TPTP Problem File: ITP108^2.p
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%------------------------------------------------------------------------------
% File : ITP108^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Localization problem prob_758__8980450_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 : Localization/prob_758__8980450_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 325 ( 88 unt; 56 typ; 0 def)
% Number of atoms : 692 ( 264 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 9320 ( 20 ~; 0 |; 8 &;8955 @)
% ( 0 <=>; 337 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 62 ( 62 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 53 usr; 10 con; 0-6 aty)
% Number of variables : 709 ( 18 ^; 631 !; 4 ?; 709 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:24:15.941
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_t_Congruence_Opartial__object_Opartial__object__ext,type,
partia1265057091ct_ext: $tType > $tType > $tType ).
thf(ty_t_Group_Omonoid_Omonoid__ext,type,
monoid_ext: $tType > $tType > $tType ).
thf(ty_t_Ring_Oring_Oring__ext,type,
ring_ext: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (48)
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere623563068d_diff:
!>[A: $tType] : $o ).
thf(sy_c_AbelCoset_Oa__l__coset,type,
a_l_coset:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_AbelCoset_Oadditive__subgroup,type,
additive_subgroup:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_AbelCoset_Oset__add,type,
set_add:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Congruence_Opartial__object_Ocarrier,type,
partial_carrier:
!>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ Z ) > ( set @ A ) ) ).
thf(sy_c_Group_Omonoid_Omult,type,
mult:
!>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A > A > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac,type,
locali1648044335f_frac:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__crng,type,
locali7914339f_crng:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__rng,type,
locali1402578320of_rng:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Osubmonoid,type,
locali1441642228monoid:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Ring_Oa__inv,type,
a_inv:
!>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A ) ).
thf(sy_c_Ring_Oa__minus,type,
a_minus:
!>[A: $tType,M: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ) ) > A > A > A ) ).
thf(sy_c_Ring_Oadd__pow,type,
add_pow:
!>[A: $tType,C: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ C ) ) ) > B > A > A ) ).
thf(sy_c_Ring_Oring_Oadd,type,
add:
!>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A > A > A ) ).
thf(sy_c_Ring_Oring_Ozero,type,
zero:
!>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
thf(sy_c_Ring_Osemiring,type,
semiring:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_R,type,
r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
thf(sy_v_S,type,
s: set @ a ).
thf(sy_v_r,type,
r2: a ).
thf(sy_v_r_H,type,
r3: a ).
thf(sy_v_s,type,
s2: a ).
thf(sy_v_s_H,type,
s3: a ).
thf(sy_v_t_H____,type,
t: a ).
thf(sy_v_t____,type,
t2: a ).
thf(sy_v_x_H____,type,
x: product_prod @ a @ a ).
thf(sy_v_y_H____,type,
y: product_prod @ a @ a ).
% Relevant facts (256)
thf(fact_0_f18,axiom,
member @ a @ t @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f18
thf(fact_1_f17,axiom,
member @ a @ t2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f17
thf(fact_2_right__add__eq,axiom,
! [A2: a,B2: a,C2: a] :
( ( A2 = B2 )
=> ( ( add @ a @ b @ r @ C2 @ A2 )
= ( add @ a @ b @ r @ C2 @ B2 ) ) ) ).
% right_add_eq
thf(fact_3_local_Oright__minus__eq,axiom,
! [A2: a,B2: a,C2: a] :
( ( A2 = B2 )
=> ( ( a_minus @ a @ b @ r @ C2 @ A2 )
= ( a_minus @ a @ b @ r @ C2 @ B2 ) ) ) ).
% local.right_minus_eq
thf(fact_4_m__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ Z2 )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% m_assoc
thf(fact_5_m__comm,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_6_m__lcomm,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) ) ) ) ) ) ).
% m_lcomm
thf(fact_7_add_Om__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
= ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) ) ) ) ) ) ).
% add.m_assoc
thf(fact_8_add_Om__comm,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ X @ Y )
= ( add @ a @ b @ r @ Y @ X ) ) ) ) ).
% add.m_comm
thf(fact_9_add_Om__lcomm,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ Y @ Z2 ) )
= ( add @ a @ b @ r @ Y @ ( add @ a @ b @ r @ X @ Z2 ) ) ) ) ) ) ).
% add.m_lcomm
thf(fact_10_f20,axiom,
member @ a @ s3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f20
thf(fact_11_f19,axiom,
member @ a @ s2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f19
thf(fact_12_f41,axiom,
member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f41
thf(fact_13_f42,axiom,
member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f42
thf(fact_14_l__distr,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add @ a @ b @ r @ X @ Y ) @ Z2 )
= ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% l_distr
thf(fact_15_r__distr,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ ( add @ a @ b @ r @ X @ Y ) )
= ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ X ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Z2 @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_16_four__elem__comm,axiom,
! [A2: a,B2: a,C2: a,D: a] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ D @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ A2 @ C2 ) @ B2 ) @ D )
= ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) @ C2 ) @ D ) ) ) ) ) ) ).
% four_elem_comm
thf(fact_17_right__inv__add,axiom,
! [A2: a,B2: a,C2: a] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_minus @ a @ b @ r @ ( a_minus @ a @ b @ r @ C2 @ A2 ) @ B2 )
= ( a_minus @ a @ b @ r @ C2 @ ( add @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% right_inv_add
thf(fact_18_f21,axiom,
member @ a @ ( product_snd @ a @ a @ y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f21
thf(fact_19_f40,axiom,
( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
= ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
% f40
thf(fact_20_f22,axiom,
member @ a @ ( product_fst @ a @ a @ x ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f22
thf(fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
( ( member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) )
= ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x') \<in> carrier R; snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y') \<in> carrier R; t \<otimes> t' \<in> carrier R\<rbrakk> \<Longrightarrow> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x') \<oplus> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))\<close>
thf(fact_22_f15,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f15
thf(fact_23__092_060open_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% \<open>snd x' \<otimes> snd y' \<in> carrier R\<close>
thf(fact_24_f28,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f28
thf(fact_25_f11,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f11
thf(fact_26_f10,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f10
thf(fact_27_f27,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f27
thf(fact_28_f35,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) )
= ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% f35
thf(fact_29__092_060open_062t_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
= ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% \<open>t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = t \<otimes> (snd x' \<otimes> r) \<ominus> t \<otimes> (s \<otimes> fst x')\<close>
thf(fact_30__092_060open_062t_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_At_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Afst_Ay_H_J_092_060close_062,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
= ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% \<open>t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = t' \<otimes> (snd y' \<otimes> r') \<ominus> t' \<otimes> (s' \<otimes> fst y')\<close>
thf(fact_31_f16,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f16
thf(fact_32_f24,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) ) ).
% f24
thf(fact_33_f9,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f9
thf(fact_34__092_060open_062t_A_092_060otimes_062_As_A_092_060otimes_062_Asnd_Ax_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% \<open>t \<otimes> s \<otimes> snd x' \<in> carrier R\<close>
thf(fact_35_f23,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ).
% f23
thf(fact_36_f34,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) )
= ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) ).
% f34
thf(fact_37_f36,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) )
= ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% f36
thf(fact_38_f13,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f13
thf(fact_39__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_061_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) )
= ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r)) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) = t' \<otimes> s' \<otimes> snd y' \<otimes> t \<otimes> (snd x' \<otimes> r) \<ominus> t' \<otimes> s' \<otimes> snd y' \<otimes> t \<otimes> (s \<otimes> fst x')\<close>
thf(fact_40_f12,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) )
= ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ t2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) ).
% f12
thf(fact_41_f25,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) )
= ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) ) ).
% f25
thf(fact_42_f32,axiom,
( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) ) ).
% f32
thf(fact_43_f26,axiom,
( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) )
= ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ).
% f26
thf(fact_44_f29,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% f29
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_f30,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) )
= ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% f30
thf(fact_49_f33,axiom,
( ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% f33
thf(fact_50__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_092_060close_062,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) )
= ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) ).
% \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r')) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y') \<otimes> (s' \<otimes> r) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y') \<otimes> (s \<otimes> r')\<close>
thf(fact_51_f14,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f14
thf(fact_52__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) )
= ( a_minus @ a @ b @ r @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ).
% \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y')) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r)) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r')) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))\<close>
thf(fact_53_f31,axiom,
( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
= ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
% f31
thf(fact_54_f37,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f37
thf(fact_55_f39,axiom,
member @ a @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f39
thf(fact_56_f38,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% f38
thf(fact_57_f5,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
= ( zero @ a @ b @ r ) ) ).
% f5
thf(fact_58_f7,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
= ( zero @ a @ b @ r ) ) ).
% f7
thf(fact_59_minus__closed,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( a_minus @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% minus_closed
thf(fact_60_add_Om__closed,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( add @ a @ b @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% add.m_closed
thf(fact_61_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( add @ a @ b @ r @ Y @ X )
= ( add @ a @ b @ r @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_62_semiring__simprules_I3_J,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% semiring_simprules(3)
thf(fact_63_f4,axiom,
member @ a @ t2 @ s ).
% f4
thf(fact_64_f6,axiom,
member @ a @ t @ s ).
% f6
thf(fact_65_f8,axiom,
member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ s ).
% f8
thf(fact_66_local_Osemiring__axioms,axiom,
semiring @ a @ b @ r ).
% local.semiring_axioms
thf(fact_67_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus @ a @ b @ r @ X @ Y )
= ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_68_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_69_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ~ ! [X3: B,Y2: A] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_70_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_71_local_Ominus__unique,axiom,
! [Y: a,X: a,Y3: a] :
( ( ( add @ a @ b @ r @ Y @ X )
= ( zero @ a @ b @ r ) )
=> ( ( ( add @ a @ b @ r @ X @ Y3 )
= ( zero @ a @ b @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_72_add_Or__inv__ex,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ? [X3: a] :
( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
& ( ( add @ a @ b @ r @ X @ X3 )
= ( zero @ a @ b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_73_add_Oone__unique,axiom,
! [U: a] :
( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ! [X3: a] :
( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( zero @ a @ b @ r ) ) ) ) ).
% add.one_unique
thf(fact_74_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ? [X3: a] :
( ( member @ a @ X3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
& ( ( add @ a @ b @ r @ X3 @ X )
= ( zero @ a @ b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_75_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add @ a @ b @ r @ X @ Y )
= ( zero @ a @ b @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ Y @ X )
= ( zero @ a @ b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_76_r__minus,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( a_inv @ a @ b @ r @ Y ) )
= ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_77_l__minus,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( a_inv @ a @ b @ r @ X ) @ Y )
= ( a_inv @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_78_r__neg2,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ X @ ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_79_r__neg1,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( add @ a @ b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_80_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
= ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_81_add_Oinv__solve__right_H,axiom,
! [A2: a,B2: a,C2: a] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) )
= A2 )
= ( B2
= ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_82_add_Oinv__solve__right,axiom,
! [A2: a,B2: a,C2: a] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A2
= ( add @ a @ b @ r @ B2 @ ( a_inv @ a @ b @ r @ C2 ) ) )
= ( B2
= ( add @ a @ b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_83_add_Oinv__solve__left_H,axiom,
! [A2: a,B2: a,C2: a] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 )
= A2 )
= ( C2
= ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_84_add_Oinv__solve__left,axiom,
! [A2: a,B2: a,C2: a] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A2
= ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ B2 ) @ C2 ) )
= ( C2
= ( add @ a @ b @ r @ B2 @ A2 ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_85_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ X @ Y ) )
= ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ Y ) @ ( a_inv @ a @ b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_86_a__transpose__inv,axiom,
! [X: a,Y: a,Z2: a] :
( ( ( add @ a @ b @ r @ X @ Y )
= Z2 )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ Z2 )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_87_eq__obj__rng__of__frac__axioms,axiom,
locali1648044335f_frac @ a @ b @ r @ s ).
% eq_obj_rng_of_frac_axioms
thf(fact_88_sum__zero__eq__neg,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( add @ a @ b @ r @ X @ Y )
= ( zero @ a @ b @ r ) )
=> ( X
= ( a_inv @ a @ b @ r @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_89_r__neg,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ X @ ( a_inv @ a @ b @ r @ X ) )
= ( zero @ a @ b @ r ) ) ) ).
% r_neg
thf(fact_90_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add @ a @ b @ r @ Y @ X )
= ( zero @ a @ b @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_inv @ a @ b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_91_l__neg,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) @ X )
= ( zero @ a @ b @ r ) ) ) ).
% l_neg
thf(fact_92_inv__add,axiom,
! [A2: a,B2: a] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_inv @ a @ b @ r @ ( add @ a @ b @ r @ A2 @ B2 ) )
= ( a_minus @ a @ b @ r @ ( a_inv @ a @ b @ r @ A2 ) @ B2 ) ) ) ) ).
% inv_add
thf(fact_93__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062t_A_092_060in_062_AS_059_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T: a] :
( ( member @ a @ T @ s )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ T @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) )
!= ( zero @ a @ b @ r ) ) ) ).
% \<open>\<And>thesis. (\<And>t. \<lbrakk>t \<in> S; t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = \<zero>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_O_A_092_060lbrakk_062t_H_A_092_060in_062_AS_059_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T2: a] :
( ( member @ a @ T2 @ s )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ T2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) )
!= ( zero @ a @ b @ r ) ) ) ).
% \<open>\<And>thesis. (\<And>t'. \<lbrakk>t' \<in> S; t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = \<zero>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_95_zero__closed,axiom,
member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% zero_closed
thf(fact_96_local_Ominus__minus,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_inv @ a @ b @ r @ ( a_inv @ a @ b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_97_add_Oinv__closed,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( a_inv @ a @ b @ r @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% add.inv_closed
thf(fact_98_local_Ominus__zero,axiom,
( ( a_inv @ a @ b @ r @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ) ).
% local.minus_zero
thf(fact_99_m__closed,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ s )
=> ( ( member @ a @ Y @ s )
=> ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ s ) ) ) ).
% m_closed
thf(fact_100_r__null,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ) ) ).
% r_null
thf(fact_101_l__null,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ X )
= ( zero @ a @ b @ r ) ) ) ).
% l_null
thf(fact_102_r__zero,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ X @ ( zero @ a @ b @ r ) )
= X ) ) ).
% r_zero
thf(fact_103_l__zero,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add @ a @ b @ r @ ( zero @ a @ b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_104_add_Or__cancel__one_H,axiom,
! [X: a,A2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( X
= ( add @ a @ b @ r @ A2 @ X ) )
= ( A2
= ( zero @ a @ b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_105_add_Or__cancel__one,axiom,
! [X: a,A2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( add @ a @ b @ r @ A2 @ X )
= X )
= ( A2
= ( zero @ a @ b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_106_add_Ol__cancel__one_H,axiom,
! [X: a,A2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( X
= ( add @ a @ b @ r @ X @ A2 ) )
= ( A2
= ( zero @ a @ b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_107_add_Ol__cancel__one,axiom,
! [X: a,A2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( add @ a @ b @ r @ X @ A2 )
= X )
= ( A2
= ( zero @ a @ b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_108_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( a_inv @ a @ b @ r @ X )
= ( zero @ a @ b @ r ) )
= ( X
= ( zero @ a @ b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_109_mult__submonoid__of__rng__axioms,axiom,
locali1402578320of_rng @ a @ b @ r @ s ).
% mult_submonoid_of_rng_axioms
thf(fact_110_mult__submonoid__of__crng__axioms,axiom,
locali7914339f_crng @ a @ b @ r @ s ).
% mult_submonoid_of_crng_axioms
thf(fact_111_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y4: product_prod @ A @ B,Z3: product_prod @ A @ B] : Y4 = Z3 )
= ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_112_submonoid__axioms,axiom,
locali1441642228monoid @ a @ ( ring_ext @ a @ b ) @ r @ s ).
% submonoid_axioms
thf(fact_113_add_Oint__pow__inv,axiom,
! [X: a,I: int] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add_pow @ a @ b @ int @ r @ I @ ( a_inv @ a @ b @ r @ X ) )
= ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% add.int_pow_inv
thf(fact_114_semiring_Osemiring__simprules_I11_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( add @ A @ B @ R @ X @ ( zero @ A @ B @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_115_semiring_Osemiring__simprules_I6_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( add @ A @ B @ R @ ( zero @ A @ B @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_116_semiring_Ol__null,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( zero @ A @ B @ R ) @ X )
= ( zero @ A @ B @ R ) ) ) ) ).
% semiring.l_null
thf(fact_117_semiring_Or__null,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ ( zero @ A @ B @ R ) )
= ( zero @ A @ B @ R ) ) ) ) ).
% semiring.r_null
thf(fact_118_semiring_Or__distr,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ ( add @ A @ B @ R @ X @ Y ) )
= ( add @ A @ B @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ X ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Z2 @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_119_add__pow__ldistr__int,axiom,
! [A2: a,B2: a,K: int] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ int @ r @ K @ A2 ) @ B2 )
= ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_120_add__pow__rdistr__int,axiom,
! [A2: a,B2: a,K: int] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ int @ r @ K @ B2 ) )
= ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_121_add_Oint__pow__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
= ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_122_add_Oint__pow__mult__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( ( add @ a @ b @ r @ X @ Y )
= ( add @ a @ b @ r @ Y @ X ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add_pow @ a @ b @ int @ r @ I @ ( add @ a @ b @ r @ X @ Y ) )
= ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ I @ Y ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_123_add_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% add.int_pow_closed
thf(fact_124_add_Oint__pow__one,axiom,
! [Z2: int] :
( ( add_pow @ a @ b @ int @ r @ Z2 @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ) ).
% add.int_pow_one
thf(fact_125_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid,axiom,
! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A] :
( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
=> ( locali1441642228monoid @ A @ B @ M2 @ S2 ) ) ).
% Localization_Mirabelle_ojffvtlkod.submonoid.is_submonoid
thf(fact_126_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
( ( locali7914339f_crng @ A @ B @ R @ S2 )
=> ( locali1402578320of_rng @ A @ B @ R @ S2 ) ) ).
% mult_submonoid_of_crng.axioms(2)
thf(fact_127_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
( ( locali1402578320of_rng @ A @ B @ R @ S2 )
=> ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R @ S2 ) ) ).
% mult_submonoid_of_rng.axioms(2)
thf(fact_128_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S2: set @ A] :
( ( locali1648044335f_frac @ A @ B @ R @ S2 )
=> ( locali7914339f_crng @ A @ B @ R @ S2 ) ) ).
% eq_obj_rng_of_frac.axioms(2)
thf(fact_129_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed,axiom,
! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A,X: A,Y: A] :
( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( member @ A @ ( mult @ A @ B @ M2 @ X @ Y ) @ S2 ) ) ) ) ).
% Localization_Mirabelle_ojffvtlkod.submonoid.m_closed
thf(fact_130_a__minus__def,axiom,
! [M: $tType,A: $tType] :
( ( a_minus @ A @ M )
= ( ^ [R2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ M ) ),X2: A,Y5: A] : ( add @ A @ M @ R2 @ X2 @ ( a_inv @ A @ M @ R2 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_131_semiring_Osemiring__simprules_I8_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Y ) @ Z2 )
= ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_132_semiring_Osemiring__simprules_I3_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_133_semiring_Osemiring__simprules_I12_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( add @ A @ B @ R @ X @ ( add @ A @ B @ R @ Y @ Z2 ) )
= ( add @ A @ B @ R @ Y @ ( add @ A @ B @ R @ X @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_134_semiring_Osemiring__simprules_I7_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( add @ A @ B @ R @ X @ Y )
= ( add @ A @ B @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_135_semiring_Osemiring__simprules_I5_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( add @ A @ B @ R @ ( add @ A @ B @ R @ X @ Y ) @ Z2 )
= ( add @ A @ B @ R @ X @ ( add @ A @ B @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_136_semiring_Osemiring__simprules_I1_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( member @ A @ ( add @ A @ B @ R @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_137_semiring_Osemiring__simprules_I2_J,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R )
=> ( member @ A @ ( zero @ A @ B @ R ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_138_semiring_Ol__distr,axiom,
! [B: $tType,A: $tType,R: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
( ( semiring @ A @ B @ R )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R @ ( add @ A @ B @ R @ X @ Y ) @ Z2 )
= ( add @ A @ B @ R @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ X @ Z2 ) @ ( mult @ A @ ( ring_ext @ A @ B ) @ R @ Y @ Z2 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_139_add_Oint__pow__diff,axiom,
! [X: a,N: int,M3: int] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add_pow @ a @ b @ int @ r @ ( minus_minus @ int @ N @ M3 ) @ X )
= ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ N @ X ) @ ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ M3 @ X ) ) ) ) ) ).
% add.int_pow_diff
thf(fact_140_add_Oint__pow__neg,axiom,
! [X: a,I: int] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add_pow @ a @ b @ int @ r @ ( uminus_uminus @ int @ I ) @ X )
= ( a_inv @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) ) ) ) ).
% add.int_pow_neg
thf(fact_141_add_Oint__pow__mult,axiom,
! [X: a,I: int,J: int] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( add_pow @ a @ b @ int @ r @ ( plus_plus @ int @ I @ J ) @ X )
= ( add @ a @ b @ r @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( add_pow @ a @ b @ int @ r @ J @ X ) ) ) ) ).
% add.int_pow_mult
thf(fact_142_subset,axiom,
ord_less_eq @ ( set @ a ) @ s @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% subset
thf(fact_143_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset,axiom,
! [B: $tType,A: $tType,M2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S2: set @ A] :
( ( locali1441642228monoid @ A @ B @ M2 @ S2 )
=> ( ord_less_eq @ ( set @ A ) @ S2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ M2 ) ) ) ).
% Localization_Mirabelle_ojffvtlkod.submonoid.subset
thf(fact_144_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_145_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% uminus_add_conv_diff
thf(fact_146_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% le_add_diff_inverse
thf(fact_147_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% le_add_diff_inverse2
thf(fact_148_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_149_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_150_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_151_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_152_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_153_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_154_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_155_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_156_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_157_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_158_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_159_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_160_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_161_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_162_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_163_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_164_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_165_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_166_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_167_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_168_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% add.commute
thf(fact_169_group__add__class_Oadd_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% group_add_class.add.right_cancel
thf(fact_170_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_171_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_172_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K: A,B2: A,A2: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_173_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_174_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_176_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_177_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( A2 = B2 )
= ( C2 = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_178_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_179_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_180_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_181_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_182_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
? [C3: A] :
( B3
= ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_183_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_184_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( B2
!= ( plus_plus @ A @ A2 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_185_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_186_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_mono
thf(fact_187_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_188_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_189_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_190_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_191_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_192_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_193_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_194_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_195_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_196_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_197_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_198_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_add
thf(fact_199_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_200_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_201_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_202_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_203_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_204_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_205_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_206_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_207_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_208_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_209_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_210_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_211_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_212_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_214_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_215_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_216_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_217_le__add__diff,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_218_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_219_diff__add,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% diff_add
thf(fact_220_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_221_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_222_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_223_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_224_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_225_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B4: A,K: A,B2: A,A2: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_226_add_Oone__in__subset,axiom,
! [H: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( H
!= ( bot_bot @ ( set @ a ) ) )
=> ( ! [X3: a] :
( ( member @ a @ X3 @ H )
=> ( member @ a @ ( a_inv @ a @ b @ r @ X3 ) @ H ) )
=> ( ! [X3: a] :
( ( member @ a @ X3 @ H )
=> ! [Xa: a] :
( ( member @ a @ Xa @ H )
=> ( member @ a @ ( add @ a @ b @ r @ X3 @ Xa ) @ H ) ) )
=> ( member @ a @ ( zero @ a @ b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_227_a__lcos__mult__one,axiom,
! [M2: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ M2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_l_coset @ a @ b @ r @ ( zero @ a @ b @ r ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_228_a__lcos__m__assoc,axiom,
! [M2: set @ a,G: a,H2: a] :
( ( ord_less_eq @ ( set @ a ) @ M2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ G @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ H2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_l_coset @ a @ b @ r @ G @ ( a_l_coset @ a @ b @ r @ H2 @ M2 ) )
= ( a_l_coset @ a @ b @ r @ ( add @ a @ b @ r @ G @ H2 ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_229_carrier__not__empty,axiom,
( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
!= ( bot_bot @ ( set @ a ) ) ) ).
% carrier_not_empty
thf(fact_230_a__l__coset__subset__G,axiom,
! [H: set @ a,X: a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( a_l_coset @ a @ b @ r @ X @ H ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_231_setadd__subset__G,axiom,
! [H: set @ a,K2: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ H @ K2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_232_set__add__closed,axiom,
! [A3: set @ a,B4: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A3 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A3 @ B4 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% set_add_closed
thf(fact_233_add__additive__subgroups,axiom,
! [H: set @ a,K2: set @ a] :
( ( additive_subgroup @ a @ b @ H @ r )
=> ( ( additive_subgroup @ a @ b @ K2 @ r )
=> ( additive_subgroup @ a @ b @ ( set_add @ a @ b @ r @ H @ K2 ) @ r ) ) ) ).
% add_additive_subgroups
thf(fact_234_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_235_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_236_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_237_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_238_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_239_subsetI,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% subsetI
thf(fact_240_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_241_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_242_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_243_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_244_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_245_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_246_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_247_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_248_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_249_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_250_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_251_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_252_in__mono,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_253_subsetD,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_254_equalityE,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_255_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B5 ) ) ) ) ).
% subset_eq
% Type constructors (12)
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( add @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ s3 ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ ( product_fst @ a @ a @ x ) ) ) ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ s2 ) @ ( product_snd @ a @ a @ x ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ ( product_fst @ a @ a @ y ) ) ) ) ) )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ t2 @ t ) @ ( add @ a @ b @ r @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ r2 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ y ) @ ( product_fst @ a @ a @ x ) ) ) ) @ ( a_minus @ a @ b @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_snd @ a @ a @ y ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ r3 ) ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( product_snd @ a @ a @ x ) @ ( product_fst @ a @ a @ y ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------