aleR_a @ U @ A ) )
= ( plus_plus_a @ A @ ( real_V1035702895aleR_a @ U @ B ) ) )
= ( ( A = B )
| ( U = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_237_scale__zero__left,axiom,
! [X: real] :
( ( real_V453051771R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% scale_zero_left
thf(fact_238_scale__zero__left,axiom,
! [X: a] :
( ( real_V1035702895aleR_a @ zero_zero_real @ X )
= zero_zero_a ) ).
% scale_zero_left
thf(fact_239_scale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( real_V453051771R_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% scale_eq_0_iff
thf(fact_240_scale__eq__0__iff,axiom,
! [A: real,X: a] :
( ( ( real_V1035702895aleR_a @ A @ X )
= zero_zero_a )
= ( ( A = zero_zero_real )
| ( X = zero_zero_a ) ) ) ).
% scale_eq_0_iff
thf(fact_241_scaleR__one,axiom,
! [X: a] :
( ( real_V1035702895aleR_a @ one_one_real @ X )
= X ) ).
% scaleR_one
thf(fact_242_scale__cancel__left,axiom,
! [A: real,X: a,Y: a] :
( ( ( real_V1035702895aleR_a @ A @ X )
= ( real_V1035702895aleR_a @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% scale_cancel_left
thf(fact_243_scaleR__left__le__one__le,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ X ) ) ) ).
% scaleR_left_le_one_le
thf(fact_244_scale__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( real_V453051771R_real @ A @ X )
= ( real_V453051771R_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% scale_cancel_right
thf(fact_245_scale__cancel__right,axiom,
! [A: real,X: a,B: real] :
( ( ( real_V1035702895aleR_a @ A @ X )
= ( real_V1035702895aleR_a @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_a ) ) ) ).
% scale_cancel_right
thf(fact_246_scale__zero__right,axiom,
! [A: real] :
( ( real_V453051771R_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% scale_zero_right
thf(fact_247_scale__zero__right,axiom,
! [A: real] :
( ( real_V1035702895aleR_a @ A @ zero_zero_a )
= zero_zero_a ) ).
% scale_zero_right
thf(fact_248_split__scaleR__neg__le,axiom,
! [A: real,X: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ X @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ X ) ) )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ).
% split_scaleR_neg_le
thf(fact_249_split__scaleR__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) ) ) ).
% split_scaleR_pos_le
thf(fact_250_set__plus__mono2__b,axiom,
! [C2: set_Product_prod_a_a,D2: set_Product_prod_a_a,E: set_Product_prod_a_a,F: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ C2 @ D2 )
=> ( ( ord_le1824328871od_a_a @ E @ F )
=> ( ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ C2 @ E ) )
=> ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ D2 @ F ) ) ) ) ) ).
% set_plus_mono2_b
thf(fact_251_set__zero__plus2,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( member449909584od_a_a @ zero_z950819678od_a_a @ A3 )
=> ( ord_le1824328871od_a_a @ B4 @ ( plus_p634297534od_a_a @ A3 @ B4 ) ) ) ).
% set_zero_plus2
thf(fact_252_set__zero__plus2,axiom,
! [A3: set_real,B4: set_real] :
( ( member_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_set_real @ B4 @ ( plus_plus_set_real @ A3 @ B4 ) ) ) ).
% set_zero_plus2
thf(fact_253_scale__left__commute,axiom,
! [A: real,B: real,X: a] :
( ( real_V1035702895aleR_a @ A @ ( real_V1035702895aleR_a @ B @ X ) )
= ( real_V1035702895aleR_a @ B @ ( real_V1035702895aleR_a @ A @ X ) ) ) ).
% scale_left_commute
thf(fact_254_scale__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( real_V453051771R_real @ A @ X )
= ( real_V453051771R_real @ B @ X ) )
=> ( A = B ) ) ) ).
% scale_right_imp_eq
thf(fact_255_scale__right__imp__eq,axiom,
! [X: a,A: real,B: real] :
( ( X != zero_zero_a )
=> ( ( ( real_V1035702895aleR_a @ A @ X )
= ( real_V1035702895aleR_a @ B @ X ) )
=> ( A = B ) ) ) ).
% scale_right_imp_eq
thf(fact_256_scaleR__add__right,axiom,
! [A: real,X: real,Y: real] :
( ( real_V453051771R_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ A @ Y ) ) ) ).
% scaleR_add_right
thf(fact_257_scaleR__add__right,axiom,
! [A: real,X: a,Y: a] :
( ( real_V1035702895aleR_a @ A @ ( plus_plus_a @ X @ Y ) )
= ( plus_plus_a @ ( real_V1035702895aleR_a @ A @ X ) @ ( real_V1035702895aleR_a @ A @ Y ) ) ) ).
% scaleR_add_right
thf(fact_258_scale__left__imp__eq,axiom,
! [A: real,X: a,Y: a] :
( ( A != zero_zero_real )
=> ( ( ( real_V1035702895aleR_a @ A @ X )
= ( real_V1035702895aleR_a @ A @ Y ) )
=> ( X = Y ) ) ) ).
% scale_left_imp_eq
thf(fact_259_scaleR__left_Oadd,axiom,
! [X: real,Y: real,Xa: real] :
( ( real_V453051771R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
= ( plus_plus_real @ ( real_V453051771R_real @ X @ Xa ) @ ( real_V453051771R_real @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_260_scaleR__left_Oadd,axiom,
! [X: real,Y: real,Xa: a] :
( ( real_V1035702895aleR_a @ ( plus_plus_real @ X @ Y ) @ Xa )
= ( plus_plus_a @ ( real_V1035702895aleR_a @ X @ Xa ) @ ( real_V1035702895aleR_a @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_261_scaleR__add__left,axiom,
! [A: real,B: real,X: real] :
( ( real_V453051771R_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ X ) ) ) ).
% scaleR_add_left
thf(fact_262_scaleR__add__left,axiom,
! [A: real,B: real,X: a] :
( ( real_V1035702895aleR_a @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_a @ ( real_V1035702895aleR_a @ A @ X ) @ ( real_V1035702895aleR_a @ B @ X ) ) ) ).
% scaleR_add_left
thf(fact_263_scaleR__right__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ X ) ) ) ) ).
% scaleR_right_mono
thf(fact_264_scaleR__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ C ) @ ( real_V453051771R_real @ B @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_265_scaleR__left__mono,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ A @ Y ) ) ) ) ).
% scaleR_left_mono
thf(fact_266_scaleR__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_267_scaleR__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ C ) @ ( real_V453051771R_real @ B @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_268_scaleR__mono,axiom,
! [A: real,B: real,X: real,Y: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_269_scaleR__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_270_scaleR__nonpos__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_271_scaleR__nonneg__nonpos,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_272_scaleR__nonneg__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ X ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_273_Pair__le,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
= ( ( ord_less_eq_real @ A @ C )
& ( ord_less_eq_real @ B @ D ) ) ) ).
% Pair_le
thf(fact_274_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_275_zero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_le_one
thf(fact_276_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_277_Pair__mono,axiom,
! [X: real,X5: real,Y: real,Y4: real] :
( ( ord_less_eq_real @ X @ X5 )
=> ( ( ord_less_eq_real @ Y @ Y4 )
=> ( ord_le1342644953l_real @ ( produc705216881l_real @ X @ Y ) @ ( produc705216881l_real @ X5 @ Y4 ) ) ) ) ).
% Pair_mono
thf(fact_278_weak__convexI,axiom,
! [Pr: set_Pr147102617l_real] :
( ! [X4: real,Y3: real,Alpha: real,Beta: real] :
( ( member1068169442l_real @ ( produc705216881l_real @ X4 @ Y3 ) @ Pr )
=> ( ( ( plus_plus_real @ Alpha @ Beta )
= one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ Alpha )
=> ( ( ord_less_real @ zero_zero_real @ Beta )
=> ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha @ X4 ) @ ( real_V453051771R_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) ) ) )
=> ( prefer1247792113f_real @ Pr ) ) ).
% weak_convexI
thf(fact_279_weak__convexI,axiom,
! [Pr: set_Product_prod_a_a] :
( ! [X4: a,Y3: a,Alpha: real,Beta: real] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Pr )
=> ( ( ( plus_plus_real @ Alpha @ Beta )
= one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ Alpha )
=> ( ( ord_less_real @ zero_zero_real @ Beta )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha @ X4 ) @ ( real_V1035702895aleR_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) ) ) )
=> ( prefer529818233pref_a @ Pr ) ) ).
% weak_convexI
thf(fact_280_subsetI,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member449909584od_a_a @ X4 @ A3 )
=> ( member449909584od_a_a @ X4 @ B4 ) )
=> ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_281_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_282_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_283_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_284_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_285_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_286_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_287_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_288_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_289_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_290_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_291_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_292_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_293_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_294_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_295_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_296_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_297_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_298_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_299_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_300_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_301_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_302_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_303_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_304_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_305_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_306_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_307_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_308_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_309_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_310_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_311_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_312_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_313_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_314_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_315_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_316_scaleR__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_317_scaleR__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_318_scaleR__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_319_subset__iff,axiom,
( ord_le1824328871od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member449909584od_a_a @ T @ A6 )
=> ( member449909584od_a_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_320_subset__eq,axiom,
( ord_le1824328871od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A6 )
=> ( member449909584od_a_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_321_subsetD,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B4 )
=> ( ( member449909584od_a_a @ C @ A3 )
=> ( member449909584od_a_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_322_in__mono,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B4 )
=> ( ( member449909584od_a_a @ X @ A3 )
=> ( member449909584od_a_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_323_zero__le__scaleR__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( A = zero_zero_real ) ) ) ).
% zero_le_scaleR_iff
thf(fact_324_scaleR__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( real_V453051771R_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( A = zero_zero_real ) ) ) ).
% scaleR_le_0_iff
thf(fact_325_weak__convexD,axiom,
! [Pr: set_Pr147102617l_real,X: real,Y: real,U: real,V: real] :
( ( prefer1247792113f_real @ Pr )
=> ( ( member1068169442l_real @ ( produc705216881l_real @ X @ Y ) @ Pr )
=> ( ( ord_less_real @ zero_zero_real @ U )
=> ( ( ord_less_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ U @ X ) @ ( real_V453051771R_real @ V @ Y ) ) @ Y ) @ Pr ) ) ) ) ) ) ).
% weak_convexD
thf(fact_326_weak__convexD,axiom,
! [Pr: set_Product_prod_a_a,X: a,Y: a,U: real,V: real] :
( ( prefer529818233pref_a @ Pr )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr )
=> ( ( ord_less_real @ zero_zero_real @ U )
=> ( ( ord_less_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ X ) @ ( real_V1035702895aleR_a @ V @ Y ) ) @ Y ) @ Pr ) ) ) ) ) ) ).
% weak_convexD
thf(fact_327_weak__convex__pref__def,axiom,
( prefer1247792113f_real
= ( ^ [Pr2: set_Pr147102617l_real] :
! [X3: real,Y5: real] :
( ( member1068169442l_real @ ( produc705216881l_real @ X3 @ Y5 ) @ Pr2 )
=> ! [Alpha2: real,Beta2: real] :
( ( ( ( plus_plus_real @ Alpha2 @ Beta2 )
= one_one_real )
& ( ord_less_real @ zero_zero_real @ Alpha2 )
& ( ord_less_real @ zero_zero_real @ Beta2 ) )
=> ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha2 @ X3 ) @ ( real_V453051771R_real @ Beta2 @ Y5 ) ) @ Y5 ) @ Pr2 ) ) ) ) ) ).
% weak_convex_pref_def
thf(fact_328_weak__convex__pref__def,axiom,
( prefer529818233pref_a
= ( ^ [Pr2: set_Product_prod_a_a] :
! [X3: a,Y5: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y5 ) @ Pr2 )
=> ! [Alpha2: real,Beta2: real] :
( ( ( ( plus_plus_real @ Alpha2 @ Beta2 )
= one_one_real )
& ( ord_less_real @ zero_zero_real @ Alpha2 )
& ( ord_less_real @ zero_zero_real @ Beta2 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha2 @ X3 ) @ ( real_V1035702895aleR_a @ Beta2 @ Y5 ) ) @ Y5 ) @ Pr2 ) ) ) ) ) ).
% weak_convex_pref_def
thf(fact_329_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_330_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_331_psubsetD,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le108336051od_a_a @ A3 @ B4 )
=> ( ( member449909584od_a_a @ C @ A3 )
=> ( member449909584od_a_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_332_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_333_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_334_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_335_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_336_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_337_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_338_add__scaleR__degen,axiom,
! [U: real,B: real,V: real,A: real] :
( ( ( plus_plus_real @ ( real_V453051771R_real @ U @ B ) @ ( real_V453051771R_real @ V @ A ) )
= ( plus_plus_real @ ( real_V453051771R_real @ U @ A ) @ ( real_V453051771R_real @ V @ B ) ) )
=> ( ( U != V )
=> ( A = B ) ) ) ).
% add_scaleR_degen
thf(fact_339_add__scaleR__degen,axiom,
! [U: real,B: a,V: real,A: a] :
( ( ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ B ) @ ( real_V1035702895aleR_a @ V @ A ) )
= ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ A ) @ ( real_V1035702895aleR_a @ V @ B ) ) )
=> ( ( U != V )
=> ( A = B ) ) ) ).
% add_scaleR_degen
thf(fact_340_order__subst1,axiom,
! [A: real,F2: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_341_order__subst2,axiom,
! [A: real,B: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_342_ord__eq__le__subst,axiom,
! [A: real,F2: real > real,B: real,C: real] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_343_ord__le__eq__subst,axiom,
! [A: real,B: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_344_eq__iff,axiom,
( ( ^ [Y6: real,Z2: real] : Y6 = Z2 )
= ( ^ [X3: real,Y5: real] :
( ( ord_less_eq_real @ X3 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_345_antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_346_linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linear
thf(fact_347_eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% eq_refl
thf(fact_348_le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% le_cases
thf(fact_349_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_350_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_351_antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_352_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y6: real,Z2: real] : Y6 = Z2 )
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
% Conjectures (1)
thf(conj_0,conjecture,
member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ).
%------------------------------------------------------------------------------