TPTP Problem File: ITP109^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP109^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Lower_Semicontinuous problem prob_1098__6259092_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 : Lower_Semicontinuous/prob_1098__6259092_1 [Des21]
% Status : Theorem
% Rating : 0.80 v8.2.0, 0.69 v8.1.0, 0.82 v7.5.0
% Syntax : Number of formulae : 418 ( 209 unt; 59 typ; 0 def)
% Number of atoms : 942 ( 525 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 3880 ( 121 ~; 12 |; 114 &;3253 @)
% ( 0 <=>; 380 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 297 ( 297 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 51 usr; 9 con; 0-3 aty)
% Number of variables : 1161 ( 141 ^; 993 !; 27 ?;1161 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:43:58.270
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (51)
thf(sy_c_Convex_Oconvex_001tf__b,type,
convex_b: set_b > $o ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__b_M_Eo_J,type,
minus_minus_b_o: ( b > $o ) > ( b > $o ) > b > $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_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__b,type,
minus_minus_b: b > b > b ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
plus_p2118002693_ereal: extended_ereal > extended_ereal > extended_ereal ).
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_Itf__b_J,type,
plus_plus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__b,type,
plus_plus_b: b > b > b ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups1296524820l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Nat__Onat,type,
groups769445524_a_nat: ( a > nat ) > set_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Real__Oreal,type,
groups1862963056a_real: ( a > real ) > set_a > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Set__Oset_Itf__b_J,type,
groups919362075_set_b: ( a > set_b ) > set_a > set_b ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001tf__b,type,
groups1199149371um_a_b: ( a > b ) > set_a > b ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__b_001t__Nat__Onat,type,
groups2098481813_b_nat: ( b > nat ) > set_b > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__b_001t__Real__Oreal,type,
groups583146225b_real: ( b > real ) > set_b > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__b_001t__Set__Oset_Itf__b_J,type,
groups448962650_set_b: ( b > set_b ) > set_b > set_b ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__b_001tf__b,type,
groups2026918778um_b_b: ( b > b ) > set_b > b ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_If_001tf__b,type,
if_b: $o > b > b > b ).
thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_001tf__b,type,
lower_311861425x_on_b: set_b > ( b > extended_ereal ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
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__b,type,
real_V1035702896aleR_b: real > b > b ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_C,type,
c: set_b ).
thf(sy_v_a,type,
a2: a > real ).
thf(sy_v_aa____,type,
aa: a > real ).
thf(sy_v_f,type,
f: b > extended_ereal ).
thf(sy_v_i____,type,
i: a ).
thf(sy_v_s,type,
s: set_a ).
thf(sy_v_sa____,type,
sa: set_a ).
thf(sy_v_y,type,
y: a > b ).
% Relevant facts (353)
thf(fact_0_insert_Ohyps_I3_J,axiom,
~ ( member_a @ i @ sa ) ).
% insert.hyps(3)
thf(fact_1_asm,axiom,
( ( aa @ i )
!= one_one_real ) ).
% asm
thf(fact_2__092_060open_062_I_092_060Sum_062x_092_060in_062s_O_Aa_Ax_A_K_092_060_094sub_062R_Ay_Ax_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_J_A_061_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_P_092_060_094sub_062R_A_I1_A_N_Aa_Ai_J_092_060close_062,axiom,
( ( groups1199149371um_a_b
@ ^ [X: a] : ( real_V1035702896aleR_b @ ( inverse_inverse_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) ) @ ( real_V1035702896aleR_b @ ( aa @ X ) @ ( y @ X ) ) )
@ sa )
= ( real_V1035702896aleR_b @ ( inverse_inverse_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) )
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( aa @ J ) @ ( y @ J ) )
@ sa ) ) ) ).
% \<open>(\<Sum>x\<in>s. a x *\<^sub>R y x /\<^sub>R (1 - a i)) = (\<Sum>j\<in>s. a j *\<^sub>R y j) /\<^sub>R (1 - a i)\<close>
thf(fact_3_insert_Ohyps_I2_J,axiom,
sa != bot_bot_set_a ).
% insert.hyps(2)
thf(fact_4_insert_Ohyps_I1_J,axiom,
finite_finite_a @ sa ).
% insert.hyps(1)
thf(fact_5__092_060open_062f_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_A_061_Af_A_I_I_I1_A_N_Aa_Ai_J_A_K_Ainverse_A_I1_A_N_Aa_Ai_J_J_A_K_092_060_094sub_062R_A_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062,axiom,
( ( f
@ ( plus_plus_b
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( aa @ J ) @ ( y @ J ) )
@ sa )
@ ( real_V1035702896aleR_b @ ( aa @ i ) @ ( y @ i ) ) ) )
= ( f
@ ( plus_plus_b
@ ( real_V1035702896aleR_b @ ( times_times_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) @ ( inverse_inverse_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) ) )
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( aa @ J ) @ ( y @ J ) )
@ sa ) )
@ ( real_V1035702896aleR_b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% \<open>f ((\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i) = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i)\<close>
thf(fact_6_calculation,axiom,
( ( f
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( aa @ J ) @ ( y @ J ) )
@ ( insert_a @ i @ sa ) ) )
= ( f
@ ( plus_plus_b
@ ( real_V1035702896aleR_b @ ( times_times_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) @ ( inverse_inverse_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) ) )
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( aa @ J ) @ ( y @ J ) )
@ sa ) )
@ ( real_V1035702896aleR_b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% calculation
thf(fact_7__092_060open_062f_A_I_092_060Sum_062j_092_060in_062insert_Ai_As_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_061_Af_A_I_I_092_060Sum_062j_092_060in_062s_O_Aa_Aj_A_K_092_060_094sub_062R_Ay_Aj_J_A_L_Aa_Ai_A_K_092_060_094sub_062R_Ay_Ai_J_092_060close_062,axiom,
( ( f
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( aa @ J ) @ ( y @ J ) )
@ ( insert_a @ i @ sa ) ) )
= ( f
@ ( plus_plus_b
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( aa @ J ) @ ( y @ J ) )
@ sa )
@ ( real_V1035702896aleR_b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
% \<open>f (\<Sum>j\<in>insert i s. a j *\<^sub>R y j) = f ((\<Sum>j\<in>s. a j *\<^sub>R y j) + a i *\<^sub>R y i)\<close>
thf(fact_8_scaleR__collapse,axiom,
! [U: real,A: b] :
( ( plus_plus_b @ ( real_V1035702896aleR_b @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1035702896aleR_b @ U @ A ) )
= A ) ).
% scaleR_collapse
thf(fact_9_scaleR__collapse,axiom,
! [U: real,A: real] :
( ( plus_plus_real @ ( real_V453051771R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V453051771R_real @ U @ A ) )
= A ) ).
% scaleR_collapse
thf(fact_10_scaleR__eq__iff,axiom,
! [B: b,U: real,A: b] :
( ( ( plus_plus_b @ B @ ( real_V1035702896aleR_b @ U @ A ) )
= ( plus_plus_b @ A @ ( real_V1035702896aleR_b @ U @ B ) ) )
= ( ( A = B )
| ( U = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_11_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_12_yai_I1_J,axiom,
member_b @ ( y @ i ) @ c ).
% yai(1)
thf(fact_13_scaleR__scaleR,axiom,
! [A: real,B: real,X2: b] :
( ( real_V1035702896aleR_b @ A @ ( real_V1035702896aleR_b @ B @ X2 ) )
= ( real_V1035702896aleR_b @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% scaleR_scaleR
thf(fact_14_scaleR__scaleR,axiom,
! [A: real,B: real,X2: real] :
( ( real_V453051771R_real @ A @ ( real_V453051771R_real @ B @ X2 ) )
= ( real_V453051771R_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% scaleR_scaleR
thf(fact_15_scaleR__one,axiom,
! [X2: b] :
( ( real_V1035702896aleR_b @ one_one_real @ X2 )
= X2 ) ).
% scaleR_one
thf(fact_16_scaleR__one,axiom,
! [X2: real] :
( ( real_V453051771R_real @ one_one_real @ X2 )
= X2 ) ).
% scaleR_one
thf(fact_17__092_060open_062sum_Aa_As_A_061_A1_A_N_Aa_Ai_092_060close_062,axiom,
( ( groups1862963056a_real @ aa @ sa )
= ( minus_minus_real @ one_one_real @ ( aa @ i ) ) ) ).
% \<open>sum a s = 1 - a i\<close>
thf(fact_18_inverse__eq__1__iff,axiom,
! [X2: real] :
( ( ( inverse_inverse_real @ X2 )
= one_one_real )
= ( X2 = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_19_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_20_inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_21_mult__scaleR__left,axiom,
! [A: real,X2: real,Y: real] :
( ( times_times_real @ ( real_V453051771R_real @ A @ X2 ) @ Y )
= ( real_V453051771R_real @ A @ ( times_times_real @ X2 @ Y ) ) ) ).
% mult_scaleR_left
thf(fact_22_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_23_assms_I2_J,axiom,
s != bot_bot_set_a ).
% assms(2)
thf(fact_24_assms_I7_J,axiom,
! [I: a] :
( ( member_a @ I @ s )
=> ( member_b @ ( y @ I ) @ c ) ) ).
% assms(7)
thf(fact_25_insert_Oprems_I3_J,axiom,
! [I: a] :
( ( member_a @ I @ ( insert_a @ i @ sa ) )
=> ( member_b @ ( y @ I ) @ c ) ) ).
% insert.prems(3)
thf(fact_26_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_27_inverse__eq__iff__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_28_insert_Oprems_I1_J,axiom,
( ( groups1862963056a_real @ aa @ ( insert_a @ i @ sa ) )
= one_one_real ) ).
% insert.prems(1)
thf(fact_29_assms_I3_J,axiom,
lower_311861425x_on_b @ c @ f ).
% assms(3)
thf(fact_30_fis,axiom,
finite_finite_a @ ( insert_a @ i @ sa ) ).
% fis
thf(fact_31_mult__scaleR__right,axiom,
! [X2: real,A: real,Y: real] :
( ( times_times_real @ X2 @ ( real_V453051771R_real @ A @ Y ) )
= ( real_V453051771R_real @ A @ ( times_times_real @ X2 @ Y ) ) ) ).
% mult_scaleR_right
thf(fact_32_assms_I4_J,axiom,
convex_b @ c ).
% assms(4)
thf(fact_33_assms_I5_J,axiom,
( ( groups1862963056a_real @ a2 @ s )
= one_one_real ) ).
% assms(5)
thf(fact_34_real__scaleR__def,axiom,
real_V453051771R_real = times_times_real ).
% real_scaleR_def
thf(fact_35_scale__left__commute,axiom,
! [A: real,B: real,X2: b] :
( ( real_V1035702896aleR_b @ A @ ( real_V1035702896aleR_b @ B @ X2 ) )
= ( real_V1035702896aleR_b @ B @ ( real_V1035702896aleR_b @ A @ X2 ) ) ) ).
% scale_left_commute
thf(fact_36_scale__left__commute,axiom,
! [A: real,B: real,X2: real] :
( ( real_V453051771R_real @ A @ ( real_V453051771R_real @ B @ X2 ) )
= ( real_V453051771R_real @ B @ ( real_V453051771R_real @ A @ X2 ) ) ) ).
% scale_left_commute
thf(fact_37_inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_38_scaleR__add__right,axiom,
! [A: real,X2: b,Y: b] :
( ( real_V1035702896aleR_b @ A @ ( plus_plus_b @ X2 @ Y ) )
= ( plus_plus_b @ ( real_V1035702896aleR_b @ A @ X2 ) @ ( real_V1035702896aleR_b @ A @ Y ) ) ) ).
% scaleR_add_right
thf(fact_39_scaleR__add__right,axiom,
! [A: real,X2: real,Y: real] :
( ( real_V453051771R_real @ A @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_real @ ( real_V453051771R_real @ A @ X2 ) @ ( real_V453051771R_real @ A @ Y ) ) ) ).
% scaleR_add_right
thf(fact_40_scaleR__add__left,axiom,
! [A: real,B: real,X2: b] :
( ( real_V1035702896aleR_b @ ( plus_plus_real @ A @ B ) @ X2 )
= ( plus_plus_b @ ( real_V1035702896aleR_b @ A @ X2 ) @ ( real_V1035702896aleR_b @ B @ X2 ) ) ) ).
% scaleR_add_left
thf(fact_41_scaleR__add__left,axiom,
! [A: real,B: real,X2: real] :
( ( real_V453051771R_real @ ( plus_plus_real @ A @ B ) @ X2 )
= ( plus_plus_real @ ( real_V453051771R_real @ A @ X2 ) @ ( real_V453051771R_real @ B @ X2 ) ) ) ).
% scaleR_add_left
thf(fact_42_scaleR__left_Oadd,axiom,
! [X2: real,Y: real,Xa: b] :
( ( real_V1035702896aleR_b @ ( plus_plus_real @ X2 @ Y ) @ Xa )
= ( plus_plus_b @ ( real_V1035702896aleR_b @ X2 @ Xa ) @ ( real_V1035702896aleR_b @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_43_scaleR__left_Oadd,axiom,
! [X2: real,Y: real,Xa: real] :
( ( real_V453051771R_real @ ( plus_plus_real @ X2 @ Y ) @ Xa )
= ( plus_plus_real @ ( real_V453051771R_real @ X2 @ Xa ) @ ( real_V453051771R_real @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_44_scale__right__diff__distrib,axiom,
! [A: real,X2: b,Y: b] :
( ( real_V1035702896aleR_b @ A @ ( minus_minus_b @ X2 @ Y ) )
= ( minus_minus_b @ ( real_V1035702896aleR_b @ A @ X2 ) @ ( real_V1035702896aleR_b @ A @ Y ) ) ) ).
% scale_right_diff_distrib
thf(fact_45_scale__right__diff__distrib,axiom,
! [A: real,X2: real,Y: real] :
( ( real_V453051771R_real @ A @ ( minus_minus_real @ X2 @ Y ) )
= ( minus_minus_real @ ( real_V453051771R_real @ A @ X2 ) @ ( real_V453051771R_real @ A @ Y ) ) ) ).
% scale_right_diff_distrib
thf(fact_46_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: real,X2: real] :
( ( ( times_times_real @ Y @ X2 )
= ( times_times_real @ X2 @ Y ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X2 )
= ( times_times_real @ X2 @ ( inverse_inverse_real @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_47_scale__sum__right,axiom,
! [A: real,F: a > b,A2: set_a] :
( ( real_V1035702896aleR_b @ A @ ( groups1199149371um_a_b @ F @ A2 ) )
= ( groups1199149371um_a_b
@ ^ [X: a] : ( real_V1035702896aleR_b @ A @ ( F @ X ) )
@ A2 ) ) ).
% scale_sum_right
thf(fact_48_scale__sum__right,axiom,
! [A: real,F: a > real,A2: set_a] :
( ( real_V453051771R_real @ A @ ( groups1862963056a_real @ F @ A2 ) )
= ( groups1862963056a_real
@ ^ [X: a] : ( real_V453051771R_real @ A @ ( F @ X ) )
@ A2 ) ) ).
% scale_sum_right
thf(fact_49_scale__sum__left,axiom,
! [F: a > real,A2: set_a,X2: b] :
( ( real_V1035702896aleR_b @ ( groups1862963056a_real @ F @ A2 ) @ X2 )
= ( groups1199149371um_a_b
@ ^ [A3: a] : ( real_V1035702896aleR_b @ ( F @ A3 ) @ X2 )
@ A2 ) ) ).
% scale_sum_left
thf(fact_50_scale__sum__left,axiom,
! [F: a > real,A2: set_a,X2: real] :
( ( real_V453051771R_real @ ( groups1862963056a_real @ F @ A2 ) @ X2 )
= ( groups1862963056a_real
@ ^ [A3: a] : ( real_V453051771R_real @ ( F @ A3 ) @ X2 )
@ A2 ) ) ).
% scale_sum_left
thf(fact_51_scaleR__right_Osum,axiom,
! [A: real,G: a > b,A2: set_a] :
( ( real_V1035702896aleR_b @ A @ ( groups1199149371um_a_b @ G @ A2 ) )
= ( groups1199149371um_a_b
@ ^ [X: a] : ( real_V1035702896aleR_b @ A @ ( G @ X ) )
@ A2 ) ) ).
% scaleR_right.sum
thf(fact_52_scaleR__right_Osum,axiom,
! [A: real,G: a > real,A2: set_a] :
( ( real_V453051771R_real @ A @ ( groups1862963056a_real @ G @ A2 ) )
= ( groups1862963056a_real
@ ^ [X: a] : ( real_V453051771R_real @ A @ ( G @ X ) )
@ A2 ) ) ).
% scaleR_right.sum
thf(fact_53_scaleR__left_Osum,axiom,
! [G: a > real,A2: set_a,X2: b] :
( ( real_V1035702896aleR_b @ ( groups1862963056a_real @ G @ A2 ) @ X2 )
= ( groups1199149371um_a_b
@ ^ [X: a] : ( real_V1035702896aleR_b @ ( G @ X ) @ X2 )
@ A2 ) ) ).
% scaleR_left.sum
thf(fact_54_scaleR__left_Osum,axiom,
! [G: a > real,A2: set_a,X2: real] :
( ( real_V453051771R_real @ ( groups1862963056a_real @ G @ A2 ) @ X2 )
= ( groups1862963056a_real
@ ^ [X: a] : ( real_V453051771R_real @ ( G @ X ) @ X2 )
@ A2 ) ) ).
% scaleR_left.sum
thf(fact_55_inverse__unique,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= one_one_real )
=> ( ( inverse_inverse_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_56_scale__left__diff__distrib,axiom,
! [A: real,B: real,X2: b] :
( ( real_V1035702896aleR_b @ ( minus_minus_real @ A @ B ) @ X2 )
= ( minus_minus_b @ ( real_V1035702896aleR_b @ A @ X2 ) @ ( real_V1035702896aleR_b @ B @ X2 ) ) ) ).
% scale_left_diff_distrib
thf(fact_57_scale__left__diff__distrib,axiom,
! [A: real,B: real,X2: real] :
( ( real_V453051771R_real @ ( minus_minus_real @ A @ B ) @ X2 )
= ( minus_minus_real @ ( real_V453051771R_real @ A @ X2 ) @ ( real_V453051771R_real @ B @ X2 ) ) ) ).
% scale_left_diff_distrib
thf(fact_58_scaleR__left_Odiff,axiom,
! [X2: real,Y: real,Xa: b] :
( ( real_V1035702896aleR_b @ ( minus_minus_real @ X2 @ Y ) @ Xa )
= ( minus_minus_b @ ( real_V1035702896aleR_b @ X2 @ Xa ) @ ( real_V1035702896aleR_b @ Y @ Xa ) ) ) ).
% scaleR_left.diff
thf(fact_59_scaleR__left_Odiff,axiom,
! [X2: real,Y: real,Xa: real] :
( ( real_V453051771R_real @ ( minus_minus_real @ X2 @ Y ) @ Xa )
= ( minus_minus_real @ ( real_V453051771R_real @ X2 @ Xa ) @ ( real_V453051771R_real @ Y @ Xa ) ) ) ).
% scaleR_left.diff
thf(fact_60_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_inverse__scaleR__distrib,axiom,
! [A: real,X2: real] :
( ( inverse_inverse_real @ ( real_V453051771R_real @ A @ X2 ) )
= ( real_V453051771R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X2 ) ) ) ).
% inverse_scaleR_distrib
thf(fact_65_asum,axiom,
( member_b
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( divide_divide_real @ ( aa @ J ) @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) ) @ ( y @ J ) )
@ sa )
@ c ) ).
% asum
thf(fact_66_sum_Oinsert,axiom,
! [A2: set_b,X2: b,G: b > b] :
( ( finite_finite_b @ A2 )
=> ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups2026918778um_b_b @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_b @ ( G @ X2 ) @ ( groups2026918778um_b_b @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_67_sum_Oinsert,axiom,
! [A2: set_b,X2: b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups583146225b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups583146225b_real @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_68_sum_Oinsert,axiom,
! [A2: set_a,X2: a,G: a > b] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups1199149371um_a_b @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_b @ ( G @ X2 ) @ ( groups1199149371um_a_b @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_69_sum_Oinsert,axiom,
! [A2: set_a,X2: a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups1862963056a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups1862963056a_real @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_70_affine__hull__finite__step,axiom,
! [S: set_b,A: b,W: real,Y: b] :
( ( finite_finite_b @ S )
=> ( ( ? [U2: b > real] :
( ( ( groups583146225b_real @ U2 @ ( insert_b @ A @ S ) )
= W )
& ( ( groups2026918778um_b_b
@ ^ [X: b] : ( real_V1035702896aleR_b @ ( U2 @ X ) @ X )
@ ( insert_b @ A @ S ) )
= Y ) ) )
= ( ? [V: real,U2: b > real] :
( ( ( groups583146225b_real @ U2 @ S )
= ( minus_minus_real @ W @ V ) )
& ( ( groups2026918778um_b_b
@ ^ [X: b] : ( real_V1035702896aleR_b @ ( U2 @ X ) @ X )
@ S )
= ( minus_minus_b @ Y @ ( real_V1035702896aleR_b @ V @ A ) ) ) ) ) ) ) ).
% affine_hull_finite_step
thf(fact_71_affine__hull__finite__step,axiom,
! [S: set_real,A: real,W: real,Y: real] :
( ( finite_finite_real @ S )
=> ( ( ? [U2: real > real] :
( ( ( groups1296524820l_real @ U2 @ ( insert_real @ A @ S ) )
= W )
& ( ( groups1296524820l_real
@ ^ [X: real] : ( real_V453051771R_real @ ( U2 @ X ) @ X )
@ ( insert_real @ A @ S ) )
= Y ) ) )
= ( ? [V: real,U2: real > real] :
( ( ( groups1296524820l_real @ U2 @ S )
= ( minus_minus_real @ W @ V ) )
& ( ( groups1296524820l_real
@ ^ [X: real] : ( real_V453051771R_real @ ( U2 @ X ) @ X )
@ S )
= ( minus_minus_real @ Y @ ( real_V453051771R_real @ V @ A ) ) ) ) ) ) ) ).
% affine_hull_finite_step
thf(fact_72_a1,axiom,
( ( groups1862963056a_real
@ ^ [J: a] : ( divide_divide_real @ ( aa @ J ) @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) )
@ sa )
= one_one_real ) ).
% a1
thf(fact_73__092_060open_062sum_Aa_As_A_P_A_I1_A_N_Aa_Ai_J_A_061_A1_092_060close_062,axiom,
( ( divide_divide_real @ ( groups1862963056a_real @ aa @ sa ) @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) )
= one_one_real ) ).
% \<open>sum a s / (1 - a i) = 1\<close>
thf(fact_74_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X: a] : X = A )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_75_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y2: a,Z: a] : Y2 = Z
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_76_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_77_sum__diff1,axiom,
! [A2: set_b,A: b,F: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ A @ A2 )
=> ( ( groups583146225b_real @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( minus_minus_real @ ( groups583146225b_real @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( groups583146225b_real @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( groups583146225b_real @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_78_sum__diff1,axiom,
! [A2: set_a,A: a,F: a > b] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ A2 )
=> ( ( groups1199149371um_a_b @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( minus_minus_b @ ( groups1199149371um_a_b @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( groups1199149371um_a_b @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( groups1199149371um_a_b @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_79_sum__diff1,axiom,
! [A2: set_a,A: a,F: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ A2 )
=> ( ( groups1862963056a_real @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( minus_minus_real @ ( groups1862963056a_real @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( groups1862963056a_real @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( groups1862963056a_real @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_80_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_81_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_82_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_83_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_84_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X: b] :
~ ( member_b @ X @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_85_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_86_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_87_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_88_insert__absorb2,axiom,
! [X2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A2 ) )
= ( insert_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_89_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_90_insert__iff,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
= ( ( A = B )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_91_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_92_insertCI,axiom,
! [A: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A @ B2 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_93_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_94_Diff__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ~ ( member_b @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_95_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_96_DiffI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_97_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_98_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_99_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_100_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_101_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_102_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_103_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_104_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_105_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_106_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_107_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_108_insert__Diff1,axiom,
! [X2: a,B2: set_a,A2: set_a] :
( ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_109_insert__Diff1,axiom,
! [X2: b,B2: set_b,A2: set_b] :
( ( member_b @ X2 @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_110_Diff__insert0,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_111_Diff__insert0,axiom,
! [X2: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ B2 ) )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_112_inverse__divide,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ B @ A ) ) ).
% inverse_divide
thf(fact_113_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_114_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_115_sum__diff1__nat,axiom,
! [A: b,A2: set_b,F: b > nat] :
( ( ( member_b @ A @ A2 )
=> ( ( groups2098481813_b_nat @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( minus_minus_nat @ ( groups2098481813_b_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( groups2098481813_b_nat @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( groups2098481813_b_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_116_sum__diff1__nat,axiom,
! [A: a,A2: set_a,F: a > nat] :
( ( ( member_a @ A @ A2 )
=> ( ( groups769445524_a_nat @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( groups769445524_a_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( groups769445524_a_nat @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( groups769445524_a_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_117_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A4: set_a,B3: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X: a] : ( member_a @ X @ A4 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_118_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A4: set_b,B3: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X: b] : ( member_b @ X @ A4 )
@ ^ [X: b] : ( member_b @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_119_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A4: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
& ~ ( member_a @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_120_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A4: set_b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A4 )
& ~ ( member_b @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_121_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_122_DiffD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( member_b @ C @ B2 ) ) ).
% DiffD2
thf(fact_123_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_124_DiffD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% DiffD1
thf(fact_125_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_126_DiffE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% DiffE
thf(fact_127_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_128_insert__Diff__if,axiom,
! [X2: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_129_insert__Diff__if,axiom,
! [X2: b,B2: set_b,A2: set_b] :
( ( ( member_b @ X2 @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) )
& ( ~ ( member_b @ X2 @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B2 )
= ( insert_b @ X2 @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_130_sum__divide__distrib,axiom,
! [F: a > real,A2: set_a,R: real] :
( ( divide_divide_real @ ( groups1862963056a_real @ F @ A2 ) @ R )
= ( groups1862963056a_real
@ ^ [N: a] : ( divide_divide_real @ ( F @ N ) @ R )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_131_convex__ereal__add,axiom,
! [S2: set_b,F: b > extended_ereal,G: b > extended_ereal] :
( ( lower_311861425x_on_b @ S2 @ F )
=> ( ( lower_311861425x_on_b @ S2 @ G )
=> ( lower_311861425x_on_b @ S2
@ ^ [X: b] : ( plus_p2118002693_ereal @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% convex_ereal_add
thf(fact_132_times__divide__times__eq,axiom,
! [X2: real,Y: real,Z2: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_133_divide__divide__times__eq,axiom,
! [X2: real,Y: real,Z2: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_134_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_135_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_136_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_137_Diff__insert__absorb,axiom,
! [X2: b,A2: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_138_Diff__insert__absorb,axiom,
! [X2: a,A2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_139_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_140_insert__Diff,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_141_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_142_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_143_finite__empty__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: b,A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ( member_b @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_b @ A6 @ ( insert_b @ A5 @ bot_bot_set_b ) ) ) ) ) )
=> ( P @ bot_bot_set_b ) ) ) ) ).
% finite_empty_induct
thf(fact_144_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_145_infinite__coinduct,axiom,
! [X3: set_a > $o,A2: set_a] :
( ( X3 @ A2 )
=> ( ! [A6: set_a] :
( ( X3 @ A6 )
=> ? [X4: a] :
( ( member_a @ X4 @ A6 )
& ( ( X3 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_146_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_147_field__class_Ofield__divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B4: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B4 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_148_divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B4: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B4 ) ) ) ) ).
% divide_inverse
thf(fact_149_divide__inverse__commute,axiom,
( divide_divide_real
= ( ^ [A3: real,B4: real] : ( times_times_real @ ( inverse_inverse_real @ B4 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_150_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_151_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X: b] : ( member_b @ X @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_152_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X: a] : ( member_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_153_equals0I,axiom,
! [A2: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_154_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_155_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_156_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_157_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_158_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_159_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_160_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B5: set_b] :
( ( A2
= ( insert_b @ A @ B5 ) )
& ~ ( member_b @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_161_insert__commute,axiom,
! [X2: a,Y: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_162_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_a] :
( ( A2
= ( insert_a @ B @ C2 ) )
& ~ ( member_a @ B @ C2 )
& ( B2
= ( insert_a @ A @ C2 ) )
& ~ ( member_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_163_insert__eq__iff,axiom,
! [A: b,A2: set_b,B: b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B @ B2 )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_b] :
( ( A2
= ( insert_b @ B @ C2 ) )
& ~ ( member_b @ B @ C2 )
& ( B2
= ( insert_b @ A @ C2 ) )
& ~ ( member_b @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_164_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_165_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_166_insert__ident,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ~ ( member_a @ X2 @ B2 )
=> ( ( ( insert_a @ X2 @ A2 )
= ( insert_a @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_167_insert__ident,axiom,
! [X2: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ~ ( member_b @ X2 @ B2 )
=> ( ( ( insert_b @ X2 @ A2 )
= ( insert_b @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_168_Set_Oset__insert,axiom,
! [X2: a,A2: set_a] :
( ( member_a @ X2 @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X2 @ B5 ) )
=> ( member_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_169_Set_Oset__insert,axiom,
! [X2: b,A2: set_b] :
( ( member_b @ X2 @ A2 )
=> ~ ! [B5: set_b] :
( ( A2
= ( insert_b @ X2 @ B5 ) )
=> ( member_b @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_170_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_171_insertI2,axiom,
! [A: b,B2: set_b,B: b] :
( ( member_b @ A @ B2 )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_172_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_173_insertI1,axiom,
! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% insertI1
thf(fact_174_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_175_insertE,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_176_sum_Ocong,axiom,
! [A2: set_a,B2: set_a,G: a > b,H: a > b] :
( ( A2 = B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ B2 )
=> ( ( G @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups1199149371um_a_b @ G @ A2 )
= ( groups1199149371um_a_b @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_177_sum_Ocong,axiom,
! [A2: set_a,B2: set_a,G: a > real,H: a > real] :
( ( A2 = B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ B2 )
=> ( ( G @ X5 )
= ( H @ X5 ) ) )
=> ( ( groups1862963056a_real @ G @ A2 )
= ( groups1862963056a_real @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_178_sum_Oeq__general,axiom,
! [B2: set_a,A2: set_b,H: b > a,Gamma: a > b,Phi: b > b] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: b] :
( ( member_b @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ( ( member_a @ ( H @ X5 ) @ B2 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups2026918778um_b_b @ Phi @ A2 )
= ( groups1199149371um_a_b @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_179_sum_Oeq__general,axiom,
! [B2: set_a,A2: set_b,H: b > a,Gamma: a > real,Phi: b > real] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: b] :
( ( member_b @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ( ( member_a @ ( H @ X5 ) @ B2 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups583146225b_real @ Phi @ A2 )
= ( groups1862963056a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_180_sum_Oeq__general,axiom,
! [B2: set_b,A2: set_a,H: a > b,Gamma: b > b,Phi: a > b] :
( ! [Y3: b] :
( ( member_b @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_b @ ( H @ X5 ) @ B2 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups1199149371um_a_b @ Phi @ A2 )
= ( groups2026918778um_b_b @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_181_sum_Oeq__general,axiom,
! [B2: set_a,A2: set_a,H: a > a,Gamma: a > b,Phi: a > b] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_a @ ( H @ X5 ) @ B2 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups1199149371um_a_b @ Phi @ A2 )
= ( groups1199149371um_a_b @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_182_sum_Oeq__general,axiom,
! [B2: set_b,A2: set_a,H: a > b,Gamma: b > real,Phi: a > real] :
( ! [Y3: b] :
( ( member_b @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_b @ ( H @ X5 ) @ B2 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups1862963056a_real @ Phi @ A2 )
= ( groups583146225b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_183_sum_Oeq__general,axiom,
! [B2: set_a,A2: set_a,H: a > a,Gamma: a > real,Phi: a > real] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_a @ ( H @ X5 ) @ B2 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups1862963056a_real @ Phi @ A2 )
= ( groups1862963056a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_184_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > b,A2: set_b,H: b > a,Gamma: a > b,Phi: b > b] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_b @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ( ( member_a @ ( H @ X5 ) @ B2 )
& ( ( K @ ( H @ X5 ) )
= X5 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups2026918778um_b_b @ Phi @ A2 )
= ( groups1199149371um_a_b @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_185_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > b,A2: set_b,H: b > a,Gamma: a > real,Phi: b > real] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_b @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ( ( member_a @ ( H @ X5 ) @ B2 )
& ( ( K @ ( H @ X5 ) )
= X5 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups583146225b_real @ Phi @ A2 )
= ( groups1862963056a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_186_sum_Oeq__general__inverses,axiom,
! [B2: set_b,K: b > a,A2: set_a,H: a > b,Gamma: b > b,Phi: a > b] :
( ! [Y3: b] :
( ( member_b @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_b @ ( H @ X5 ) @ B2 )
& ( ( K @ ( H @ X5 ) )
= X5 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups1199149371um_a_b @ Phi @ A2 )
= ( groups2026918778um_b_b @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_187_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > a,A2: set_a,H: a > a,Gamma: a > b,Phi: a > b] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_a @ ( H @ X5 ) @ B2 )
& ( ( K @ ( H @ X5 ) )
= X5 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups1199149371um_a_b @ Phi @ A2 )
= ( groups1199149371um_a_b @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_188_sum_Oeq__general__inverses,axiom,
! [B2: set_b,K: b > a,A2: set_a,H: a > b,Gamma: b > real,Phi: a > real] :
( ! [Y3: b] :
( ( member_b @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_b @ ( H @ X5 ) @ B2 )
& ( ( K @ ( H @ X5 ) )
= X5 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups1862963056a_real @ Phi @ A2 )
= ( groups583146225b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_189_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K: a > a,A2: set_a,H: a > a,Gamma: a > real,Phi: a > real] :
( ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( ( member_a @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_a @ ( H @ X5 ) @ B2 )
& ( ( K @ ( H @ X5 ) )
= X5 )
& ( ( Gamma @ ( H @ X5 ) )
= ( Phi @ X5 ) ) ) )
=> ( ( groups1862963056a_real @ Phi @ A2 )
= ( groups1862963056a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_190_sum_Oreindex__bij__witness,axiom,
! [S: set_b,I2: a > b,J2: b > a,T: set_a,H: a > b,G: b > b] :
( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( I2 @ ( J2 @ A5 ) )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( member_a @ ( J2 @ A5 ) @ T ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( ( J2 @ ( I2 @ B6 ) )
= B6 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( member_b @ ( I2 @ B6 ) @ S ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( H @ ( J2 @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups2026918778um_b_b @ G @ S )
= ( groups1199149371um_a_b @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_191_sum_Oreindex__bij__witness,axiom,
! [S: set_b,I2: a > b,J2: b > a,T: set_a,H: a > real,G: b > real] :
( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( I2 @ ( J2 @ A5 ) )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( member_a @ ( J2 @ A5 ) @ T ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( ( J2 @ ( I2 @ B6 ) )
= B6 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( member_b @ ( I2 @ B6 ) @ S ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( H @ ( J2 @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups583146225b_real @ G @ S )
= ( groups1862963056a_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_192_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: b > a,J2: a > b,T: set_b,H: b > b,G: a > b] :
( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( I2 @ ( J2 @ A5 ) )
= A5 ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( member_b @ ( J2 @ A5 ) @ T ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( ( J2 @ ( I2 @ B6 ) )
= B6 ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( member_a @ ( I2 @ B6 ) @ S ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( H @ ( J2 @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups1199149371um_a_b @ G @ S )
= ( groups2026918778um_b_b @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_193_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: a > a,J2: a > a,T: set_a,H: a > b,G: a > b] :
( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( I2 @ ( J2 @ A5 ) )
= A5 ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( member_a @ ( J2 @ A5 ) @ T ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( ( J2 @ ( I2 @ B6 ) )
= B6 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( member_a @ ( I2 @ B6 ) @ S ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( H @ ( J2 @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups1199149371um_a_b @ G @ S )
= ( groups1199149371um_a_b @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_194_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: b > a,J2: a > b,T: set_b,H: b > real,G: a > real] :
( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( I2 @ ( J2 @ A5 ) )
= A5 ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( member_b @ ( J2 @ A5 ) @ T ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( ( J2 @ ( I2 @ B6 ) )
= B6 ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( member_a @ ( I2 @ B6 ) @ S ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( H @ ( J2 @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups1862963056a_real @ G @ S )
= ( groups583146225b_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_195_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I2: a > a,J2: a > a,T: set_a,H: a > real,G: a > real] :
( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( I2 @ ( J2 @ A5 ) )
= A5 ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( member_a @ ( J2 @ A5 ) @ T ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( ( J2 @ ( I2 @ B6 ) )
= B6 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( member_a @ ( I2 @ B6 ) @ S ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( H @ ( J2 @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups1862963056a_real @ G @ S )
= ( groups1862963056a_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_196_if__smult,axiom,
! [P: $o,X2: real,Y: real,V2: b] :
( ( P
=> ( ( real_V1035702896aleR_b @ ( if_real @ P @ X2 @ Y ) @ V2 )
= ( real_V1035702896aleR_b @ X2 @ V2 ) ) )
& ( ~ P
=> ( ( real_V1035702896aleR_b @ ( if_real @ P @ X2 @ Y ) @ V2 )
= ( real_V1035702896aleR_b @ Y @ V2 ) ) ) ) ).
% if_smult
thf(fact_197_if__smult,axiom,
! [P: $o,X2: real,Y: real,V2: real] :
( ( P
=> ( ( real_V453051771R_real @ ( if_real @ P @ X2 @ Y ) @ V2 )
= ( real_V453051771R_real @ X2 @ V2 ) ) )
& ( ~ P
=> ( ( real_V453051771R_real @ ( if_real @ P @ X2 @ Y ) @ V2 )
= ( real_V453051771R_real @ Y @ V2 ) ) ) ) ).
% if_smult
thf(fact_198_sum_Oinsert__remove,axiom,
! [A2: set_a,G: a > b,X2: a] :
( ( finite_finite_a @ A2 )
=> ( ( groups1199149371um_a_b @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_b @ ( G @ X2 ) @ ( groups1199149371um_a_b @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_199_sum_Oinsert__remove,axiom,
! [A2: set_a,G: a > real,X2: a] :
( ( finite_finite_a @ A2 )
=> ( ( groups1862963056a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups1862963056a_real @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_200_sum_Oremove,axiom,
! [A2: set_b,X2: b,G: b > b] :
( ( finite_finite_b @ A2 )
=> ( ( member_b @ X2 @ A2 )
=> ( ( groups2026918778um_b_b @ G @ A2 )
= ( plus_plus_b @ ( G @ X2 ) @ ( groups2026918778um_b_b @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_201_sum_Oremove,axiom,
! [A2: set_b,X2: b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( member_b @ X2 @ A2 )
=> ( ( groups583146225b_real @ G @ A2 )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups583146225b_real @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_202_sum_Oremove,axiom,
! [A2: set_a,X2: a,G: a > b] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( groups1199149371um_a_b @ G @ A2 )
= ( plus_plus_b @ ( G @ X2 ) @ ( groups1199149371um_a_b @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_203_sum_Oremove,axiom,
! [A2: set_a,X2: a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( groups1862963056a_real @ G @ A2 )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups1862963056a_real @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_204_sum_Odelta__remove,axiom,
! [S: set_b,A: b,B: b > b,C: b > b] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups2026918778um_b_b
@ ^ [K2: b] : ( if_b @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_b @ ( B @ A ) @ ( groups2026918778um_b_b @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups2026918778um_b_b
@ ^ [K2: b] : ( if_b @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups2026918778um_b_b @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_205_sum_Odelta__remove,axiom,
! [S: set_b,A: b,B: b > real,C: b > real] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups583146225b_real
@ ^ [K2: b] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_real @ ( B @ A ) @ ( groups583146225b_real @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups583146225b_real
@ ^ [K2: b] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups583146225b_real @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_206_sum_Odelta__remove,axiom,
! [S: set_a,A: a,B: a > b,C: a > b] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups1199149371um_a_b
@ ^ [K2: a] : ( if_b @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_b @ ( B @ A ) @ ( groups1199149371um_a_b @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups1199149371um_a_b
@ ^ [K2: a] : ( if_b @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups1199149371um_a_b @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_207_sum_Odelta__remove,axiom,
! [S: set_a,A: a,B: a > real,C: a > real] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups1862963056a_real
@ ^ [K2: a] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_real @ ( B @ A ) @ ( groups1862963056a_real @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups1862963056a_real
@ ^ [K2: a] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups1862963056a_real @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_208_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_209_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_a,R2: b > a > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ B2 )
& ( R2 @ X5 @ Xa2 ) ) )
=> ? [X5: a] :
( ( member_a @ X5 @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R2 @ A3 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_210_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_a,R2: a > a > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ B2 )
& ( R2 @ X5 @ Xa2 ) ) )
=> ? [X5: a] :
( ( member_a @ X5 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( R2 @ A3 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_211_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_212_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_213_insert__compr,axiom,
( insert_a
= ( ^ [A3: a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( X = A3 )
| ( member_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_214_insert__compr,axiom,
( insert_b
= ( ^ [A3: b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( X = A3 )
| ( member_b @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_215_sum__delta__notmem_I4_J,axiom,
! [X2: a,S2: set_a,P: a > b,Q: a > b] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups1199149371um_a_b
@ ^ [Y4: a] : ( if_b @ ( X2 = Y4 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1199149371um_a_b @ Q @ S2 ) ) ) ).
% sum_delta_notmem(4)
thf(fact_216_sum__delta__notmem_I4_J,axiom,
! [X2: a,S2: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups1862963056a_real
@ ^ [Y4: a] : ( if_real @ ( X2 = Y4 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1862963056a_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(4)
thf(fact_217_sum__delta__notmem_I3_J,axiom,
! [X2: a,S2: set_a,P: a > b,Q: a > b] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups1199149371um_a_b
@ ^ [Y4: a] : ( if_b @ ( Y4 = X2 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1199149371um_a_b @ Q @ S2 ) ) ) ).
% sum_delta_notmem(3)
thf(fact_218_sum__delta__notmem_I3_J,axiom,
! [X2: a,S2: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups1862963056a_real
@ ^ [Y4: a] : ( if_real @ ( Y4 = X2 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1862963056a_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(3)
thf(fact_219_sum__delta__notmem_I2_J,axiom,
! [X2: a,S2: set_a,P: a > b,Q: a > b] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups1199149371um_a_b
@ ^ [Y4: a] : ( if_b @ ( X2 = Y4 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1199149371um_a_b @ Q @ S2 ) ) ) ).
% sum_delta_notmem(2)
thf(fact_220_sum__delta__notmem_I2_J,axiom,
! [X2: a,S2: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups1862963056a_real
@ ^ [Y4: a] : ( if_real @ ( X2 = Y4 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1862963056a_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(2)
thf(fact_221_sum__delta__notmem_I1_J,axiom,
! [X2: a,S2: set_a,P: a > b,Q: a > b] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups1199149371um_a_b
@ ^ [Y4: a] : ( if_b @ ( Y4 = X2 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1199149371um_a_b @ Q @ S2 ) ) ) ).
% sum_delta_notmem(1)
thf(fact_222_sum__delta__notmem_I1_J,axiom,
! [X2: a,S2: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups1862963056a_real
@ ^ [Y4: a] : ( if_real @ ( Y4 = X2 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups1862963056a_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(1)
thf(fact_223_sum_Oswap,axiom,
! [G: a > a > b,B2: set_a,A2: set_a] :
( ( groups1199149371um_a_b
@ ^ [I3: a] : ( groups1199149371um_a_b @ ( G @ I3 ) @ B2 )
@ A2 )
= ( groups1199149371um_a_b
@ ^ [J: a] :
( groups1199149371um_a_b
@ ^ [I3: a] : ( G @ I3 @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_224_sum_Oswap,axiom,
! [G: a > a > real,B2: set_a,A2: set_a] :
( ( groups1862963056a_real
@ ^ [I3: a] : ( groups1862963056a_real @ ( G @ I3 ) @ B2 )
@ A2 )
= ( groups1862963056a_real
@ ^ [J: a] :
( groups1862963056a_real
@ ^ [I3: a] : ( G @ I3 @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_225_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_226_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_227_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_228_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_229_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_230_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_231_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_232_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_233_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_234_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_235_sum__product,axiom,
! [F: a > real,A2: set_a,G: a > real,B2: set_a] :
( ( times_times_real @ ( groups1862963056a_real @ F @ A2 ) @ ( groups1862963056a_real @ G @ B2 ) )
= ( groups1862963056a_real
@ ^ [I3: a] :
( groups1862963056a_real
@ ^ [J: a] : ( times_times_real @ ( F @ I3 ) @ ( G @ J ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_236_sum__distrib__left,axiom,
! [R: real,F: a > real,A2: set_a] :
( ( times_times_real @ R @ ( groups1862963056a_real @ F @ A2 ) )
= ( groups1862963056a_real
@ ^ [N: a] : ( times_times_real @ R @ ( F @ N ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_237_sum__distrib__right,axiom,
! [F: a > real,A2: set_a,R: real] :
( ( times_times_real @ ( groups1862963056a_real @ F @ A2 ) @ R )
= ( groups1862963056a_real
@ ^ [N: a] : ( times_times_real @ ( F @ N ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_238_sum_Odistrib,axiom,
! [G: a > b,H: a > b,A2: set_a] :
( ( groups1199149371um_a_b
@ ^ [X: a] : ( plus_plus_b @ ( G @ X ) @ ( H @ X ) )
@ A2 )
= ( plus_plus_b @ ( groups1199149371um_a_b @ G @ A2 ) @ ( groups1199149371um_a_b @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_239_sum_Odistrib,axiom,
! [G: a > real,H: a > real,A2: set_a] :
( ( groups1862963056a_real
@ ^ [X: a] : ( plus_plus_real @ ( G @ X ) @ ( H @ X ) )
@ A2 )
= ( plus_plus_real @ ( groups1862963056a_real @ G @ A2 ) @ ( groups1862963056a_real @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_240_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_241_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_242_sum__subtractf,axiom,
! [F: a > b,G: a > b,A2: set_a] :
( ( groups1199149371um_a_b
@ ^ [X: a] : ( minus_minus_b @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_b @ ( groups1199149371um_a_b @ F @ A2 ) @ ( groups1199149371um_a_b @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_243_sum__subtractf,axiom,
! [F: a > real,G: a > real,A2: set_a] :
( ( groups1862963056a_real
@ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_real @ ( groups1862963056a_real @ F @ A2 ) @ ( groups1862963056a_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_244_sum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_a,G: b > a > b,R2: b > a > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups2026918778um_b_b
@ ^ [X: b] :
( groups1199149371um_a_b @ ( G @ X )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B2 )
& ( R2 @ X @ Y4 ) ) ) )
@ A2 )
= ( groups1199149371um_a_b
@ ^ [Y4: a] :
( groups2026918778um_b_b
@ ^ [X: b] : ( G @ X @ Y4 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R2 @ X @ Y4 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_245_sum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_a,G: b > a > real,R2: b > a > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups583146225b_real
@ ^ [X: b] :
( groups1862963056a_real @ ( G @ X )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B2 )
& ( R2 @ X @ Y4 ) ) ) )
@ A2 )
= ( groups1862963056a_real
@ ^ [Y4: a] :
( groups583146225b_real
@ ^ [X: b] : ( G @ X @ Y4 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R2 @ X @ Y4 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_246_sum_Oswap__restrict,axiom,
! [A2: set_a,B2: set_b,G: a > b > b,R2: a > b > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups1199149371um_a_b
@ ^ [X: a] :
( groups2026918778um_b_b @ ( G @ X )
@ ( collect_b
@ ^ [Y4: b] :
( ( member_b @ Y4 @ B2 )
& ( R2 @ X @ Y4 ) ) ) )
@ A2 )
= ( groups2026918778um_b_b
@ ^ [Y4: b] :
( groups1199149371um_a_b
@ ^ [X: a] : ( G @ X @ Y4 )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( R2 @ X @ Y4 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_247_sum_Oswap__restrict,axiom,
! [A2: set_a,B2: set_a,G: a > a > b,R2: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups1199149371um_a_b
@ ^ [X: a] :
( groups1199149371um_a_b @ ( G @ X )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B2 )
& ( R2 @ X @ Y4 ) ) ) )
@ A2 )
= ( groups1199149371um_a_b
@ ^ [Y4: a] :
( groups1199149371um_a_b
@ ^ [X: a] : ( G @ X @ Y4 )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( R2 @ X @ Y4 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_248_sum_Oswap__restrict,axiom,
! [A2: set_a,B2: set_b,G: a > b > real,R2: a > b > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups1862963056a_real
@ ^ [X: a] :
( groups583146225b_real @ ( G @ X )
@ ( collect_b
@ ^ [Y4: b] :
( ( member_b @ Y4 @ B2 )
& ( R2 @ X @ Y4 ) ) ) )
@ A2 )
= ( groups583146225b_real
@ ^ [Y4: b] :
( groups1862963056a_real
@ ^ [X: a] : ( G @ X @ Y4 )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( R2 @ X @ Y4 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_249_sum_Oswap__restrict,axiom,
! [A2: set_a,B2: set_a,G: a > a > real,R2: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups1862963056a_real
@ ^ [X: a] :
( groups1862963056a_real @ ( G @ X )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B2 )
& ( R2 @ X @ Y4 ) ) ) )
@ A2 )
= ( groups1862963056a_real
@ ^ [Y4: a] :
( groups1862963056a_real
@ ^ [X: a] : ( G @ X @ Y4 )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( R2 @ X @ Y4 ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_250_infinite__finite__induct,axiom,
! [P: set_b > $o,A2: set_b] :
( ! [A6: set_b] :
( ~ ( finite_finite_b @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X5: b,F2: set_b] :
( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ X5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_b @ X5 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_251_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X5: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X5 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_252_finite__ne__induct,axiom,
! [F3: set_b,P: set_b > $o] :
( ( finite_finite_b @ F3 )
=> ( ( F3 != bot_bot_set_b )
=> ( ! [X5: b] : ( P @ ( insert_b @ X5 @ bot_bot_set_b ) )
=> ( ! [X5: b,F2: set_b] :
( ( finite_finite_b @ F2 )
=> ( ( F2 != bot_bot_set_b )
=> ( ~ ( member_b @ X5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_b @ X5 @ F2 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_253_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X5: a] : ( P @ ( insert_a @ X5 @ bot_bot_set_a ) )
=> ( ! [X5: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X5 @ F2 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_254_finite_Oinducts,axiom,
! [X2: set_a,P: set_a > $o] :
( ( finite_finite_a @ X2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a,A5: a] :
( ( finite_finite_a @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( insert_a @ A5 @ A6 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finite.inducts
thf(fact_255_finite__induct,axiom,
! [F3: set_b,P: set_b > $o] :
( ( finite_finite_b @ F3 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X5: b,F2: set_b] :
( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ X5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_b @ X5 @ F2 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_256_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X5: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X5 @ F2 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_257_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A4: set_a,B4: a] :
( ( A3
= ( insert_a @ B4 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_258_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A5: a] :
( A
= ( insert_a @ A5 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_259_prod_Ofinite__Collect__op,axiom,
! [I4: set_b,X2: b > real,Y: b > real] :
( ( finite_finite_b
@ ( collect_b
@ ^ [I3: b] :
( ( member_b @ I3 @ I4 )
& ( ( X2 @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [I3: b] :
( ( member_b @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [I3: b] :
( ( member_b @ I3 @ I4 )
& ( ( times_times_real @ ( X2 @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_260_prod_Ofinite__Collect__op,axiom,
! [I4: set_a,X2: a > real,Y: a > real] :
( ( finite_finite_a
@ ( collect_a
@ ^ [I3: a] :
( ( member_a @ I3 @ I4 )
& ( ( X2 @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [I3: a] :
( ( member_a @ I3 @ I4 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [I3: a] :
( ( member_a @ I3 @ I4 )
& ( ( times_times_real @ ( X2 @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_261_sum_Oinsert__if,axiom,
! [A2: set_b,X2: b,G: b > b] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ X2 @ A2 )
=> ( ( groups2026918778um_b_b @ G @ ( insert_b @ X2 @ A2 ) )
= ( groups2026918778um_b_b @ G @ A2 ) ) )
& ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups2026918778um_b_b @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_b @ ( G @ X2 ) @ ( groups2026918778um_b_b @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_262_sum_Oinsert__if,axiom,
! [A2: set_b,X2: b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ X2 @ A2 )
=> ( ( groups583146225b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( groups583146225b_real @ G @ A2 ) ) )
& ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups583146225b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups583146225b_real @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_263_sum_Oinsert__if,axiom,
! [A2: set_a,X2: a,G: a > b] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X2 @ A2 )
=> ( ( groups1199149371um_a_b @ G @ ( insert_a @ X2 @ A2 ) )
= ( groups1199149371um_a_b @ G @ A2 ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups1199149371um_a_b @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_b @ ( G @ X2 ) @ ( groups1199149371um_a_b @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_264_sum_Oinsert__if,axiom,
! [A2: set_a,X2: a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X2 @ A2 )
=> ( ( groups1862963056a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( groups1862963056a_real @ G @ A2 ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups1862963056a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups1862963056a_real @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_265_convex__singleton,axiom,
! [A: b] : ( convex_b @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% convex_singleton
thf(fact_266_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_267_convex__empty,axiom,
convex_b @ bot_bot_set_b ).
% convex_empty
thf(fact_268_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_269_add__diff__cancel,axiom,
! [A: b,B: b] :
( ( minus_minus_b @ ( plus_plus_b @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_270_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_271_diff__add__cancel,axiom,
! [A: b,B: b] :
( ( plus_plus_b @ ( minus_minus_b @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_272_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_273_add__right__cancel,axiom,
! [B: b,A: b,C: b] :
( ( ( plus_plus_b @ B @ A )
= ( plus_plus_b @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_274_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_275_add__left__cancel,axiom,
! [A: b,B: b,C: b] :
( ( ( plus_plus_b @ A @ B )
= ( plus_plus_b @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_276_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_277_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_278_mult_Oleft__neutral,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult.left_neutral
thf(fact_279_add__diff__cancel__right_H,axiom,
! [A: b,B: b] :
( ( minus_minus_b @ ( plus_plus_b @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_280_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_281_add__diff__cancel__right,axiom,
! [A: b,C: b,B: b] :
( ( minus_minus_b @ ( plus_plus_b @ A @ C ) @ ( plus_plus_b @ B @ C ) )
= ( minus_minus_b @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_282_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_283_add__diff__cancel__left_H,axiom,
! [A: b,B: b] :
( ( minus_minus_b @ ( plus_plus_b @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_284_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_285_add__diff__cancel__left,axiom,
! [C: b,A: b,B: b] :
( ( minus_minus_b @ ( plus_plus_b @ C @ A ) @ ( plus_plus_b @ C @ B ) )
= ( minus_minus_b @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_286_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_287_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_288_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B4: real] : ( times_times_real @ B4 @ A3 ) ) ) ).
% mult.commute
thf(fact_289_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_290_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_291_add__right__imp__eq,axiom,
! [B: b,A: b,C: b] :
( ( ( plus_plus_b @ B @ A )
= ( plus_plus_b @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_292_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_293_add__left__imp__eq,axiom,
! [A: b,B: b,C: b] :
( ( ( plus_plus_b @ A @ B )
= ( plus_plus_b @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_294_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_295_add_Oleft__commute,axiom,
! [B: b,A: b,C: b] :
( ( plus_plus_b @ B @ ( plus_plus_b @ A @ C ) )
= ( plus_plus_b @ A @ ( plus_plus_b @ B @ C ) ) ) ).
% add.left_commute
thf(fact_296_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_297_add_Ocommute,axiom,
( plus_plus_b
= ( ^ [A3: b,B4: b] : ( plus_plus_b @ B4 @ A3 ) ) ) ).
% add.commute
thf(fact_298_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B4: real] : ( plus_plus_real @ B4 @ A3 ) ) ) ).
% add.commute
thf(fact_299_add_Oright__cancel,axiom,
! [B: b,A: b,C: b] :
( ( ( plus_plus_b @ B @ A )
= ( plus_plus_b @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_300_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_301_add_Oleft__cancel,axiom,
! [A: b,B: b,C: b] :
( ( ( plus_plus_b @ A @ B )
= ( plus_plus_b @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_302_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_303_add_Oassoc,axiom,
! [A: b,B: b,C: b] :
( ( plus_plus_b @ ( plus_plus_b @ A @ B ) @ C )
= ( plus_plus_b @ A @ ( plus_plus_b @ B @ C ) ) ) ).
% add.assoc
thf(fact_304_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_305_group__cancel_Oadd2,axiom,
! [B2: b,K: b,B: b,A: b] :
( ( B2
= ( plus_plus_b @ K @ B ) )
=> ( ( plus_plus_b @ A @ B2 )
= ( plus_plus_b @ K @ ( plus_plus_b @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_306_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_307_group__cancel_Oadd1,axiom,
! [A2: b,K: b,A: b,B: b] :
( ( A2
= ( plus_plus_b @ K @ A ) )
=> ( ( plus_plus_b @ A2 @ B )
= ( plus_plus_b @ K @ ( plus_plus_b @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_308_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_309_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J2: real,K: real,L: real] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_310_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: b,B: b,C: b] :
( ( plus_plus_b @ ( plus_plus_b @ A @ B ) @ C )
= ( plus_plus_b @ A @ ( plus_plus_b @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_311_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_312_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_313_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_314_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_315_convex__set__plus,axiom,
! [S: set_b,T: set_b] :
( ( convex_b @ S )
=> ( ( convex_b @ T )
=> ( convex_b @ ( plus_plus_set_b @ S @ T ) ) ) ) ).
% convex_set_plus
thf(fact_316_convex__set__sum,axiom,
! [A2: set_a,B2: a > set_b] :
( ! [I5: a] :
( ( member_a @ I5 @ A2 )
=> ( convex_b @ ( B2 @ I5 ) ) )
=> ( convex_b @ ( groups919362075_set_b @ B2 @ A2 ) ) ) ).
% convex_set_sum
thf(fact_317_convex__set__sum,axiom,
! [A2: set_b,B2: b > set_b] :
( ! [I5: b] :
( ( member_b @ I5 @ A2 )
=> ( convex_b @ ( B2 @ I5 ) ) )
=> ( convex_b @ ( groups448962650_set_b @ B2 @ A2 ) ) ) ).
% convex_set_sum
thf(fact_318_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_319_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_320_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_321_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_322_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_323_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_324_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_325_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_326_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_327_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_328_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_329_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_330_add__implies__diff,axiom,
! [C: b,B: b,A: b] :
( ( ( plus_plus_b @ C @ B )
= A )
=> ( C
= ( minus_minus_b @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_331_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_332_diff__diff__add,axiom,
! [A: b,B: b,C: b] :
( ( minus_minus_b @ ( minus_minus_b @ A @ B ) @ C )
= ( minus_minus_b @ A @ ( plus_plus_b @ B @ C ) ) ) ).
% diff_diff_add
thf(fact_333_diff__diff__add,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_add
thf(fact_334_diff__add__eq__diff__diff__swap,axiom,
! [A: b,B: b,C: b] :
( ( minus_minus_b @ A @ ( plus_plus_b @ B @ C ) )
= ( minus_minus_b @ ( minus_minus_b @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_335_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_336_diff__add__eq,axiom,
! [A: b,B: b,C: b] :
( ( plus_plus_b @ ( minus_minus_b @ A @ B ) @ C )
= ( minus_minus_b @ ( plus_plus_b @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_337_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_338_diff__diff__eq2,axiom,
! [A: b,B: b,C: b] :
( ( minus_minus_b @ A @ ( minus_minus_b @ B @ C ) )
= ( minus_minus_b @ ( plus_plus_b @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_339_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_340_add__diff__eq,axiom,
! [A: b,B: b,C: b] :
( ( plus_plus_b @ A @ ( minus_minus_b @ B @ C ) )
= ( minus_minus_b @ ( plus_plus_b @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_341_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_342_eq__diff__eq,axiom,
! [A: b,C: b,B: b] :
( ( A
= ( minus_minus_b @ C @ B ) )
= ( ( plus_plus_b @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_343_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_344_diff__eq__eq,axiom,
! [A: b,B: b,C: b] :
( ( ( minus_minus_b @ A @ B )
= C )
= ( A
= ( plus_plus_b @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_345_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_346_group__cancel_Osub1,axiom,
! [A2: b,K: b,A: b,B: b] :
( ( A2
= ( plus_plus_b @ K @ A ) )
=> ( ( minus_minus_b @ A2 @ B )
= ( plus_plus_b @ K @ ( minus_minus_b @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_347_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_348_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_349_square__diff__square__factored,axiom,
! [X2: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X2 @ Y ) @ ( minus_minus_real @ X2 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_350_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_351_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_352_ai1,axiom,
ord_less_eq_real @ ( aa @ i ) @ one_one_real ).
% ai1
% Helper facts (5)
thf(help_If_2_1_If_001tf__b_T,axiom,
! [X2: b,Y: b] :
( ( if_b @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__b_T,axiom,
! [X2: b,Y: b] :
( ( if_b @ $true @ X2 @ Y )
= 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,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( f
@ ( plus_plus_b
@ ( real_V1035702896aleR_b @ ( times_times_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) @ ( inverse_inverse_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) ) )
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( aa @ J ) @ ( y @ J ) )
@ sa ) )
@ ( real_V1035702896aleR_b @ ( aa @ i ) @ ( y @ i ) ) ) )
= ( f
@ ( plus_plus_b
@ ( real_V1035702896aleR_b @ ( minus_minus_real @ one_one_real @ ( aa @ i ) )
@ ( groups1199149371um_a_b
@ ^ [J: a] : ( real_V1035702896aleR_b @ ( times_times_real @ ( aa @ J ) @ ( inverse_inverse_real @ ( minus_minus_real @ one_one_real @ ( aa @ i ) ) ) ) @ ( y @ J ) )
@ sa ) )
@ ( real_V1035702896aleR_b @ ( aa @ i ) @ ( y @ i ) ) ) ) ) ).
%------------------------------------------------------------------------------